CRYPTOGRAPHY AND NETWORK SECURITY PRINCIPLES AND PRACTICE SEVENTH EDITION GLOBAL EDITION
William Stallings
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3
CONTENTS Notation 10
Preface 12
About the Author 18
PART ONE: BACKGROUND 19
Chapter 1 Computer and Network Security Concepts 19
1.1 Computer Security Concepts 21 1.2 The OSI Security Architecture 26 1.3 Security Attacks 27 1.4 Security Services 29 1.5 Security Mechanisms 32 1.6 Fundamental Security Design Principles 34 1.7 Attack Surfaces and Attack Trees 37 1.8 A Model for Network Security 41 1.9 Standards 43 1.10 Key Terms, Review Questions, and Problems 44
Chapter 2 Introduction to Number Theory 46
2.1 Divisibility and the Division Algorithm 47 2.2 The Euclidean Algorithm 49 2.3 Modular Arithmetic 53 2.4 Prime Numbers 61 2.5 Fermat’s and Euler’s Theorems 64 2.6 Testing for Primality 68 2.7 The Chinese Remainder Theorem 71 2.8 Discrete Logarithms 73 2.9 Key Terms, Review Questions, and Problems 78 Appendix 2A The Meaning of Mod 82
PART TWO: SYMMETRIC CIPHERS 85
Chapter 3 Classical Encryption Techniques 85
3.1 Symmetric Cipher Model 86 3.2 Substitution Techniques 92 3.3 Transposition Techniques 107 3.4 Rotor Machines 108 3.5 Steganography 110 3.6 Key Terms, Review Questions, and Problems 112
Chapter 4 Block Ciphers and the Data Encryption Standard 118
4.1 Traditional Block Cipher Structure 119 4.2 The Data Encryption Standard 129 4.3 A DES Example 131 4.4 The Strength of DES 134
4 CONTENTS
4.5 Block Cipher Design Principles 135 4.6 Key Terms, Review Questions, and Problems 137
Chapter 5 Finite Fields 141
5.1 Groups 143 5.2 Rings 145 5.3 Fields 146 5.4 Finite Fields of the Form GF( p) 147 5.5 Polynomial Arithmetic 151 5.6 Finite Fields of the Form GF(2n) 157 5.7 Key Terms, Review Questions, and Problems 169
Chapter 6 Advanced Encryption Standard 171
6.1 Finite Field Arithmetic 172 6.2 AES Structure 174 6.3 AES Transformation Functions 179 6.4 AES Key Expansion 190 6.5 An AES Example 193 6.6 AES Implementation 197 6.7 Key Terms, Review Questions, and Problems 202 Appendix 6A Polynomials with Coefficients in GF(28) 203
Chapter 7 Block Cipher Operation 207
7.1 Multiple Encryption and Triple DES 208 7.2 Electronic Codebook 213 7.3 Cipher Block Chaining Mode 216 7.4 Cipher Feedback Mode 218 7.5 Output Feedback Mode 220 7.6 Counter Mode 222 7.7 XTS-AES Mode for Block-Oriented Storage Devices 224 7.8 Format-Preserving Encryption 231 7.9 Key Terms, Review Questions, and Problems 245
Chapter 8 Random Bit Generation and Stream Ciphers 250
8.1 Principles of Pseudorandom Number Generation 252 8.2 Pseudorandom Number Generators 258 8.3 Pseudorandom Number Generation Using a Block Cipher 261 8.4 Stream Ciphers 267 8.5 RC4 269 8.6 True Random Number Generators 271 8.7 Key Terms, Review Questions, and Problems 280
PART THREE: ASYMMETRIC CIPHERS 283
Chapter 9 Public-Key Cryptography and RSA 283
9.1 Principles of Public-Key Cryptosystems 285 9.2 The RSA Algorithm 294 9.3 Key Terms, Review Questions, and Problems 308
CONTENTS 5
Chapter 10 Other Public-Key Cryptosystems 313
10.1 Diffie-Hellman Key Exchange 314 10.2 Elgamal Cryptographic System 318 10.3 Elliptic Curve Arithmetic 321 10.4 Elliptic Curve Cryptography 330 10.5 Pseudorandom Number Generation Based on an Asymmetric Cipher 334 10.6 Key Terms, Review Questions, and Problems 336
PART FOUR: CRYPTOGRAPHIC DATA INTEGRITY ALGORITHMS 339
Chapter 11 Cryptographic Hash Functions 339
11.1 Applications of Cryptographic Hash Functions 341 11.2 Two Simple Hash Functions 346 11.3 Requirements and Security 348 11.4 Hash Functions Based on Cipher Block Chaining 354 11.5 Secure Hash Algorithm (SHA) 355 11.6 SHA-3 365 11.7 Key Terms, Review Questions, and Problems 377
Chapter 12 Message Authentication Codes 381
12.1 Message Authentication Requirements 382 12.2 Message Authentication Functions 383 12.3 Requirements for Message Authentication Codes 391 12.4 Security of MACs 393 12.5 MACs Based on Hash Functions: HMAC 394 12.6 MACs Based on Block Ciphers: DAA and CMAC 399 12.7 Authenticated Encryption: CCM and GCM 402 12.8 Key Wrapping 408 12.9 Pseudorandom Number Generation Using Hash Functions and MACs 413 12.10 Key Terms, Review Questions, and Problems 416
Chapter 13 Digital Signatures 419
13.1 Digital Signatures 421 13.2 Elgamal Digital Signature Scheme 424 13.3 Schnorr Digital Signature Scheme 425 13.4 NIST Digital Signature Algorithm 426 13.5 Elliptic Curve Digital Signature Algorithm 430 13.6 RSA-PSS Digital Signature Algorithm 433 13.7 Key Terms, Review Questions, and Problems 438
PART FIVE: MUTUAL TRUST 441
Chapter 14 Key Management and Distribution 441
14.1 Symmetric Key Distribution Using Symmetric Encryption 442 14.2 Symmetric Key Distribution Using Asymmetric Encryption 451 14.3 Distribution of Public Keys 454 14.4 X.509 Certificates 459
6 CONTENTS
14.5 Public-Key Infrastructure 467 14.6 Key Terms, Review Questions, and Problems 469
Chapter 15 User Authentication 473
15.1 Remote User-Authentication Principles 474 15.2 Remote User-Authentication Using Symmetric Encryption 478 15.3 Kerberos 482 15.4 Remote User-Authentication Using Asymmetric Encryption 500 15.5 Federated Identity Management 502 15.6 Personal Identity Verification 508 15.7 Key Terms, Review Questions, and Problems 515
PART SIX: NETWORK AND INTERNET SECURITY 519
Chapter 16 Network Access Control and Cloud Security 519
16.1 Network Access Control 520 16.2 Extensible Authentication Protocol 523 16.3 IEEE 802.1X Port-Based Network Access Control 527 16.4 Cloud Computing 529 16.5 Cloud Security Risks and Countermeasures 535 16.6 Data Protection in the Cloud 537 16.7 Cloud Security as a Service 541 16.8 Addressing Cloud Computing Security Concerns 544 16.9 Key Terms, Review Questions, and Problems 545
Chapter 17 Transport-Level Security 546
17.1 Web Security Considerations 547 17.2 Transport Layer Security 549 17.3 HTTPS 566 17.4 Secure Shell (SSH) 567 17.5 Key Terms, Review Questions, and Problems 579
Chapter 18 Wireless Network Security 581
18.1 Wireless Security 582 18.2 Mobile Device Security 585 18.3 IEEE 802.11 Wireless LAN Overview 589 18.4 IEEE 802.11i Wireless LAN Security 595 18.5 Key Terms, Review Questions, and Problems 610
Chapter 19 Electronic Mail Security 612
19.1 Internet Mail Architecture 613 19.2 Email Formats 617 19.3 Email Threats and Comprehensive Email Security 625 19.4 S/MIME 627 19.5 Pretty Good Privacy 638 19.6 DNSSEC 639 19.7 DNS-Based Authentication of Named Entities 643 19.8 Sender Policy Framework 645 19.9 DomainKeys Identified Mail 648
CONTENTS 7
19.10 Domain-Based Message Authentication, Reporting, and Conformance 654 19.11 Key Terms, Review Questions, and Problems 659
Chapter 20 IP Security 661
20.1 IP Security Overview 662 20.2 IP Security Policy 668 20.3 Encapsulating Security Payload 673 20.4 Combining Security Associations 681 20.5 Internet Key Exchange 684 20.6 Cryptographic Suites 692 20.7 Key Terms, Review Questions, and Problems 694
APPENDICES 696
Appendix A Projects for Teaching Cryptography and Network Security 696
A.1 Sage Computer Algebra Projects 697 A.2 Hacking Project 698 A.3 Block Cipher Projects 699 A.4 Laboratory Exercises 699 A.5 Research Projects 699 A.6 Programming Projects 700 A.7 Practical Security Assessments 700 A.8 Firewall Projects 701 A.9 Case Studies 701 A.10 Writing Assignments 701 A.11 Reading/Report Assignments 702 A.12 Discussion Topics 702
Appendix B Sage Examples 703
B.1 Linear Algebra and Matrix Functionality 704 B.2 Chapter 2: Number Theory 705 B.3 Chapter 3: Classical Encryption 710 B.4 Chapter 4: Block Ciphers and the Data Encryption Standard 713 B.5 Chapter 5: Basic Concepts in Number Theory and Finite Fields 717 B.6 Chapter 6: Advanced Encryption Standard 724 B.7 Chapter 8: Pseudorandom Number Generation and Stream Ciphers 729 B.8 Chapter 9: Public-Key Cryptography and RSA 731 B.9 Chapter 10: Other Public-Key Cryptosystems 734 B.10 Chapter 11: Cryptographic Hash Functions 739 B.11 Chapter 13: Digital Signatures 741
References 744
Credits 753
Index 754
8 CONTENTS
ONLINE CHAPTERS AND APPENDICES1
PART SEVEN: SYSTEM SECURITY
Chapter 21 Malicious Software
21.1 Types of Malicious Software (Malware) 21.2 Advanced Persistent Threat 21.3 Propagation—Infected Content—Viruses 21.4 Propagation—Vulnerability Exploit—Worms 21.5 Propagation—Social Engineering—Spam E-mail, Trojans 21.6 Payload—System Corruption 21.7 Payload—Attack Agent—Zombie, Bots 21.8 Payload—Information Theft—Keyloggers, Phishing, Spyware 21.9 Payload—Stealthing—Backdoors, Rootkits 21.10 Countermeasures 21.11 Distributed Denial of Service Attacks 21.12 References 21.13 Key Terms, Review Questions, and Problems
Chapter 22 Intruders
22.1 Intruders 22.2 Intrusion Detection 22.3 Password Management 22.4 References 22.5 Key Terms, Review Questions, and Problems
Chapter 23 Firewalls
23.1 The Need for Firewalls 23.2 Firewall Characteristics and Access Policy 23.3 Types of Firewalls 23.4 Firewall Basing 23.5 Firewall Location and Configurations 23.6 References 23.7 Key Terms, Review Questions, and Problems
PART EIGHT: LEGAL AND ETHICAL ISSUES
Chapter 24 Legal and Ethical Aspects
24.1 Cybercrime and Computer Crime 24.2 Intellectual Property 24.3 Privacy 24.4 Ethical Issues 24.5 Recommended Reading 24.6 References 24.7 Key Terms, Review Questions, and Problems 24.A Information Privacy
1Online chapters, appendices, and other documents are at the Companion Website, available via the access card at the front of this book.
CONTENTS 9
Appendix C Sage Exercises
Appendix D Standards and Standard-Setting Organizations
Appendix E Basic Concepts from Linear Algebra
Appendix F Measures of Secrecy and Security
Appendix G Simplified DES
Appendix H Evaluation Criteria for AES
Appendix I Simplified AES
Appendix J The Knapsack Algorithm
Appendix K Proof of the Digital Signature Algorithm
Appendix L TCP/IP and OSI
Appendix M Java Cryptographic APIs
Appendix N MD5 Hash Function
Appendix O Data Compression Using ZIP
Appendix P PGP
Appendix Q The International Reference Alphabet
Appendix R Proof of the RSA Algorithm
Appendix S Data Encryption Standard
Appendix T Kerberos Encryption Techniques
Appendix U Mathematical Basis of the Birthday Attack
Appendix V Evaluation Criteria for SHA-3
Appendix W The Complexity of Algorithms
Appendix X Radix-64 Conversion
Appendix Y The Base Rate Fallacy
Glossary
NOTATION
Symbol Expression Meaning
D, K D(K, Y) Symmetric decryption of ciphertext Y using secret key K
D, PRa D(PRa, Y) Asymmetric decryption of ciphertext Y using A’s private key PRa
D, PUa D(PUa, Y) Asymmetric decryption of ciphertext Y using A’s public key PUa
E, K E(K, X) Symmetric encryption of plaintext X using secret key K
E, PRa E(PRa, X) Asymmetric encryption of plaintext X using A’s private key PRa
E, PUa E(PUa, X) Asymmetric encryption of plaintext X using A’s public key PUa
K Secret key
PRa Private key of user A
PUa Public key of user A
MAC, K MAC(K, X) Message authentication code of message X using secret key K
GF(p) The finite field of order p, where p is prime.The field is defined as the set Zp together with the arithmetic operations modulo p.
GF(2n) The finite field of order 2n
Zn Set of nonnegative integers less than n
gcd gcd(i, j) Greatest common divisor; the largest positive integer that
divides both i and j with no remainder on division.
mod a mod m Remainder after division of a by m
mod, K a K b (mod m) a mod m = b mod m
mod, [ a [ b (mod m) a mod m ≠ b mod m
dlog dloga,p(b) Discrete logarithm of the number b for the base a (mod p)
w f(n) The number of positive integers less than n and relatively prime to n. This is Euler’s totient function.
Σ a n
i = 1 ai
a1 + a2 + g + an
Π q n
i = 1 ai
a1 * a2 * g * an
� i� j i divides j, which means that there is no remainder when j is divided by i
� , � �a� Absolute value of a
10
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NOTATION 11
Symbol Expression Meaning
} x} y x concatenated with y
≈ x ≈ y x is approximately equal to y
⊕ x ⊕ y Exclusive-OR of x and y for single-bit variables; Bitwise exclusive-OR of x and y for multiple-bit variables
:, ; :x; The largest integer less than or equal to x ∈ x ∈ S The element x is contained in the set S.
· A · (a1, a2, c ak)
The integer A corresponds to the sequence of integers
(a1, a2, c ak)
PREFACE
WHAT’S NEW IN THE SEVENTH EDITION
In the four years since the sixth edition of this book was published, the field has seen contin-
ued innovations and improvements. In this new edition, I try to capture these changes while
maintaining a broad and comprehensive coverage of the entire field. To begin this process of
revision, the sixth edition of this book was extensively reviewed by a number of professors
who teach the subject and by professionals working in the field. The result is that, in many
places, the narrative has been clarified and tightened, and illustrations have been improved.
Beyond these refinements to improve pedagogy and user-friendliness, there have been
substantive changes throughout the book. Roughly the same chapter organization has been
retained, but much of the material has been revised and new material has been added. The
most noteworthy changes are as follows:
■ Fundamental security design principles: Chapter 1 includes a new section discussing the security design principles listed as fundamental by the National Centers of Academic
Excellence in Information Assurance/Cyber Defense, which is jointly sponsored by the
U.S. National Security Agency and the U.S. Department of Homeland Security.
■ Attack surfaces and attack trees: Chapter 1 includes a new section describing these two concepts, which are useful in evaluating and classifying security threats.
■ Number theory coverage: The material on number theory has been consolidated into a single chapter, Chapter 2. This makes for a convenient reference. The relevant
portions of Chapter 2 can be assigned as needed.
■ Finite fields: The chapter on finite fields has been revised and expanded with addi- tional text and new figures to enhance understanding.
■ Format-preserving encryption: This relatively new mode of encryption is enjoying increasing commercial success. A new section in Chapter 7 covers this method.
■ Conditioning and health testing for true random number generators: Chapter 8 now provides coverage of these important topics.
■ User authentication model: Chapter 15 includes a new description of a general model for user authentication, which helps to unify the discussion of the various approaches
to user authentication.
■ Cloud security: The material on cloud security in Chapter 16 has been updated and expanded to reflect its importance and recent developments.
■ Transport Layer Security (TLS): The treatment of TLS in Chapter 17 has been updated, reorganized to improve clarity, and now includes a discussion of the new TLS version 1.3.
■ Email Security: Chapter 19 has been completely rewritten to provide a comprehensive and up-to-date discussion of email security. It includes:
— New: discussion of email threats and a comprehensive approach to email security.
— New: discussion of STARTTLS, which provides confidentiality and authentication
for SMTP. 12
PREFACE 13
— Revised: treatment of S/MIME has been updated to reflect the latest version 3.2.
— New: discussion of DNSSEC and its role in supporting email security.
— New: discussion of DNS-based Authentication of Named Entities (DANE) and the
use of this approach to enhance security for certificate use in SMTP and S/MIME.
— New: discussion of Sender Policy Framework (SPF), which is the standardized way
for a sending domain to identify and assert the mail senders for a given domain.
— Revised: discussion of DomainKeys Identified Mail (DKIM) has been revised.
— New: discussion of Domain-based Message Authentication, Reporting, and Confor-
mance (DMARC) allows email senders to specify policy on how their mail should
be handled, the types of reports that receivers can send back, and the frequency
those reports should be sent.
OBJECTIVES
The subject, and therefore this book, draws on a variety of disciplines. In particular,
it is impossible to appreciate the significance of some of the techniques discussed in this
book without a basic understanding of number theory and some results from probability
theory. Nevertheless, an attempt has been made to make the book self-contained. The book
not only presents the basic mathematical results that are needed but provides the reader
with an intuitive understanding of those results. Such background material is introduced
as needed. This approach helps to motivate the material that is introduced, and the author
considers this preferable to simply presenting all of the mathematical material in a lump at
the beginning of the book.
SUPPORT OF ACM/IEEE COMPUTER SCIENCE CURRICULA 2013
The book is intended for both academic and professional audiences. As a textbook, it is
intended as a one-semester undergraduate course in cryptography and network security for
computer science, computer engineering, and electrical engineering majors. The changes to
this edition are intended to provide support of the ACM/IEEE Computer Science Curricula
2013 (CS2013). CS2013 adds Information Assurance and Security (IAS) to the curriculum rec-
ommendation as one of the Knowledge Areas in the Computer Science Body of Knowledge.
The document states that IAS is now part of the curriculum recommendation because of the
critical role of IAS in computer science education. CS2013 divides all course work into three
categories: Core-Tier 1 (all topics should be included in the curriculum), Core-Tier-2 (all or
almost all topics should be included), and elective (desirable to provide breadth and depth).
In the IAS area, CS2013 recommends topics in Fundamental Concepts and Network Security
It is the purpose of this book to provide a practical survey of both the principles and practice
of cryptography and network security. In the first part of the book, the basic issues to be
addressed by a network security capability are explored by providing a tutorial and survey
of cryptography and network security technology. The latter part of the book deals with the
practice of network security: practical applications that have been implemented and are in
use to provide network security.
14 PREFACE
in Tier 1 and Tier 2, and Cryptography topics as elective. This text covers virtually all of the
topics listed by CS2013 in these three categories.
The book also serves as a basic reference volume and is suitable for self-study.
PLAN OF THE TEXT
The book is divided into eight parts.
■ Background
■ Symmetric Ciphers
■ Asymmetric Ciphers
■ Cryptographic Data Integrity Algorithms
■ Mutual Trust
■ Network and Internet Security
■ System Security
■ Legal and Ethical Issues
The book includes a number of pedagogic features, including the use of the computer
algebra system Sage and numerous figures and tables to clarify the discussions. Each chap-
ter includes a list of key words, review questions, homework problems, and suggestions
for further reading. The book also includes an extensive glossary, a list of frequently used
acronyms, and a bibliography. In addition, a test bank is available to instructors.
INSTRUCTOR SUPPORT MATERIALS
The major goal of this text is to make it as effective a teaching tool for this exciting and
fast-moving subject as possible. This goal is reflected both in the structure of the book and in
the supporting material. The text is accompanied by the following supplementary material
that will aid the instructor:
■ Solutions manual: Solutions to all end-of-chapter Review Questions and Problems.
■ Projects manual: Suggested project assignments for all of the project categories listed below.
■ PowerPoint slides: A set of slides covering all chapters, suitable for use in lecturing.
■ PDF files: Reproductions of all figures and tables from the book.
■ Test bank: A chapter-by-chapter set of questions with a separate file of answers.
■ Sample syllabuses: The text contains more material than can be conveniently covered in one semester. Accordingly, instructors are provided with several sample syllabuses
that guide the use of the text within limited time.
All of these support materials are available at the Instructor Resource Center (IRC) for this textbook, which can be reached through the publisher’s Web site www.pearsonglobaleditions.com/stallings. To gain access to the IRC, please contact your
local Pearson sales representative.
PREFACE 15
PROJECTS AND OTHER STUDENT EXERCISES
For many instructors, an important component of a cryptography or network security course
is a project or set of projects by which the student gets hands-on experience to reinforce
concepts from the text. This book provides an unparalleled degree of support, including a
projects component in the course. The IRC not only includes guidance on how to assign and
structure the projects, but also includes a set of project assignments that covers a broad range
of topics from the text:
■ Sage projects: Described in the next section.
■ Hacking project: Exercise designed to illuminate the key issues in intrusion detection and prevention.
■ Block cipher projects: A lab that explores the operation of the AES encryption algo- rithm by tracing its execution, computing one round by hand, and then exploring the
various block cipher modes of use. The lab also covers DES. In both cases, an online
Java applet is used (or can be downloaded) to execute AES or DES.
■ Lab exercises: A series of projects that involve programming and experimenting with concepts from the book.
■ Research projects: A series of research assignments that instruct the student to research a particular topic on the Internet and write a report.
■ Programming projects: A series of programming projects that cover a broad range of topics and that can be implemented in any suitable language on any platform.
■ Practical security assessments: A set of exercises to examine current infrastructure and practices of an existing organization.
■ Firewall projects: A portable network firewall visualization simulator, together with exercises for teaching the fundamentals of firewalls.
■ Case studies: A set of real-world case studies, including learning objectives, case description, and a series of case discussion questions.
■ Writing assignments: A set of suggested writing assignments, organized by chapter.
■ Reading/report assignments: A list of papers in the literature—one for each chapter— that can be assigned for the student to read and then write a short report.
This diverse set of projects and other student exercises enables the instructor to use
the book as one component in a rich and varied learning experience and to tailor a course
plan to meet the specific needs of the instructor and students. See Appendix A in this book
for details.
THE SAGE COMPUTER ALGEBRA SYSTEM
One of the most important features of this book is the use of Sage for cryptographic examples
and homework assignments. Sage is an open-source, multiplatform, freeware package that
implements a very powerful, flexible, and easily learned mathematics and computer algebra
system. Unlike competing systems (such as Mathematica, Maple, and MATLAB), there are
16 PREFACE
no licensing agreements or fees involved. Thus, Sage can be made available on computers
and networks at school, and students can individually download the software to their own
personal computers for use at home. Another advantage of using Sage is that students learn
a powerful, flexible tool that can be used for virtually any mathematical application, not
just cryptography.
The use of Sage can make a significant difference to the teaching of the mathematics
of cryptographic algorithms. This book provides a large number of examples of the use of
Sage covering many cryptographic concepts in Appendix B, which is included in this book.
Appendix C lists exercises in each of these topic areas to enable the student to gain
hands-on experience with cryptographic algorithms. This appendix is available to instruc-
tors at the IRC for this book. Appendix C includes a section on how to download and get
started with Sage, a section on programming with Sage, and exercises that can be assigned to
students in the following categories:
■ Chapter 2—Number Theory and Finite Fields: Euclidean and extended Euclidean algorithms, polynomial arithmetic, GF(24), Euler’s Totient function, Miller–Rabin, fac-
toring, modular exponentiation, discrete logarithm, and Chinese remainder theorem.
■ Chapter 3—Classical Encryption: Affine ciphers and the Hill cipher.
■ Chapter 4—Block Ciphers and the Data Encryption Standard: Exercises based on SDES.
■ Chapter 6—Advanced Encryption Standard: Exercises based on SAES.
■ Chapter 8—Pseudorandom Number Generation and Stream Ciphers: Blum Blum Shub, linear congruential generator, and ANSI X9.17 PRNG.
■ Chapter 9—Public-Key Cryptography and RSA: RSA encrypt/decrypt and signing.
■ Chapter 10—Other Public-Key Cryptosystems: Diffie–Hellman, elliptic curve.
■ Chapter 11—Cryptographic Hash Functions: Number-theoretic hash function.
■ Chapter 13—Digital Signatures: DSA.
ONLINE DOCUMENTS FOR STUDENTS
For this new edition, a tremendous amount of original supporting material for students has
been made available online.
Purchasing this textbook new also grants the reader six months of access to the
Companion Website, which includes the following materials:
■ Online chapters: To limit the size and cost of the book, four chapters of the book are provided in PDF format. This includes three chapters on computer security and one on
legal and ethical issues. The chapters are listed in this book’s table of contents.
■ Online appendices: There are numerous interesting topics that support material found in the text but whose inclusion is not warranted in the printed text. A total of 20 online
appendices cover these topics for the interested student. The appendices are listed in
this book’s table of contents.
PREFACE 17
■ Homework problems and solutions: To aid the student in understanding the material, a separate set of homework problems with solutions are available.
■ Key papers: A number of papers from the professional literature, many hard to find, are provided for further reading.
■ Supporting documents: A variety of other useful documents are referenced in the text and provided online.
■ Sage code: The Sage code from the examples in Appendix B is useful in case the student wants to play around with the examples.
To access the Companion Website, follow the instructions for “digital resources for
students” found in the front of this book.
ACKNOWLEDGMENTS
This new edition has benefited from review by a number of people who gave generously
of their time and expertise. The following professors reviewed all or a large part of the
manuscript: Hossein Beyzavi (Marymount University), Donald F. Costello (University of
Nebraska–Lincoln), James Haralambides (Barry University), Anand Seetharam (California
State University at Monterey Bay), Marius C. Silaghi (Florida Institute of Technology),
Shambhu Upadhyaya (University at Buffalo), Zhengping Wu (California State University
at San Bernardino), Liangliang Xiao (Frostburg State University), Seong-Moo (Sam) Yoo
(The University of Alabama in Huntsville), and Hong Zhang (Armstrong State University).
Thanks also to the people who provided detailed technical reviews of one or more
chapters: Dino M. Amaral, Chris Andrew, Prof. (Dr). C. Annamalai, Andrew Bain, Riccardo
Bernardini, Olivier Blazy, Zervopoulou Christina, Maria Christofi, Dhananjoy Dey, Mario
Emmanuel, Mike Fikuart, Alexander Fries, Pierpaolo Giacomin, Pedro R. M. Inácio,
Daniela Tamy Iwassa, Krzysztof Janowski, Sergey Katsev, Adnan Kilic, Rob Knox, Mina
Pourdashty, Yuri Poeluev, Pritesh Prajapati, Venkatesh Ramamoorthy, Andrea Razzini,
Rami Rosen, Javier Scodelaro, Jamshid Shokrollahi, Oscar So, and David Tillemans.
In addition, I was fortunate to have reviews of individual topics by “subject-area
gurus,” including Jesse Walker of Intel (Intel’s Digital Random Number Generator), Russ
Housley of Vigil Security (key wrapping), Joan Daemen (AES), Edward F. Schaefer of
Santa Clara University (Simplified AES), Tim Mathews, formerly of RSA Laboratories
(S/MIME), Alfred Menezes of the University of Waterloo (elliptic curve cryptography),
William Sutton, Editor/Publisher of The Cryptogram (classical encryption), Avi Rubin of Johns Hopkins University (number theory), Michael Markowitz of Information Security
Corporation (SHA and DSS), Don Davis of IBM Internet Security Systems (Kerberos),
Steve Kent of BBN Technologies (X.509), and Phil Zimmerman (PGP).
Nikhil Bhargava (IIT Delhi) developed the set of online homework problems and
solutions. Dan Shumow of Microsoft and the University of Washington developed all of
the Sage examples and assignments in Appendices B and C. Professor Sreekanth Malladi of
Dakota State University developed the hacking exercises. Lawrie Brown of the Australian
Defence Force Academy provided the AES/DES block cipher projects and the security
assessment assignments.
18 PREFACE
Sanjay Rao and Ruben Torres of Purdue University developed the laboratory exercises
that appear in the IRC. The following people contributed project assignments that appear in
the instructor’s supplement: Henning Schulzrinne (Columbia University); Cetin Kaya Koc
(Oregon State University); and David Balenson (Trusted Information Systems and George
Washington University). Kim McLaughlin developed the test bank.
Finally, I thank the many people responsible for the publication of this book, all of
whom did their usual excellent job. This includes the staff at Pearson, particularly my editor
Tracy Johnson, program manager Carole Snyder, and production manager Bob Engelhardt.
Thanks also to the marketing and sales staffs at Pearson, without whose efforts this book
would not be in front of you.
ACKNOWLEDGMENTS FOR THE GLOBAL EDITION
Pearson would like to thank and acknowledge Somitra Kumar Sanadhya (Indraprastha
Institute of Information Technology Delhi), and Somanath Tripathy (Indian Institute of
Technology Patna) for contributing to the Global Edition, and Anwitaman Datta (Nanyang
Technological University Singapore), Atul Kahate (Pune University), Goutam Paul (Indian
Statistical Institute Kolkata), and Khyat Sharma for reviewing the Global Edition.
ABOUT THE AUTHOR
Dr. William Stallings has authored 18 titles, and counting revised editions, over 40 books on computer security, computer networking, and computer architecture. His writings have
appeared in numerous publications, including the Proceedings of the IEEE, ACM Computing Reviews, and Cryptologia.
He has 13 times received the award for the best Computer Science textbook of the
year from the Text and Academic Authors Association.
In over 30 years in the field, he has been a technical contributor, technical manager,
and an executive with several high-technology firms. He has designed and implemented
both TCP/IP-based and OSI-based protocol suites on a variety of computers and operating
systems, ranging from microcomputers to mainframes. As a consultant, he has advised gov-
ernment agencies, computer and software vendors, and major users on the design, selection,
and use of networking software and products.
He created and maintains the Computer Science Student Resource Site at ComputerScienceStudent.com. This site provides documents and links on a variety of
subjects of general interest to computer science students (and professionals). He is a member
of the editorial board of Cryptologia, a scholarly journal devoted to all aspects of cryptology. Dr. Stallings holds a PhD from MIT in computer science and a BS from Notre Dame
in electrical engineering.
19
PART ONE: BACKGROUND
CHAPTER
Computer and Network Security Concepts
1.1 Computer Security Concepts
A Definition of Computer Security
Examples
The Challenges of Computer Security
1.2 The OSI Security Architecture
1.3 Security Attacks
Passive Attacks
Active Attacks
1.4 Security Services
Authentication
Access Control
Data Confidentiality
Data Integrity
Nonrepudiation
Availability Service
1.5 Security Mechanisms
1.6 Fundamental Security Design Principles
1.7 Attack Surfaces and Attack Trees
Attack Surfaces
Attack Trees
1.8 A Model for Network Security
1.9 Standards
1.10 Key Terms, Review Questions, and Problems
19
Hiva-Network.Com
20 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS
This book focuses on two broad areas: cryptographic algorithms and protocols, which
have a broad range of applications; and network and Internet security, which rely
heavily on cryptographic techniques.
Cryptographic algorithms and protocols can be grouped into four main areas:
■ Symmetric encryption: Used to conceal the contents of blocks or streams of data of any size, including messages, files, encryption keys, and passwords.
■ Asymmetric encryption: Used to conceal small blocks of data, such as encryp- tion keys and hash function values, which are used in digital signatures.
■ Data integrity algorithms: Used to protect blocks of data, such as messages, from alteration.
■ Authentication protocols: These are schemes based on the use of crypto- graphic algorithms designed to authenticate the identity of entities.
The field of network and Internet security consists of measures to deter, prevent, detect, and correct security violations that involve the transmission of information.
That is a broad statement that covers a host of possibilities. To give you a feel for the
areas covered in this book, consider the following examples of security violations:
1. User A transmits a file to user B. The file contains sensitive information (e.g., payroll records) that is to be protected from disclosure. User C, who is
not authorized to read the file, is able to monitor the transmission and capture
a copy of the file during its transmission.
2. A network manager, D, transmits a message to a computer, E, under its man- agement. The message instructs computer E to update an authorization file to
include the identities of a number of new users who are to be given access to
that computer. User F intercepts the message, alters its contents to add or delete
entries, and then forwards the message to computer E, which accepts the mes-
sage as coming from manager D and updates its authorization file accordingly.
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Describe the key security requirements of confidentiality, integrity, and availability.
◆ Describe the X.800 security architecture for OSI.
◆ Discuss the types of security threats and attacks that must be dealt with and give examples of the types of threats and attacks that apply to differ-
ent categories of computer and network assets.
◆ Explain the fundamental security design principles.
◆ Discuss the use of attack surfaces and attack trees.
◆ List and briefly describe key organizations involved in cryptography standards.
1.1 / COMPUTER SECURITY CONCEPTS 21
3. Rather than intercept a message, user F constructs its own message with the desired entries and transmits that message to computer E as if it had come
from manager D. Computer E accepts the message as coming from manager D
and updates its authorization file accordingly.
4. An employee is fired without warning. The personnel manager sends a mes- sage to a server system to invalidate the employee’s account. When the invali-
dation is accomplished, the server is to post a notice to the employee’s file as
confirmation of the action. The employee is able to intercept the message and
delay it long enough to make a final access to the server to retrieve sensitive
information. The message is then forwarded, the action taken, and the confir-
mation posted. The employee’s action may go unnoticed for some consider-
able time.
5. A message is sent from a customer to a stockbroker with instructions for vari- ous transactions. Subsequently, the investments lose value and the customer
denies sending the message.
Although this list by no means exhausts the possible types of network security viola-
tions, it illustrates the range of concerns of network security.
1.1 COMPUTER SECURITY CONCEPTS
A Definition of Computer Security
The NIST Computer Security Handbook [NIST95] defines the term computer secu- rity as follows:
Computer Security: The protection afforded to an automated information system in order to attain the applicable objectives of preserving the integrity, availability,
and confidentiality of information system resources (includes hardware, software,
firmware, information/data, and telecommunications).
This definition introduces three key objectives that are at the heart of com-
puter security:
■ Confidentiality: This term covers two related concepts:
Data1 confidentiality: Assures that private or confidential information is not made available or disclosed to unauthorized individuals.
Privacy: Assures that individuals control or influence what information re- lated to them may be collected and stored and by whom and to whom that
information may be disclosed.
1RFC 4949 defines information as “facts and ideas, which can be represented (encoded) as various forms of data,” and data as “information in a specific physical representation, usually a sequence of symbols that have meaning; especially a representation of information that can be processed or produced by a computer.” Security literature typically does not make much of a distinction, nor does this book.
22 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS
■ Integrity: This term covers two related concepts:
Data integrity: Assures that information (both stored and in transmit- ted packets) and programs are changed only in a specified and authorized
manner.
System integrity: Assures that a system performs its intended function in an unimpaired manner, free from deliberate or inadvertent unauthorized
manipulation of the system.
■ Availability: Assures that systems work promptly and service is not denied to authorized users.
These three concepts form what is often referred to as the CIA triad. The three concepts embody the fundamental security objectives for both data and for
information and computing services. For example, the NIST standard FIPS 199
(Standards for Security Categorization of Federal Information and Information Systems) lists confidentiality, integrity, and availability as the three security objec- tives for information and for information systems. FIPS 199 provides a useful char-
acterization of these three objectives in terms of requirements and the definition of
a loss of security in each category:
■ Confidentiality: Preserving authorized restrictions on information access and disclosure, including means for protecting personal privacy and propri-
etary information. A loss of confidentiality is the unauthorized disclosure of
information.
■ Integrity: Guarding against improper information modification or destruc- tion, including ensuring information nonrepudiation and authenticity. A loss
of integrity is the unauthorized modification or destruction of information.
■ Availability: Ensuring timely and reliable access to and use of information. A loss of availability is the disruption of access to or use of information or an
information system.
Although the use of the CIA triad to define security objectives is well estab-
lished, some in the security field feel that additional concepts are needed to present a
complete picture (Figure 1.1). Two of the most commonly mentioned are as follows:
Figure 1.1 Essential Network and Computer Security Requirements
Data and
services
Availability
Integrity
A ccountability
A ut
he nt
ic ity
Co nfi
den tia
lity
1.1 / COMPUTER SECURITY CONCEPTS 23
■ Authenticity: The property of being genuine and being able to be verified and trusted; confidence in the validity of a transmission, a message, or message
originator. This means verifying that users are who they say they are and that
each input arriving at the system came from a trusted source.
■ Accountability: The security goal that generates the requirement for actions of an entity to be traced uniquely to that entity. This supports nonrepudia-
tion, deterrence, fault isolation, intrusion detection and prevention, and after-
action recovery and legal action. Because truly secure systems are not yet an
achievable goal, we must be able to trace a security breach to a responsible
party. Systems must keep records of their activities to permit later forensic
analysis to trace security breaches or to aid in transaction disputes.
Examples
We now provide some examples of applications that illustrate the requirements just
enumerated.2 For these examples, we use three levels of impact on organizations or
individuals should there be a breach of security (i.e., a loss of confidentiality, integ-
rity, or availability). These levels are defined in FIPS PUB 199:
■ Low: The loss could be expected to have a limited adverse effect on organi- zational operations, organizational assets, or individuals. A limited adverse
effect means that, for example, the loss of confidentiality, integrity, or avail-
ability might (i) cause a degradation in mission capability to an extent and
duration that the organization is able to perform its primary functions, but the
effectiveness of the functions is noticeably reduced; (ii) result in minor dam-
age to organizational assets; (iii) result in minor financial loss; or (iv) result in
minor harm to individuals.
■ Moderate: The loss could be expected to have a serious adverse effect on organizational operations, organizational assets, or individuals. A serious
adverse effect means that, for example, the loss might (i) cause a signifi-
cant degradation in mission capability to an extent and duration that the
organization is able to perform its primary functions, but the effectiveness
of the functions is significantly reduced; (ii) result in significant damage to
organizational assets; (iii) result in significant financial loss; or (iv) result in
significant harm to individuals that does not involve loss of life or serious,
life-threatening injuries.
■ High: The loss could be expected to have a severe or catastrophic adverse effect on organizational operations, organizational assets, or individuals.
A severe or catastrophic adverse effect means that, for example, the loss
might (i) cause a severe degradation in or loss of mission capability to an
extent and duration that the organization is not able to perform one or more
of its primary functions; (ii) result in major damage to organizational assets;
(iii) result in major financial loss; or (iv) result in severe or catastrophic harm
to individuals involving loss of life or serious, life-threatening injuries.
2These examples are taken from a security policy document published by the Information Technology Security and Privacy Office at Purdue University.
24 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS
CONFIDENTIALITY Student grade information is an asset whose confidentiality is considered to be highly important by students. In the United States, the release of
such information is regulated by the Family Educational Rights and Privacy Act
(FERPA). Grade information should only be available to students, their parents,
and employees that require the information to do their job. Student enrollment
information may have a moderate confidentiality rating. While still covered by
FERPA, this information is seen by more people on a daily basis, is less likely to be
targeted than grade information, and results in less damage if disclosed. Directory
information, such as lists of students or faculty or departmental lists, may be as-
signed a low confidentiality rating or indeed no rating. This information is typically
freely available to the public and published on a school’s Web site.
INTEGRITY Several aspects of integrity are illustrated by the example of a hospital patient’s allergy information stored in a database. The doctor should be able to
trust that the information is correct and current. Now suppose that an employee
(e.g., a nurse) who is authorized to view and update this information deliberately
falsifies the data to cause harm to the hospital. The database needs to be restored
to a trusted basis quickly, and it should be possible to trace the error back to the
person responsible. Patient allergy information is an example of an asset with a high
requirement for integrity. Inaccurate information could result in serious harm or
death to a patient and expose the hospital to massive liability.
An example of an asset that may be assigned a moderate level of integrity
requirement is a Web site that offers a forum to registered users to discuss some
specific topic. Either a registered user or a hacker could falsify some entries or
deface the Web site. If the forum exists only for the enjoyment of the users, brings
in little or no advertising revenue, and is not used for something important such
as research, then potential damage is not severe. The Web master may experience
some data, financial, and time loss.
An example of a low integrity requirement is an anonymous online poll. Many
Web sites, such as news organizations, offer these polls to their users with very few
safeguards. However, the inaccuracy and unscientific nature of such polls is well
understood.
AVAILABILITY The more critical a component or service, the higher is the level of availability required. Consider a system that provides authentication services for
critical systems, applications, and devices. An interruption of service results in the
inability for customers to access computing resources and staff to access the re-
sources they need to perform critical tasks. The loss of the service translates into a
large financial loss in lost employee productivity and potential customer loss.
An example of an asset that would typically be rated as having a moderate
availability requirement is a public Web site for a university; the Web site provides
information for current and prospective students and donors. Such a site is not a
critical component of the university’s information system, but its unavailability will
cause some embarrassment.
An online telephone directory lookup application would be classified as a low
availability requirement. Although the temporary loss of the application may be
an annoyance, there are other ways to access the information, such as a hardcopy
directory or the operator.
1.1 / COMPUTER SECURITY CONCEPTS 25
The Challenges of Computer Security
Computer and network security is both fascinating and complex. Some of the
reasons follow:
1. Security is not as simple as it might first appear to the novice. The require- ments seem to be straightforward; indeed, most of the major requirements for
security services can be given self-explanatory, one-word labels: confidential-
ity, authentication, nonrepudiation, or integrity. But the mechanisms used to
meet those requirements can be quite complex, and understanding them may
involve rather subtle reasoning.
2. In developing a particular security mechanism or algorithm, one must always consider potential attacks on those security features. In many cases, successful
attacks are designed by looking at the problem in a completely different way,
therefore exploiting an unexpected weakness in the mechanism.
3. Because of point 2, the procedures used to provide particular services are often counterintuitive. Typically, a security mechanism is complex, and it is not
obvious from the statement of a particular requirement that such elaborate
measures are needed. It is only when the various aspects of the threat are con-
sidered that elaborate security mechanisms make sense.
4. Having designed various security mechanisms, it is necessary to decide where to use them. This is true both in terms of physical placement (e.g., at what points
in a network are certain security mechanisms needed) and in a logical sense
(e.g., at what layer or layers of an architecture such as TCP/IP [Transmission
Control Protocol/Internet Protocol] should mechanisms be placed).
5. Security mechanisms typically involve more than a particular algorithm or protocol. They also require that participants be in possession of some secret in-
formation (e.g., an encryption key), which raises questions about the creation,
distribution, and protection of that secret information. There also may be a re-
liance on communications protocols whose behavior may complicate the task
of developing the security mechanism. For example, if the proper functioning
of the security mechanism requires setting time limits on the transit time of a
message from sender to receiver, then any protocol or network that introduces
variable, unpredictable delays may render such time limits meaningless.
6. Computer and network security is essentially a battle of wits between a per- petrator who tries to find holes and the designer or administrator who tries to
close them. The great advantage that the attacker has is that he or she need
only find a single weakness, while the designer must find and eliminate all
weaknesses to achieve perfect security.
7. There is a natural tendency on the part of users and system managers to per- ceive little benefit from security investment until a security failure occurs.
8. Security requires regular, even constant, monitoring, and this is difficult in today’s short-term, overloaded environment.
9. Security is still too often an afterthought to be incorporated into a system after the design is complete rather than being an integral part of the design
process.
26 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS
10. Many users and even security administrators view strong security as an impediment to efficient and user-friendly operation of an information system
or use of information.
The difficulties just enumerated will be encountered in numerous ways as we
examine the various security threats and mechanisms throughout this book.
1.2 THE OSI SECURITY ARCHITECTURE
To assess effectively the security needs of an organization and to evaluate and
choose various security products and policies, the manager responsible for security
needs some systematic way of defining the requirements for security and character-
izing the approaches to satisfying those requirements. This is difficult enough in a
centralized data processing environment; with the use of local and wide area net-
works, the problems are compounded.
ITU-T3 Recommendation X.800, Security Architecture for OSI, defines such a systematic approach.4 The OSI security architecture is useful to managers as a way
of organizing the task of providing security. Furthermore, because this architecture
was developed as an international standard, computer and communications vendors
have developed security features for their products and services that relate to this
structured definition of services and mechanisms.
For our purposes, the OSI security architecture provides a useful, if abstract,
overview of many of the concepts that this book deals with. The OSI security archi-
tecture focuses on security attacks, mechanisms, and services. These can be defined
briefly as
■ Security attack: Any action that compromises the security of information owned by an organization.
■ Security mechanism: A process (or a device incorporating such a process) that is designed to detect, prevent, or recover from a security attack.
■ Security service: A processing or communication service that enhances the security of the data processing systems and the information transfers of an
organization. The services are intended to counter security attacks, and they
make use of one or more security mechanisms to provide the service.
In the literature, the terms threat and attack are commonly used to mean more or less the same thing. Table 1.1 provides definitions taken from RFC 4949, Internet Security Glossary.
3The International Telecommunication Union (ITU) Telecommunication Standardization Sector (ITU-T) is a United Nations-sponsored agency that develops standards, called Recommendations, relating to tele- communications and to open systems interconnection (OSI). 4The OSI security architecture was developed in the context of the OSI protocol architecture, which is described in Appendix L. However, for our purposes in this chapter, an understanding of the OSI proto- col architecture is not required.
1.3 / SECURITY ATTACKS 27
1.3 SECURITY ATTACKS
A useful means of classifying security attacks, used both in X.800 and RFC 4949, is
in terms of passive attacks and active attacks (Figure 1.2). A passive attack attempts to learn or make use of information from the system but does not affect system re-
sources. An active attack attempts to alter system resources or affect their operation.
Passive Attacks
Passive attacks (Figure 1.2a) are in the nature of eavesdropping on, or monitoring
of, transmissions. The goal of the opponent is to obtain information that is being
transmitted. Two types of passive attacks are the release of message contents and
traffic analysis.
The release of message contents is easily understood. A telephone conver- sation, an electronic mail message, and a transferred file may contain sensitive or
confidential information. We would like to prevent an opponent from learning the
contents of these transmissions.
A second type of passive attack, traffic analysis, is subtler. Suppose that we had a way of masking the contents of messages or other information traffic so that
opponents, even if they captured the message, could not extract the information
from the message. The common technique for masking contents is encryption. If we
had encryption protection in place, an opponent might still be able to observe the
pattern of these messages. The opponent could determine the location and identity
of communicating hosts and could observe the frequency and length of messages
being exchanged. This information might be useful in guessing the nature of the
communication that was taking place.
Passive attacks are very difficult to detect, because they do not involve any
alteration of the data. Typically, the message traffic is sent and received in an appar-
ently normal fashion, and neither the sender nor receiver is aware that a third party
has read the messages or observed the traffic pattern. However, it is feasible to pre-
vent the success of these attacks, usually by means of encryption. Thus, the empha-
sis in dealing with passive attacks is on prevention rather than detection.
Active Attacks
Active attacks (Figure 1.2b) involve some modification of the data stream or the
creation of a false stream and can be subdivided into four categories: masquerade,
replay, modification of messages, and denial of service.
Threat A potential for violation of security, which exists when there is a circumstance, capability, action,
or event that could breach security and cause harm. That is, a threat is a possible danger that might
exploit a vulnerability.
Attack An assault on system security that derives from an intelligent threat; that is, an intelligent act that
is a deliberate attempt (especially in the sense of a method or technique) to evade security services
and violate the security policy of a system.
Table 1.1 Threats and Attacks (RFC 4949)
28 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS
A masquerade takes place when one entity pretends to be a different entity (path 2 of Figure 1.2b is active). A masquerade attack usually includes one of the
other forms of active attack. For example, authentication sequences can be captured
and replayed after a valid authentication sequence has taken place, thus enabling an
authorized entity with few privileges to obtain extra privileges by impersonating an
entity that has those privileges.
Replay involves the passive capture of a data unit and its subsequent retrans- mission to produce an unauthorized effect (paths 1, 2, and 3 active).
Modification of messages simply means that some portion of a legitimate mes- sage is altered, or that messages are delayed or reordered, to produce an unauthor-
ized effect (paths 1 and 2 active). For example, a message meaning “Allow John
Smith to read confidential file accounts” is modified to mean “Allow Fred Brown to read confidential file accounts.”
Figure 1.2 Security Attacks
(a) Passive attacks
Alice
(b) Active attacks
Bob
Darth
Bob
Darth
Alice
Internet or other communications facility
Internet or other communications facility
1 2 3
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1.4 / SECURITY SERVICES 29
The denial of service prevents or inhibits the normal use or management of communications facilities (path 3 active). This attack may have a specific target; for
example, an entity may suppress all messages directed to a particular destination
(e.g., the security audit service). Another form of service denial is the disruption of
an entire network, either by disabling the network or by overloading it with mes-
sages so as to degrade performance.
Active attacks present the opposite characteristics of passive attacks. Whereas
passive attacks are difficult to detect, measures are available to prevent their success.
On the other hand, it is quite difficult to prevent active attacks absolutely because
of the wide variety of potential physical, software, and network vulnerabilities.
Instead, the goal is to detect active attacks and to recover from any disruption or
delays caused by them. If the detection has a deterrent effect, it may also contribute
to prevention.
1.4 SECURITY SERVICES
X.800 defines a security service as a service that is provided by a protocol layer of
communicating open systems and that ensures adequate security of the systems or
of data transfers. Perhaps a clearer definition is found in RFC 4949, which provides
the following definition: a processing or communication service that is provided by
a system to give a specific kind of protection to system resources; security services
implement security policies and are implemented by security mechanisms.
X.800 divides these services into five categories and fourteen specific services
(Table 1.2). We look at each category in turn.5
Authentication
The authentication service is concerned with assuring that a communication is au-
thentic. In the case of a single message, such as a warning or alarm signal, the function
of the authentication service is to assure the recipient that the message is from the
source that it claims to be from. In the case of an ongoing interaction, such as the con-
nection of a terminal to a host, two aspects are involved. First, at the time of connec-
tion initiation, the service assures that the two entities are authentic, that is, that each
is the entity that it claims to be. Second, the service must assure that the connection is
not interfered with in such a way that a third party can masquerade as one of the two
legitimate parties for the purposes of unauthorized transmission or reception.
Two specific authentication services are defined in X.800:
■ Peer entity authentication: Provides for the corroboration of the identity of a peer entity in an association. Two entities are considered peers if they imple-
ment to same protocol in different systems; for example two TCP modules
in two communicating systems. Peer entity authentication is provided for
5There is no universal agreement about many of the terms used in the security literature. For example, the term integrity is sometimes used to refer to all aspects of information security. The term authentication is sometimes used to refer both to verification of identity and to the various functions listed under integrity in this chapter. Our usage here agrees with both X.800 and RFC 4949.
30 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS
AUTHENTICATION
The assurance that the communicating entity is the
one that it claims to be.
Peer Entity Authentication Used in association with a logical connection to
provide confidence in the identity of the entities
connected.
Data-Origin Authentication In a connectionless transfer, provides assurance that
the source of received data is as claimed.
ACCESS CONTROL
The prevention of unauthorized use of a resource
(i.e., this service controls who can have access to a
resource, under what conditions access can occur,
and what those accessing the resource are allowed
to do).
DATA CONFIDENTIALITY
The protection of data from unauthorized
disclosure.
Connection Confidentiality The protection of all user data on a connection.
Connectionless Confidentiality The protection of all user data in a single data block.
Selective-Field Confidentiality The confidentiality of selected fields within the user
data on a connection or in a single data block.
Traffic-Flow Confidentiality The protection of the information that might be
derived from observation of traffic flows.
DATA INTEGRITY
The assurance that data received are exactly as
sent by an authorized entity (i.e., contain no modi-
fication, insertion, deletion, or replay).
Connection Integrity with Recovery Provides for the integrity of all user data on a connec-
tion and detects any modification, insertion, deletion,
or replay of any data within an entire data sequence,
with recovery attempted.
Connection Integrity without Recovery As above, but provides only detection without
recovery.
Selective-Field Connection Integrity Provides for the integrity of selected fields within the
user data of a data block transferred over a connec-
tion and takes the form of determination of whether
the selected fields have been modified, inserted,
deleted, or replayed.
Connectionless Integrity Provides for the integrity of a single connectionless
data block and may take the form of detection of
data modification. Additionally, a limited form of
replay detection may be provided.
Selective-Field Connectionless Integrity Provides for the integrity of selected fields within a
single connectionless data block; takes the form of
determination of whether the selected fields have
been modified.
NONREPUDIATION
Provides protection against denial by one of the
entities involved in a communication of having par-
ticipated in all or part of the communication.
Nonrepudiation, Origin Proof that the message was sent by the specified
party.
Nonrepudiation, Destination Proof that the message was received by the specified
party.
Table 1.2 Security Services (X.800)
use at the establishment of, or at times during the data transfer phase of, a
connection. It attempts to provide confidence that an entity is not performing
either a masquerade or an unauthorized replay of a previous connection.
■ Data origin authentication: Provides for the corroboration of the source of a data unit. It does not provide protection against the duplication or modifica-
tion of data units. This type of service supports applications like electronic mail,
where there are no prior interactions between the communicating entities.
1.4 / SECURITY SERVICES 31
Access Control
In the context of network security, access control is the ability to limit and control
the access to host systems and applications via communications links. To achieve
this, each entity trying to gain access must first be identified, or authenticated,
so that access rights can be tailored to the individual.
Data Confidentiality
Confidentiality is the protection of transmitted data from passive attacks. With re-
spect to the content of a data transmission, several levels of protection can be iden-
tified. The broadest service protects all user data transmitted between two users
over a period of time. For example, when a TCP connection is set up between two
systems, this broad protection prevents the release of any user data transmitted over
the TCP connection. Narrower forms of this service can also be defined, including
the protection of a single message or even specific fields within a message. These
refinements are less useful than the broad approach and may even be more complex
and expensive to implement.
The other aspect of confidentiality is the protection of traffic flow from
analysis. This requires that an attacker not be able to observe the source and desti-
nation, frequency, length, or other characteristics of the traffic on a communications
facility.
Data Integrity
As with confidentiality, integrity can apply to a stream of messages, a single mes-
sage, or selected fields within a message. Again, the most useful and straightforward
approach is total stream protection.
A connection-oriented integrity service, one that deals with a stream of mes-
sages, assures that messages are received as sent with no duplication, insertion,
modification, reordering, or replays. The destruction of data is also covered under
this service. Thus, the connection-oriented integrity service addresses both mes-
sage stream modification and denial of service. On the other hand, a connection-
less integrity service, one that deals with individual messages without regard to any
larger context, generally provides protection against message modification only.
We can make a distinction between service with and without recovery. Because
the integrity service relates to active attacks, we are concerned with detection rather
than prevention. If a violation of integrity is detected, then the service may simply
report this violation, and some other portion of software or human intervention is
required to recover from the violation. Alternatively, there are mechanisms avail-
able to recover from the loss of integrity of data, as we will review subsequently. The
incorporation of automated recovery mechanisms is, in general, the more attractive
alternative.
Nonrepudiation
Nonrepudiation prevents either sender or receiver from denying a transmitted mes-
sage. Thus, when a message is sent, the receiver can prove that the alleged sender in
fact sent the message. Similarly, when a message is received, the sender can prove
that the alleged receiver in fact received the message.
32 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS
Availability Service
Both X.800 and RFC 4949 define availability to be the property of a system or a
system resource being accessible and usable upon demand by an authorized system
entity, according to performance specifications for the system (i.e., a system is avail-
able if it provides services according to the system design whenever users request
them). A variety of attacks can result in the loss of or reduction in availability. Some
of these attacks are amenable to automated countermeasures, such as authentica-
tion and encryption, whereas others require some sort of physical action to prevent
or recover from loss of availability of elements of a distributed system.
X.800 treats availability as a property to be associated with various security
services. However, it makes sense to call out specifically an availability service. An
availability service is one that protects a system to ensure its availability. This ser-
vice addresses the security concerns raised by denial-of-service attacks. It depends
on proper management and control of system resources and thus depends on access
control service and other security services.
1.5 SECURITY MECHANISMS
Table 1.3 lists the security mechanisms defined in X.800. The mechanisms are
divided into those that are implemented in a specific protocol layer, such as TCP or
an application-layer protocol, and those that are not specific to any particular pro-
tocol layer or security service. These mechanisms will be covered in the appropriate
SPECIFIC SECURITY MECHANISMS May be incorporated into the appropriate protocol
layer in order to provide some of the OSI security
services.
Encipherment The use of mathematical algorithms to transform
data into a form that is not readily intelligible. The
transformation and subsequent recovery of the data
depend on an algorithm and zero or more encryption
keys.
Digital Signature Data appended to, or a cryptographic transformation
of, a data unit that allows a recipient of the data unit
to prove the source and integrity of the data unit and
protect against forgery (e.g., by the recipient).
Access Control A variety of mechanisms that enforce access rights to
resources.
Data Integrity A variety of mechanisms used to assure the integrity
of a data unit or stream of data units.
PERVASIVE SECURITY MECHANISMS
Mechanisms that are not specific to any particular
OSI security service or protocol layer.
Trusted Functionality That which is perceived to be correct with respect
to some criteria (e.g., as established by a security
policy).
Security Label The marking bound to a resource (which may be a
data unit) that names or designates the security attri-
butes of that resource.
Event Detection Detection of security-relevant events.
Security Audit Trail Data collected and potentially used to facilitate a
security audit, which is an independent review and
examination of system records and activities.
Security Recovery Deals with requests from mechanisms, such as event
handling and management functions, and takes
recovery actions.
Table 1.3 Security Mechanisms (X.800)
1.5 / SECURITY MECHANISMS 33
places in the book. So we do not elaborate now, except to comment on the defini-
tion of encipherment. X.800 distinguishes between reversible encipherment mech-
anisms and irreversible encipherment mechanisms. A reversible encipherment
mechanism is simply an encryption algorithm that allows data to be encrypted and
subsequently decrypted. Irreversible encipherment mechanisms include hash algo-
rithms and message authentication codes, which are used in digital signature and
message authentication applications.
Table 1.4, based on one in X.800, indicates the relationship between security
services and security mechanisms.
SPECIFIC SECURITY MECHANISMS
Authentication Exchange A mechanism intended to ensure the identity of an
entity by means of information exchange.
Traffic Padding The insertion of bits into gaps in a data stream to
frustrate traffic analysis attempts.
Routing Control Enables selection of particular physically secure
routes for certain data and allows routing changes,
especially when a breach of security is suspected.
Notarization The use of a trusted third party to assure certain
properties of a data exchange.
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Peer entity authentication
SERVICE
MECHANISM
En cip
he rm
en t
Di git
al sig
na tu
re
Ac ce
ss co
nt ro
l
Da ta
int eg
rit y
Au th
en tic
ati on
ex ch
an ge
Tr affi
c p ad
din g
Ro ut
ing co
nt ro
l
No tar
iza tio
n
Data origin authentication
Access control
Confidentiality
Traffic flow confidentiality
Data integrity
Nonrepudiation
Availability
Table 1.4 Relationship Between Security Services and Mechanisms
34 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS
1.6 FUNDAMENTAL SECURITY DESIGN PRINCIPLES
Despite years of research and development, it has not been possible to develop
security design and implementation techniques that systematically exclude security
flaws and prevent all unauthorized actions. In the absence of such foolproof tech-
niques, it is useful to have a set of widely agreed design principles that can guide
the development of protection mechanisms. The National Centers of Academic
Excellence in Information Assurance/Cyber Defense, which is jointly sponsored by
the U.S. National Security Agency and the U.S. Department of Homeland Security,
list the following as fundamental security design principles [NCAE13]:
■ Economy of mechanism
■ Fail-safe defaults
■ Complete mediation
■ Open design
■ Separation of privilege
■ Least privilege
■ Least common mechanism
■ Psychological acceptability
■ Isolation
■ Encapsulation
■ Modularity
■ Layering
■ Least astonishment
The first eight listed principles were first proposed in [SALT75] and have withstood
the test of time. In this section, we briefly discuss each principle.
Economy of mechanism means that the design of security measures embod- ied in both hardware and software should be as simple and small as possible.
The motivation for this principle is that relatively simple, small design is eas-
ier to test and verify thoroughly. With a complex design, there are many more
opportunities for an adversary to discover subtle weaknesses to exploit that may
be difficult to spot ahead of time. The more complex the mechanism, the more
likely it is to possess exploitable flaws. Simple mechanisms tend to have fewer
exploitable flaws and require less maintenance. Further, because configuration
management issues are simplified, updating or replacing a simple mechanism
becomes a less intensive process. In practice, this is perhaps the most difficult
principle to honor. There is a constant demand for new features in both hard-
ware and software, complicating the security design task. The best that can be
done is to keep this principle in mind during system design to try to eliminate
unnecessary complexity.
Fail-safe defaults means that access decisions should be based on permission rather than exclusion. That is, the default situation is lack of access, and the protec-
tion scheme identifies conditions under which access is permitted. This approach
1.6 / FUNDAMENTAL SECURITY DESIGN PRINCIPLES 35
exhibits a better failure mode than the alternative approach, where the default is
to permit access. A design or implementation mistake in a mechanism that gives
explicit permission tends to fail by refusing permission, a safe situation that can
be quickly detected. On the other hand, a design or implementation mistake in a
mechanism that explicitly excludes access tends to fail by allowing access, a failure
that may long go unnoticed in normal use. Most file access systems and virtually all
protected services on client/server systems use fail-safe defaults.
Complete mediation means that every access must be checked against the access control mechanism. Systems should not rely on access decisions retrieved
from a cache. In a system designed to operate continuously, this principle requires
that, if access decisions are remembered for future use, careful consideration be
given to how changes in authority are propagated into such local memories. File
access systems appear to provide an example of a system that complies with this
principle. However, typically, once a user has opened a file, no check is made to see
if permissions change. To fully implement complete mediation, every time a user
reads a field or record in a file, or a data item in a database, the system must exercise
access control. This resource-intensive approach is rarely used.
Open design means that the design of a security mechanism should be open rather than secret. For example, although encryption keys must be secret, encryption
algorithms should be open to public scrutiny. The algorithms can then be reviewed
by many experts, and users can therefore have high confidence in them. This is the
philosophy behind the National Institute of Standards and Technology (NIST)
program of standardizing encryption and hash algorithms, and has led to the wide-
spread adoption of NIST-approved algorithms.
Separation of privilege is defined in [SALT75] as a practice in which mul- tiple privilege attributes are required to achieve access to a restricted resource.
A good example of this is multifactor user authentication, which requires the use of
multiple techniques, such as a password and a smart card, to authorize a user. The
term is also now applied to any technique in which a program is divided into parts
that are limited to the specific privileges they require in order to perform a specific
task. This is used to mitigate the potential damage of a computer security attack.
One example of this latter interpretation of the principle is removing high privilege
operations to another process and running that process with the higher privileges
required to perform its tasks. Day-to-day interfaces are executed in a lower privi-
leged process.
Least privilege means that every process and every user of the system should operate using the least set of privileges necessary to perform the task. A good
example of the use of this principle is role-based access control. The system security
policy can identify and define the various roles of users or processes. Each role is
assigned only those permissions needed to perform its functions. Each permission
specifies a permitted access to a particular resource (such as read and write access
to a specified file or directory, connect access to a given host and port). Unless a
permission is granted explicitly, the user or process should not be able to access the
protected resource. More generally, any access control system should allow each
user only the privileges that are authorized for that user. There is also a temporal
aspect to the least privilege principle. For example, system programs or administra-
tors who have special privileges should have those privileges only when necessary;
36 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS
when they are doing ordinary activities the privileges should be withdrawn. Leaving
them in place just opens the door to accidents.
Least common mechanism means that the design should minimize the func- tions shared by different users, providing mutual security. This principle helps
reduce the number of unintended communication paths and reduces the amount of
hardware and software on which all users depend, thus making it easier to verify if
there are any undesirable security implications.
Psychological acceptability implies that the security mechanisms should not interfere unduly with the work of users, while at the same time meeting the needs of
those who authorize access. If security mechanisms hinder the usability or accessibil-
ity of resources, then users may opt to turn off those mechanisms. Where possible,
security mechanisms should be transparent to the users of the system or at most
introduce minimal obstruction. In addition to not being intrusive or burdensome,
security procedures must reflect the user’s mental model of protection. If the protec-
tion procedures do not make sense to the user or if the user must translate his image
of protection into a substantially different protocol, the user is likely to make errors.
Isolation is a principle that applies in three contexts. First, public access sys- tems should be isolated from critical resources (data, processes, etc.) to prevent dis-
closure or tampering. In cases where the sensitivity or criticality of the information
is high, organizations may want to limit the number of systems on which that data is
stored and isolate them, either physically or logically. Physical isolation may include
ensuring that no physical connection exists between an organization’s public access
information resources and an organization’s critical information. When implement-
ing logical isolation solutions, layers of security services and mechanisms should be
established between public systems and secure systems responsible for protecting
critical resources. Second, the processes and files of individual users should be iso-
lated from one another except where it is explicitly desired. All modern operating
systems provide facilities for such isolation, so that individual users have separate,
isolated process space, memory space, and file space, with protections for prevent-
ing unauthorized access. And finally, security mechanisms should be isolated in the
sense of preventing access to those mechanisms. For example, logical access control
may provide a means of isolating cryptographic software from other parts of the
host system and for protecting cryptographic software from tampering and the keys
from replacement or disclosure.
Encapsulation can be viewed as a specific form of isolation based on object- oriented functionality. Protection is provided by encapsulating a collection of pro-
cedures and data objects in a domain of its own so that the internal structure of a
data object is accessible only to the procedures of the protected subsystem, and the
procedures may be called only at designated domain entry points.
Modularity in the context of security refers both to the development of security functions as separate, protected modules and to the use of a modular architecture for
mechanism design and implementation. With respect to the use of separate security
modules, the design goal here is to provide common security functions and services,
such as cryptographic functions, as common modules. For example, numerous proto-
cols and applications make use of cryptographic functions. Rather than implement-
ing such functions in each protocol or application, a more secure design is provided
by developing a common cryptographic module that can be invoked by numerous
1.7 / ATTACK SURFACES AND ATTACK TREES 37
protocols and applications. The design and implementation effort can then focus on
the secure design and implementation of a single cryptographic module and includ-
ing mechanisms to protect the module from tampering. With respect to the use of a
modular architecture, each security mechanism should be able to support migration
to new technology or upgrade of new features without requiring an entire system
redesign. The security design should be modular so that individual parts of the secu-
rity design can be upgraded without the requirement to modify the entire system.
Layering refers to the use of multiple, overlapping protection approaches addressing the people, technology, and operational aspects of information systems.
By using multiple, overlapping protection approaches, the failure or circumven-
tion of any individual protection approach will not leave the system unprotected.
We will see throughout this book that a layering approach is often used to provide
multiple barriers between an adversary and protected information or services. This
technique is often referred to as defense in depth. Least astonishment means that a program or user interface should always
respond in the way that is least likely to astonish the user. For example, the mechanism
for authorization should be transparent enough to a user that the user has a good intui-
tive understanding of how the security goals map to the provided security mechanism.
1.7 ATTACK SURFACES AND ATTACK TREES
In Section 1.3, we provided an overview of the spectrum of security threats and
attacks facing computer and network systems. Section 22.1 goes into more detail
about the nature of attacks and the types of adversaries that present security threats.
In this section, we elaborate on two concepts that are useful in evaluating and clas-
sifying threats: attack surfaces and attack trees.
Attack Surfaces
An attack surface consists of the reachable and exploitable vulnerabilities in a sys-
tem [MANA11, HOWA03]. Examples of attack surfaces are the following:
■ Open ports on outward facing Web and other servers, and code listening on
those ports
■ Services available on the inside of a firewall
■ Code that processes incoming data, email, XML, office documents, and indus-
try-specific custom data exchange formats
■ Interfaces, SQL, and Web forms
■ An employee with access to sensitive information vulnerable to a social
engineering attack
Attack surfaces can be categorized as follows:
■ Network attack surface: This category refers to vulnerabilities over an enterprise network, wide-area network, or the Internet. Included in this category are net-
work protocol vulnerabilities, such as those used for a denial-of-service attack,
disruption of communications links, and various forms of intruder attacks.
Hiva-Network.Com
38 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS
■ Software attack surface: This refers to vulnerabilities in application, utility, or operating system code. A particular focus in this category is Web server
software.
■ Human attack surface: This category refers to vulnerabilities created by personnel or outsiders, such as social engineering, human error, and trusted
insiders.
An attack surface analysis is a useful technique for assessing the scale and
severity of threats to a system. A systematic analysis of points of vulnerability
makes developers and security analysts aware of where security mechanisms are
required. Once an attack surface is defined, designers may be able to find ways to
make the surface smaller, thus making the task of the adversary more difficult. The
attack surface also provides guidance on setting priorities for testing, strengthening
security measures, and modifying the service or application.
As illustrated in Figure 1.3, the use of layering, or defense in depth, and attack
surface reduction complement each other in mitigating security risk.
Attack Trees
An attack tree is a branching, hierarchical data structure that represents a set of poten-
tial techniques for exploiting security vulnerabilities [MAUW05, MOOR01, SCHN99].
The security incident that is the goal of the attack is represented as the root node of
the tree, and the ways that an attacker could reach that goal are iteratively and incre-
mentally represented as branches and subnodes of the tree. Each subnode defines a
subgoal, and each subgoal may have its own set of further subgoals, and so on. The
final nodes on the paths outward from the root, that is, the leaf nodes, represent differ-
ent ways to initiate an attack. Each node other than a leaf is either an AND-node or an
OR-node. To achieve the goal represented by an AND-node, the subgoals represented
by all of that node’s subnodes must be achieved; and for an OR-node, at least one of
the subgoals must be achieved. Branches can be labeled with values representing dif-
ficulty, cost, or other attack attributes, so that alternative attacks can be compared.
Figure 1.3 Defense in Depth and Attack Surface
Attack surface
Medium security risk
High security risk
Low security riskD
ee p
L ay
er in
g
Sh al
lo w
Small Large
Medium security risk
1.7 / ATTACK SURFACES AND ATTACK TREES 39
The motivation for the use of attack trees is to effectively exploit the infor-
mation available on attack patterns. Organizations such as CERT publish security
advisories that have enabled the development of a body of knowledge about both
general attack strategies and specific attack patterns. Security analysts can use the
attack tree to document security attacks in a structured form that reveals key vul-
nerabilities. The attack tree can guide both the design of systems and applications,
and the choice and strength of countermeasures.
Figure 1.4, based on a figure in [DIMI07], is an example of an attack tree
analysis for an Internet banking authentication application. The root of the tree is
the objective of the attacker, which is to compromise a user’s account. The shaded
boxes on the tree are the leaf nodes, which represent events that comprise the
attacks. Note that in this tree, all the nodes other than leaf nodes are OR-nodes.
The analysis to generate this tree considered the three components involved in
authentication:
Figure 1.4 An Attack Tree for Internet Banking Authentication
Bank account compromise
User credential compromise
User credential guessing
UT/U1a User surveillance
UT/U1b Theft of token and handwritten notes
Malicious software installation
Vulnerability exploit
UT/U2a Hidden code
UT/U2b Worms
UT/U3a Smartcard analyzers
UT/U2c Emails with malicious code
UT/U3b Smartcard reader manipulator
UT/U3c Brute force attacks with PIN calculators
CC2 Sniffing
UT/U4a Social engineering
IBS3 Web site manipulation
UT/U4b Web page obfuscation
CC1 Pharming
Redirection of communication toward fraudulent site
CC3 Active man-in-the middle attacks
IBS1 Brute force attacks
User communication with attacker
Injection of commands
Use of known authenticated session by attacker
Normal user authentication with specified session ID
CC4 Pre-defined session IDs (session hijacking)
IBS2 Security policy violation
40 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS
■ User terminal and user (UT/U): These attacks target the user equipment, including the tokens that may be involved, such as smartcards or other pass-
word generators, as well as the actions of the user.
■ Communications channel (CC): This type of attack focuses on communica- tion links.
■ Internet banking server (IBS): These types of attacks are offline attacks against the servers that host the Internet banking application.
Five overall attack strategies can be identified, each of which exploits one or
more of the three components. The five strategies are as follows:
■ User credential compromise: This strategy can be used against many ele- ments of the attack surface. There are procedural attacks, such as monitoring
a user’s action to observe a PIN or other credential, or theft of the user’s
token or handwritten notes. An adversary may also compromise token
information using a variety of token attack tools, such as hacking the smart-
card or using a brute force approach to guess the PIN. Another possible
strategy is to embed malicious software to compromise the user’s login and
password. An adversary may also attempt to obtain credential information
via the communication channel (sniffing). Finally, an adversary may use
various means to engage in communication with the target user, as shown
in Figure 1.4.
■ Injection of commands: In this type of attack, the attacker is able to intercept communication between the UT and the IBS. Various schemes can be used
to be able to impersonate the valid user and so gain access to the banking
system.
■ User credential guessing: It is reported in [HILT06] that brute force attacks against some banking authentication schemes are feasible by sending ran-
dom usernames and passwords. The attack mechanism is based on distributed
zombie personal computers, hosting automated programs for username- or
password-based calculation.
■ Security policy violation: For example, violating the bank’s security policy in combination with weak access control and logging mechanisms, an em-
ployee may cause an internal security incident and expose a customer’s
account.
■ Use of known authenticated session: This type of attack persuades or forces the user to connect to the IBS with a preset session ID. Once the user authen-
ticates to the server, the attacker may utilize the known session ID to send
packets to the IBS, spoofing the user’s identity.
Figure 1.4 provides a thorough view of the different types of attacks on an
Internet banking authentication application. Using this tree as a starting point, secu-
rity analysts can assess the risk of each attack and, using the design principles out-
lined in the preceding section, design a comprehensive security facility. [DIMO07]
provides a good account of the results of this design effort.
1.8 / A MODEL FOR NETWORK SECURITY 41
1.8 A MODEL FOR NETWORK SECURITY
A model for much of what we will be discussing is captured, in very general terms, in
Figure 1.5. A message is to be transferred from one party to another across some sort
of Internet service. The two parties, who are the principals in this transaction, must cooperate for the exchange to take place. A logical information channel is established
by defining a route through the Internet from source to destination and by the coop-
erative use of communication protocols (e.g., TCP/IP) by the two principals.
Security aspects come into play when it is necessary or desirable to protect the
information transmission from an opponent who may present a threat to confidentiality,
authenticity, and so on. All the techniques for providing security have two components:
■ A security-related transformation on the information to be sent. Examples
include the encryption of the message, which scrambles the message so that it
is unreadable by the opponent, and the addition of a code based on the con-
tents of the message, which can be used to verify the identity of the sender.
■ Some secret information shared by the two principals and, it is hoped,
unknown to the opponent. An example is an encryption key used in conjunc-
tion with the transformation to scramble the message before transmission
and unscramble it on reception.6
A trusted third party may be needed to achieve secure transmission. For
example, a third party may be responsible for distributing the secret information
6Part Two discusses a form of encryption, known as a symmetric encryption, in which only one of the two principals needs to have the secret information.
Figure 1.5 Model for Network Security
Information channelSecurity-related
transformation
Sender
Secret information
M es
sa ge
M es
sa ge
Se cu
re m
es sa
ge
Se cu
re m
es sa
ge
Recipient
Opponent
Trusted third party (e.g., arbiter, distributer of secret information)
Security-related transformation
Secret information
42 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS
to the two principals while keeping it from any opponent. Or a third party may be
needed to arbitrate disputes between the two principals concerning the authenticity
of a message transmission.
This general model shows that there are four basic tasks in designing a par-
ticular security service:
1. Design an algorithm for performing the security-related transformation. The algorithm should be such that an opponent cannot defeat its purpose.
2. Generate the secret information to be used with the algorithm.
3. Develop methods for the distribution and sharing of the secret information.
4. Specify a protocol to be used by the two principals that makes use of the security algorithm and the secret information to achieve a particular security
service.
Parts One through Five of this book concentrate on the types of security
mechanisms and services that fit into the model shown in Figure 1.5. However,
there are other security-related situations of interest that do not neatly fit this
model but are considered in this book. A general model of these other situations
is illustrated in Figure 1.6, which reflects a concern for protecting an information
system from unwanted access. Most readers are familiar with the concerns caused
by the existence of hackers, who attempt to penetrate systems that can be accessed
over a network. The hacker can be someone who, with no malign intent, simply gets
satisfaction from breaking and entering a computer system. The intruder can be a
disgruntled employee who wishes to do damage or a criminal who seeks to exploit
computer assets for financial gain (e.g., obtaining credit card numbers or perform-
ing illegal money transfers).
Another type of unwanted access is the placement in a computer system of
logic that exploits vulnerabilities in the system and that can affect application pro-
grams as well as utility programs, such as editors and compilers. Programs can pres-
ent two kinds of threats:
■ Information access threats: Intercept or modify data on behalf of users who should not have access to that data.
■ Service threats: Exploit service flaws in computers to inhibit use by legitimate users.
Figure 1.6 Network Access Security Model
Computing resources (processor, memory, I/O)
Data
Processes
Software
Internal security controls
Information system
Gatekeeper function
Opponent —human (e.g., hacker) —software (e.g., virus, worm)
Access channel
1.9 / STANDARDS 43
Viruses and worms are two examples of software attacks. Such attacks can be
introduced into a system by means of a disk that contains the unwanted logic con-
cealed in otherwise useful software. They can also be inserted into a system across a
network; this latter mechanism is of more concern in network security.
The security mechanisms needed to cope with unwanted access fall into two
broad categories (see Figure 1.6). The first category might be termed a gatekeeper
function. It includes password-based login procedures that are designed to deny
access to all but authorized users and screening logic that is designed to detect and
reject worms, viruses, and other similar attacks. Once either an unwanted user
or unwanted software gains access, the second line of defense consists of a vari-
ety of internal controls that monitor activity and analyze stored information in an
attempt to detect the presence of unwanted intruders. These issues are explored
in Part Six.
1.9 STANDARDS
Many of the security techniques and applications described in this book have been
specified as standards. Additionally, standards have been developed to cover man-
agement practices and the overall architecture of security mechanisms and services.
Throughout this book, we describe the most important standards in use or that are
being developed for various aspects of cryptography and network security. Various
organizations have been involved in the development or promotion of these stan-
dards. The most important (in the current context) of these organizations are as
follows:
■ National Institute of Standards and Technology: NIST is a U.S. federal agency that deals with measurement science, standards, and technology related to
U.S. government use and to the promotion of U.S. private-sector innovation.
Despite its national scope, NIST Federal Information Processing Standards
(FIPS) and Special Publications (SP) have a worldwide impact.
■ Internet Society: ISOC is a professional membership society with world- wide organizational and individual membership. It provides leadership in
addressing issues that confront the future of the Internet and is the organiza-
tion home for the groups responsible for Internet infrastructure standards,
including the Internet Engineering Task Force (IETF) and the Internet
Architecture Board (IAB). These organizations develop Internet stan-
dards and related specifications, all of which are published as Requests for
Comments (RFCs).
■ ITU-T: The International Telecommunication Union (ITU) is an interna- tional organization within the United Nations System in which governments
and the private sector coordinate global telecom networks and services. The
ITU Telecommunication Standardization Sector (ITU-T) is one of the three
sectors of the ITU. ITU-T’s mission is the development of technical standards
covering all fields of telecommunications. ITU-T standards are referred to as
Recommendations.
44 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS
■ ISO: The International Organization for Standardization (ISO)7 is a world- wide federation of national standards bodies from more than 140 countries,
one from each country. ISO is a nongovernmental organization that promotes
the development of standardization and related activities with a view to fa-
cilitating the international exchange of goods and services and to developing
cooperation in the spheres of intellectual, scientific, technological, and eco-
nomic activity. ISO’s work results in international agreements that are pub-
lished as International Standards.
A more detailed discussion of these organizations is contained in Appendix D.
1.10 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
7ISO is not an acronym (in which case it would be IOS), but it is a word, derived from the Greek, mean- ing equal.
Key Terms
access control
active attack
authentication
authenticity
availability
data confidentiality
data integrity
denial of service
encryption
integrity
intruder
masquerade
nonrepudiation
OSI security architecture
passive attack
replay
security attacks
security mechanisms
security services
traffic analysis
Review Questions
1.1 What is the OSI security architecture? 1.2 List and briefly define the three key objectives of computer security. 1.3 List and briefly define categories of passive and active security attacks. 1.4 List and briefly define categories of security services. 1.5 List and briefly define categories of security mechanisms. 1.6 List and briefly define the fundamental security design principles. 1.7 Explain the difference between an attack surface and an attack tree.
Problems
1.1 Consider an automated cash deposit machine in which users provide a card or an ac- count number to deposit cash. Give examples of confidentiality, integrity, and avail- ability requirements associated with the system, and, in each case, indicate the degree of importance of the requirement.
1.2 Repeat Problem 1.1 for a payment gateway system where a user pays for an item using their account via the payment gateway.
1.10 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 45
1.3 Consider a financial report publishing system used to produce reports for various organizations. a. Give an example of a type of publication in which confidentiality of the stored
data is the most important requirement. b. Give an example of a type of publication in which data integrity is the most im-
portant requirement. c. Give an example in which system availability is the most important requirement.
1.4 For each of the following assets, assign a low, moderate, or high impact level for the loss of confidentiality, availability, and integrity, respectively. Justify your answers. a. A student maintaining a blog to post public information. b. An examination section of a university that is managing sensitive information
about exam papers. c. An information system in a pathological laboratory maintaining the patient’s data. d. A student information system used for maintaining student data in a university
that contains both personal, academic information and routine administrative in- formation (not privacy related). Assess the impact for the two data sets separately and the information system as a whole.
e. A University library contains a library management system which controls the distribution of books amongst the students of various departments. The library management system contains both the student data and the book data. Assess the impact for the two data sets separately and the information system as a whole.
1.5 Draw a matrix similar to Table 1.4 that shows the relationship between security ser- vices and attacks.
1.6 Draw a matrix similar to Table 1.4 that shows the relationship between security mechanisms and attacks.
1.7 Develop an attack tree for gaining access to the contents of a physical safe. 1.8 Consider a company whose operations are housed in two buildings on the same prop-
erty; one building is headquarters, the other building contains network and computer services. The property is physically protected by a fence around the perimeter, and the only entrance to the property is through this fenced perimeter. In addition to the perimeter fence, physical security consists of a guarded front gate. The local net- works are split between the Headquarters’ LAN and the Network Services’ LAN. Internet users connect to the Web server through a firewall. Dial-up users get access to a particular server on the Network Services’ LAN. Develop an attack tree in which the root node represents disclosure of proprietary secrets. Include physical, social engineering, and technical attacks. The tree may contain both AND and OR nodes. Develop a tree that has at least 15 leaf nodes.
1.9 Read all of the classic papers cited in the Recommended Reading section for this chapter, available at the Author Web site at WilliamStallings.com/Cryptography. The papers are available at box.com/Crypto7e. Compose a 500–1000 word paper (or 8–12 slide PowerPoint presentation) that summarizes the key concepts that emerge from these papers, emphasizing concepts that are common to most or all of the papers.
4646
2.1 Divisibility and The Division Algorithm Divisibility
The Division Algorithm
2.2 The Euclidean Algorithm Greatest Common Divisor
Finding the Greatest Common Divisor
2.3 Modular Arithmetic The Modulus
Properties of Congruences
Modular Arithmetic Operations
Properties of Modular Arithmetic
Euclidean Algorithm Revisited
The Extended Euclidean Algorithm
2.4 Prime Numbers
2.5 Fermat’s and Euler’s Theorems
Fermat’s Theorem
Euler’s Totient Function
Euler’s Theorem
2.6 Testing for Primality
Miller–Rabin Algorithm
A Deterministic Primality Algorithm
Distribution of Primes
2.7 The Chinese Remainder Theorem
2.8 Discrete Logarithms
The Powers of an Integer, Modulo n Logarithms for Modular Arithmetic
Calculation of Discrete Logarithms
2.9 Key Terms, Review Questions, and Problems
Appendix 2A The Meaning of Mod
CHAPTER
Introduction to Number Theory
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2.1 / DIVISIBILITY AND THE DIVISION ALGORITHM 47
Number theory is pervasive in cryptographic algorithms. This chapter provides
sufficient breadth and depth of coverage of relevant number theory topics for under-
standing the wide range of applications in cryptography. The reader familiar with these
topics can safely skip this chapter.
The first three sections introduce basic concepts from number theory that are
needed for understanding finite fields; these include divisibility, the Euclidian algo-
rithm, and modular arithmetic. The reader may study these sections now or wait until
ready to tackle Chapter 5 on finite fields.
Sections 2.4 through 2.8 discuss aspects of number theory related to prime num-
bers and discrete logarithms. These topics are fundamental to the design of asymmetric
(public-key) cryptographic algorithms. The reader may study these sections now or
wait until ready to read Part Three.
The concepts and techniques of number theory are quite abstract, and it is often
difficult to grasp them intuitively without examples. Accordingly, this chapter includes
a number of examples, each of which is highlighted in a shaded box.
2.1 DIVISIBILITY AND THE DIVISION ALGORITHM
Divisibility
We say that a nonzero b divides a if a = mb for some m, where a, b, and m are integers. That is, b divides a if there is no remainder on division. The notation b�a is commonly used to mean b divides a. Also, if b�a, we say that b is a divisor of a.
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Understand the concept of divisibility and the division algorithm.
◆ Understand how to use the Euclidean algorithm to find the greatest com- mon divisor.
◆ Present an overview of the concepts of modular arithmetic.
◆ Explain the operation of the extended Euclidean algorithm.
◆ Discuss key concepts relating to prime numbers.
◆ Understand Fermat’s theorem.
◆ Understand Euler’s theorem.
◆ Define Euler’s totient function.
◆ Make a presentation on the topic of testing for primality.
◆ Explain the Chinese remainder theorem.
◆ Define discrete logarithms.
48 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
Subsequently, we will need some simple properties of divisibility for integers,
which are as follows:
■ If a�1, then a = {1. ■ If a�b and b�a, then a = {b. ■ Any b ≠ 0 divides 0. ■ If a�b and b�c, then a�c:
The positive divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
13�182; - 5�30; 17�289; - 3�33; 17�0
11�66 and 66�198 1 11�198
b = 7; g = 14; h = 63; m = 3; n = 2 7�14 and 7�63. To show 7�(3 * 14 + 2 * 63), we have (3 * 14 + 2 * 63) = 7(3 * 2 + 2 * 9), and it is obvious that 7�(7(3 * 2 + 2 * 9)).
■ If b�g and b�h, then b�(mg + nh) for arbitrary integers m and n.
To see this last point, note that
■ If b�g, then g is of the form g = b * g1 for some integer g1. ■ If b�h, then h is of the form h = b * h1 for some integer h1.
So
mg + nh = mbg1 + nbh1 = b * (mg1 + nh1)
and therefore b divides mg + nh.
The Division Algorithm
Given any positive integer n and any nonnegative integer a, if we divide a by n, we get an integer quotient q and an integer remainder r that obey the following relationship:
a = qn + r 0 … r 6 n; q = :a/n; (2.1) where :x; is the largest integer less than or equal to x. Equation (2.1) is referred to as the division algorithm.1
1Equation (2.1) expresses a theorem rather than an algorithm, but by tradition, this is referred to as the division algorithm.
2.2 / THE EUCLIDEAN ALGORITHM 49
Figure 2.1a demonstrates that, given a and positive n, it is always possible to find q and r that satisfy the preceding relationship. Represent the integers on the number line; a will fall somewhere on that line (positive a is shown, a similar dem- onstration can be made for negative a). Starting at 0, proceed to n, 2n, up to qn, such that qn … a and (q + 1)n 7 a. The distance from qn to a is r, and we have found the unique values of q and r. The remainder r is often referred to as a residue.
a = 11; n = 7; 11 = 1 * 7 + 4; r = 4 q = 1 a = - 11; n = 7; - 11 = ( - 2) * 7 + 3; r = 3 q = - 2
Figure 2.1b provides another example.
Figure 2.1 The Relationship a = qn + r; 0 … r 6 n
0
n 2n 3n qn (q + 1)na
n
r(a) General relationship
0 15
15
10
30 = 2 × 15
70
(b) Example: 70 = (4 × 15) + 10
45 = 3 × 15
60 = 4 × 15
75 = 5 × 15
2.2 THE EUCLIDEAN ALGORITHM
One of the basic techniques of number theory is the Euclidean algorithm, which
is a simple procedure for determining the greatest common divisor of two positive
integers. First, we need a simple definition: Two integers are relatively prime if and only if their only common positive integer factor is 1.
Greatest Common Divisor
Recall that nonzero b is defined to be a divisor of a if a = mb for some m, where a, b, and m are integers. We will use the notation gcd(a, b) to mean the greatest common divisor of a and b. The greatest common divisor of a and b is the largest integer that divides both a and b. We also define gcd(0, 0) = 0.
50 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
More formally, the positive integer c is said to be the greatest common divisor of a and b if
1. c is a divisor of a and of b.
2. any divisor of a and b is a divisor of c.
An equivalent definition is the following:
gcd(a, b) = max[k, such that k�a and k�b]
Because we require that the greatest common divisor be positive, gcd(a, b) = gcd(a, - b) = gcd( - a, b) = gcd( - a, - b). In general, gcd(a, b) = gcd(�a� , �b�).
gcd(60, 24) = gcd(60, - 24) = 12
8 and 15 are relatively prime because the positive divisors of 8 are 1, 2, 4, and 8, and
the positive divisors of 15 are 1, 3, 5, and 15. So 1 is the only integer on both lists.
Also, because all nonzero integers divide 0, we have gcd(a, 0) = �a� . We stated that two integers a and b are relatively prime if and only if their
only common positive integer factor is 1. This is equivalent to saying that a and b are relatively prime if gcd(a, b) = 1.
Finding the Greatest Common Divisor
We now describe an algorithm credited to Euclid for easily finding the greatest
common divisor of two integers (Figure 2.2). This algorithm has broad significance
in cryptography. The explanation of the algorithm can be broken down into the fol-
lowing points:
1. Suppose we wish to determine the greatest common divisor d of the integers a and b; that is determine d = gcd(a, b). Because gcd(�a� , �b�) = gcd(a, b), there is no harm in assuming a Ú b 7 0.
2. Dividing a by b and applying the division algorithm, we can state:
a = q1b + r1 0 … r1 6 b (2.2)
3. First consider the case in which r1 = 0. Therefore b divides a and clearly no larger number divides both b and a, because that number would be larger than b. So we have d = gcd(a, b) = b.
4. The other possibility from Equation (2.2) is r1 ≠ 0. For this case, we can state that d�r1. This is due to the basic properties of divisibility: the relations d�a and d�b together imply that d�(a - q1b), which is the same as d�r1.
5. Before proceeding with the Euclidian algorithm, we need to answer the ques- tion: What is the gcd(b, r1)? We know that d�b and d�r1. Now take any arbi- trary integer c that divides both b and r1. Therefore, c�(q1b + r1) = a. Because c divides both a and b, we must have c … d, which is the greatest common divisor of a and b. Therefore d = gcd(b, r1).
2.2 / THE EUCLIDEAN ALGORITHM 51
Let us now return to Equation (2.2) and assume that r1 ≠ 0. Because b 7 r1, we can divide b by r1 and apply the division algorithm to obtain:
b = q2r1 + r2 0 … r2 6 r1
As before, if r2 = 0, then d = r1 and if r2 ≠ 0, then d = gcd(r1, r2). Note that the remainders form a descending series of nonnegative values and so must terminate
when the remainder is zero. This happens, say, at the (n + 1)th stage where rn - 1 is divided by rn. The result is the following system of equations:
a = q1b + r1 0 6 r1 6 b b = q2r1 + r2 0 6 r2 6 r1 r1 = q3r2 + r3 0 6 r3 6 r2
~ ~
~ ~
~ ~
rn - 2 = qnrn - 1 + rn 0 6 rn 6 rn - 1 rn - 1 = qn + 1rn + 0 d = gcd(a, b) = rn
w (2.3) At each iteration, we have d = gcd(ri, ri + 1) until finally d = gcd(rn, 0) = rn.
Thus, we can find the greatest common divisor of two integers by repetitive appli-
cation of the division algorithm. This scheme is known as the Euclidean algorithm.
Figure 2.3 illustrates a simple example.
We have essentially argued from the top down that the final result is the
gcd(a, b). We can also argue from the bottom up. The first step is to show that rn divides a and b. It follows from the last division in Equation (2.3) that rn divides rn - 1. The next to last division shows that rn divides rn - 2 because it divides both
Figure 2.2 Euclidean Algorithm
No
No Yes a > b?
r > 0? Swap
a and b
Replace b with r
Replace a with b
Divide a by b, calling the
remainder r
GCD is the final
value of b
START
END Figure 2.3 Euclidean Algorithm Example: gcd(710, 310)
710 = 2 × 310 + 90
310 = 3 × 90 + 40
90 = 2 × 40 + 10
40 = 4 × 10
GCDGCD
Same GCD
52 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
terms on the right. Successively, one sees that rn divides all ri>s and finally a and b. It remains to show that rn is the largest divisor that divides a and b. If we take any arbitrary integer that divides a and b, it must also divide r1, as explained previously. We can follow the sequence of equations in Equation (2.3) down and show that c must divide all ri>s. Therefore c must divide rn, so that rn = gcd(a, b).
Let us now look at an example with relatively large numbers to see the power
of this algorithm:
To find d = gcd(a, b) = gcd(1160718174, 316258250)
a = q1b + r1 1160718174 = 3 * 316258250 + 211943424 d = gcd(316258250, 211943424) b = q2r1 + r2 316258250 = 1 * 211943424 + 104314826 d = gcd(211943424, 104314826) r1 = q3r2 + r3 211943424 = 2 * 104314826 + 3313772 d = gcd(104314826, 3313772) r2 = q4r3 + r4 104314826 = 31 * 3313772 + 1587894 d = gcd(3313772, 1587894) r3 = q5r4 + r5 3313772 = 2 * 1587894 + 137984 d = gcd(1587894, 137984) r4 = q6r5 + r6 1587894 = 11 * 137984 + 70070 d = gcd(137984, 70070) r5 = q7r6 + r7 137984 = 1 * 70070 + 67914 d = gcd(70070, 67914) r6 = q8r7 + r8 70070 = 1 * 67914 + 2156 d = gcd(67914, 2156) r7 = q9r8 + r9 67914 = 31 * 2156 + 1078 d = gcd(2156, 1078) r8 = q10r9 + r10 2156 = 2 * 1078 + 0 d = gcd(1078, 0) = 1078 Therefore, d = gcd(1160718174, 316258250) = 1078
In this example, we begin by dividing 1160718174 by 316258250, which gives 3
with a remainder of 211943424. Next we take 316258250 and divide it by 211943424.
The process continues until we get a remainder of 0, yielding a result of 1078.
It will be helpful in what follows to recast the above computation in tabular
form. For every step of the iteration, we have ri - 2 = qiri - 1 + ri, where ri - 2 is the dividend, ri - 1 is the divisor, qi is the quotient, and ri is the remainder. Table 2.1 sum- marizes the results.
Dividend Divisor Quotient Remainder
a = 1160718174 b = 316258250 q1 = 3 r1 = 211943424
b = 316258250 r1 = 211943434 q2 = 1 r2 = 104314826
r1 = 211943424 r2 = 104314826 q3 = 2 r3 = 3313772
r2 = 104314826 r3 = 3313772 q4 = 31 r4 = 1587894
r3 = 3313772 r4 = 1587894 q5 = 2 r5 = 137984
r4 = 1587894 r5 = 137984 q6 = 11 r6 = 70070
r5 = 137984 r6 = 70070 q7 = 1 r7 = 67914
r6 = 70070 r7 = 67914 q8 = 1 r8 = 2156
r7 = 67914 r8 = 2156 q9 = 31 r9 = 1078
r8 = 2156 r9 = 1078 q10 = 2 r10 = 0
Table 2.1 Euclidean Algorithm Example
2.3 / MODULAR ARITHMETIC 53
2.3 MODULAR ARITHMETIC
The Modulus
If a is an integer and n is a positive integer, we define a mod n to be the remainder when a is divided by n. The integer n is called the modulus. Thus, for any integer a, we can rewrite Equation (2.1) as follows:
a = qn + r 0 … r 6 n; q = :a/n; a = :a/n; * n + (a mod n)
11 mod 7 = 4; - 11 mod 7 = 3
73 K 4 (mod 23); 21 K - 9 (mod 10)
Two integers a and b are said to be congruent modulo n, if (a mod n) = (b mod n). This is written as a K b (mod n).2
2We have just used the operator mod in two different ways: first as a binary operator that produces a re- mainder, as in the expression a mod b; second as a congruence relation that shows the equivalence of two integers, as in the expression a K b (mod n). See Appendix 2A for a discussion.
Note that if a K 0 (mod n), then n�a.
Properties of Congruences
Congruences have the following properties:
1. a K b (mod n) if n�(a - b). 2. a K b (mod n) implies b K a (mod n). 3. a K b (mod n) and b K c (mod n) imply a K c (mod n).
To demonstrate the first point, if n�(a - b), then (a - b) = kn for some k. So we can write a = b + kn. Therefore, (a mod n) = (remainder when b + kn is divided by n) = (remainder when b is divided by n) = (b mod n).
23 K 8 (mod 5) because 23 - 8 = 15 = 5 * 3 - 11 K 5 (mod 8) because - 11 - 5 = - 16 = 8 * ( - 2) 81 K 0 (mod 27) because 81 - 0 = 81 = 27 * 3
The remaining points are as easily proved.
54 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
Modular Arithmetic Operations
Note that, by definition (Figure 2.1), the (mod n) operator maps all integers into the set of integers {0, 1, c , (n - 1)}. This suggests the question: Can we perform arithmetic operations within the confines of this set? It turns out that we can; this
technique is known as modular arithmetic. Modular arithmetic exhibits the following properties:
1. [(a mod n) + (b mod n)] mod n = (a + b) mod n 2. [(a mod n) - (b mod n)] mod n = (a - b) mod n 3. [(a mod n) * (b mod n)] mod n = (a * b) mod n
We demonstrate the first property. Define (a mod n) = ra and (b mod n) = rb. Then we can write a = ra + jn for some integer j and b = rb + kn for some integer k. Then
(a + b) mod n = (ra + jn + rb + kn) mod n = (ra + rb + (k + j)n) mod n = (ra + rb) mod n = [(a mod n) + (b mod n)] mod n
The remaining properties are proven as easily. Here are examples of the three
properties:
11 mod 8 = 3; 15 mod 8 = 7 [(11 mod 8) + (15 mod 8)] mod 8 = 10 mod 8 = 2 (11 + 15) mod 8 = 26 mod 8 = 2 [(11 mod 8) - (15 mod 8)] mod 8 = - 4 mod 8 = 4 (11 - 15) mod 8 = - 4 mod 8 = 4 [(11 mod 8) * (15 mod 8)] mod 8 = 21 mod 8 = 5 (11 * 15) mod 8 = 165 mod 8 = 5
To find 117 mod 13, we can proceed as follows:
112 = 121 K 4 (mod 13) 114 = (112)2 K 42 K 3 (mod 13) 117 = 11 * 112 * 114
117 K 11 * 4 * 3 K 132 K 2 (mod 13)
Exponentiation is performed by repeated multiplication, as in ordinary
arithmetic.
Thus, the rules for ordinary arithmetic involving addition, subtraction, and
multiplication carry over into modular arithmetic.
2.3 / MODULAR ARITHMETIC 55
Table 2.2 provides an illustration of modular addition and multiplication
modulo 8. Looking at addition, the results are straightforward, and there is a reg-
ular pattern to the matrix. Both matrices are symmetric about the main diagonal
in conformance to the commutative property of addition and multiplication. As in
ordinary addition, there is an additive inverse, or negative, to each integer in modu-
lar arithmetic. In this case, the negative of an integer x is the integer y such that (x + y) mod 8 = 0. To find the additive inverse of an integer in the left-hand col- umn, scan across the corresponding row of the matrix to find the value 0; the integer
at the top of that column is the additive inverse; thus, (2 + 6) mod 8 = 0. Similarly, the entries in the multiplication table are straightforward. In modular arithmetic mod
8, the multiplicative inverse of x is the integer y such that (x * y) mod 8 = 1 mod 8. Now, to find the multiplicative inverse of an integer from the multiplication table,
scan across the matrix in the row for that integer to find the value 1; the integer at
the top of that column is the multiplicative inverse; thus, (3 * 3) mod 8 = 1. Note that not all integers mod 8 have a multiplicative inverse; more about that later.
Properties of Modular Arithmetic
Define the set Zn as the set of nonnegative integers less than n:
Zn = {0, 1, c , (n - 1)}
Table 2.2 Arithmetic Modulo 8 + 0 1 2 3 4 5 6 7
0 0 1 2 3 4 5 6 7
1 1 2 3 4 5 6 7 0
2 2 3 4 5 6 7 0 1
3 3 4 5 6 7 0 1 2
4 4 5 6 7 0 1 2 3
5 5 6 7 0 1 2 3 4
6 6 7 0 1 2 3 4 5
7 7 0 1 2 3 4 5 6
(a) Addition modulo 8
* 0 1 2 3 4 5 6 7
0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7
2 0 2 4 6 0 2 4 6
3 0 3 6 1 4 7 2 5
4 0 4 0 4 0 4 0 4
5 0 5 2 7 4 1 6 3
6 0 6 4 2 0 6 4 2
7 0 7 6 5 4 3 2 1
(b) Multiplication modulo 8
w - w w -1
0 0 —
1 7 1
2 6 —
3 5 3
4 4 —
5 3 5
6 2 —
7 1 7
(c) Additive and multiplicative
inverse modulo 8
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56 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
This is referred to as the set of residues, or residue classes (mod n). To be more pre- cise, each integer in Zn represents a residue class. We can label the residue classes
(mod n) as [0], [1], [2], c , [n - 1], where
[r] = {a: a is an integer, a K r (mod n)}
The residue classes (mod 4) are
[0] = { c , - 16, - 12, - 8, - 4, 0, 4, 8, 12, 16, c } [1] = { c , - 15, - 11, - 7, - 3, 1, 5, 9, 13, 17, c } [2] = { c , - 14, - 10, - 6, - 2, 2, 6, 10, 14, 18, c } [3] = { c , - 13, - 9, - 5, - 1, 3, 7, 11, 15, 19, c }
Property Expression
Commutative Laws (w + x) mod n = (x + w) mod n (w * x) mod n = (x * w) mod n
Associative Laws [(w + x) + y] mod n = [w + (x + y)] mod n [(w * x) * y] mod n = [w * (x * y)] mod n
Distributive Law [w * (x + y)] mod n = [(w * x) + (w * y)] mod n
Identities (0 + w) mod n = w mod n (1 * w) mod n = w mod n
Additive Inverse ( - w) For each w ∈ Zn, there exists a z such that w + z K 0 mod n
Table 2.3 Properties of Modular Arithmetic for Integers in Zn
Of all the integers in a residue class, the smallest nonnegative integer is the
one used to represent the residue class. Finding the smallest nonnegative integer to
which k is congruent modulo n is called reducing k modulo n. If we perform modular arithmetic within Zn, the properties shown in Table 2.3
hold for integers in Zn. We show in the next section that this implies that Zn is a
commutative ring with a multiplicative identity element.
There is one peculiarity of modular arithmetic that sets it apart from ordinary
arithmetic. First, observe that (as in ordinary arithmetic) we can write the following:
if (a + b) K (a + c) (mod n) then b K c (mod n) (2.4)
(5 + 23) K (5 + 7)(mod 8); 23 K 7(mod 8)
Equation (2.4) is consistent with the existence of an additive inverse. Adding
the additive inverse of a to both sides of Equation (2.4), we have
(( - a) + a + b) K (( - a) + a + c)(mod n) b K c (mod n)
2.3 / MODULAR ARITHMETIC 57
However, the following statement is true only with the attached condition:
if (a * b) K (a * c)(mod n) then b K c(mod n) if a is relatively prime to n (2.5)
Recall that two integers are relatively prime if their only common positive integer factor is 1. Similar to the case of Equation (2.4), we can say that Equation (2.5) is
consistent with the existence of a multiplicative inverse. Applying the multiplicative
inverse of a to both sides of Equation (2.5), we have
((a-1)ab) K ((a-1)ac)(mod n) b K c(mod n)
To see this, consider an example in which the condition of Equation (2.5) does not
hold. The integers 6 and 8 are not relatively prime, since they have the common
factor 2. We have the following:
6 * 3 = 18 K 2(mod 8) 6 * 7 = 42 K 2(mod 8)
Yet 3 [ 7 (mod 8).
The reason for this strange result is that for any general modulus n, a multi- plier a that is applied in turn to the integers 0 through (n - 1) will fail to produce a complete set of residues if a and n have any factors in common.
With a = 6 and n = 8,
Z8 0 1 2 3 4 5 6 7
Multiply by 6 0 6 12 18 24 30 36 42
Residues 0 6 4 2 0 6 4 2
Because we do not have a complete set of residues when multiplying by
6, more than one integer in Z8 maps into the same residue. Specifically,
6 * 0 mod 8 = 6 * 4 mod 8; 6 * 1 mod 8 = 6 * 5 mod 8; and so on. Because this is a many-to-one mapping, there is not a unique inverse to the multiply
operation.
However, if we take a = 5 and n = 8, whose only common factor is 1,
Z8 0 1 2 3 4 5 6 7
Multiply by 5 0 5 10 15 20 25 30 35
Residues 0 5 2 7 4 1 6 3
The line of residues contains all the integers in Z8, in a different order.
58 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
In general, an integer has a multiplicative inverse in Zn if and only if that inte-
ger is relatively prime to n. Table 2.2c shows that the integers 1, 3, 5, and 7 have a multiplicative inverse in Z8; but 2, 4, and 6 do not.
Euclidean Algorithm Revisited
The Euclidean algorithm can be based on the following theorem: For any integers
a, b, with a Ú b Ú 0,
gcd(a, b) = gcd(b, a mod b) (2.6)
gcd(55, 22) = gcd(22, 55 mod 22) = gcd(22, 11) = 11
gcd(18, 12) = gcd(12, 6) = gcd(6, 0) = 6 gcd(11, 10) = gcd(10, 1) = gcd(1, 0) = 1
To see that Equation (2.6) works, let d = gcd(a, b). Then, by the definition of gcd, d�a and d�b. For any positive integer b, we can express a as
a = kb + r K r (mod b) a mod b = r
with k, r integers. Therefore, (a mod b) = a - kb for some integer k. But because d�b, it also divides kb. We also have d�a. Therefore, d�(a mod b). This shows that d is a common divisor of b and (a mod b). Conversely, if d is a common divisor of b and (a mod b), then d�kb and thus d�[kb + (a mod b)], which is equivalent to d�a. Thus, the set of common divisors of a and b is equal to the set of common divisors of b and (a mod b). Therefore, the gcd of one pair is the same as the gcd of the other pair, proving the theorem.
Equation (2.6) can be used repetitively to determine the greatest common divisor.
This is the same scheme shown in Equation (2.3), which can be rewritten in
the following way.
Euclidean Algorithm
Calculate Which satisfies
r1 = a mod b a = q1b + r1 r2 = b mod r1 b = q2r1 + r2 r3 = r1 mod r2 r1 = q3r2 + r3
~
~
~
~
~
~
rn = rn - 2 mod rn - 1 rn - 2 = qnrn - 1 + rn rn + 1 = rn - 1 mod rn = 0 rn - 1 = qn + 1rn + 0
d = gcd(a, b) = rn
We can define the Euclidean algorithm concisely as the following recursive
function.
2.3 / MODULAR ARITHMETIC 59
Euclid(a,b) if (b=0) then return a; else return Euclid(b, a mod b);
The Extended Euclidean Algorithm
We now proceed to look at an extension to the Euclidean algorithm that will be
important for later computations in the area of finite fields and in encryption algo-
rithms, such as RSA. For given integers a and b, the extended Euclidean algorithm not only calculates the greatest common divisor d but also two additional integers x and y that satisfy the following equation.
ax + by = d = gcd(a, b) (2.7)
It should be clear that x and y will have opposite signs. Before examining the algorithm, let us look at some of the values of x and y when a = 42 and b = 30. Note that gcd(42, 30) = 6. Here is a partial table of values3 for 42x + 30y.
x − 3 − 2 − 1 0 1 2 3
y
- 3 - 216 - 174 - 132 - 90 - 48 - 6 36 - 2 - 186 - 144 - 102 - 60 - 18 24 66 - 1 - 156 - 114 - 72 - 30 12 54 96
0 - 126 - 84 - 42 0 42 84 126 1 - 96 - 54 - 12 30 72 114 156 2 - 66 - 24 18 60 102 144 186 3 - 36 6 48 90 132 174 216
Observe that all of the entries are divisible by 6. This is not surpris-
ing, because both 42 and 30 are divisible by 6, so every number of the form
42x + 30y = 6(7x + 5y) is a multiple of 6. Note also that gcd(42, 30) = 6 appears in the table. In general, it can be shown that for given integers a and b, the smallest positive value of ax + by is equal to gcd(a, b).
Now let us show how to extend the Euclidean algorithm to determine (x, y, d) given a and b. We again go through the sequence of divisions indicated in Equation (2.3), and we assume that at each step i we can find integers xi and yi that satisfy ri = axi + byi. We end up with the following sequence.
a = q1b + r1 r1 = ax1 + by1 b = q2r1 + r2 r2 = ax2 + by2 r1 = q3r2 + r3 r3 = ax3 + by3
f f rn - 2 = qnrn - 1 + rn rn = axn + byn rn - 1 = qn + 1rn + 0
3This example is taken from [SILV06].
60 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
Now, observe that we can rearrange terms to write
ri = ri - 2 - ri - 1qi (2.8)
Also, in rows i - 1 and i - 2, we find the values
ri - 2 = axi - 2 + byi - 2 and ri - 1 = axi - 1 + byi - 1
Substituting into Equation (2.8), we have
ri = (axi - 2 + byi - 2) - (axi - 1 + byi - 1)qi = a(xi - 2 - qixi - 1) + b(yi - 2 - qiyi - 1)
But we have already assumed that ri = axi + byi. Therefore,
xi = xi - 2 - qixi - 1 and yi = yi - 2 - qiyi - 1
We now summarize the calculations:
Extended Euclidean Algorithm
Calculate Which satisfies Calculate Which satisfies
r-1 = a x-1 = 1; y-1 = 0 a = ax-1 + by-1 r0 = b x0 = 0; y0 = 1 b = ax0 + by0 r1 = a mod b q1 = :a/b;
a = q1b + r1 x1 = x-1 - q1x0 = 1 y1 = y-1 - q1y0 = - q1
r1 = ax1 + by1
r2 = b mod r1 q2 = :b/r1;
b = q2r1 + r2 x2 = x0 - q2x1 y2 = y0 - q2y1
r2 = ax2 + by2
r3 = r1 mod r2 q3 = :r1/r2;
r1 = q3r2 + r3 x3 = x1 - q3x2 y3 = y1 - q3y2
r3 = ax3 + by3
~
~
~
~
~
~
~
~
~
~
~
~
rn = rn - 2 mod rn - 1 qn = :rn - 2/rn - 1;
rn - 2 = qnrn - 1 + rn xn = xn - 2 - qnxn - 1 yn = yn - 2 - qnyn - 1
rn = axn + byn
rn + 1 = rn - 1 mod rn = 0 qn + 1 = :rn - 1/rn;
rn - 1 = qn + 1rn + 0 d = gcd(a, b) = rn x = xn; y = yn
We need to make several additional comments here. In each row, we calculate
a new remainder ri based on the remainders of the previous two rows, namely ri - 1 and ri - 2. To start the algorithm, we need values for r0 and r-1, which are just a and b. It is then straightforward to determine the required values for x-1, y-1, x0, and y0.
We know from the original Euclidean algorithm that the process ends
with a remainder of zero and that the greatest common divisor of a and b is d = gcd(a, b) = rn. But we also have determined that d = rn = axn + byn. Therefore, in Equation (2.7), x = xn and y = yn.
As an example, let us use a = 1759 and b = 550 and solve for 1759x + 550y = gcd(1759, 550). The results are shown in Table 2.4. Thus, we have 1759 * ( - 111) + 550 * 355 = - 195249 + 195250 = 1.
2.4 / PRIME NUMBERS 61
2.4 PRIME NUMBERS4
A central concern of number theory is the study of prime numbers. Indeed, whole
books have been written on the subject (e.g., [CRAN01], [RIBE96]). In this section,
we provide an overview relevant to the concerns of this book.
An integer p 7 1 is a prime number if and only if its only divisors5 are {1 and {p. Prime numbers play a critical role in number theory and in the techniques dis- cussed in this chapter. Table 2.5 shows the primes less than 2000. Note the way the
primes are distributed. In particular, note the number of primes in each range of
100 numbers.
Any integer a 7 1 can be factored in a unique way as
a = p1 a1 * p2a2 * g * ptat (2.9)
where p1 6 p2 6 c 6 pt are prime numbers and where each ai is a positive inte- ger. This is known as the fundamental theorem of arithmetic; a proof can be found
in any text on number theory.
4In this section, unless otherwise noted, we deal only with the nonnegative integers. The use of negative integers would introduce no essential differences. 5Recall from Section 2.1 that integer a is said to be a divisor of integer b if there is no remainder on division. Equivalently, we say that a divides b.
i ri qi xi yi
- 1 1759 1 0
0 550 0 1
1 109 3 1 - 3
2 5 5 - 5 16
3 4 21 106 - 339
4 1 1 - 111 355
5 0 4
Result: d = 1; x = - 111; y = 355
Table 2.4 Extended Euclidean Algorithm Example
91 = 7 * 13 3600 = 24 * 32 * 52
11011 = 7 * 112 * 13
It is useful for what follows to express this another way. If P is the set of
all prime numbers, then any positive integer a can be written uniquely in the following form:
a = q p∈P
pap where each ap Ú 0
62 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
2 1 0 1
2 1 1
3 0 7
4 0 1
5 0 3
6 0 1
7 0 1
8 0 9
9 0 7
1 0 0 9
1 1 0 3
1 2 0 1
1 3 0 1
1 4 0 9
1 5 1 1
1 6 0 1
1 7 0 9
1 8 0 1
1 9 0 1
3 1
0 3
2 2 3
3 1 1
4 0 9
5 0 9
6 0 7
7 0 9
8 1 1
9 1 1
1 0 1 3
1 1 0 9
1 2 1 3
1 3 0 3
1 4 2 3
1 5 2 3
1 6 0 7
1 7 2 1
1 8 1 1
1 9 0 7
5 1 0 7
2 2 7
3 1 3
4 1 9
5 2 1
6 1 3
7 1 9
8 2 1
9 1 9
1 0 1 9
1 1 1 7
1 2 1 7
1 3 0 7
1 4 2 7
1 5 3 1
1 6 0 9
1 7 2 3
1 8 2 3
1 9 1 3
7 1
0 9
2 2 9
3 1 7
4 2 1
5 2 3
6 1 7
7 2 7
8 2 3
9 2 9
1 0 2 1
1 1 2 3
1 2 2 3
1 3 1 9
1 4 2 9
1 5 4 3
1 6 1 3
1 7 3 3
1 8 3 1
1 9 3 1
1 1
1 1 3
2 3 3
3 3 1
4 3 1
5 4 1
6 1 9
7 3 3
8 2 7
9 3 7
1 0 3 1
1 1 2 9
1 2 2 9
1 3 2 1
1 4 3 3
1 5 4 9
1 6 1 9
1 7 4 1
1 8 4 7
1 9 3 3
1 3
1 2 7
2 3 9
3 3 7
4 3 3
5 4 7
6 3 1
7 3 9
8 2 9
9 4 1
1 0 3 3
1 1 5 1
1 2 3 1
1 3 2 7
1 4 3 9
1 5 5 3
1 6 2 1
1 7 4 7
1 8 6 1
1 9 4 9
1 7
1 3 1
2 4 1
3 4 7
4 3 9
5 5 7
6 4 1
7 4 3
8 3 9
9 4 7
1 0 3 9
1 1 5 3
1 2 3 7
1 3 6 1
1 4 4 7
1 5 5 9
1 6 2 7
1 7 5 3
1 8 6 7
1 9 5 1
1 9
1 3 7
2 5 1
3 4 9
4 4 3
5 6 3
6 4 3
7 5 1
8 5 3
9 5 3
1 0 4 9
1 1 6 3
1 2 4 9
1 3 6 7
1 4 5 1
1 5 6 7
1 6 3 7
1 7 5 9
1 8 7 1
1 9 7 3
2 3
1 3 9
2 5 7
3 5 3
4 4 9
5 6 9
6 4 7
7 5 7
8 5 7
9 6 7
1 0 5 1
1 1 7 1
1 2 5 9
1 3 7 3
1 4 5 3
1 5 7 1
1 6 5 7
1 7 7 7
1 8 7 3
1 9 7 9
2 9
1 4 9
2 6 3
3 5 9
4 5 7
5 7 1
6 5 3
7 6 1
8 5 9
9 7 1
1 0 6 1
1 1 8 1
1 2 7 7
1 3 8 1
1 4 5 9
1 5 7 9
1 6 6 3
1 7 8 3
1 8 7 7
1 9 8 7
3 1
1 5 1
2 6 9
3 6 7
4 6 1
5 7 7
6 5 9
7 6 9
8 6 3
9 7 7
1 0 6 3
1 1 8 7
1 2 7 9
1 3 9 9
1 4 7 1
1 5 8 3
1 6 6 7
1 7 8 7
1 8 7 9
1 9 9 3
3 7
1 5 7
2 7 1
3 7 3
4 6 3
5 8 7
6 6 1
7 7 3
8 7 7
9 8 3
1 0 6 9
1 1 9 3
1 2 8 3
1 4 8 1
1 5 9 7
1 6 6 9
1 7 8 9
1 8 8 9
1 9 9 7
4 1
1 6 3
2 7 7
3 7 9
4 6 7
5 9 3
6 7 3
7 8 7
8 8 1
9 9 1
1 0 8 7
1 2 8 9
1 4 8 3
1 6 9 3
1 9 9 9
4 3
1 6 7
2 8 1
3 8 3
4 7 9
5 9 9
6 7 7
7 9 7
8 8 3
9 9 7
1 0 9 1
1 2 9 1
1 4 8 7
1 6 9 7
4 7
1 7 3
2 8 3
3 8 9
4 8 7
6 8 3
8 8 7
1 0 9 3
1 2 9 7
1 4 8 9
1 6 9 9
5 3
1 7 9
2 9 3
3 9 7
4 9 1
6 9 1
1 0 9 7
1 4 9 3
5 9
1 8 1
4 9 9
1 4 9 9
6 1
1 9 1
6 7
1 9 3
7 1
1 9 7
7 3
1 9 9
7 9
8 3
8 9
9 7
T ab
le 2
.5
P ri
m e s
U n
d e r
2 0
0 0
2.4 / PRIME NUMBERS 63
The right-hand side is the product over all possible prime numbers p; for any par- ticular value of a, most of the exponents ap will be 0.
The value of any given positive integer can be specified by simply listing all the
nonzero exponents in the foregoing formulation.
The integer 12 is represented by {a2 = 2, a3 = 1}. The integer 18 is represented by {a2 = 1, a3 = 2}. The integer 91 is represented by {a7 = 1, a13 = 1}.
Multiplication of two numbers is equivalent to adding the corresponding
exponents. Given a = q p∈P
pap, b = q p∈P
pbp. Define k = ab. We know that the inte-
ger k can be expressed as the product of powers of primes: k = q p∈P
pkp. It follows that kp = ap + bp for all p ∈ P.
k = 12 * 18 = (22 * 3) * (2 * 32) = 216 k2 = 2 + 1 = 3; k3 = 1 + 2 = 3 216 = 23 * 33 = 8 * 27
a = 12; b = 36; 12�36 12 = 22 * 3; 36 = 22 * 32
a2 = 2 = b2 a3 = 1 … 2 = b3 Thus, the inequality ap … bp is satisfied for all prime numbers.
What does it mean, in terms of the prime factors of a and b, to say that a divides b? Any integer of the form pn can be divided only by an integer that is of a lesser or equal power of the same prime number, pj with j … n. Thus, we can say the following.
Given
a = q p∈P
pap, b = q p∈P
pbp
If a�b, then ap … bp for all p.
It is easy to determine the greatest common divisor of two positive integers if
we express each integer as the product of primes.
64 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
The following relationship always holds:
If k = gcd(a, b), then kp = min(ap, bp) for all p.
Determining the prime factors of a large number is no easy task, so the pre-
ceding relationship does not directly lead to a practical method of calculating the
greatest common divisor.
2.5 FERMAT’S AND EULER’S THEOREMS
Two theorems that play important roles in public-key cryptography are Fermat’s
theorem and Euler’s theorem.
Fermat’s Theorem6
Fermat’s theorem states the following: If p is prime and a is a positive integer not divisible by p, then
ap - 1 K 1 (mod p) (2.10)
Proof: Consider the set of positive integers less than p: {1, 2, c , p - 1} and mul- tiply each element by a, modulo p, to get the set X = {a mod p, 2a mod p, c , (p - 1)a mod p}. None of the elements of X is equal to zero because p does not divide a. Furthermore, no two of the integers in X are equal. To see this, assume that ja K ka(mod p)), where 1 … j 6 k … p - 1. Because a is relatively prime7 to p, we can eliminate a from both sides of the equation [see Equation (2.3)] resulting in j K k(mod p). This last equality is impossible, because j and k are both positive inte- gers less than p. Therefore, we know that the (p - 1) elements of X are all positive integers with no two elements equal. We can conclude the X consists of the set of integers {1, 2, c , p - 1} in some order. Multiplying the numbers in both sets (p and X) and taking the result mod p yields
a * 2a * g * (p - 1)a K [(1 * 2 * g * (p - 1)](mod p) ap - 1(p - 1)! K (p - 1)! (mod p)
We can cancel the (p - 1)! term because it is relatively prime to p [see Equation (2.5)]. This yields Equation (2.10), which completes the proof.
6This is sometimes referred to as Fermat’s little theorem. 7Recall from Section 2.2 that two numbers are relatively prime if they have no prime factors in common; that is, their only common divisor is 1. This is equivalent to saying that two numbers are relatively prime if their greatest common divisor is 1.
300 = 22 * 31 * 52
18 = 21 * 32
gcd(18,300) = 21 * 31 * 50 = 6
Hiva-Network.Com
2.5 / FERMAT’S AND EULER’S THEOREMS 65
An alternative form of Fermat’s theorem is also useful: If p is prime and a is a positive integer, then
ap K a(mod p) (2.11)
Note that the first form of the theorem [Equation (2.10)] requires that a be rela- tively prime to p, but this form does not.
a = 7, p = 19 72 = 49 K 11 (mod 19) 74 K 121 K 7 (mod 19) 78 K 49 K 11 (mod 19) 716 K 121 K 7 (mod 19) ap - 1 = 718 = 716 * 72 K 7 * 11 K 1 (mod 19)
p = 5, a = 3 ap = 35 = 243 K 3(mod 5) = a(mod p) p = 5, a = 10 ap = 105 = 100000 K 10(mod 5) K 0(mod 5) = a(mod p)
Euler’s Totient Function
Before presenting Euler’s theorem, we need to introduce an important quantity in
number theory, referred to as Euler’s totient function. This function, written f(n), is defined as the number of positive integers less than n and relatively prime to n. By convention, f(1) = 1.
Determine f(37) and f(35).
Because 37 is prime, all of the positive integers from 1 through 36 are relatively
prime to 37. Thus f(37) = 36. To determine f(35), we list all of the positive integers less than 35 that are
relatively prime to it:
1, 2, 3, 4, 6, 8, 9, 11, 12, 13, 16, 17, 18
19, 22, 23, 24, 26, 27, 29, 31, 32, 33, 34
There are 24 numbers on the list, so f(35) = 24.
Table 2.6 lists the first 30 values of f(n). The value f(1) is without meaning but is defined to have the value 1.
It should be clear that, for a prime number p,
f(p) = p - 1
Now suppose that we have two prime numbers p and q with p ≠ q. Then we can show that, for n = pq,
66 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
f(n) = f(pq) = f(p) * f(q) = (p - 1) * (q - 1)
To see that f(n) = f(p) * f(q), consider that the set of positive integers less than n is the set {1, c , (pq - 1)}. The integers in this set that are not relatively prime to n are the set {p, 2p, c , (q - 1)p} and the set {q, 2q, c , (p - 1)q}. To see this, consider that any integer that divides n must divide either of the prime num- bers p or q. Therefore, any integer that does not contain either p or q as a factor is relatively prime to n. Further note that the two sets just listed are non-overlapping: Because p and q are prime, we can state that none of the integers in the first set can be written as a multiple of q, and none of the integers in the second set can be writ- ten as a multiple of p. Thus the total number of unique integers in the two sets is (q - 1) + (p - 1). Accordingly,
f(n) = (pq - 1) - [(q - 1) + (p - 1)] = pq - (p + q) + 1 = (p - 1) * (q - 1) = f(p) * f(q)
f(21) = f(3) * f(7) = (3 - 1) * (7 - 1) = 2 * 6 = 12 where the 12 integers are {1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20}.
Table 2.6 Some Values of Euler’s Totient Function f(n)
n f(n)
1 1
2 1
3 2
4 2
5 4
6 2
7 6
8 4
9 6
10 4
n f(n)
11 10
12 4
13 12
14 6
15 8
16 8
17 16
18 6
19 18
20 8
n f(n)
21 12
22 10
23 22
24 8
25 20
26 12
27 18
28 12
29 28
30 8
Euler’s Theorem
Euler’s theorem states that for every a and n that are relatively prime:
af(n) K 1(mod n) (2.12)
Proof: Equation (2.12) is true if n is prime, because in that case, f(n) = (n - 1) and Fermat’s theorem holds. However, it also holds for any integer n. Recall that
2.5 / FERMAT’S AND EULER’S THEOREMS 67
f(n) is the number of positive integers less than n that are relatively prime to n. Consider the set of such integers, labeled as
R = {x1, x2, c , xf(n)}
That is, each element xi of R is a unique positive integer less than n with gcd(xi, n) = 1. Now multiply each element by a, modulo n:
S = {(ax1 mod n), (ax2 mod n), c , (axf(n) mod n)}
The set S is a permutation8 of R , by the following line of reasoning:
1. Because a is relatively prime to n and xi is relatively prime to n, axi must also be relatively prime to n. Thus, all the members of S are integers that are less than n and that are relatively prime to n.
2. There are no duplicates in S. Refer to Equation (2.5). If axi mod n = axj mod n, then xi = xj.
Therefore,
q f(n)
i = 1 (axi mod n) = q
f(n)
i = 1 xi
q f(n)
i = 1 axi K q
f(n)
i = 1 xi (mod n)
af(n) * J qf(n) i = 1
xi R K qf(n) i = 1
xi (mod n)
af(n) K 1 (mod n)
which completes the proof. This is the same line of reasoning applied to the proof
of Fermat’s theorem.
8A permutation of a finite set of elements S is an ordered sequence of all the elements of S, with each element appearing exactly once.
a = 3; n = 10; f(10) = 4; af(n) = 34 = 81 = 1(mod 10) = 1(mod n) a = 2; n = 11; f(11) = 10; af(n) = 210 = 1024 = 1(mod 11) = 1(mod n)
As is the case for Fermat’s theorem, an alternative form of the theorem is also
useful:
af(n) + 1 K a(mod n) (2.13)
Again, similar to the case with Fermat’s theorem, the first form of Euler’s theorem
[Equation (2.12)] requires that a be relatively prime to n, but this form does not.
68 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
2.6 TESTING FOR PRIMALITY
For many cryptographic algorithms, it is necessary to select one or more very large
prime numbers at random. Thus, we are faced with the task of determining whether
a given large number is prime. There is no simple yet efficient means of accomplish-
ing this task.
In this section, we present one attractive and popular algorithm. You may be
surprised to learn that this algorithm yields a number that is not necessarily a prime.
However, the algorithm can yield a number that is almost certainly a prime. This will
be explained presently. We also make reference to a deterministic algorithm for find-
ing primes. The section closes with a discussion concerning the distribution of primes.
Miller–Rabin Algorithm9
The algorithm due to Miller and Rabin [MILL75, RABI80] is typically used to test
a large number for primality. Before explaining the algorithm, we need some back-
ground. First, any positive odd integer n Ú 3 can be expressed as
n - 1 = 2kq with k 7 0, q odd
To see this, note that n - 1 is an even integer. Then, divide (n - 1) by 2 until the result is an odd number q, for a total of k divisions. If n is expressed as a binary number, then the result is achieved by shifting the number to the right until the
rightmost digit is a 1, for a total of k shifts. We now develop two properties of prime numbers that we will need.
TWO PROPERTIES OF PRIME NUMBERS The first property is stated as follows: If p is prime and a is a positive integer less than p, then a2 mod p = 1 if and only if either a mod p = 1 or a mod p = - 1 mod p = p - 1. By the rules of modular arithmetic (a mod p) (a mod p) = a2 mod p. Thus, if either a mod p = 1 or a mod p = - 1, then a2 mod p = 1. Conversely, if a2 mod p = 1, then (a mod p)2 = 1, which is true only for a mod p = 1 or a mod p = - 1.
The second property is stated as follows: Let p be a prime number greater than 2. We can then write p - 1 = 2kq with k 7 0, q odd. Let a be any integer in the range 1 6 a 6 p - 1. Then one of the two following conditions is true.
1. aq is congruent to 1 modulo p. That is, aq mod p = 1, or equivalently, aq K 1(mod p).
2. One of the numbers aq, a2q, a4q, c , a2 k - 1q is congruent to - 1 mod-
ulo p. That is, there is some number j in the range (1 … j … k) such that a2
j - 1q mod p = - 1 mod p = p - 1 or equivalently, a2 j - 1q K - 1(mod p).
Proof: Fermat’s theorem [Equation (2.10)] states that an - 1 K 1(mod n) if n is prime. We have p - 1 = 2kq. Thus, we know that ap - 1 mod p = a2
kq mod p = 1. Thus, if we look at the sequence of numbers
aq mod p, a2q mod p, a4q mod p, c , a2 k - 1q mod p, a2
kq mod p (2.14)
9Also referred to in the literature as the Rabin-Miller algorithm, or the Rabin-Miller test, or the Miller– Rabin test.
2.6 / TESTING FOR PRIMALITY 69
we know that the last number in the list has value 1. Further, each number in the list
is the square of the previous number. Therefore, one of the following possibilities
must be true.
1. The first number on the list, and therefore all subsequent numbers on the list, equals 1.
2. Some number on the list does not equal 1, but its square mod p does equal 1. By virtue of the first property of prime numbers defined above, we know that
the only number that satisfies this condition is p - 1. So, in this case, the list contains an element equal to p - 1.
This completes the proof.
DETAILS OF THE ALGORITHM These considerations lead to the conclusion that, if n is prime, then either the first element in the list of residues, or remainders, (aq, a2q, c , a2
k - 1q, a2 kq) modulo n equals 1; or some element in the list equals
(n - 1); otherwise n is composite (i.e., not a prime). On the other hand, if the condition is met, that does not necessarily mean that n is prime. For example, if n = 2047 = 23 * 89, then n - 1 = 2 * 1023. We compute 21023 mod 2047 = 1, so that 2047 meets the condition but is not prime.
We can use the preceding property to devise a test for primality. The procedure
TEST takes a candidate integer n as input and returns the result composite if n is definitely not a prime, and the result inconclusive if n may or may not be a prime.
TEST (n) 1. Find integers k, q, with k > 0, q odd, so that
(n − 1 = 2k q); 2. Select a random integer a, 1 < a < n - 1; 3. if aq mod n = 1 then return(”inconclusive”); 4. for j = 0 to k - 1 do 5. if a2
j qmod n = n - 1 then return(”inconclusive”);
6. return(”composite”);
Let us apply the test to the prime number n = 29. We have (n - 1) = 28 = 22(7) = 2kq. First, let us try a = 10. We compute 107 mod 29 = 17, which is neither 1 nor 28, so we continue the test. The next calculation finds that (107)2 mod 29 = 28, and the test returns inconclusive (i.e., 29 may be prime). Let’s try again with a = 2. We have the following calculations: 27 mod 29 = 12; 214 mod 29 = 28; and the test again returns inconclusive. If we perform the test for all integers a in the range 1 through 28, we get the same inconclusive result, which is compatible with n being a prime number.
Now let us apply the test to the composite number n = 13 * 17 = 221. Then (n - 1) = 220 = 22(55) = 2kq. Let us try a = 5. Then we have 555 mod 221 = 112, which is neither 1 nor 220(555)2 mod 221 = 168. Because we have used all values of j (i.e., j = 0 and j = 1) in line 4 of the TEST algorithm, the test returns composite, indi- cating that 221 is definitely a composite number. But suppose we had selected a = 21. Then we have 2155 mod 221 = 200; (2155)2 mod 221 = 220; and the test returns inconclusive, indicating that 221 may be prime. In fact, of the 218 integers from 2 through 219, four of these will return an inconclusive result, namely 21, 47, 174, and 200.
70 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
REPEATED USE OF THE MILLER–RABIN ALGORITHM How can we use the Miller–Rabin algorithm to determine with a high degree of confidence whether or not an integer
is prime? It can be shown [KNUT98] that given an odd number n that is not prime and a randomly chosen integer, a with 1 6 a 6 n - 1, the probability that TEST will return inconclusive (i.e., fail to detect that n is not prime) is less than 1/4. Thus, if t different values of a are chosen, the probability that all of them will pass TEST (return inconclusive) for n is less than (1/4)t. For example, for t = 10, the probability that a nonprime number will pass all ten tests is less than 10-6. Thus,
for a sufficiently large value of t , we can be confident that n is prime if Miller’s test always returns inconclusive.
This gives us a basis for determining whether an odd integer n is prime with a reasonable degree of confidence. The procedure is as follows: Repeatedly invoke
TEST (n) using randomly chosen values for a. If, at any point, TEST returns composite, then n is determined to be nonprime. If TEST continues to return inconclusive for t tests, then for a sufficiently large value of t, assume that n is prime.
A Deterministic Primality Algorithm
Prior to 2002, there was no known method of efficiently proving the primality of
very large numbers. All of the algorithms in use, including the most popular (Miller–
Rabin), produced a probabilistic result. In 2002 (announced in 2002, published
in 2004), Agrawal, Kayal, and Saxena [AGRA04] developed a relatively simple
deterministic algorithm that efficiently determines whether a given large number
is a prime. The algorithm, known as the AKS algorithm, does not appear to be as
efficient as the Miller–Rabin algorithm. Thus far, it has not supplanted this older,
probabilistic technique.
Distribution of Primes
It is worth noting how many numbers are likely to be rejected before a prime num-
ber is found using the Miller–Rabin test, or any other test for primality. A result
from number theory, known as the prime number theorem, states that the primes
near n are spaced on the average one every ln (n) integers. Thus, on average, one would have to test on the order of ln(n) integers before a prime is found. Because all even integers can be immediately rejected, the correct figure is 0.5 ln(n). For example, if a prime on the order of magnitude of 2200 were sought, then about
0.5 ln(2200) = 69 trials would be needed to find a prime. However, this figure is just an average. In some places along the number line, primes are closely packed, and in
other places there are large gaps.
The two consecutive odd integers 1,000,000,000,061 and 1,000,000,000,063
are both prime. On the other hand, 1001! + 2, 1001! + 3, c , 1001! + 1000, 1001! + 1001 is a sequence of 1000 consecutive composite integers.
2.7 / THE CHINESE REMAINDER THEOREM 71
2.7 THE CHINESE REMAINDER THEOREM
One of the most useful results of number theory is the Chinese remainder theorem (CRT).10 In essence, the CRT says it is possible to reconstruct integers in a certain
range from their residues modulo a set of pairwise relatively prime moduli.
10The CRT is so called because it is believed to have been discovered by the Chinese mathematician Sun-Tsu in around 100 A.D.
The 10 integers in Z10, that is the integers 0 through 9, can be reconstructed from
their two residues modulo 2 and 5 (the relatively prime factors of 10). Say the
known residues of a decimal digit x are r2 = 0 and r5 = 3; that is, x mod 2 = 0 and x mod 5 = 3. Therefore, x is an even integer in Z10 whose remainder, on divi- sion by 5, is 3. The unique solution is x = 8.
The CRT can be stated in several ways. We present here a formulation that is most
useful from the point of view of this text. An alternative formulation is explored in
Problem 2.33. Let
M = q k
i = 1 mi
where the mi are pairwise relatively prime; that is, gcd(mi, mj) = 1 for 1 … i, j … k, and i ≠ j. We can represent any integer A in ZM by a k-tuple whose elements are in Zmi using the following correspondence:
A 4 (a1, a2, c , ak) (2.15)
where A ∈ ZM, ai ∈ Zmi, and ai = A mod mi for 1 … i … k. The CRT makes two assertions.
1. The mapping of Equation (2.15) is a one-to-one correspondence (called a bijection) between ZM and the Cartesian product Zm1 * Zm2 * c * Zmk. That is, for every integer A such that 0 … A 6 M, there is a unique k- tuple (a1, a2, c , ak) with 0 … ai 6 mi that represents it, and for every such k- tuple (a1, a2, c , ak), there is a unique integer A in ZM.
2. Operations performed on the elements of ZM can be equivalently performed on the corresponding k-tuples by performing the operation independently in each coordinate position in the appropriate system.
Let us demonstrate the first assertion. The transformation from A to (a1, a2, c , ak), is obviously unique; that is, each ai is uniquely calculated as ai = A mod mi. Computing A from (a1, a2, c , ak) can be done as follows. Let
72 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
Mi = M/mi for 1 … i … k. Note that Mi = m1 * m2 * c * mi - 1 * mi + 1 * c * mk, so that Mi K 0 (mod mj) for all j ≠ i. Then let
ci = Mi * (Mi-1 mod mi) for 1 … i … k (2.16)
By the definition of Mi, it is relatively prime to mi and therefore has a unique multi- plicative inverse mod mi. So Equation (2.16) is well defined and produces a unique value ci. We can now compute
A K ¢ ak i = 1
aici≤(mod M) (2.17) To show that the value of A produced by Equation (2.17) is correct, we must
show that ai = A mod mi for 1 … i … k. Note that cj K Mj K 0 (mod mi) if j ≠ i, and that ci K 1 (mod mi). It follows that ai = A mod mi.
The second assertion of the CRT, concerning arithmetic operations, follows from the rules for modular arithmetic. That is, the second assertion can be stated as
follows: If
A 4 (a1, a2, c , ak) B 4 (b1, b2, c , bk)
then
(A + B) mod M 4 ((a1 + b1) mod m1, c , (ak + bk) mod mk) (A - B) mod M 4 ((a1 - b1) mod m1, c , (ak - bk) mod mk) (A * B) mod M 4 ((a1 * b1) mod m1, c , (ak * bk) mod mk)
One of the useful features of the Chinese remainder theorem is that it provides
a way to manipulate (potentially very large) numbers mod M in terms of tuples of smaller numbers. This can be useful when M is 150 digits or more. However, note that it is necessary to know beforehand the factorization of M.
To represent 973 mod 1813 as a pair of numbers mod 37 and 49, define
m1 = 37 m2 = 49 M = 1813 A = 973
We also have M1 = 49 and M2 = 37. Using the extended Euclidean algorithm, we compute M1
-1 = 34 mod m1 and M2 -1 = 4 mod m2. (Note that we only need
to compute each Mi and each Mi -1 once.) Taking residues modulo 37 and 49, our
representation of 973 is (11, 42), because 973 mod 37 = 11 and 973 mod 49 = 42. Now suppose we want to add 678 to 973. What do we do to (11, 42)? First
we compute (678) 4 (678 mod 37, 678 mod 49) = (12, 41). Then we add the tuples element-wise and reduce (11 + 12 mod 37, 42 + 41 mod 49) = (23, 34). To verify that this has the correct effect, we compute
2.8 / DISCRETE LOGARITHMS 73
2.8 DISCRETE LOGARITHMS
Discrete logarithms are fundamental to a number of public-key algorithms, includ-
ing Diffie–Hellman key exchange and the digital signature algorithm (DSA). This
section provides a brief overview of discrete logarithms. For the interested reader,
more detailed developments of this topic can be found in [ORE67] and [LEVE90].
The Powers of an Integer, Modulo n
Recall from Euler’s theorem [Equation (2.12)] that, for every a and n that are rela- tively prime,
af(n) K 1 (mod n)
where f(n), Euler’s totient function, is the number of positive integers less than n and relatively prime to n. Now consider the more general expression:
am K 1 (mod n) (2.18)
If a and n are relatively prime, then there is at least one integer m that satisfies Equation (2.18), namely, m = f(n). The least positive exponent m for which Equation (2.18) holds is referred to in several ways:
■ The order of a (mod n)
■ The exponent to which a belongs (mod n)
■ The length of the period generated by a
(23, 34) 4 a1M1M1-1 + a2M2M2-1 mod M = [(23)(49)(34) + (34)(37)(4)] mod 1813 = 43350 mod 1813 = 1651
and check that it is equal to (973 + 678) mod 1813 = 1651. Remember that in the above derivation, Mi
-1 is the multiplicative inverse of M1 modulo m1 and M2 -1
is the multiplicative inverse of M2 modulo m2. Suppose we want to multiply 1651 (mod 1813) by 73. We multiply (23, 34)
by 73 and reduce to get (23 * 73 mod 37, 34 * 73 mod 49) = (14, 32). It is eas- ily verified that
(14, 32) 4 [(14)(49)(34) + (32)(37)(4)] mod 1813 = 865 = 1651 * 73 mod 1813
Hiva-Network.Com
74 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
Table 2.7 shows all the powers of a, modulo 19 for all positive a 6 19. The length of the sequence for each base value is indicated by shading. Note the
following:
1. All sequences end in 1. This is consistent with the reasoning of the preceding few paragraphs.
2. The length of a sequence divides f(19) = 18. That is, an integral number of sequences occur in each row of the table.
3. Some of the sequences are of length 18. In this case, it is said that the base inte- ger a generates (via powers) the set of nonzero integers modulo 19. Each such integer is called a primitive root of the modulus 19.
More generally, we can say that the highest possible exponent to which a num-
ber can belong (mod n) is f(n). If a number is of this order, it is referred to as a primitive root of n. The importance of this notion is that if a is a primitive root of n, then its powers
a, a2, c , af(n)
are distinct (mod n) and are all relatively prime to n. In particular, for a prime num- ber p, if a is a primitive root of p, then
a, a2, c , ap - 1
are distinct (mod p). For the prime number 19, its primitive roots are 2, 3, 10, 13, 14, and 15.
Not all integers have primitive roots. In fact, the only integers with primitive
roots are those of the form 2, 4, pa, and 2pa, where p is any odd prime and a is a positive integer. The proof is not simple but can be found in many number theory
books, including [ORE76].
To see this last point, consider the powers of 7, modulo 19:
71 K 7 (mod 19) 72 = 49 = 2 * 19 + 11 K 11 (mod 19) 73 = 343 = 18 * 19 + 1 K 1 (mod 19) 74 = 2401 = 126 * 19 + 7 K 7 (mod 19) 75 = 16807 = 884 * 19 + 11 K 11 (mod 19)
There is no point in continuing because the sequence is repeating. This can be
proven by noting that 73 K 1(mod 19), and therefore, 73 + j K 737j K 7j(mod 19), and hence, any two powers of 7 whose exponents differ by 3 (or a multiple of 3)
are congruent to each other (mod 19). In other words, the sequence is periodic,
and the length of the period is the smallest positive exponent m such that 7m K 1(mod 19).
2.8 / DISCRETE LOGARITHMS 75
Logarithms for Modular Arithmetic
With ordinary positive real numbers, the logarithm function is the inverse of expo-
nentiation. An analogous function exists for modular arithmetic.
Let us briefly review the properties of ordinary logarithms. The logarithm of a
number is defined to be the power to which some positive base (except 1) must be
raised in order to equal the number. That is, for base x and for a value y,
y = xlogx(y)
The properties of logarithms include
logx(1) = 0 logx(x) = 1
logx(yz) = logx(y) + log x(z) (2.19)
logx(y r) = r * log x(y) (2.20)
Consider a primitive root a for some prime number p (the argument can be developed for nonprimes as well). Then we know that the powers of a from
a a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 4 8 16 13 7 14 9 18 17 15 11 3 6 12 5 10 1
3 9 8 5 15 7 2 6 18 16 10 11 14 4 12 17 13 1
4 16 7 9 17 11 6 5 1 4 16 7 9 17 11 6 5 1
5 6 11 17 9 7 16 4 1 5 6 11 17 9 7 16 4 1
6 17 7 4 5 11 9 16 1 6 17 7 4 5 11 9 16 1
7 11 1 7 11 1 7 11 1 7 11 1 7 11 1 7 11 1
8 7 18 11 12 1 8 7 18 11 12 1 8 7 18 11 12 1
9 5 7 6 16 11 4 17 1 9 5 7 6 16 11 4 17 1
10 5 12 6 3 11 15 17 18 9 14 7 13 16 8 4 2 1
11 7 1 11 7 1 11 7 1 11 7 1 11 7 1 11 7 1
12 11 18 7 8 1 12 11 18 7 8 1 12 11 18 7 8 1
13 17 12 4 14 11 10 16 18 6 2 7 15 5 8 9 3 1
14 6 8 17 10 7 3 4 18 5 13 11 2 9 12 16 15 1
15 16 12 9 2 11 13 5 18 4 3 7 10 17 8 6 14 1
16 9 11 5 4 7 17 6 1 16 9 11 5 4 7 17 6 1
17 4 11 16 6 7 5 9 1 17 4 11 16 6 7 5 9 1
18 1 18 1 18 1 18 1 18 1 18 1 18 1 18 1 18 1
Table 2.7 Powers of Integers, Modulo 19
76 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
1 through (p - 1) produce each integer from 1 through (p - 1) exactly once. We also know that any integer b satisfies
b K r (mod p) for some r, where 0 … r … (p - 1)
by the definition of modular arithmetic. It follows that for any integer b and a primi- tive root a of prime number p, we can find a unique exponent i such that
b K ai(mod p) where 0 … i … (p - 1)
This exponent i is referred to as the discrete logarithm of the number b for the base a (mod p). We denote this value as dloga,p(b).
11
Note the following:
dloga,p(1) = 0 because a 0 mod p = 1 mod p = 1 (2.21)
dloga,p(a) = 1 because a 1 mod p = a (2.22)
11Many texts refer to the discrete logarithm as the index. There is no generally agreed notation for this concept, much less an agreed name.
Here is an example using a nonprime modulus, n = 9. Here f(n) = 6 and a = 2 is a primitive root. We compute the various powers of a and find
20 = 1 24 K 7 (mod 9) 21 = 2 25 K 5 (mod 9) 22 = 4 26 K 1 (mod 9) 23 = 8
This gives us the following table of the numbers with given discrete logarithms
(mod 9) for the root a = 2:
Logarithm 0 1 2 3 4 5
Number 1 2 4 8 7 5
To make it easy to obtain the discrete logarithms of a given number, we rearrange
the table:
Number 1 2 4 5 7 8
Logarithm 0 1 2 5 4 3
Now consider
x = adloga, p(x) mod p y = adloga, p(y) mod p xy = adloga, p(xy) mod p
2.8 / DISCRETE LOGARITHMS 77
Using the rules of modular multiplication,
xy mod p = [(x mod p)(y mod p)] mod p
adloga, p(xy) mod p = [(adloga, p(x) mod p)(adloga, p(y) mod p)] mod p
= (adloga, p(x) + dloga, p(y)) mod p
But now consider Euler’s theorem, which states that, for every a and n that are relatively prime,
af(n) K 1(mod n)
Any positive integer z can be expressed in the form z = q + kf(n), with 0 … q 6 f(n). Therefore, by Euler’s theorem,
az K aq(mod n) if z K q mod f(n)
Applying this to the foregoing equality, we have
dloga, p(xy) K [dlog a, p(x) + dlog a, p(y)](mod f(p))
and generalizing,
dloga, p(y r) K [r * dloga, p(y)](mod f(p))
This demonstrates the analogy between true logarithms and discrete logarithms.
Keep in mind that unique discrete logarithms mod m to some base a exist only if a is a primitive root of m.
Table 2.8, which is directly derived from Table 2.7, shows the sets of discrete
logarithms that can be defined for modulus 19.
Calculation of Discrete Logarithms
Consider the equation
y = gx mod p
Given g, x, and p, it is a straightforward matter to calculate y. At the worst, we must perform x repeated multiplications, and algorithms exist for achieving greater effi- ciency (see Chapter 9).
However, given y, g, and p, it is, in general, very difficult to calculate x (take the discrete logarithm). The difficulty seems to be on the same order of magnitude
as that of factoring primes required for RSA. At the time of this writing, the asymp-
totically fastest known algorithm for taking discrete logarithms modulo a prime
number is on the order of [BETH91]:
e((ln p) 1/3(ln(ln p))2/3)
which is not feasible for large primes.
78 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
2.9 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
(a) Discrete logarithms to the base 2, modulo 19
a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
log2,19(a) 18 1 13 2 16 14 6 3 8 17 12 15 5 7 11 4 10 9
(b) Discrete logarithms to the base 3, modulo 19
a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
log3,19(a) 18 7 1 14 4 8 6 3 2 11 12 15 17 13 5 10 16 9
(c) Discrete logarithms to the base 10, modulo 19
a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
log10,19(a) 18 17 5 16 2 4 12 15 10 1 6 3 13 11 7 14 8 9
(d) Discrete logarithms to the base 13, modulo 19
a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
log13,19(a) 18 11 17 4 14 10 12 15 16 7 6 3 1 5 13 8 2 9
(e) Discrete logarithms to the base 14, modulo 19
a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
log14,19(a) 18 13 7 8 10 2 6 3 14 5 12 15 11 1 17 16 4 9
(f) Discrete logarithms to the base 15, modulo 19
a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
log15,19(a) 18 5 11 10 8 16 12 15 4 13 6 3 7 17 1 2 14 9
Table 2.8 Tables of Discrete Logarithms, Modulo 19
Key Terms
bijection
composite number
commutative
Chinese remainder theorem
discrete logarithm
divisor
Euclidean algorithm
Euler’s theorem
Euler’s totient function
Fermat’s theorem
greatest common divisor
identity element
index
modular arithmetic
modulus
order
prime number
primitive root
relatively prime
residue
2.9 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 79
Review Questions
2.1 What does it mean to say that b is a divisor of a? 2.2 What is the meaning of the expression a divides b? 2.3 What is the difference between modular arithmetic and ordinary arithmetic? 2.4 What is a prime number? 2.5 What is Euler’s totient function? 2.6 The Miller–Rabin test can determine if a number is not prime but cannot determine
if a number is prime. How can such an algorithm be used to test for primality?
2.7 What is a primitive root of a number? 2.8 What is the difference between an index and a discrete logarithm?
Problems
2.1 Reformulate Equation (2.1), removing the restriction that a is a nonnegative integer. That is, let a be any integer.
2.2 Draw a figure similar to Figure 2.1 for a 6 0. 2.3 For each of the following equations, find an integer x that satisfies the equation.
a. 4 x K 2 (m od 3 ) b. 7 x K 4 (m od 9 ) c. 5 x K 3 (m od 1 1 )
2.4 In this text, we assume that the modulus is a positive integer. But the definition of the expression a mod n also makes perfect sense if n is negative. Determine the following: a. 7 mod 4 b. 7 mod - 4 c. - 7 mod 4 d. - 7 m od - 4
2.5 A modulus of 0 does not fit the definition but is defined by convention as follows: a mod 0 = a. With this definition in mind, what does the following expression mean: a K b (mod 0)?
2.6 In Section 2.3, we define the congruence relationship as follows: Two integers a and b are said to be congruent modulo n if (a mod n) = (b mod n). We then proved that a K b (mod n) if n�(a - b). Some texts on number theory use this latter relation- ship as the definition of congruence: Two integers a and b are said to be congruent modulo n if n�(a - b). Using this latter definition as the starting point, prove that, if (a mod n) = (b mod n), then n divides (a - b).
2.7 What is the smallest positive integer that has exactly k divisors? Provide answers for values for 1 … k … 8.
2.8 Prove the following: a. a K b (mod n) implies b K a (mod n) b. a K b (mod n) and b K c (mod n) imply a K c (mod n)
2.9 Prove the following: a. [(a mod n) - (b mod n)] mod n = (a - b) mod n b. [(a mod n) * (b mod n)] mod n = (a * b) mod n
2.10 Find the multiplicative inverse of each nonzero element in Z5. 2.11 Show that an integer N is congruent modulo 9 to the sum of its decimal digits. For
example, 7 2 3 K 7 + 2 + 3 K 1 2 K 1 + 2 K 3 (m od 9 ). This is the basis for the familiar procedure of “casting out 9’s” when checking computations in arithmetic.
80 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
2.12 a. Determine gcd(72345, 43215) b. Determine gcd(3486, 10292)
2.13 The purpose of this problem is to set an upper bound on the number of iterations of the Euclidean algorithm. a. Suppose that m = qn + r with q 7 0 and 0 … r 6 n. Show that m/2 7 r. b. Let Ai be the value of A in the Euclidean algorithm after the ith iteration. Show that
Ai + 2 6 Ai 2
c. Show that if m, n, and N are integers with (1 … m, n, … 2N), then the Euclidean algorithm takes at most 2N steps to find gcd(m, n).
2.14 The Euclidean algorithm has been known for over 2000 years and has always been a favorite among number theorists. After these many years, there is now a potential competitor, invented by J. Stein in 1961. Stein’s algorithms is as follows: Determine gcd(A, B) with A, B Ú 1. STEP 1 Set A1 = A, B1 = B, C1 = 1 STEP 2 For n > 1, (1) If An = Bn, stop. gcd(A, B) = AnCn
(2) If An and Bn are both even, set An + 1 = An/2, Bn + 1 = Bn/2, Cn + 1 = 2Cn
(3) If An is even and Bn is odd, set An + 1 = An/2, Bn + 1 = Bn, Cn + 1 = Cn
(4) If An is odd and Bn is even, set An + 1 = An, Bn + 1 = Bn/2, Cn + 1 = Cn
(5) If An and Bn are both odd, set An + 1 = �An - Bn� , Bn + 1 = min (Bn, An), Cn + 1 = Cn
Continue to step n + 1. a. To get a feel for the two algorithms, compute gcd(6150, 704) using both the Euclid-
ean and Stein’s algorithm. b. What is the apparent advantage of Stein’s algorithm over the Euclidean algorithm?
2.15 a. Show that if Stein’s algorithm does not stop before the nth step, then
Cn + 1 * gcd(An + 1, Bn + 1) = Cn * gcd(An, Bn)
b. Show that if the algorithm does not stop before step (n - 1), then
An + 2Bn + 2 … AnBn
2
c. Show that if 1 … A, B … 2N, then Stein’s algorithm takes at most 4N steps to find gcd(m, n). Thus, Stein’s algorithm works in roughly the same number of steps as the Euclidean algorithm.
d. Demonstrate that Stein’s algorithm does indeed return gcd(A, B). 2.16 Using the extended Euclidean algorithm, find the multiplicative inverse of
a. 135 mod 61 b. 7465 mod 2464 c. 42828 mod 6407
2.17 The purpose of this problem is to determine how many prime numbers there are. Suppose there are a total of n prime numbers, and we list these in order: p1 = 2 6 p2 = 3 6 p3 = 5 6 c 6 pn. a. Define X = 1 + p1p2 c pn. That is, X is equal to one plus the product of all the
primes. Can we find a prime number Pm that divides X? b. What can you say about m? c. Deduce that the total number of primes cannot be finite. d. Show that Pn + 1 … 1 + p1p2 c pn.
2.9 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 81
2.18 The purpose of this problem is to demonstrate that the probability that two random numbers are relatively prime is about 0.6. a. Let P = Pr[gcd(a, b) = 1]. Show that P = Pr[gcd(a, b) = d] = P/d2. Hint:
Consider the quantity gcd aa d
, b d b.
b. The sum of the result of part (a) over all possible values of d is 1. That is Σd Ú 1Pr[gcd(a, b) = d] = 1. Use this equality to determine the value of P. Hint:
Use the identity a ∞
i = 1
1
i2 = p2
6 .
2.19 Why is gcd(n, n + 1) = 1 for two consecutive integers n and n + 1? 2.20 Using Fermat’s theorem, find 4 2 2 5 mod 13. 2.21 Use Fermat’s theorem to find a number a between 0 and 92 with a congruent to 71013
modulo 93.
2.22 Use Fermat’s theorem to find a number x between 0 and 37 with x 7 3 congruent to 4 modulo 37. (You should not need to use any brute-force searching.)
2.23 Use Euler’s theorem to find a number a between 0 and 9 such that a is congruent to 9 1 0 1 modulo 10. (Note: This is the same as the last digit of the decimal expansion of 9 1 0 0 . )
2.24 Use Euler’s theorem to find a number x between 0 and 14 with x 6 1 congruent to 7 modulo 15. (You should not need to use any brute-force searching.)
2.25 Notice in Table 2.6 that f(n) is even for n 7 2. This is true for all n 7 2. Give a con- cise argument why this is so.
2.26 Prove the following: If p is prime, then f(pi) = pi - pi - 1. Hint: What numbers have a factor in common with pi?
2.27 It can be shown (see any book on number theory) that if gcd(m, n) = 1 then f(mn) = f(m)f(n). Using this property, the property developed in the preceding problem, and the property that f(p) = p - 1 for p prime, it is straightforward to determine the value of f(n) for any n. Determine the following: a. f(29) b. f(51) c. f(455) d. f(616)
2.28 It can also be shown that for arbitrary positive integer a, f(a) is given by
f(a) = q t
i = 1 [pi
ai - 1(pi - 1)]
where a is given by Equation (2.9), namely: a = P1 a1P2
a2 c Pt at. Demonstrate this result.
2.29 Consider the function: f(n) = number of elements in the set {a: 0 … a 6 n and gcd(a, n) = 1}. What is this function?
2.30 Although ancient Chinese mathematicians did good work coming up with their remainder theorem, they did not always get it right. They had a test for primality. The test said that n is prime if and only if n divides (2n - 2). a. Give an example that satisfies the condition using an odd prime. b. The condition is obviously true for n = 2. Prove that the condition is true if n is an
odd prime (proving the if condition). c. Give an example of an odd n that is not prime and that does not satisfy the condi-
tion. You can do this with nonprime numbers up to a very large value. This misled the Chinese mathematicians into thinking that if the condition is true then n is prime.
d. Unfortunately, the ancient Chinese never tried n = 341, which is nonprime (341 = 11 * 31), yet 341 divides 2341 - 2 without remainder. Demonstrate that 2341 K 2 (mod 341) (disproving the only if condition). Hint: It is not necessary to calculate 2341; play around with the congruences instead.
82 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
2.31 Show that, if n is an odd composite integer, then the Miller–Rabin test will return inconclusive for a = 1 and a = (n - 1).
2.32 If n is composite and passes the Miller–Rabin test for the base a, then n is called a strong pseudoprime to the base a. Show that 2047 is a strong pseudoprime to the base 2.
2.33 A common formulation of the Chinese remainder theorem (CRT) is as follows: Let m1, c , mk be integers that are pairwise relatively prime for 1 … i, j … k, and i ≠ j. Define M to be the product of all the mi>s. Let a1, c , ak be integers. Then the set of congruences:
x K a1(mod m1) x K a2(mod m2)
~
~
~
x K ak(mod mk)
has a unique solution modulo M. Show that the theorem stated in this form is true. 2.34 The example used by Sun-Tsu to illustrate the CRT was
x K 2 (mod 3); x K 3 (mod 5); x K 2 (mod 7)
Solve for x. 2.35 Six professors begin courses on Monday, Tuesday, Wednesday, Thursday, Friday,
and Saturday, respectively, and announce their intentions of lecturing at intervals of 3, 2, 5, 6, 1, and 4 days, respectively. The regulations of the university forbid Sunday lectures (so that a Sunday lecture must be omitted). When first will all six professors find themselves compelled to omit a lecture? Hint: Use the CRT.
2.36 Find all primitive roots of 37. 2.37 Given 5 as a primitive root of 23, construct a table of discrete logarithms, and use it to
solve the following congruences. a. 3x5 K 2 (mod 23) b. 7x10 + 1 K 0 (mod 23) c. 5x K 6 (mod 23)
Programming Problems
2.1 Write a computer program that implements fast exponentiation (successive squaring) modulo n.
2.2 Write a computer program that implements the Miller–Rabin algorithm for a user- specified n. The program should allow the user two choices: (1) specify a possible witness a to test using the Witness procedure or (2) specify a number s of random witnesses for the Miller–Rabin test to check.
APPENDIX 2A THE MEANING OF MOD
The operator mod is used in this book and in the literature in two different ways: as
a binary operator and as a congruence relation. This appendix explains the distinc-
tion and precisely defines the notation used in this book regarding parentheses. This
notation is common but, unfortunately, not universal.
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APPENDIX 2A / THE MEANING OF MOD 83
The Binary Operator mod
If a is an integer and n is a positive integer, we define a mod n to be the remainder when a is divided by n. The integer n is called the modulus, and the remainder is called the residue. Thus, for any integer a, we can always write
a = :a/n; * n + (a mod n) Formally, we define the operator mod as
a mod n = a - :a/n; * n for n ≠ 0 As a binary operation, mod takes two integer arguments and returns the re-
mainder. For example, 7 mod 3 = 1. The arguments may be integers, integer vari- ables, or integer variable expressions. For example, all of the following are valid,
with the obvious meanings:
7 mod 3
7 mod m
x mod 3
x mod m
(x2 + y + 1) mod (2m + n)
where all of the variables are integers. In each case, the left-hand term is divided by
the right-hand term, and the resulting value is the remainder. Note that if either the
left- or right-hand argument is an expression, the expression is parenthesized. The
operator mod is not inside parentheses.
In fact, the mod operation also works if the two arguments are arbitrary real num-
bers, not just integers. In this book, we are concerned only with the integer operation.
The Congruence Relation mod
As a congruence relation, mod expresses that two arguments have the same remain-
der with respect to a given modulus. For example, 7 K 4 (mod 3) expresses the fact that both 7 and 4 have a remainder of 1 when divided by 3. The following two
expressions are equivalent:
a K b (mod m) 3 a mod m = b mod m
Another way of expressing it is to say that the expression a K b (mod m) is the same as saying that a - b is an integral multiple of m. Again, all the arguments may be integers, integer variables, or integer variable expressions. For example, all of
the following are valid, with the obvious meanings:
7 K 4 (mod 3) x K y (mod m) (x2 + y + 1) K (a + 1)(mod [m + n])
where all of the variables are integers. Two conventions are used. The congruence
sign is K. The modulus for the relation is defined by placing the mod operator fol- lowed by the modulus in parentheses.
84 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY
The congruence relation is used to define residue classes. Those numbers that have the same remainder r when divided by m form a residue class (mod m). There are m residue classes (mod m). For a given remainder r, the residue class to which it belongs consists of the numbers
r, r { m, r { 2m, c
According to our definition, the congruence
a K b (mod m)
signifies that the numbers a and b differ by a multiple of m. Consequently, the con- gruence can also be expressed in the terms that a and b belong to the same residue class (mod m).
85
PART TWO: SYMMETRIC CIPHERS
CHAPTER
Classical Encryption Techniques 3.1 Symmetric Cipher Model
Cryptography
Cryptanalysis and Brute-Force Attack
3.2 Substitution Techniques
Caesar Cipher
Monoalphabetic Ciphers
Playfair Cipher
Hill Cipher
Polyalphabetic Ciphers
One-Time Pad
3.3 Transposition Techniques
3.4 Rotor Machines
3.5 Steganography
3.6 Key Terms, Review Questions, and Problems
86 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES
Symmetric encryption, also referred to as conventional encryption or single-key
encryption, was the only type of encryption in use prior to the development of public-
key encryption in the 1970s. It remains by far the most widely used of the two types
of encryption. Part One examines a number of symmetric ciphers. In this chapter, we
begin with a look at a general model for the symmetric encryption process; this will
enable us to understand the context within which the algorithms are used. Next, we
examine a variety of algorithms in use before the computer era. Finally, we look briefly
at a different approach known as steganography. Chapters 4 and 6 introduce the two
most widely used symmetric cipher: DES and AES.
Before beginning, we define some terms. An original message is known as the
plaintext, while the coded message is called the ciphertext. The process of convert- ing from plaintext to ciphertext is known as enciphering or encryption; restoring the plaintext from the ciphertext is deciphering or decryption. The many schemes used for encryption constitute the area of study known as cryptography. Such a scheme is known as a cryptographic system or a cipher. Techniques used for deciphering a message without any knowledge of the enciphering details fall into the area of crypt- analysis. Cryptanalysis is what the layperson calls “breaking the code.” The areas of cryptography and cryptanalysis together are called cryptology.
3.1 SYMMETRIC CIPHER MODEL
A symmetric encryption scheme has five ingredients (Figure 3.1):
■ Plaintext: This is the original intelligible message or data that is fed into the algorithm as input.
■ Encryption algorithm: The encryption algorithm performs various substitu- tions and transformations on the plaintext.
■ Secret key: The secret key is also input to the encryption algorithm. The key is a value independent of the plaintext and of the algorithm. The algorithm will
produce a different output depending on the specific key being used at the
time. The exact substitutions and transformations performed by the algorithm
depend on the key.
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Present an overview of the main concepts of symmetric cryptography.
◆ Explain the difference between cryptanalysis and brute-force attack.
◆ Understand the operation of a monoalphabetic substitution cipher.
◆ Understand the operation of a polyalphabetic cipher.
◆ Present an overview of the Hill cipher.
◆ Describe the operation of a rotor machine.
3.1 / SYMMETRIC CIPHER MODEL 87
■ Ciphertext: This is the scrambled message produced as output. It depends on the plaintext and the secret key. For a given message, two different keys will
produce two different ciphertexts. The ciphertext is an apparently random
stream of data and, as it stands, is unintelligible.
■ Decryption algorithm: This is essentially the encryption algorithm run in reverse. It takes the ciphertext and the secret key and produces the original
plaintext.
There are two requirements for secure use of conventional encryption:
1. We need a strong encryption algorithm. At a minimum, we would like the algo- rithm to be such that an opponent who knows the algorithm and has access to
one or more ciphertexts would be unable to decipher the ciphertext or figure
out the key. This requirement is usually stated in a stronger form: The oppo-
nent should be unable to decrypt ciphertext or discover the key even if he or
she is in possession of a number of ciphertexts together with the plaintext that
produced each ciphertext.
2. Sender and receiver must have obtained copies of the secret key in a secure fashion and must keep the key secure. If someone can discover the key and
knows the algorithm, all communication using this key is readable.
We assume that it is impractical to decrypt a message on the basis of the
ciphertext plus knowledge of the encryption/decryption algorithm. In other words, we do not need to keep the algorithm secret; we need to keep only the key secret.
This feature of symmetric encryption is what makes it feasible for widespread use.
The fact that the algorithm need not be kept secret means that manufacturers can
and have developed low-cost chip implementations of data encryption algorithms.
These chips are widely available and incorporated into a number of products. With
the use of symmetric encryption, the principal security problem is maintaining the
secrecy of the key.
Let us take a closer look at the essential elements of a symmetric encryp-
tion scheme, using Figure 3.2. A source produces a message in plaintext,
X = [X1, X2, c , XM]. The M elements of X are letters in some finite alphabet. Traditionally, the alphabet usually consisted of the 26 capital letters. Nowadays,
Figure 3.1 Simplified Model of Symmetric Encryption
Plaintext input
Y = E(K, X ) X = D(K, Y )
X
KK
Transmitted ciphertext
Plaintext output
Secret key shared by sender and recipient
Secret key shared by sender and recipient
Encryption algorithm (e.g., AES)
Decryption algorithm (reverse of encryption
algorithm)
88 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES
the binary alphabet {0, 1} is typically used. For encryption, a key of the form
K = [K1, K2, c , KJ] is generated. If the key is generated at the message source, then it must also be provided to the destination by means of some secure channel.
Alternatively, a third party could generate the key and securely deliver it to both
source and destination.
With the message X and the encryption key K as input, the encryption algo- rithm forms the ciphertext Y = [Y1, Y2, c , YN]. We can write this as
Y = E(K, X)
This notation indicates that Y is produced by using encryption algorithm E as a function of the plaintext X, with the specific function determined by the value of the key K.
The intended receiver, in possession of the key, is able to invert the
transformation:
X = D(K, Y)
An opponent, observing Y but not having access to K or X, may attempt to recover X or K or both X and K. It is assumed that the opponent knows the encryp- tion (E) and decryption (D) algorithms. If the opponent is interested in only this
particular message, then the focus of the effort is to recover X by generating a plain- text estimate Xn . Often, however, the opponent is interested in being able to read future messages as well, in which case an attempt is made to recover K by generat- ing an estimate Kn .
Figure 3.2 Model of Symmetric Cryptosystem
Message source
Cryptanalyst
Key source
Destination X X
X
K
Y = E(K, X )
Secure channel
K
Encryption algorithm
Decryption algorithm
3.1 / SYMMETRIC CIPHER MODEL 89
Cryptography
Cryptographic systems are characterized along three independent dimensions:
1. The type of operations used for transforming plaintext to ciphertext. All encryption algorithms are based on two general principles: substitution,
in which each element in the plaintext (bit, letter, group of bits or letters)
is mapped into another element, and transposition, in which elements
in the plaintext are rearranged. The fundamental requirement is that no
information be lost (i.e., that all operations are reversible). Most systems,
referred to as product systems, involve multiple stages of substitutions and transpositions.
2. The number of keys used. If both sender and receiver use the same key, the system is referred to as symmetric, single-key, secret-key, or conventional
encryption. If the sender and receiver use different keys, the system is referred
to as asymmetric, two-key, or public-key encryption.
3. The way in which the plaintext is processed. A block cipher processes the input one block of elements at a time, producing an output block for each input
block. A stream cipher processes the input elements continuously, producing output one element at a time, as it goes along.
Cryptanalysis and Brute-Force Attack
Typically, the objective of attacking an encryption system is to recover the key in
use rather than simply to recover the plaintext of a single ciphertext. There are two
general approaches to attacking a conventional encryption scheme:
■ Cryptanalysis: Cryptanalytic attacks rely on the nature of the algorithm plus perhaps some knowledge of the general characteristics of the plaintext or even
some sample plaintext–ciphertext pairs. This type of attack exploits the charac-
teristics of the algorithm to attempt to deduce a specific plaintext or to deduce
the key being used.
■ Brute-force attack: The attacker tries every possible key on a piece of cipher- text until an intelligible translation into plaintext is obtained. On average, half
of all possible keys must be tried to achieve success.
If either type of attack succeeds in deducing the key, the effect is catastrophic:
All future and past messages encrypted with that key are compromised.
We first consider cryptanalysis and then discuss brute-force attacks.
Table 3.1 summarizes the various types of cryptanalytic attacks based on the amount of information known to the cryptanalyst. The most difficult problem is
presented when all that is available is the ciphertext only. In some cases, not even the encryption algorithm is known, but in general, we can assume that the opponent
does know the algorithm used for encryption. One possible attack under these cir-
cumstances is the brute-force approach of trying all possible keys. If the key space
is very large, this becomes impractical. Thus, the opponent must rely on an analysis
of the ciphertext itself, generally applying various statistical tests to it. To use this
90 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES
approach, the opponent must have some general idea of the type of plaintext that
is concealed, such as English or French text, an EXE file, a Java source listing, an
accounting file, and so on.
The ciphertext-only attack is the easiest to defend against because the oppo-
nent has the least amount of information to work with. In many cases, however,
the analyst has more information. The analyst may be able to capture one or more
plaintext messages as well as their encryptions. Or the analyst may know that certain
plaintext patterns will appear in a message. For example, a file that is encoded in the
Postscript format always begins with the same pattern, or there may be a standard-
ized header or banner to an electronic funds transfer message, and so on. All these
are examples of known plaintext. With this knowledge, the analyst may be able to deduce the key on the basis of the way in which the known plaintext is transformed.
Closely related to the known-plaintext attack is what might be referred to as a
probable-word attack. If the opponent is working with the encryption of some gen-
eral prose message, he or she may have little knowledge of what is in the message.
However, if the opponent is after some very specific information, then parts of the
message may be known. For example, if an entire accounting file is being transmit-
ted, the opponent may know the placement of certain key words in the header of the
file. As another example, the source code for a program developed by Corporation
X might include a copyright statement in some standardized position.
If the analyst is able somehow to get the source system to insert into the sys-
tem a message chosen by the analyst, then a chosen-plaintext attack is possible. An example of this strategy is differential cryptanalysis, explored in Appendix S.
Type of Attack Known to Cryptanalyst
Ciphertext Only ■ Encryption algorithm
■ Ciphertext
Known Plaintext ■ Encryption algorithm
■ Ciphertext
■ One or more plaintext–ciphertext pairs formed with the secret key
Chosen Plaintext ■ Encryption algorithm
■ Ciphertext
■ Plaintext message chosen by cryptanalyst, together with its corresponding
ciphertext generated with the secret key
Chosen Ciphertext ■ Encryption algorithm
■ Ciphertext
■ Ciphertext chosen by cryptanalyst, together with its corresponding decrypted
plaintext generated with the secret key
Chosen Text ■ Encryption algorithm
■ Ciphertext
■ Plaintext message chosen by cryptanalyst, together with its corresponding
ciphertext generated with the secret key
■ Ciphertext chosen by cryptanalyst, together with its corresponding decrypted
plaintext generated with the secret key
Table 3.1 Types of Attacks on Encrypted Messages
3.1 / SYMMETRIC CIPHER MODEL 91
In general, if the analyst is able to choose the messages to encrypt, the analyst may
deliberately pick patterns that can be expected to reveal the structure of the key.
Table 3.1 lists two other types of attack: chosen ciphertext and chosen text.
These are less commonly employed as cryptanalytic techniques but are nevertheless
possible avenues of attack.
Only relatively weak algorithms fail to withstand a ciphertext-only attack.
Generally, an encryption algorithm is designed to withstand a known-plaintext
attack.
Two more definitions are worthy of note. An encryption scheme is
unconditionally secure if the ciphertext generated by the scheme does not contain enough information to determine uniquely the corresponding plaintext, no matter
how much ciphertext is available. That is, no matter how much time an opponent
has, it is impossible for him or her to decrypt the ciphertext simply because the
required information is not there. With the exception of a scheme known as the
one-time pad (described later in this chapter), there is no encryption algorithm that
is unconditionally secure. Therefore, all that the users of an encryption algorithm
can strive for is an algorithm that meets one or both of the following criteria:
■ The cost of breaking the cipher exceeds the value of the encrypted information.
■ The time required to break the cipher exceeds the useful lifetime of the
information.
An encryption scheme is said to be computationally secure if either of the foregoing two criteria are met. Unfortunately, it is very difficult to estimate the
amount of effort required to cryptanalyze ciphertext successfully.
All forms of cryptanalysis for symmetric encryption schemes are designed
to exploit the fact that traces of structure or pattern in the plaintext may survive
encryption and be discernible in the ciphertext. This will become clear as we exam-
ine various symmetric encryption schemes in this chapter. We will see in Part Two
that cryptanalysis for public-key schemes proceeds from a fundamentally different
premise, namely, that the mathematical properties of the pair of keys may make it
possible for one of the two keys to be deduced from the other.
A brute-force attack involves trying every possible key until an intelligible translation of the ciphertext into plaintext is obtained. On average, half of all pos-
sible keys must be tried to achieve success. That is, if there are X different keys, on average an attacker would discover the actual key after X/2 tries. It is important to note that there is more to a brute-force attack than simply running through all pos-
sible keys. Unless known plaintext is provided, the analyst must be able to recognize
plaintext as plaintext. If the message is just plain text in English, then the result pops
out easily, although the task of recognizing English would have to be automated. If
the text message has been compressed before encryption, then recognition is more
difficult. And if the message is some more general type of data, such as a numeri-
cal file, and this has been compressed, the problem becomes even more difficult to
automate. Thus, to supplement the brute-force approach, some degree of knowl-
edge about the expected plaintext is needed, and some means of automatically dis-
tinguishing plaintext from garble is also needed.
Hiva-Network.Com
92 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES
3.2 SUBSTITUTION TECHNIQUES
In this section and the next, we examine a sampling of what might be called classical
encryption techniques. A study of these techniques enables us to illustrate the basic
approaches to symmetric encryption used today and the types of cryptanalytic at-
tacks that must be anticipated.
The two basic building blocks of all encryption techniques are substitution
and transposition. We examine these in the next two sections. Finally, we discuss a
system that combines both substitution and transposition.
A substitution technique is one in which the letters of plaintext are replaced
by other letters or by numbers or symbols.1 If the plaintext is viewed as a sequence
of bits, then substitution involves replacing plaintext bit patterns with ciphertext bit
patterns.
Caesar Cipher
The earliest known, and the simplest, use of a substitution cipher was by Julius
Caesar. The Caesar cipher involves replacing each letter of the alphabet with the
letter standing three places further down the alphabet. For example,
plain: meet me after the toga party cipher: PHHW PH DIWHU WKH WRJD SDUWB
Note that the alphabet is wrapped around, so that the letter following Z is A.
We can define the transformation by listing all possibilities, as follows:
plain: a b c d e f g h i j k l m n o p q r s t u v w x y z cipher: D E F G H I J K L M N O P Q R S T U V W X Y Z A B C
Let us assign a numerical equivalent to each letter:
a b c d e f g h i j k l m
0 1 2 3 4 5 6 7 8 9 10 11 12
n o p q r s t u v w x y z
13 14 15 16 17 18 19 20 21 22 23 24 25
Then the algorithm can be expressed as follows. For each plaintext letter p, substi- tute the ciphertext letter C:2
C = E(3, p) = (p + 3) mod 26
A shift may be of any amount, so that the general Caesar algorithm is
C = E(k, p) = (p + k) mod 26 (3.1)
1When letters are involved, the following conventions are used in this book. Plaintext is always in lowercase; ciphertext is in uppercase; key values are in italicized lowercase. 2We define a mod n to be the remainder when a is divided by n. For example, 11 mod 7 = 4. See Chapter 2 for a further discussion of modular arithmetic.
3.2 / SUBSTITUTION TECHNIQUES 93
where k takes on a value in the range 1 to 25. The decryption algorithm is simply
p = D(k, C) = (C - k) mod 26 (3.2)
If it is known that a given ciphertext is a Caesar cipher, then a brute-force
cryptanalysis is easily performed: simply try all the 25 possible keys. Figure 3.3
shows the results of applying this strategy to the example ciphertext. In this case, the
plaintext leaps out as occupying the third line.
Three important characteristics of this problem enabled us to use a brute-
force cryptanalysis:
1. The encryption and decryption algorithms are known.
2. There are only 25 keys to try.
3. The language of the plaintext is known and easily recognizable.
In most networking situations, we can assume that the algorithms are known.
What generally makes brute-force cryptanalysis impractical is the use of an algo-
rithm that employs a large number of keys. For example, the triple DES algorithm,
Figure 3.3 Brute-Force Cryptanalysis of Caesar Cipher
PHHW PH DIWHU WKH WRJD SDUWB KEY
1 oggv og chvgt vjg vqic rctva
2 nffu nf bgufs uif uphb qbsuz
3 meet me after the toga party
4 ldds ld zesdq sgd snfz ozqsx
5 kccr kc ydrcp rfc rmey nyprw
6 jbbq jb xcqbo qeb qldx mxoqv
7 iaap ia wbpan pda pkcw lwnpu
8 hzzo hz vaozm ocz ojbv kvmot
9 gyyn gy uznyl nby niau julns
10 fxxm fx tymxk max mhzt itkmr
11 ewwl ew sxlwj lzw lgys hsjlq
12 dvvk dv rwkvi kyv kfxr grikp
13 cuuj cu qvjuh jxu jewq fqhjo
14 btti bt puitg iwt idvp epgin
15 assh as othsf hvs hcuo dofhm
16 zrrg zr nsgre gur gbtn cnegl
17 yqqf yq mrfqd ftq fasm bmdfk
18 xppe xp lqepc esp ezrl alcej
19 wood wo kpdob dro dyqk zkbdi
20 vnnc vn jocna cqn cxpj yjach
21 ummb um inbmz bpm bwoi xizbg
22 tlla tl hmaly aol avnh whyaf
23 skkz sk glzkx znk zumg vgxze
24 rjjy rj fkyjw ymj ytlf ufwyd
25 qiix qi ejxiv xli xske tevxc
94 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES
examined in Chapter 7, makes use of a 168-bit key, giving a key space of 2168 or
greater than 3.7 * 1050 possible keys. The third characteristic is also significant. If the language of the plaintext is
unknown, then plaintext output may not be recognizable. Furthermore, the input
may be abbreviated or compressed in some fashion, again making recognition dif-
ficult. For example, Figure 3.4 shows a portion of a text file compressed using an
algorithm called ZIP. If this file is then encrypted with a simple substitution cipher
(expanded to include more than just 26 alphabetic characters), then the plaintext
may not be recognized when it is uncovered in the brute-force cryptanalysis.
Monoalphabetic Ciphers
With only 25 possible keys, the Caesar cipher is far from secure. A dramatic increase
in the key space can be achieved by allowing an arbitrary substitution. Before pro-
ceeding, we define the term permutation. A permutation of a finite set of elements S is an ordered sequence of all the elements of S, with each element appearing exactly once. For example, if S = {a, b, c}, there are six permutations of S:
abc, acb, bac, bca, cab, cba
In general, there are n! permutations of a set of n elements, because the first element can be chosen in one of n ways, the second in n - 1 ways, the third in n - 2 ways, and so on.
Recall the assignment for the Caesar cipher:
plain: a b c d e f g h i j k l m n o p q r s t u v w x y z cipher: D E F G H I J K L M N O P Q R S T U V W X Y Z A B C
If, instead, the “cipher” line can be any permutation of the 26 alphabetic characters,
then there are 26! or greater than 4 * 1026 possible keys. This is 10 orders of mag- nitude greater than the key space for DES and would seem to eliminate brute-force
techniques for cryptanalysis. Such an approach is referred to as a monoalphabetic substitution cipher, because a single cipher alphabet (mapping from plain alphabet to cipher alphabet) is used per message.
There is, however, another line of attack. If the cryptanalyst knows the nature
of the plaintext (e.g., noncompressed English text), then the analyst can exploit the
regularities of the language. To see how such a cryptanalysis might proceed, we give
a partial example here that is adapted from one in [SINK09]. The ciphertext to be
solved is
Figure 3.4 Sample of Compressed Text
3.2 / SUBSTITUTION TECHNIQUES 95
UZQSOVUOHXMOPVGPOZPEVSGZWSZOPFPESXUDBMETSXAIZ VUEPHZHMDZSHZOWSFPAPPDTSVPQUZWYMXUZUHSX EPYEPOPDZSZUFPOMBZWPFUPZHMDJUDTMOHMQ
As a first step, the relative frequency of the letters can be determined and
compared to a standard frequency distribution for English, such as is shown in
Figure 3.5 (based on [LEWA00]). If the message were long enough, this technique
alone might be sufficient, but because this is a relatively short message, we cannot
expect an exact match. In any case, the relative frequencies of the letters in the
ciphertext (in percentages) are as follows:
P 13.33 H 5.83 F 3.33 B 1.67 C 0.00
Z 11.67 D 5.00 W 3.33 G 1.67 K 0.00
S 8.33 E 5.00 Q 2.50 Y 1.67 L 0.00
U 8.33 V 4.17 T 2.50 I 0.83 N 0.00
O 7.50 X 4.17 A 1.67 J 0.83 R 0.00
M 6.67
Comparing this breakdown with Figure 3.5, it seems likely that cipher letters
P and Z are the equivalents of plain letters e and t, but it is not certain which is which.
The letters S, U, O, M, and H are all of relatively high frequency and probably
Figure 3.5 Relative Frequency of Letters in English Text
0
2
4
6
8
10
12
14
A
8. 16
7
1. 49
2
2. 78
2
4. 25
3
12 .7
02
2. 22
8
2. 01
5
6. 09
4 6 .9
96
0. 15
3 0. 77
2
4. 02
5
2. 40
6
6. 74
9 7. 50
7
1. 92
9
0. 09
5
5. 98
7
6. 32
7
9. 05
6
2. 75
8
0. 97
8
2. 36
0
0. 15
0
1. 97
4
0. 07
4
B C D E F G H I J K L M N
R el
at iv
e fr
eq ue
nc y
(% )
O P Q R S T U V W X Y Z
96 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES
correspond to plain letters from the set {a, h, i, n, o, r, s}. The letters with the lowest
frequencies (namely, A, B, G, Y, I, J) are likely included in the set {b, j, k, q, v, x, z}.
There are a number of ways to proceed at this point. We could make some
tentative assignments and start to fill in the plaintext to see if it looks like a rea-
sonable “skeleton” of a message. A more systematic approach is to look for other
regularities. For example, certain words may be known to be in the text. Or we
could look for repeating sequences of cipher letters and try to deduce their plaintext
equivalents.
A powerful tool is to look at the frequency of two-letter combinations, known
as digrams. A table similar to Figure 3.5 could be drawn up showing the relative fre- quency of digrams. The most common such digram is th. In our ciphertext, the most
common digram is ZW, which appears three times. So we make the correspondence
of Z with t and W with h. Then, by our earlier hypothesis, we can equate P with e.
Now notice that the sequence ZWP appears in the ciphertext, and we can translate
that sequence as “the.” This is the most frequent trigram (three-letter combination)
in English, which seems to indicate that we are on the right track.
Next, notice the sequence ZWSZ in the first line. We do not know that these
four letters form a complete word, but if they do, it is of the form th_t. If so, S
equates with a.
So far, then, we have
UZQSOVUOHXMOPVGPOZPEVSGZWSZOPFPESXUDBMETSXAIZ t a e e te a that e e a a VUEPHZHMDZSHZOWSFPAPPDTSVPQUZWYMXUZUHSX
e t ta t ha e ee a e th t a EPYEPOPDZSZUFPOMBZWPFUPZHMDJUDTMOHMQ e e e tat e the t
Only four letters have been identified, but already we have quite a bit of the
message. Continued analysis of frequencies plus trial and error should easily yield a
solution from this point. The complete plaintext, with spaces added between words,
follows:
it was disclosed yesterday that several informal but direct contacts have been made with political representatives of the viet cong in moscow
Monoalphabetic ciphers are easy to break because they reflect the frequency
data of the original alphabet. A countermeasure is to provide multiple substi-
tutes, known as homophones, for a single letter. For example, the letter e could
be assigned a number of different cipher symbols, such as 16, 74, 35, and 21, with
each homophone assigned to a letter in rotation or randomly. If the number of
symbols assigned to each letter is proportional to the relative frequency of that let-
ter, then single-letter frequency information is completely obliterated. The great
mathematician Carl Friedrich Gauss believed that he had devised an unbreak-
able cipher using homophones. However, even with homophones, each element
of plaintext affects only one element of ciphertext, and multiple-letter patterns
3.2 / SUBSTITUTION TECHNIQUES 97
(e.g., digram frequencies) still survive in the ciphertext, making cryptanalysis rela-
tively straightforward.
Two principal methods are used in substitution ciphers to lessen the extent to
which the structure of the plaintext survives in the ciphertext: One approach is to
encrypt multiple letters of plaintext, and the other is to use multiple cipher alpha-
bets. We briefly examine each.
Playfair Cipher
The best-known multiple-letter encryption cipher is the Playfair, which treats di-
grams in the plaintext as single units and translates these units into ciphertext
digrams.3
The Playfair algorithm is based on the use of a 5 * 5 matrix of letters con- structed using a keyword. Here is an example, solved by Lord Peter Wimsey in
Dorothy Sayers’s Have His Carcase:4
M O N A R
C H Y B D
E F G I/J K
L P Q S T
U V W X Z
In this case, the keyword is monarchy. The matrix is constructed by filling in the letters of the keyword (minus duplicates) from left to right and from top to
bottom, and then filling in the remainder of the matrix with the remaining letters in
alphabetic order. The letters I and J count as one letter. Plaintext is encrypted two
letters at a time, according to the following rules:
1. Repeating plaintext letters that are in the same pair are separated with a filler letter, such as x, so that balloon would be treated as ba lx lo on.
2. Two plaintext letters that fall in the same row of the matrix are each replaced by the letter to the right, with the first element of the row circularly following
the last. For example, ar is encrypted as RM.
3. Two plaintext letters that fall in the same column are each replaced by the let- ter beneath, with the top element of the column circularly following the last.
For example, mu is encrypted as CM.
4. Otherwise, each plaintext letter in a pair is replaced by the letter that lies in its own row and the column occupied by the other plaintext letter. Thus, hs
becomes BP and ea becomes IM (or JM, as the encipherer wishes).
The Playfair cipher is a great advance over simple monoalphabetic ciphers.
For one thing, whereas there are only 26 letters, there are 26 * 26 = 676 digrams,
3This cipher was actually invented by British scientist Sir Charles Wheatstone in 1854, but it bears the name of his friend Baron Playfair of St. Andrews, who championed the cipher at the British foreign office. 4The book provides an absorbing account of a probable-word attack.
98 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES
so that identification of individual digrams is more difficult. Furthermore, the rela-
tive frequencies of individual letters exhibit a much greater range than that of
digrams, making frequency analysis much more difficult. For these reasons, the
Playfair cipher was for a long time considered unbreakable. It was used as the stan-
dard field system by the British Army in World War I and still enjoyed considerable
use by the U.S. Army and other Allied forces during World War II.
Despite this level of confidence in its security, the Playfair cipher is relatively
easy to break, because it still leaves much of the structure of the plaintext language
intact. A few hundred letters of ciphertext are generally sufficient.
One way of revealing the effectiveness of the Playfair and other ciphers is
shown in Figure 3.6. The line labeled plaintext plots a typical frequency distribution of the 26 alphabetic characters (no distinction between upper and lower case) in
ordinary text. This is also the frequency distribution of any monoalphabetic substi-
tution cipher, because the frequency values for individual letters are the same, just
with different letters substituted for the original letters. The plot is developed in the
following way: The number of occurrences of each letter in the text is counted and
divided by the number of occurrences of the most frequently used letter. Using the
results of Figure 3.5, we see that e is the most frequently used letter. As a result, e
has a relative frequency of 1, t of 9.056/12.702 ≈ 0.72, and so on. The points on the horizontal axis correspond to the letters in order of decreasing frequency.
Figure 3.6 also shows the frequency distribution that results when the text is
encrypted using the Playfair cipher. To normalize the plot, the number of occur-
rences of each letter in the ciphertext was again divided by the number of occur-
rences of e in the plaintext. The resulting plot therefore shows the extent to which
the frequency distribution of letters, which makes it trivial to solve substitution
Figure 3.6 Relative Frequency of Occurrence of Letters
0 1 2 3 4 5 6 1 7 8 9 10 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Plaintext
Playfair
Vigenère
Random polyalphabetic
Frequency ranked letters (decreasing frequency)
N or
m al
iz ed
r el
at iv
e fr
eq ue
nc y
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
3.2 / SUBSTITUTION TECHNIQUES 99
ciphers, is masked by encryption. If the frequency distribution information were
totally concealed in the encryption process, the ciphertext plot of frequencies would
be flat, and cryptanalysis using ciphertext only would be effectively impossible. As
the figure shows, the Playfair cipher has a flatter distribution than does plaintext,
but nevertheless, it reveals plenty of structure for a cryptanalyst to work with. The
plot also shows the Vigenère cipher, discussed subsequently. The Hill and Vigenère
curves on the plot are based on results reported in [SIMM93].
Hill Cipher5
Another interesting multiletter cipher is the Hill cipher, developed by the math-
ematician Lester Hill in 1929.
CONCEPTS FROM LINEAR ALGEBRA Before describing the Hill cipher, let us briefly review some terminology from linear algebra. In this discussion, we are concerned
with matrix arithmetic modulo 26. For the reader who needs a refresher on matrix
multiplication and inversion, see Appendix E.
We define the inverse M-1 of a square matrix M by the equation M(M-1) = M-1M = I, where I is the identity matrix. I is a square matrix that is all zeros except for ones along the main diagonal from upper left to lower right. The inverse of a
matrix does not always exist, but when it does, it satisfies the preceding equation.
For example,
A = ¢ 5 8 17 3
≤ A-1 mod 26 = ¢9 2 1 15
≤ AA-1 = ¢ (5 * 9) + (8 * 1) (5 * 2) + (8 * 15)
(17 * 9) + (3 * 1) (17 * 2) + (3 * 15) ≤
= ¢ 53 130 156 79
≤ mod 26 = ¢1 0 0 1
≤ To explain how the inverse of a matrix is computed, we begin with the concept
of determinant. For any square matrix (m * m), the determinant equals the sum of all the products that can be formed by taking exactly one element from each row
and exactly one element from each column, with certain of the product terms pre-
ceded by a minus sign. For a 2 * 2 matrix,
¢k11 k12 k21 k22
≤ the determinant is k11k22 - k12k21. For a 3 * 3 matrix, the value of the determinant is k11k22k33 + k21k32k13 + k31k12k23 - k31k22k13 - k21k12k33 - k11k32k23. If a square
5This cipher is somewhat more difficult to understand than the others in this chapter, but it illustrates an important point about cryptanalysis that will be useful later on. This subsection can be skipped on a first reading.
100 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES
matrix A has a nonzero determinant, then the inverse of the matrix is computed as [A-1]ij = (det A)
-1( - 1)i + j(Dji), where (Dji) is the subdeterminant formed by deleting the jth row and the ith column of A, det(A) is the determinant of A, and (det A)-1 is the multiplicative inverse of (det A) mod 26.
Continuing our example,
det ¢ 5 8 17 3
≤ = (5 * 3) - (8 * 17) = - 121 mod 26 = 9 We can show that 9-1 mod 26 = 3, because 9 * 3 = 27 mod 26 = 1 (see
Chapter 2 or Appendix E). Therefore, we compute the inverse of A as
A = ¢ 5 8 17 3
≤ A-1 mod 26 = 3¢ 3 - 8
- 17 5 ≤ = 3¢3 18
9 5 ≤ = ¢ 9 54
27 15 ≤ = ¢9 2
1 15 ≤
THE HILL ALGORITHM This encryption algorithm takes m successive plaintext let- ters and substitutes for them m ciphertext letters. The substitution is determined by m linear equations in which each character is assigned a numerical value (a = 0, b = 1, c , z = 25). For m = 3, the system can be described as
c1 = (k11p1 + k21p2 + k31p3) mod 26
c2 = (k12p1 + k22p2 + k32p3) mod 26
c3 = (k13p1 + k23p2 + k33p3) mod 26
This can be expressed in terms of row vectors and matrices:6
(c1 c2 c3) = (p1 p2 p3)£ k11 k12 k13k21 k22 k23 k31 k32 k33
≥ mod 26 or
C = PK mod 26
where C and P are row vectors of length 3 representing the plaintext and ciphertext, and K is a 3 * 3 matrix representing the encryption key. Operations are performed mod 26.
6Some cryptography books express the plaintext and ciphertext as column vectors, so that the column vector is placed after the matrix rather than the row vector placed before the matrix. Sage uses row vec- tors, so we adopt that convention.
Hiva-Network.Com
3.2 / SUBSTITUTION TECHNIQUES 101
For example, consider the plaintext “paymoremoney” and use the encryption key
K = £ 17 17 521 18 21 2 2 19
≥ The first three letters of the plaintext are represented by the vector (15 0 24).
Then (15 0 24)K = (303 303 531) mod 26 = (17 17 11) = RRL. Continuing in this fashion, the ciphertext for the entire plaintext is RRLMWBKASPDH.
Decryption requires using the inverse of the matrix K. We can compute det K = 23, and therefore, (det K)-1 mod 26 = 17. We can then compute the inverse as7
K-1 = £ 4 9 1515 17 6 24 0 17
≥ This is demonstrated as
£ 17 17 521 18 21 2 2 19
≥£ 4 9 1515 17 6 24 0 17
≥ = £ 443 442 442858 495 780 494 52 365
≥ mod 26 = £ 1 0 00 1 0 0 0 1
≥ It is easily seen that if the matrix K-1 is applied to the ciphertext, then the
plaintext is recovered.
In general terms, the Hill system can be expressed as
C = E(K, P) = PK mod 26
P = D(K, C) = CK-1 mod 26 = PKK-1 = P
As with Playfair, the strength of the Hill cipher is that it completely hides
single-letter frequencies. Indeed, with Hill, the use of a larger matrix hides more
frequency information. Thus, a 3 * 3 Hill cipher hides not only single-letter but also two-letter frequency information.
Although the Hill cipher is strong against a ciphertext-only attack, it is easily
broken with a known plaintext attack. For an m * m Hill cipher, suppose we have m plaintext–ciphertext pairs, each of length m. We label the pairs Pj = (p1jp1j c pmj) and Cj = (c1jc1j c cmj) such that Cj = PjK for 1 … j … m and for some unknown key matrix K. Now define two m * m matrices X = (pij) and Y = (cij). Then we can form the matrix equation Y = XK. If X has an inverse, then we can determine K = X-1Y. If X is not invertible, then a new version of X can be formed with addi- tional plaintext–ciphertext pairs until an invertible X is obtained.
Consider this example. Suppose that the plaintext “hillcipher” is encrypted
using a 2 * 2 Hill cipher to yield the ciphertext HCRZSSXNSP. Thus, we know that (7 8)K mod 26 = (7 2); (11 11)K mod 26 = (17 25); and so on. Using the first two plaintext-ciphertext pairs, we have
7The calculations for this example are provided in detail in Appendix E.
102 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES
¢ 7 2 17 25
≤ = ¢ 7 8 11 11
≤K mod 26 The inverse of X can be computed:
¢ 7 8 11 11
≤-1 = ¢25 22 1 23
≤ so
K = ¢25 22 1 23
≤ ¢ 7 2 17 25
≤ = ¢549 600 398 577
≤ mod 26 = ¢3 2 8 5
≤ This result is verified by testing the remaining plaintext–ciphertext pairs.
Polyalphabetic Ciphers
Another way to improve on the simple monoalphabetic technique is to use differ-
ent monoalphabetic substitutions as one proceeds through the plaintext message.
The general name for this approach is polyalphabetic substitution cipher. All these techniques have the following features in common:
1. A set of related monoalphabetic substitution rules is used.
2. A key determines which particular rule is chosen for a given transformation.
VIGENÈRE CIPHER The best known, and one of the simplest, polyalphabetic ciphers is the Vigenère cipher. In this scheme, the set of related monoalphabetic substitu-
tion rules consists of the 26 Caesar ciphers with shifts of 0 through 25. Each cipher is
denoted by a key letter, which is the ciphertext letter that substitutes for the plain-
text letter a. Thus, a Caesar cipher with a shift of 3 is denoted by the key value 3.8
We can express the Vigenère cipher in the following manner. Assume a
sequence of plaintext letters P = p0, p1, p2, c , pn - 1 and a key consisting of the sequence of letters K = k0, k1, k2, c , km - 1, where typically m 6 n. The sequence of ciphertext letters C = C0, C1, C2, c , Cn - 1 is calculated as follows:
C = C0, C1, C2, c , Cn - 1 = E(K, P) = E[(k0, k1, k2, c , km - 1), (p0, p1, p2, c , pn - 1)]
= (p0 + k0) mod 26, (p1 + k1) mod 26, c ,(pm - 1 + km - 1) mod 26,
(pm + k0) mod 26, (pm + 1 + k1) mod 26, c , (p2m - 1 + km - 1) mod 26, c
Thus, the first letter of the key is added to the first letter of the plaintext, mod 26,
the second letters are added, and so on through the first m letters of the plaintext. For the next m letters of the plaintext, the key letters are repeated. This process
8To aid in understanding this scheme and also to aid in it use, a matrix known as the Vigenère tableau is often used. This tableau is discussed in a document at box.com/Crypto7e.
3.2 / SUBSTITUTION TECHNIQUES 103
continues until all of the plaintext sequence is encrypted. A general equation of the
encryption process is
Ci = (pi + ki mod m) mod 26 (3.3)
Compare this with Equation (3.1) for the Caesar cipher. In essence, each plain-
text character is encrypted with a different Caesar cipher, depending on the corre-
sponding key character. Similarly, decryption is a generalization of Equation (3.2):
pi = (Ci - ki mod m) mod 26 (3.4)
To encrypt a message, a key is needed that is as long as the message. Usually,
the key is a repeating keyword. For example, if the keyword is deceptive, the mes- sage “we are discovered save yourself” is encrypted as
key: deceptivedeceptivedeceptive plaintext: wearediscoveredsaveyourself ciphertext: ZICVTWQNGRZGVTWAVZHCQYGLMGJ
Expressed numerically, we have the following result.
key 3 4 2 4 15 19 8 21 4 3 4 2 4 15
plaintext 22 4 0 17 4 3 8 18 2 14 21 4 17 4
ciphertext 25 8 2 21 19 22 16 13 6 17 25 6 21 19
key 19 8 21 4 3 4 2 4 15 19 8 21 4
plaintext 3 18 0 21 4 24 14 20 17 18 4 11 5
ciphertext 22 0 21 25 7 2 16 24 6 11 12 6 9
The strength of this cipher is that there are multiple ciphertext letters for
each plaintext letter, one for each unique letter of the keyword. Thus, the letter fre-
quency information is obscured. However, not all knowledge of the plaintext struc-
ture is lost. For example, Figure 3.6 shows the frequency distribution for a Vigenère
cipher with a keyword of length 9. An improvement is achieved over the Playfair
cipher, but considerable frequency information remains.
It is instructive to sketch a method of breaking this cipher, because the method
reveals some of the mathematical principles that apply in cryptanalysis.
First, suppose that the opponent believes that the ciphertext was encrypted
using either monoalphabetic substitution or a Vigenère cipher. A simple test can
be made to make a determination. If a monoalphabetic substitution is used, then
the statistical properties of the ciphertext should be the same as that of the lan-
guage of the plaintext. Thus, referring to Figure 3.5, there should be one cipher let-
ter with a relative frequency of occurrence of about 12.7%, one with about 9.06%,
and so on. If only a single message is available for analysis, we would not expect
an exact match of this small sample with the statistical profile of the plaintext lan-
guage. Nevertheless, if the correspondence is close, we can assume a monoalpha-
betic substitution.
104 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES
If, on the other hand, a Vigenère cipher is suspected, then progress depends on
determining the length of the keyword, as will be seen in a moment. For now, let us
concentrate on how the keyword length can be determined. The important insight
that leads to a solution is the following: If two identical sequences of plaintext let-
ters occur at a distance that is an integer multiple of the keyword length, they will
generate identical ciphertext sequences. In the foregoing example, two instances
of the sequence “red” are separated by nine character positions. Consequently, in
both cases, r is encrypted using key letter e, e is encrypted using key letter p, and d is encrypted using key letter t. Thus, in both cases, the ciphertext sequence is VTW. We indicate this above by underlining the relevant ciphertext letters and shading
the relevant ciphertext numbers.
An analyst looking at only the ciphertext would detect the repeated sequences
VTW at a displacement of 9 and make the assumption that the keyword is either
three or nine letters in length. The appearance of VTW twice could be by chance
and may not reflect identical plaintext letters encrypted with identical key letters.
However, if the message is long enough, there will be a number of such repeated
ciphertext sequences. By looking for common factors in the displacements of the vari-
ous sequences, the analyst should be able to make a good guess of the keyword length.
Solution of the cipher now depends on an important insight. If the keyword
length is m, then the cipher, in effect, consists of m monoalphabetic substitution ciphers. For example, with the keyword DECEPTIVE, the letters in positions 1, 10,
19, and so on are all encrypted with the same monoalphabetic cipher. Thus, we can
use the known frequency characteristics of the plaintext language to attack each of
the monoalphabetic ciphers separately.
The periodic nature of the keyword can be eliminated by using a nonrepeating
keyword that is as long as the message itself. Vigenère proposed what is referred to
as an autokey system, in which a keyword is concatenated with the plaintext itself to provide a running key. For our example,
key: deceptivewearediscoveredsav plaintext: wearediscoveredsaveyourself ciphertext: ZICVTWQNGKZEIIGASXSTSLVVWLA
Even this scheme is vulnerable to cryptanalysis. Because the key and the
plaintext share the same frequency distribution of letters, a statistical technique can
be applied. For example, e enciphered by e, by Figure 3.5, can be expected to occur with a frequency of (0.127)2 ≈ 0.016, whereas t enciphered by t would occur only about half as often. These regularities can be exploited to achieve successful
cryptanalysis.9
VERNAM CIPHER The ultimate defense against such a cryptanalysis is to choose a keyword that is as long as the plaintext and has no statistical relationship to it. Such
a system was introduced by an AT&T engineer named Gilbert Vernam in 1918.
9Although the techniques for breaking a Vigenère cipher are by no means complex, a 1917 issue of Scientific American characterized this system as “impossible of translation.” This is a point worth remem- bering when similar claims are made for modern algorithms.
3.2 / SUBSTITUTION TECHNIQUES 105
His system works on binary data (bits) rather than letters. The system can be
expressed succinctly as follows (Figure 3.7):
ci = pi ⊕ ki
where
pi = ith binary digit of plaintext ki = ith binary digit of key ci = ith binary digit of ciphertext ⊕ = exclusive@or (XOR) operation
Compare this with Equation (3.3) for the Vigenère cipher.
Thus, the ciphertext is generated by performing the bitwise XOR of the plain-
text and the key. Because of the properties of the XOR, decryption simply involves
the same bitwise operation:
pi = ci ⊕ ki
which compares with Equation (3.4).
The essence of this technique is the means of construction of the key. Vernam
proposed the use of a running loop of tape that eventually repeated the key, so that
in fact the system worked with a very long but repeating keyword. Although such
a scheme, with a long key, presents formidable cryptanalytic difficulties, it can be
broken with sufficient ciphertext, the use of known or probable plaintext sequences,
or both.
One-Time Pad
An Army Signal Corp officer, Joseph Mauborgne, proposed an improvement to the
Vernam cipher that yields the ultimate in security. Mauborgne suggested using a
random key that is as long as the message, so that the key need not be repeated. In
addition, the key is to be used to encrypt and decrypt a single message, and then is
discarded. Each new message requires a new key of the same length as the new mes-
sage. Such a scheme, known as a one-time pad, is unbreakable. It produces random output that bears no statistical relationship to the plaintext. Because the ciphertext
Figure 3.7 Vernam Cipher
Key stream generator
Cryptographic bit stream (ki)
Cryptographic bit stream (ki)
Plaintext (pi)
Plaintext (pi)
Ciphertext (ci )
Key stream generator
106 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES
contains no information whatsoever about the plaintext, there is simply no way to
break the code.
An example should illustrate our point. Suppose that we are using a Vigenère
scheme with 27 characters in which the twenty-seventh character is the space
character, but with a one-time key that is as long as the message. Consider the
ciphertext
ANKYODKYUREPFJBYOJDSPLREYIUNOFDOIUERFPLUYTS
We now show two different decryptions using two different keys:
ciphertext: ANKYODKYUREPFJBYOJDSPLREYIUNOFDOIUERFPLUYTS key: pxlmvmsydofuyrvzwc tnlebnecvgdupahfzzlmnyih plaintext: mr mustard with the candlestick in the hall
ciphertext: ANKYODKYUREPFJBYOJDSPLREYIUNOFDOIUERFPLUYTS key: pftgpmiydgaxgoufhklllmhsqdqogtewbqfgyovuhwt plaintext: miss scarlet with the knife in the library
Suppose that a cryptanalyst had managed to find these two keys. Two plau-
sible plaintexts are produced. How is the cryptanalyst to decide which is the correct
decryption (i.e., which is the correct key)? If the actual key were produced in a truly
random fashion, then the cryptanalyst cannot say that one of these two keys is more
likely than the other. Thus, there is no way to decide which key is correct and there-
fore which plaintext is correct.
In fact, given any plaintext of equal length to the ciphertext, there is a key that
produces that plaintext. Therefore, if you did an exhaustive search of all possible
keys, you would end up with many legible plaintexts, with no way of knowing which
was the intended plaintext. Therefore, the code is unbreakable.
The security of the one-time pad is entirely due to the randomness of the key.
If the stream of characters that constitute the key is truly random, then the stream
of characters that constitute the ciphertext will be truly random. Thus, there are no
patterns or regularities that a cryptanalyst can use to attack the ciphertext.
In theory, we need look no further for a cipher. The one-time pad offers com-
plete security but, in practice, has two fundamental difficulties:
1. There is the practical problem of making large quantities of random keys. Any heavily used system might require millions of random characters on a regular
basis. Supplying truly random characters in this volume is a significant task.
2. Even more daunting is the problem of key distribution and protection. For every message to be sent, a key of equal length is needed by both sender and
receiver. Thus, a mammoth key distribution problem exists.
Because of these difficulties, the one-time pad is of limited utility and is useful
primarily for low-bandwidth channels requiring very high security.
The one-time pad is the only cryptosystem that exhibits what is referred to as
perfect secrecy. This concept is explored in Appendix F.
3.3 / TRANSPOSITION TECHNIQUES 107
3.3 TRANSPOSITION TECHNIQUES
All the techniques examined so far involve the substitution of a ciphertext symbol
for a plaintext symbol. A very different kind of mapping is achieved by performing
some sort of permutation on the plaintext letters. This technique is referred to as a
transposition cipher.
The simplest such cipher is the rail fence technique, in which the plaintext is written down as a sequence of diagonals and then read off as a sequence of rows.
For example, to encipher the message “meet me after the toga party” with a rail
fence of depth 2, we write the following:
m e m a t r h t g p r y e t e f e t e o a a t
The encrypted message is
MEMATRHTGPRYETEFETEOAAT
This sort of thing would be trivial to cryptanalyze. A more complex scheme is
to write the message in a rectangle, row by row, and read the message off, column
by column, but permute the order of the columns. The order of the columns then
becomes the key to the algorithm. For example,
Key: 4 3 1 2 5 6 7 Plaintext: a t t a c k p o s t p o n e d u n t i l t w o a m x y z Ciphertext: TTNAAPTMTSUOAODWCOIXKNLYPETZ
Thus, in this example, the key is 4312567. To encrypt, start with the column
that is labeled 1, in this case column 3. Write down all the letters in that column.
Proceed to column 4, which is labeled 2, then column 2, then column 1, then
columns 5, 6, and 7.
A pure transposition cipher is easily recognized because it has the same letter
frequencies as the original plaintext. For the type of columnar transposition just
shown, cryptanalysis is fairly straightforward and involves laying out the cipher-
text in a matrix and playing around with column positions. Digram and trigram fre-
quency tables can be useful.
The transposition cipher can be made significantly more secure by perform-
ing more than one stage of transposition. The result is a more complex permutation
that is not easily reconstructed. Thus, if the foregoing message is reencrypted using
the same algorithm,
108 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES
Key: 4 3 1 2 5 6 7 Input: t t n a a p t m t s u o a o d w c o i x k n l y p e t z Output: NSCYAUOPTTWLTMDNAOIEPAXTTOKZ
To visualize the result of this double transposition, designate the letters in the
original plaintext message by the numbers designating their position. Thus, with 28
letters in the message, the original sequence of letters is
01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
After the first transposition, we have
03 10 17 24 04 11 18 25 02 09 16 23 01 08 15 22 05 12 19 26 06 13 20 27 07 14 21 28
which has a somewhat regular structure. But after the second transposition, we have
17 09 05 27 24 16 12 07 10 02 22 20 03 25 15 13 04 23 19 14 11 01 26 21 18 08 06 28
This is a much less structured permutation and is much more difficult to cryptanalyze.
3.4 ROTOR MACHINES
The example just given suggests that multiple stages of encryption can produce an
algorithm that is significantly more difficult to cryptanalyze. This is as true of substi-
tution ciphers as it is of transposition ciphers. Before the introduction of DES, the
most important application of the principle of multiple stages of encryption was a
class of systems known as rotor machines.10
The basic principle of the rotor machine is illustrated in Figure 3.8. The
machine consists of a set of independently rotating cylinders through which electri-
cal pulses can flow. Each cylinder has 26 input pins and 26 output pins, with internal
wiring that connects each input pin to a unique output pin. For simplicity, only three
of the internal connections in each cylinder are shown.
If we associate each input and output pin with a letter of the alphabet, then a
single cylinder defines a monoalphabetic substitution. For example, in Figure 3.8,
if an operator depresses the key for the letter A, an electric signal is applied to
10Machines based on the rotor principle were used by both Germany (Enigma) and Japan (Purple) in World War II. The breaking of both codes by the Allies was a significant factor in the war’s outcome.
3.4 / ROTOR MACHINES 109
the first pin of the first cylinder and flows through the internal connection to the
twenty-fifth output pin.
Consider a machine with a single cylinder. After each input key is depressed,
the cylinder rotates one position, so that the internal connections are shifted accord-
ingly. Thus, a different monoalphabetic substitution cipher is defined. After 26 let-
ters of plaintext, the cylinder would be back to the initial position. Thus, we have a
polyalphabetic substitution algorithm with a period of 26.
A single-cylinder system is trivial and does not present a formidable crypt-
analytic task. The power of the rotor machine is in the use of multiple cylinders, in
which the output pins of one cylinder are connected to the input pins of the next.
Figure 3.8 shows a three-cylinder system. The left half of the figure shows a position
in which the input from the operator to the first pin (plaintext letter a) is routed
through the three cylinders to appear at the output of the second pin (ciphertext
letter B).
With multiple cylinders, the one closest to the operator input rotates one
pin position with each keystroke. The right half of Figure 3.8 shows the system’s
configuration after a single keystroke. For every complete rotation of the inner
cylinder, the middle cylinder rotates one pin position. Finally, for every complete
rotation of the middle cylinder, the outer cylinder rotates one pin position. This
is the same type of operation seen with an odometer. The result is that there are
26 * 26 * 26 = 17,576 different substitution alphabets used before the system
Figure 3.8 Three-Rotor Machine with Wiring Represented by Numbered Contacts
24 25 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
21 3
15 1
19 10 14 26 20 8
16 7
22 4
11 5
17 9
12 23 18 2
25 6
24 13
A B C D E F G H I J
K L M N O P Q R S T U V W X Y Z
26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
20 1 6 4
15 3
14 12 23 5
16 2
22 19 11 18 25 24 13 7
10 8
21 9
26 17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
8 18 26 17 20 22 10 3
13 11 4
23 5
24 9
12 25 16 19 6
15 21 2 7 1
14
A B C D E F G H I J
K L M N O P Q R S T U V W X Y Z
Direction of motion Direction of motion
Fast rotor Medium rotor Slow rotor Fast rotor Medium rotor Slow rotor (a) Initial setting (b) Setting after one keystroke
A B C D E F G H I J
K L M N O P Q R S T U V W X Y Z
23 24 25 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
13 21 3
15 1
19 10 14 26 20 8
16 7
22 4
11 5
17 9
12 23 18 2
25 6
24
26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
20 1 6 4
15 3
14 12 23 5
16 2
22 19 11 18 25 24 13 7
10 8
21 9
26 17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
8 18 26 17 20 22 10 3
13 11 4
23 5
24 9
12 25 16 19 6
15 21 2 7 1
14
A B C D E F G H I J
K L M N O P Q R S T U V W X Y Z
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110 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES
repeats. The addition of fourth and fifth rotors results in periods of 456,976 and
11,881,376 letters, respectively. Thus, a given setting of a 5-rotor machine is equiva-
lent to a Vigenère cipher with a key length of 11,881,376.
Such a scheme presents a formidable cryptanalytic challenge. If, for example,
the cryptanalyst attempts to use a letter frequency analysis approach, the analyst
is faced with the equivalent of over 11 million monoalphabetic ciphers. We might
need on the order of 50 letters in each monalphabetic cipher for a solution, which
means that the analyst would need to be in possession of a ciphertext with a length
of over half a billion letters.
The significance of the rotor machine today is that it points the way to a large
class of symmetric ciphers, of which the Data Encryption Standard (DES) is the
most prominent. DES is introduced in Chapter 4.
3.5 STEGANOGRAPHY
We conclude with a discussion of a technique that (strictly speaking), is not encryp-
tion, namely, steganography. A plaintext message may be hidden in one of two ways. The methods of
steganography conceal the existence of the message, whereas the methods of cryp- tography render the message unintelligible to outsiders by various transformations
of the text.11
A simple form of steganography, but one that is time-consuming to construct,
is one in which an arrangement of words or letters within an apparently innocuous
text spells out the real message. For example, the sequence of first letters of each
word of the overall message spells out the hidden message. Figure 3.9 shows an
example in which a subset of the words of the overall message is used to convey the
hidden message. See if you can decipher this; it’s not too hard.
Various other techniques have been used historically; some examples are the
following [MYER91]:
■ Character marking: Selected letters of printed or typewritten text are over- written in pencil. The marks are ordinarily not visible unless the paper is held
at an angle to bright light.
■ Invisible ink: A number of substances can be used for writing but leave no vis- ible trace until heat or some chemical is applied to the paper.
■ Pin punctures: Small pin punctures on selected letters are ordinarily not vis- ible unless the paper is held up in front of a light.
■ Typewriter correction ribbon: Used between lines typed with a black ribbon, the results of typing with the correction tape are visible only under a strong
light.
11Steganography was an obsolete word that was revived by David Kahn and given the meaning it has today [KAHN96].
3.5 / STEGANOGRAPHY 111
Although these techniques may seem archaic, they have contemporary equiv-
alents. [WAYN09] proposes hiding a message by using the least significant bits of
frames on a CD. For example, the Kodak Photo CD format’s maximum resolution
is 3096 * 6144 pixels, with each pixel containing 24 bits of RGB color information. The least significant bit of each 24-bit pixel can be changed without greatly affecting
the quality of the image. The result is that you can hide a 130-kB message in a single
digital snapshot. There are now a number of software packages available that take
this type of approach to steganography.
Steganography has a number of drawbacks when compared to encryption.
It requires a lot of overhead to hide a relatively few bits of information, although
using a scheme like that proposed in the preceding paragraph may make it more
effective. Also, once the system is discovered, it becomes virtually worthless. This
problem, too, can be overcome if the insertion method depends on some sort of key
(e.g., see Problem 3.22). Alternatively, a message can be first encrypted and then
hidden using steganography.
The advantage of steganography is that it can be employed by parties who
have something to lose should the fact of their secret communication (not necessar-
ily the content) be discovered. Encryption flags traffic as important or secret or may
identify the sender or receiver as someone with something to hide.
Figure 3.9 A Puzzle for Inspector Morse
(From The Silent World of Nicholas Quinn, by Colin Dexter)
112 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES
3.6 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
Key Terms
block cipher
brute-force attack
Caesar cipher
cipher
ciphertext
computationally secure
conventional encryption
cryptanalysis
cryptographic system
cryptography
cryptology
deciphering
decryption
digram
enciphering
encryption
Hill cipher
monoalphabetic cipher
one-time pad
plaintext
Playfair cipher
polyalphabetic cipher
rail fence cipher
single-key encryption
steganography
stream cipher
symmetric encryption
transposition cipher
unconditionally secure
Vigenère cipher
Review Questions
3.1 Describe the main requirements for the secure use of symmetric encryption. 3.2 What are the two basic functions used in encryption algorithms? 3.3 Differentiate between secret-key encryption and public-key encryption. 3.4 What is the difference between a block cipher and a stream cipher? 3.5 What are the two general approaches to attacking a cipher? 3.6 List and briefly define types of cryptanalytic attacks based on what is known to the
attacker.
3.7 What is the difference between an unconditionally secure cipher and a computation- ally secure cipher?
3.8 Why is the Caesar cipher substitution technique vulnerable to a brute-force cryptanalysis? 3.9 How much key space is available when a monoalphabetic substitution cipher is used
to replace plaintext with ciphertext?
3.10 What is the drawback of a Playfair cipher? 3.11 What is the difference between a monoalphabetic cipher and a polyalphabetic cipher? 3.12 What are two problems with the one-time pad? 3.13 What is a transposition cipher? 3.14 What are the drawbacks of Steganography?
Problems
3.1 A generalization of the Caesar cipher, known as the affine Caesar cipher, has the fol- lowing form: For each plaintext letter p, substitute the ciphertext letter C:
C = E([a, b], p) = (ap + b) mod 26
A basic requirement of any encryption algorithm is that it be one-to-one. That is, if p ≠ q, then E(k, p) ≠ E(k, q). Otherwise, decryption is impossible, because more than one plaintext character maps into the same ciphertext character. The affine Caesar cipher is not one-to-one for all values of a. For example, for a = 2 and b = 3, then E([a, b], 0) = E([a, b], 13) = 3.
a. Are there any limitations on the value of b? Explain why or why not. b. Determine which values of a are not allowed.
3.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 113
c. Provide a general statement of which values of a are and are not allowed. Justify your statement.
3.2 How many one-to-one affine Caesar ciphers are there? 3.3 A ciphertext has been generated with an affine cipher. The most frequent letter of
the ciphertext is “C,” and the second most frequent letter of the ciphertext is “Z.” Break this code.
3.4 The following ciphertext was generated using a simple substitution algorithm.
hzsrnqc klyy wqc flo mflwf ol zqdn nsoznj wskn lj xzsrbjnf, wzsxz gqv zqhhnf ol ozn glco zlfnco hnlhrn; nsoznj jnrqosdnc lj fnqj kjsnfbc, wzsxz sc xnjoqsfrv gljn efeceqr. zn rsdnb qrlfn sf zsc zlecn sf cqdsrrn jlw, wzsoznj flfn hnfnojqonb. q csfyrn blgncosx cekksxnb ol cnjdn zsg. zn pjnqmkqconb qfb bsfnb qo ozn xrep, qo zlejc gqozngqosxqrrv ksanb, sf ozn cqgn jllg, qo ozn cqgn oqprn, fndnj oqmsfy zsc gnqrc wsoz loznj gngpnjc, gexz rncc pjsfysfy q yenco wsoz zsg; qfb wnfo zlgn qo naqxorv gsbfsyzo, lfrv ol jnosjn qo lfxn ol pnb. zn fndnj ecnb ozn xlcv xzqgpnjc wzsxz ozn jnkljg hjldsbnc klj soc kqdlejnb gngpnjc. zn hqccnb onf zlejc leo lk ozn ownfov-klej sf cqdsrrn jlw, nsoznj sf crnnhsfy lj gqmsfy zsc olsrno.
Decrypt this message.
Hints: 1. As you know, the most frequently occurring letter in English is e. Therefore, the
first or second (or perhaps third?) most common character in the message is likely to stand for e. Also, e is often seen in pairs (e.g., meet, fleet, speed, seen, been, agree, etc.). Try to find a character in the ciphertext that decodes to e.
2. The most common word in English is “the.” Use this fact to guess the characters that stand for t and h.
3. Decipher the rest of the message by deducing additional words. Warning: The resulting message is in English but may not make much sense on a first
reading.
3.5 One way to solve the key distribution problem is to use a line from a book that both the sender and the receiver possess. Typically, at least in spy novels, the first sentence of a book serves as the key. The particular scheme discussed in this problem is from one of the best suspense novels involving secret codes, Talking to Strange Men, by Ruth Rendell. Work this problem without consulting that book!
Consider the following message:
SIDKHKDM AF HCRKIABIE SHIMC KD LFEAILA
This ciphertext was produced using the first sentence of The Other Side of Silence (a book about the spy Kim Philby):
The snow lay thick on the steps and the snowflakes driven by the wind looked black in the headlights of the cars.
A simple substitution cipher was used. a. What is the encryption algorithm? b. How secure is it? c. To make the key distribution problem simple, both parties can agree to use the first or
last sentence of a book as the key. To change the key, they simply need to agree on a new book. The use of the first sentence would be preferable to the use of the last. Why?
3.6 In one of his cases, Sherlock Holmes was confronted with the following message.
534 C2 13 127 36 31 4 17 21 41
DOUGLAS 109 293 5 37 BIRLSTONE
26 BIRLSTONE 9 127 171
114 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES
Although Watson was puzzled, Holmes was able immediately to deduce the type of cipher. Can you?
3.7 This problem uses a real-world example, from an old U.S. Special Forces manual (public domain). The document, filename SpecialForces.pdf, is available at box.com/ Crypto7e. a. Using the two keys (memory words) cryptographic and network security, encrypt
the following message:
Be at the third pillar from the left outside the lyceum theatre tonight at seven. If you are distrustful bring two friends.
Make reasonable assumptions about how to treat redundant letters and excess letters in the memory words and how to treat spaces and punctuation. Indicate what your assumptions are. Note: The message is from the Sherlock Holmes novel, The Sign of Four.
b. Decrypt the ciphertext. Show your work. c. Comment on when it would be appropriate to use this technique and what its
advantages are.
3.8 A disadvantage of the general monoalphabetic cipher is that both sender and receiver must commit the permuted cipher sequence to memory. A common technique for avoiding this is to use a keyword from which the cipher sequence can be gener- ated. For example, using the keyword CRYPTO, write out the keyword followed by unused letters in normal order and match this against the plaintext letters:
plain: a b c d e f g h i j k l m n o p q r s t u v w x y z
cipher: C R Y P T O A B D E F G H I J K L M N Q S U V W X Z
If it is felt that this process does not produce sufficient mixing, write the remain- ing letters on successive lines and then generate the sequence by reading down the columns:
C R Y P T O
A B D E F G
H I J K L M
N Q S U V W
X Z
This yields the sequence:
C A H N X R B I Q Z Y D J S P E K U T F L V O G M W
Such a system is used in the example in Section 3.2 (the one that begins “it was disclosed yesterday”). Determine the keyword.
3.9 When the PT-109 American patrol boat, under the command of Lieutenant John F. Kennedy, was sunk by a Japanese destroyer, a message was received at an Australian wireless station in Playfair code:
KXJEY UREBE ZWEHE WRYTU HEYFS
KREHE GOYFI WTTTU OLKSY CAJPO
BOTEI ZONTX BYBNT GONEY CUZWR
GDSON SXBOU YWRHE BAAHY USEDQ
The key used was royal new zealand navy. Decrypt the message. Translate TT into tt.
3.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 115
3.10 a. Construct a Playfair matrix with the key algorithm. b. Construct a Playfair matrix with the key cryptography. Make a reasonable assump-
tion about how to treat redundant letters in the key.
3.11 a. Using this Playfair matrix:
J/K C D E F
U N P Q S
Z V W X Y
R A L G O
B I T H M
Encrypt this message:
I only regret that I have but one life to give for my country.
Note: This message is by Nathan Hale, a soldier in the American Revolutionary War. b. Repeat part (a) using the Playfair matrix from Problem 3.10a. c. How do you account for the results of this problem? Can you generalize your
conclusion?
3.12 a. How many possible keys does the Playfair cipher have? Ignore the fact that some keys might produce identical encryption results. Express your answer as an approximate power of 2.
b. Now take into account the fact that some Playfair keys produce the same encryp- tion results. How many effectively unique keys does the Playfair cipher have?
3.13 What substitution system results when we use a 1 * 25 Playfair matrix? 3.14 a. Encrypt the message “meet me at the usual place at ten rather than eight o clock”
using the Hill cipher with the key ¢7 3 2 5
≤. Show your calculations and the result. b. Show the calculations for the corresponding decryption of the ciphertext to
recover the original plaintext.
3.15 We have shown that the Hill cipher succumbs to a known plaintext attack if sufficient plaintext–ciphertext pairs are provided. It is even easier to solve the Hill cipher if a chosen plaintext attack can be mounted. Describe such an attack.
3.16 It can be shown that the Hill cipher with the matrix ¢a b c d
≤ requires that (ad - bc) is relatively prime to 26; that is, the only common positive integer factor of (ad - bc) and 26 is 1. Thus, if (ad - bc) = 13 or is even, the matrix is not allowed. Determine the number of different (good) keys there are for a 2 * 2 Hill cipher without count- ing them one by one, using the following steps: a. Find the number of matrices whose determinant is even because one or both rows
are even. (A row is “even” if both entries in the row are even.) b. Find the number of matrices whose determinant is even because one or both col-
umns are even. (A column is “even” if both entries in the column are even.) c. Find the number of matrices whose determinant is even because all of the entries
are odd. d. Taking into account overlaps, find the total number of matrices whose determi-
nant is even. e. Find the number of matrices whose determinant is a multiple of 13 because the
first column is a multiple of 13.
116 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES
f. Find the number of matrices whose determinant is a multiple of 13 where the first column is not a multiple of 13 but the second column is a mul- tiple of the first modulo 13.
g. Find the total number of matrices whose determinant is a multiple of 13. h. Find the number of matrices whose determinant is a multiple of 26
because they fit cases parts (a) and (e), (b) and (e), (c) and (e), (a) and (f), and so on.
i. Find the total number of matrices whose determinant is neither a mul- tiple of 2 nor a multiple of 13.
3.17 Calculate the determinant mod 26 of
a. ¢2 3 5 1 3 7
≤ b. £ 2 1 1 3 2 55 7 1 8 3 1 4 1 2
≥ 3.18 Determine the inverse mod 26 of
a. ¢2 3 1 22
≤ b. £ 6 24 113 16 10 20 17 15
≥ 3.19 Using the Vigenère cipher, encrypt the word “cryptographic” using the word
“eng”.
3.20 This problem explores the use of a one-time pad version of the Vigenère cipher. In this scheme, the key is a stream of random numbers between 0 and 26. For example, if the key is 3 19 5 . . . , then the first letter of plaintext is encrypted with a shift of 3 letters, the second with a shift of 19 letters, the third with a shift of 5 letters, and so on. a. Encrypt the plaintext sendmoremoney with the key stream
3 11 5 7 17 21 0 11 14 8 7 13 9
b. Using the ciphertext produced in part (a), find a key so that the cipher- text decrypts to the plaintext cashnotneeded.
3.21 What is the message embedded in Figure 3.9? 3.22 In one of Dorothy Sayers’s mysteries, Lord Peter is confronted with the
message shown in Figure 3.10. He also discovers the key to the message, which is a sequence of integers:
787656543432112343456567878878765654
3432112343456567878878765654433211234
a. Decrypt the message. Hint: What is the largest integer value? b. If the algorithm is known but not the key, how secure is the scheme? c. If the key is known but not the algorithm, how secure is the scheme?
Figure 3.10 A Puzzle for Lord Peter
I thought to see the fairies in the fields, but I saw only the evil elephants with their black backs. Woe! how that sight awed me! The elves danced all around and about while I heard voices calling clearly. Ah! how I tried to see—throw off the ugly cloud—but no blind eye of a mortal was permitted to spy them. So then came minstrels, having gold trumpets, harps and drums. These played very loudly beside me, breaking that spell. So the dream vanished, whereat I thanked Heaven. I shed many tears before the thin moon rose up, frail and faint as a sickle of straw. Now though the Enchanter gnash his teeth vainly, yet shall he return as the Spring returns. Oh, wretched man! Hell gapes, Erebus now lies open. The mouths of Death wait on thy end.
3.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 117
Programming Problems
3.23 Write a program that can encrypt and decrypt using the general Caesar cipher, also known as an additive cipher.
3.24 Write a program that can encrypt and decrypt using the affine cipher described in Problem 3.1.
3.25 Write a program that can perform a letter frequency attack on an additive cipher without human intervention. Your software should produce possible plaintexts in rough order of likelihood. It would be good if your user inter- face allowed the user to specify “give me the top 10 possible plaintexts.”
3.26 Write a program that can perform a letter frequency attack on any mono- alphabetic substitution cipher without human intervention. Your software should produce possible plaintexts in rough order of likelihood. It would be good if your user interface allowed the user to specify “give me the top 10 possible plaintexts.”
3.27 Create software that can encrypt and decrypt using a 2 * 2 Hill cipher. 3.28 Create software that can perform a fast known plaintext attack on a Hill cipher,
given the dimension m. How fast are your algorithms, as a function of m?
118118
4.1 Traditional Block Cipher Structure
Stream Ciphers and Block Ciphers
Motivation for the Feistel Cipher Structure
The Feistel Cipher
4.2 The Data Encryption Standard
DES Encryption
DES Decryption
4.3 A DES Example
Results
The Avalanche Effect
4.4 The Strength of DES
The Use of 56-Bit Keys
The Nature of the DES Algorithm
Timing Attacks
4.5 Block Cipher Design Principles
Number of Rounds
Design of Function F
Key Schedule Algorithm
4.6 Key Terms, Review Questions, and Problems
CHAPTER
Block Ciphers and the Data Encryption Standard
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4.1 / TRADITIONAL BLOCK CIPHER STRUCTURE 119
The objective of this chapter is to illustrate the principles of modern symmetric
ciphers. For this purpose, we focus on the most widely used symmetric cipher: the Data
Encryption Standard (DES). Although numerous symmetric ciphers have been devel-
oped since the introduction of DES, and although it is destined to be replaced by the
Advanced Encryption Standard (AES), DES remains the most important such algo-
rithm. Furthermore, a detailed study of DES provides an understanding of the prin-
ciples used in other symmetric ciphers.
This chapter begins with a discussion of the general principles of symmetric block
ciphers, which are the principal type of symmetric ciphers studied in this book. The
other form of symmetric ciphers, stream ciphers, are discussed in Chapter 8. Next, we
cover full DES. Following this look at a specific algorithm, we return to a more general
discussion of block cipher design.
Compared to public-key ciphers, such as RSA, the structure of DES and most
symmetric ciphers is very complex and cannot be explained as easily as RSA and simi-
lar algorithms. Accordingly, the reader may wish to begin with a simplified version of
DES, which is described in Appendix G. This version allows the reader to perform
encryption and decryption by hand and gain a good understanding of the working of
the algorithm details. Classroom experience indicates that a study of this simplified
version enhances understanding of DES.1
4.1 TRADITIONAL BLOCK CIPHER STRUCTURE
Several important symmetric block encryption algorithms in current use are based
on a structure referred to as a Feistel block cipher [FEIS73]. For that reason, it is
important to examine the design principles of the Feistel cipher. We begin with a
comparison of stream ciphers and block ciphers. Then we discuss the motivation for
the Feistel block cipher structure. Finally, we discuss some of its implications.
1However, you may safely skip Appendix G, at least on a first reading. If you get lost or bogged down in the details of DES, then you can go back and start with simplified DES.
LEARNING OBJECTIVES
After studying this chapter, you should be able to
◆ Understand the distinction between stream ciphers and block ciphers.
◆ Present an overview of the Feistel cipher and explain how decryption is the inverse of encryption.
◆ Present an overview of Data Encryption Standard (DES).
◆ Explain the concept of the avalanche effect.
◆ Discuss the cryptographic strength of DES.
◆ Summarize the principal block cipher design principles.
120 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD
Stream Ciphers and Block Ciphers
A stream cipher is one that encrypts a digital data stream one bit or one byte at a time. Examples of classical stream ciphers are the autokeyed Vigenère cipher and
the Vernam cipher. In the ideal case, a one-time pad version of the Vernam cipher
would be used (Figure 3.7), in which the keystream (ki) is as long as the plaintext bit stream (pi). If the cryptographic keystream is random, then this cipher is unbreakable by any means other than acquiring the keystream. However, the keystream must be
provided to both users in advance via some independent and secure channel. This
introduces insurmountable logistical problems if the intended data traffic is very large.
Accordingly, for practical reasons, the bit-stream generator must be imple-
mented as an algorithmic procedure, so that the cryptographic bit stream can be
produced by both users. In this approach (Figure 4.1a), the bit-stream generator is
a key-controlled algorithm and must produce a bit stream that is cryptographically
strong. That is, it must be computationally impractical to predict future portions of
the bit stream based on previous portions of the bit stream. The two users need only
share the generating key, and each can produce the keystream.
A block cipher is one in which a block of plaintext is treated as a whole and used to produce a ciphertext block of equal length. Typically, a block size of 64 or
Figure 4.1 Stream Cipher and Block Cipher
Bit-stream generation algorithm
ENCRYPTION
(a) Stream cipher using algorithmic bit-stream generator
(b) Block cipher
Key ( K )
Encryption algorithm
Plaintext
b bits
b bits
Key ( K )
ki
Plaintext (pi)
Plaintext (pi)
Bit-stream generation algorithm
DECRYPTION
Key ( K )
ki
Ciphertext (ci)
Ciphertext
Decryption algorithm
Ciphertext
b bits
b bits
Key ( K )
Plaintext
4.1 / TRADITIONAL BLOCK CIPHER STRUCTURE 121
128 bits is used. As with a stream cipher, the two users share a symmetric encryption
key (Figure 4.1b). Using some of the modes of operation explained in Chapter 7, a
block cipher can be used to achieve the same effect as a stream cipher.
Far more effort has gone into analyzing block ciphers. In general, they seem
applicable to a broader range of applications than stream ciphers. The vast majority
of network-based symmetric cryptographic applications make use of block ciphers.
Accordingly, the concern in this chapter, and in our discussions throughout the
book of symmetric encryption, will primarily focus on block ciphers.
Motivation for the Feistel Cipher Structure
A block cipher operates on a plaintext block of n bits to produce a ciphertext block of n bits. There are 2n possible different plaintext blocks and, for the encryption to be reversible (i.e., for decryption to be possible), each must produce a unique
ciphertext block. Such a transformation is called reversible, or nonsingular. The fol-
lowing examples illustrate nonsingular and singular transformations for n = 2.
Reversible Mapping Irreversible Mapping
Plaintext Ciphertext Plaintext Ciphertext
00 11 00 11
01 10 01 10
10 00 10 01
11 01 11 01
In the latter case, a ciphertext of 01 could have been produced by one of two plain-
text blocks. So if we limit ourselves to reversible mappings, the number of different
transformations is 2n!.2
Figure 4.2 illustrates the logic of a general substitution cipher for n = 4. A 4-bit input produces one of 16 possible input states, which is mapped by the sub-
stitution cipher into a unique one of 16 possible output states, each of which is repre-
sented by 4 ciphertext bits. The encryption and decryption mappings can be defined
by a tabulation, as shown in Table 4.1. This is the most general form of block cipher
and can be used to define any reversible mapping between plaintext and ciphertext.
Feistel refers to this as the ideal block cipher, because it allows for the maximum number of possible encryption mappings from the plaintext block [FEIS75].
But there is a practical problem with the ideal block cipher. If a small block
size, such as n = 4, is used, then the system is equivalent to a classical substitution cipher. Such systems, as we have seen, are vulnerable to a statistical analysis of the
plaintext. This weakness is not inherent in the use of a substitution cipher but rather
results from the use of a small block size. If n is sufficiently large and an arbitrary reversible substitution between plaintext and ciphertext is allowed, then the statisti-
cal characteristics of the source plaintext are masked to such an extent that this type
of cryptanalysis is infeasible.
2The reasoning is as follows: For the first plaintext, we can choose any of 2n ciphertext blocks. For the second plaintext, we choose from among 2n - 1 remaining ciphertext blocks, and so on.
122 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD
An arbitrary reversible substitution cipher (the ideal block cipher) for a large
block size is not practical, however, from an implementation and performance
point of view. For such a transformation, the mapping itself constitutes the key.
Consider again Table 4.1, which defines one particular reversible mapping from
Figure 4.2 General n-bit-n-bit Block Substitution (shown with n = 4)
4-bit input
4 to 16 decoder
16 to 4 encoder
4-bit output
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Table 4.1 Encryption and Decryption Tables for Substitution Cipher of Figure 4.2
Plaintext Ciphertext
0000 1110
0001 0100
0010 1101
0011 0001
0100 0010
0101 1111
0110 1011
0111 1000
1000 0011
1001 1010
1010 0110
1011 1100
1100 0101
1101 1001
1110 0000
1111 0111
Ciphertext Plaintext
0000 1110
0001 0011
0010 0100
0011 1000
0100 0001
0101 1100
0110 1010
0111 1111
1000 0111
1001 1101
1010 1001
1011 0110
1100 1011
1101 0010
1110 0000
1111 0101
4.1 / TRADITIONAL BLOCK CIPHER STRUCTURE 123
plaintext to ciphertext for n = 4. The mapping can be defined by the entries in the second column, which show the value of the ciphertext for each plaintext block.
This, in essence, is the key that determines the specific mapping from among all
possible mappings. In this case, using this straightforward method of defining the
key, the required key length is (4 bits) * (16 rows) = 64 bits. In general, for an n-bit ideal block cipher, the length of the key defined in this fashion is n * 2n bits. For a 64-bit block, which is a desirable length to thwart statistical attacks, the
required key length is 64 * 264 = 270 ≈ 1021 bits. In considering these difficulties, Feistel points out that what is needed is an
approximation to the ideal block cipher system for large n, built up out of compo- nents that are easily realizable [FEIS75]. But before turning to Feistel’s approach,
let us make one other observation. We could use the general block substitution
cipher but, to make its implementation tractable, confine ourselves to a subset of
the 2n! possible reversible mappings. For example, suppose we define the mapping in terms of a set of linear equations. In the case of n = 4, we have
y1 = k11x1 + k12x2 + k13x3 + k14x4 y2 = k21x1 + k22x2 + k23x3 + k24x4 y3 = k31x1 + k32x2 + k33x3 + k34x4 y4 = k41x1 + k42x2 + k43x3 + k44x4
where the xi are the four binary digits of the plaintext block, the yi are the four bi- nary digits of the ciphertext block, the kij are the binary coefficients, and arithmetic is mod 2. The key size is just n2, in this case 16 bits. The danger with this kind of for- mulation is that it may be vulnerable to cryptanalysis by an attacker that is aware of
the structure of the algorithm. In this example, what we have is essentially the Hill
cipher discussed in Chapter 3, applied to binary data rather than characters. As we
saw in Chapter 3, a simple linear system such as this is quite vulnerable.
The Feistel Cipher
Feistel proposed [FEIS73] that we can approximate the ideal block cipher by utiliz-
ing the concept of a product cipher, which is the execution of two or more simple
ciphers in sequence in such a way that the final result or product is cryptographi-
cally stronger than any of the component ciphers. The essence of the approach is
to develop a block cipher with a key length of k bits and a block length of n bits, allowing a total of 2k possible transformations, rather than the 2n! transformations available with the ideal block cipher.
In particular, Feistel proposed the use of a cipher that alternates substitutions
and permutations, where these terms are defined as follows:
■ Substitution: Each plaintext element or group of elements is uniquely replaced by a corresponding ciphertext element or group of elements.
■ Permutation: A sequence of plaintext elements is replaced by a permutation of that sequence. That is, no elements are added or deleted or replaced in the
sequence, rather the order in which the elements appear in the sequence is
changed.
124 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD
In fact, Feistel’s is a practical application of a proposal by Claude Shannon
to develop a product cipher that alternates confusion and diffusion functions [SHAN49].3 We look next at these concepts of diffusion and confusion and then
present the Feistel cipher. But first, it is worth commenting on this remarkable fact:
The Feistel cipher structure, which dates back over a quarter century and which, in
turn, is based on Shannon’s proposal of 1945, is the structure used by a number of
significant symmetric block ciphers currently in use. In particular, the Feistel struc-
ture is used for Triple Data Encryption Algorithm (TDEA), which is one of the two
encryption algorithms (along with AES), approved for general use by the National
Institute of Standards and Technology (NIST). The Feistel structure is also used for
several schemes for format-preserving encryption, which have recently come into
prominence. In addition, the Camellia block cipher is a Feistel structure; it is one
of the possible symmetric ciphers in TLS and a number of other Internet security
protocols. Both TDEA and format-preserving encryption are covered in Chapter 7.
DIFFUSION AND CONFUSION The terms diffusion and confusion were introduced by Claude Shannon to capture the two basic building blocks for any cryptographic sys-
tem [SHAN49]. Shannon’s concern was to thwart cryptanalysis based on statisti-
cal analysis. The reasoning is as follows. Assume the attacker has some knowledge
of the statistical characteristics of the plaintext. For example, in a human-readable
message in some language, the frequency distribution of the various letters may be
known. Or there may be words or phrases likely to appear in the message (probable
words). If these statistics are in any way reflected in the ciphertext, the cryptanalyst
may be able to deduce the encryption key, part of the key, or at least a set of keys
likely to contain the exact key. In what Shannon refers to as a strongly ideal cipher,
all statistics of the ciphertext are independent of the particular key used. The arbi-
trary substitution cipher that we discussed previously (Figure 4.2) is such a cipher,
but as we have seen, it is impractical.4
Other than recourse to ideal systems, Shannon suggests two methods for
frustrating statistical cryptanalysis: diffusion and confusion. In diffusion, the sta- tistical structure of the plaintext is dissipated into long-range statistics of the
ciphertext. This is achieved by having each plaintext digit affect the value of many
ciphertext digits; generally, this is equivalent to having each ciphertext digit be
affected by many plaintext digits. An example of diffusion is to encrypt a message
M = m1, m2, m3, c of characters with an averaging operation:
yn = ¢ ak i = 1
mn + i≤ mod 26 3The paper is available at box.com/Crypto7e. Shannon’s 1949 paper appeared originally as a classified report in 1945. Shannon enjoys an amazing and unique position in the history of computer and informa- tion science. He not only developed the seminal ideas of modern cryptography but is also responsible for inventing the discipline of information theory. Based on his work in information theory, he developed a formula for the capacity of a data communications channel, which is still used today. In addition, he founded another discipline, the application of Boolean algebra to the study of digital circuits; this last he managed to toss off as a master’s thesis.
4Appendix F expands on Shannon’s concepts concerning measures of secrecy and the security of crypto- graphic algorithms.
4.1 / TRADITIONAL BLOCK CIPHER STRUCTURE 125
adding k successive letters to get a ciphertext letter yn. One can show that the sta- tistical structure of the plaintext has been dissipated. Thus, the letter frequencies in
the ciphertext will be more nearly equal than in the plaintext; the digram frequen-
cies will also be more nearly equal, and so on. In a binary block cipher, diffusion can
be achieved by repeatedly performing some permutation on the data followed by
applying a function to that permutation; the effect is that bits from different posi-
tions in the original plaintext contribute to a single bit of ciphertext.5
Every block cipher involves a transformation of a block of plaintext into a
block of ciphertext, where the transformation depends on the key. The mechanism
of diffusion seeks to make the statistical relationship between the plaintext and
ciphertext as complex as possible in order to thwart attempts to deduce the key. On
the other hand, confusion seeks to make the relationship between the statistics of the ciphertext and the value of the encryption key as complex as possible, again to
thwart attempts to discover the key. Thus, even if the attacker can get some handle
on the statistics of the ciphertext, the way in which the key was used to produce that
ciphertext is so complex as to make it difficult to deduce the key. This is achieved by
the use of a complex substitution algorithm. In contrast, a simple linear substitution
function would add little confusion.
As [ROBS95b] points out, so successful are diffusion and confusion in captur-
ing the essence of the desired attributes of a block cipher that they have become the
cornerstone of modern block cipher design.
FEISTEL CIPHER STRUCTURE The left-hand side of Figure 4.3 depicts the encryption structure proposed by Feistel. The inputs to the encryption algorithm are a plaintext
block of length 2w bits and a key K. The plaintext block is divided into two halves, LE0 and RE0. The two halves of the data pass through n rounds of processing and then combine to produce the ciphertext block. Each round i has as inputs LEi - 1 and REi - 1 derived from the previous round, as well as a subkey Ki derived from the over- all K. In general, the subkeys Ki are different from K and from each other. In Figure 4.3, 16 rounds are used, although any number of rounds could be implemented.
All rounds have the same structure. A substitution is performed on the left half of the data. This is done by applying a round function F to the right half of the data and then taking the exclusive-OR of the output of that function and the left
half of the data. The round function has the same general structure for each round
but is parameterized by the round subkey Ki. Another way to express this is to say that F is a function of right-half block of w bits and a subkey of y bits, which pro- duces an output value of length w bits: F(REi, Ki + 1). Following this substitution, a permutation is performed that consists of the interchange of the two halves of the data.6 This structure is a particular form of the substitution-permutation network
(SPN) proposed by Shannon.
5Some books on cryptography equate permutation with diffusion. This is incorrect. Permutation, by itself, does not change the statistics of the plaintext at the level of individual letters or permuted blocks. For exam- ple, in DES, the permutation swaps two 32-bit blocks, so statistics of strings of 32 bits or less are preserved. 6The final round is followed by an interchange that undoes the interchange that is part of the final round. One could simply leave both interchanges out of the diagram, at the sacrifice of some consistency of pre- sentation. In any case, the effective lack of a swap in the final round is done to simplify the implementa- tion of the decryption process, as we shall see.
126 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD
The exact realization of a Feistel network depends on the choice of the follow-
ing parameters and design features:
■ Block size: Larger block sizes mean greater security (all other things being equal) but reduced encryption/decryption speed for a given algorithm. The
greater security is achieved by greater diffusion. Traditionally, a block size of
64 bits has been considered a reasonable tradeoff and was nearly universal in
block cipher design. However, the new AES uses a 128-bit block size.
Figure 4.3 Feistel Encryption and Decryption (16 rounds)
Output (ciphertext)
K1
LD0 = RE16 RD0 = LE16
LD2 = RE14 RD2 = LE14
LD14 = RE2 RD14 = LE2
LD16 = RE0
LD17 = RE0
RD16 = LE0
RD17 = LE0
RD1 = LE15LD1 = RE15
RD15 = LE1LD15 = RE1
Input (ciphertext)
Output (plaintext)
R ou
nd 1
K1
K2
K15
K16
K2
K15
K16
F
LE0 RE0
Input (plaintext)
LE1 RE1
LE2 RE2
F
F
LE14 RE14
LE15 RE15
LE16 RE16
LE17 RE17
F
F
F
F
F
R ou
nd 2
R ou
nd 1
5 R
ou nd
1 6
R ou
nd 1
6 R
ou nd
1 5
R ou
nd 2
R ou
nd 1
4.1 / TRADITIONAL BLOCK CIPHER STRUCTURE 127
■ Key size: Larger key size means greater security but may decrease encryption/ decryption speed. The greater security is achieved by greater resistance to
brute-force attacks and greater confusion. Key sizes of 64 bits or less are now
widely considered to be inadequate, and 128 bits has become a common size.
■ Number of rounds: The essence of the Feistel cipher is that a single round offers inadequate security but that multiple rounds offer increasing security.
A typical size is 16 rounds.
■ Subkey generation algorithm: Greater complexity in this algorithm should lead to greater difficulty of cryptanalysis.
■ Round function F: Again, greater complexity generally means greater resis- tance to cryptanalysis.
There are two other considerations in the design of a Feistel cipher:
■ Fast software encryption/decryption: In many cases, encryption is embedded in applications or utility functions in such a way as to preclude a hardware im-
plementation. Accordingly, the speed of execution of the algorithm becomes a
concern.
■ Ease of analysis: Although we would like to make our algorithm as difficult as possible to cryptanalyze, there is great benefit in making the algorithm easy
to analyze. That is, if the algorithm can be concisely and clearly explained, it is
easier to analyze that algorithm for cryptanalytic vulnerabilities and therefore
develop a higher level of assurance as to its strength. DES, for example, does
not have an easily analyzed functionality.
FEISTEL DECRYPTION ALGORITHM The process of decryption with a Feistel cipher is essentially the same as the encryption process. The rule is as follows: Use the
ciphertext as input to the algorithm, but use the subkeys Ki in reverse order. That is, use Kn in the first round, Kn - 1 in the second round, and so on, until K1 is used in the last round. This is a nice feature, because it means we need not implement two
different algorithms; one for encryption and one for decryption.
To see that the same algorithm with a reversed key order produces the cor-
rect result, Figure 4.3 shows the encryption process going down the left-hand side
and the decryption process going up the right-hand side for a 16-round algorithm.
For clarity, we use the notation LEi and REi for data traveling through the encryp- tion algorithm and LDi and RDi for data traveling through the decryption algo- rithm. The diagram indicates that, at every round, the intermediate value of the
decryption process is equal to the corresponding value of the encryption process
with the two halves of the value swapped. To put this another way, let the output
of the ith encryption round be LEi ‘ REi (LEi concatenated with REi). Then the cor- responding output of the (16 - i)th decryption round is REi ‘ LEi or, equivalently, LD16 - i ‘ RD16 - i.
Let us walk through Figure 4.3 to demonstrate the validity of the preceding
assertions. After the last iteration of the encryption process, the two halves of the
output are swapped, so that the ciphertext is RE16 ‘ LE16. The output of that round is the ciphertext. Now take that ciphertext and use it as input to the same algorithm.
The input to the first round is RE16 ‘ LE16, which is equal to the 32-bit swap of the output of the sixteenth round of the encryption process.
Hiva-Network.Com
128 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD
Now we would like to show that the output of the first round of the decryption
process is equal to a 32-bit swap of the input to the sixteenth round of the encryp-
tion process. First, consider the encryption process. We see that
LE16 = RE15 RE16 = LE15 ⊕ F(RE15, K16)
On the decryption side,
LD1 = RD0 = LE16 = RE15 RD1 = LD0 ⊕ F(RD0, K16)
= RE16 ⊕ F(RE15, K16) = [LE15 ⊕ F(RE15, K16)] ⊕ F(RE15, K16)
The XOR has the following properties:
[A ⊕ B] ⊕ C = A ⊕ [B ⊕ C] D ⊕ D = 0 E ⊕ 0 = E
Thus, we have LD1 = RE15 and RD1 = LE15. Therefore, the output of the first round of the decryption process is RE15 ‘ LE15, which is the 32-bit swap of the input to the sixteenth round of the encryption. This correspondence holds all the way
through the 16 iterations, as is easily shown. We can cast this process in general
terms. For the ith iteration of the encryption algorithm,
LEi = REi - 1 REi = LEi - 1 ⊕ F(REi - 1, Ki)
Rearranging terms:
REi - 1 = LEi LEi - 1 = REi ⊕ F(REi - 1, Ki) = REi ⊕ F(LEi, Ki)
Thus, we have described the inputs to the ith iteration as a function of the outputs, and these equations confirm the assignments shown in the right-hand side of Figure 4.3.
Finally, we see that the output of the last round of the decryption process is
RE0 ‘ LE0. A 32-bit swap recovers the original plaintext, demonstrating the validity of the Feistel decryption process.
Note that the derivation does not require that F be a reversible function. To
see this, take a limiting case in which F produces a constant output (e.g., all ones)
regardless of the values of its two arguments. The equations still hold.
To help clarify the preceding concepts, let us look at a specific example
(Figure 4.4 and focus on the fifteenth round of encryption, corresponding to the sec-
ond round of decryption. Suppose that the blocks at each stage are 32 bits (two 16-bit
halves) and that the key size is 24 bits. Suppose that at the end of encryption round
fourteen, the value of the intermediate block (in hexadecimal) is DE7F03A6. Then
LE14 = DE7F and RE14 = 03A6. Also assume that the value of K15 is 12DE52. After round 15, we have LE15 = 03A6 and RE15 = F(03A6, 12DE52) ⊕ DE7F.
4.2 / THE DATA ENCRYPTION STANDARD 129
Now let’s look at the decryption. We assume that LD1 = RE15 and RD1 = LE15, as shown in Figure 4.3, and we want to demonstrate that LD2 = RE14 and RD2 = LE14. So, we start with LD1 = F(03A6, 12DE52) ⊕ DE7F and RD1 = 03A6. Then, from Figure 4.3, LD2 = 03A6 = RE14 and RD2 = F(03A6, 12DE52) ⊕ [F(03A6, 12DE52) ⊕ DE7F] = DE7F = LE14.
4.2 THE DATA ENCRYPTION STANDARD
Until the introduction of the Advanced Encryption Standard (AES) in 2001, the
Data Encryption Standard (DES) was the most widely used encryption scheme.
DES was issued in 1977 by the National Bureau of Standards, now the National
Institute of Standards and Technology (NIST), as Federal Information Processing
Standard 46 (FIPS PUB 46). The algorithm itself is referred to as the Data
Encryption Algorithm (DEA).7 For DEA, data are encrypted in 64-bit blocks using
a 56-bit key. The algorithm transforms 64-bit input in a series of steps into a 64-bit
output. The same steps, with the same key, are used to reverse the encryption.
Over the years, DES became the dominant symmetric encryption algorithm,
especially in financial applications. In 1994, NIST reaffirmed DES for federal use
for another five years; NIST recommended the use of DES for applications other
than the protection of classified information. In 1999, NIST issued a new version
of its standard (FIPS PUB 46-3) that indicated that DES should be used only
for legacy systems and that triple DES (which in essence involves repeating the
DES algorithm three times on the plaintext using two or three different keys to
produce the ciphertext) be used. We study triple DES in Chapter 7. Because the
underlying encryption and decryption algorithms are the same for DES and triple
DES, it remains important to understand the DES cipher. This section provides an
overview.For the interested reader, Appendix S provides further detail.
7The terminology is a bit confusing. Until recently, the terms DES and DEA could be used interchange- ably. However, the most recent edition of the DES document includes a specification of the DEA described here plus the triple DEA (TDEA) described in Chapter 7. Both DEA and TDEA are part of the Data Encryption Standard. Further, until the recent adoption of the official term TDEA, the triple DEA algorithm was typically referred to as triple DES and written as 3DES. For the sake of convenience, we will use the term 3DES.
Figure 4.4 Feistel Example
12DE52
12DE52
F
DE7F 03A6
Decryption roundEncryption round
03A6
6A306A30 F(03A6, 12DE52) DE7F F(03A6, 12DE52) DE7F
F(03A6, 12DE52) [F(03A6, 12DE52) DE7F]
= DE7F
FR ou
nd 1
5
R ou
nd 2
130 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD
DES Encryption
The overall scheme for DES encryption is illustrated in Figure 4.5. As with any
encryption scheme, there are two inputs to the encryption function: the plaintext to
be encrypted and the key. In this case, the plaintext must be 64 bits in length and the
key is 56 bits in length.8
Looking at the left-hand side of the figure, we can see that the processing
of the plaintext proceeds in three phases. First, the 64-bit plaintext passes through
an initial permutation (IP) that rearranges the bits to produce the permuted input.
8Actually, the function expects a 64-bit key as input. However, only 56 of these bits are ever used; the other 8 bits can be used as parity bits or simply set arbitrarily.
Figure 4.5 General Depiction of DES Encryption Algorithm
Initial permutation
Permuted choice 2Round 1
32-bit swap
Inverse initial permutation
Permuted choice 1
Round 2
Round 16
64-bit plaintext 64-bit key
K1
K2
K16
64-bit ciphertext
Left circular shift
Permuted choice 2 Left circular shift
Permuted choice 2 Left circular shift
64 56
56
56
56
48
48
48
56 64
64 bits
4.3 / A DES EXAMPLE 131
This is followed by a phase consisting of sixteen rounds of the same function, which
involves both permutation and substitution functions. The output of the last (six-
teenth) round consists of 64 bits that are a function of the input plaintext and the
key. The left and right halves of the output are swapped to produce the preoutput.
Finally, the preoutput is passed through a permutation [IP -1] that is the inverse of
the initial permutation function, to produce the 64-bit ciphertext. With the excep-
tion of the initial and final permutations, DES has the exact structure of a Feistel
cipher, as shown in Figure 4.3.
The right-hand portion of Figure 4.5 shows the way in which the 56-bit key is
used. Initially, the key is passed through a permutation function. Then, for each of
the sixteen rounds, a subkey (Ki) is produced by the combination of a left circular shift and a permutation. The permutation function is the same for each round, but a
different subkey is produced because of the repeated shifts of the key bits.
DES Decryption
As with any Feistel cipher, decryption uses the same algorithm as encryption, except
that the application of the subkeys is reversed. Additionally, the initial and final
permutations are reversed.
4.3 A DES EXAMPLE
We now work through an example and consider some of its implications. Although
you are not expected to duplicate the example by hand, you will find it informative
to study the hex patterns that occur from one step to the next.
For this example, the plaintext is a hexadecimal palindrome. The plaintext,
key, and resulting ciphertext are as follows:
Plaintext: 02468aceeca86420
Key: 0f1571c947d9e859
Ciphertext: da02ce3a89ecac3b
Results
Table 4.2 shows the progression of the algorithm. The first row shows the 32-bit
values of the left and right halves of data after the initial permutation. The next 16
rows show the results after each round. Also shown is the value of the 48-bit subkey
generated for each round. Note that Li = Ri - 1. The final row shows the left- and right-hand values after the inverse initial permutation. These two values combined
form the ciphertext.
The Avalanche Effect
A desirable property of any encryption algorithm is that a small change in either
the plaintext or the key should produce a significant change in the ciphertext. In
particular, a change in one bit of the plaintext or one bit of the key should produce
132 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD
a change in many bits of the ciphertext. This is referred to as the avalanche effect.
If the change were small, this might provide a way to reduce the size of the plaintext
or key space to be searched.
Using the example from Table 4.2, Table 4.3 shows the result when the fourth
bit of the plaintext is changed, so that the plaintext is 12468aceeca86420. The second column of the table shows the intermediate 64-bit values at the end of each
round for the two plaintexts. The third column shows the number of bits that differ
between the two intermediate values. The table shows that, after just three rounds,
18 bits differ between the two blocks. On completion, the two ciphertexts differ in
32 bit positions.
Table 4.4 shows a similar test using the original plaintext of with two keys that
differ in only the fourth bit position: the original key, 0f1571c947d9e859, and the altered key, 1f1571c947d9e859. Again, the results show that about half of the bits in the ciphertext differ and that the avalanche effect is pronounced after just
a few rounds.
Round Ki Li Ri
IP 5a005a00 3cf03c0f
1 1e030f03080d2930 3cf03c0f bad22845
2 0a31293432242318 bad22845 99e9b723
3 23072318201d0c1d 99e9b723 0bae3b9e
4 05261d3824311a20 0bae3b9e 42415649
5 3325340136002c25 42415649 18b3fa41
6 123a2d0d04262a1c 18b3fa41 9616fe23
7 021f120b1c130611 9616fe23 67117cf2
8 1c10372a2832002b 67117cf2 c11bfc09
9 04292a380c341f03 c11bfc09 887fbc6c
10 2703212607280403 887fbc6c 600f7e8b
11 2826390c31261504 600f7e8b f596506e
12 12071c241a0a0f08 f596506e 738538b8
13 300935393c0d100b 738538b8 c6a62c4e
14 311e09231321182a c6a62c4e 56b0bd75
15 283d3e0227072528 56b0bd75 75e8fd8f
16 2921080b13143025 75e8fd8f 25896490
IP−1 da02ce3a 89ecac3b
Note: DES subkeys are shown as eight 6-bit values in hex format
Table 4.2 DES Example
4.3 / A DES EXAMPLE 133
Table 4.3 Avalanche Effect in DES: Change in Plaintext
Round D
9 c11bfc09887fbc6c 99f911532eed7d94
32
10 887fbc6c600f7e8b 2eed7d94d0f23094
34
11 600f7e8bf596506e d0f23094455da9c4
37
12 f596506e738538b8 455da9c47f6e3cf3
31
13 738538b8c6a62c4e 7f6e3cf34bc1a8d9
29
14 c6a62c4e56b0bd75 4bc1a8d91e07d409
33
15 56b0bd7575e8fd8f 1e07d4091ce2e6dc
31
16 75e8fd8f25896490 1ce2e6dc365e5f59
32
IP−1 da02ce3a89ecac3b 057cde97d7683f2a
32
Round D
02468aceeca86420 12468aceeca86420
1
1 3cf03c0fbad22845 3cf03c0fbad32845
1
2 bad2284599e9b723 bad3284539a9b7a3
5
3 99e9b7230bae3b9e 39a9b7a3171cb8b3
18
4 0bae3b9e42415649 171cb8b3ccaca55e
34
5 4241564918b3fa41 ccaca55ed16c3653
37
6 18b3fa419616fe23 d16c3653cf402c68
33
7 9616fe2367117cf2 cf402c682b2cefbc
32
8 67117cf2c11bfc09 2b2cefbc99f91153
33
Table 4.4 Avalanche Effect in DES: Change in Key
Round D
02468aceeca86420 02468aceeca86420
0
1 3cf03c0fbad22845 3cf03c0f9ad628c5
3
2 bad2284599e9b723 9ad628c59939136b
11
3 99e9b7230bae3b9e 9939136b768067b7
25
4 0bae3b9e42415649 768067b75a8807c5
29
5 4241564918b3fa41 5a8807c5488dbe94
26
6 18b3fa419616fe23 488dbe94aba7fe53
26
7 9616fe2367117cf2 aba7fe53177d21e4
27
8 67117cf2c11bfc09 177d21e4548f1de4
32
Round D
9 c11bfc09887fbc6c 548f1de471f64dfd
34
10 887fbc6c600f7e8b 71f64dfd4279876c
36
11 600f7e8bf596506e 4279876c399fdc0d
32
12 f596506e738538b8 399fdc0d6d208dbb
28
13 738538b8c6a62c4e 6d208dbbb9bdeeaa
33
14 c6a62c4e56b0bd75 b9bdeeaad2c3a56f
30
15 56b0bd7575e8fd8f d2c3a56f2765c1fb
27
16 75e8fd8f25896490 2765c1fb01263dc4
30
IP−1 da02ce3a89ecac3b ee92b50606b62b0b
30
134 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD
4.4 THE STRENGTH OF DES
Since its adoption as a federal standard, there have been lingering concerns about
the level of security provided by DES. These concerns, by and large, fall into two
areas: key size and the nature of the algorithm.
The Use of 56-Bit Keys
With a key length of 56 bits, there are 256 possible keys, which is approximately
7.2 * 1016 keys. Thus, on the face of it, a brute-force attack appears impractical. Assuming that, on average, half the key space has to be searched, a single machine
performing one DES encryption per microsecond would take more than a thousand
years to break the cipher.
However, the assumption of one encryption per microsecond is overly con-
servative. As far back as 1977, Diffie and Hellman postulated that the technology
existed to build a parallel machine with 1 million encryption devices, each of which
could perform one encryption per microsecond [DIFF77]. This would bring the
average search time down to about 10 hours. The authors estimated that the cost
would be about $20 million in 1977 dollars.
With current technology, it is not even necessary to use special, purpose-built
hardware. Rather, the speed of commercial, off-the-shelf processors threaten the
security of DES. A recent paper from Seagate Technology [SEAG08] suggests that
a rate of 1 billion (109) key combinations per second is reasonable for today’s mul-
ticore computers. Recent offerings confirm this. Both Intel and AMD now offer
hardware-based instructions to accelerate the use of AES. Tests run on a contem-
porary multicore Intel machine resulted in an encryption rate of about half a bil-
lion encryptions per second [BASU12]. Another recent analysis suggests that with
contemporary supercomputer technology, a rate of 1013 encryptions per second is
reasonable [AROR12].
With these results in mind, Table 4.5 shows how much time is required for a
brute-force attack for various key sizes. As can be seen, a single PC can break DES in
about a year; if multiple PCs work in parallel, the time is drastically shortened. And
today’s supercomputers should be able to find a key in about an hour. Key sizes of
128 bits or greater are effectively unbreakable using simply a brute-force approach.
Even if we managed to speed up the attacking system by a factor of 1 trillion (1012),
it would still take over 100,000 years to break a code using a 128-bit key.
Fortunately, there are a number of alternatives to DES, the most important of
which are AES and triple DES, discussed in Chapters 6 and 7, respectively.
The Nature of the DES Algorithm
Another concern is the possibility that cryptanalysis is possible by exploiting
the characteristics of the DES algorithm. The focus of concern has been on the
eight substitution tables, or S-boxes, that are used in each iteration (described in
Appendix S). Because the design criteria for these boxes, and indeed for the entire
algorithm, were not made public, there is a suspicion that the boxes were con-
structed in such a way that cryptanalysis is possible for an opponent who knows
4.5 / BLOCK CIPHER DESIGN PRINCIPLES 135
Key Size (bits) Cipher
Number of Alternative
Keys Time Required at 109
Decryptions/s
Time Required at 1013
Decryptions/s
56 DES 256 ≈ 7.2 * 1016 255 ns = 1.125 years 1 hour
128 AES 2128 ≈ 3.4 * 1038 2127 ns = 5.3 * 1021 years 5.3 * 1017 years
168 Triple DES 2168 ≈ 3.7 * 1050 2167 ns = 5.8 * 1033 years 5.8 * 1029 years
192 AES 2192 ≈ 6.3 * 1057 2191 ns = 9.8 * 1040 years 9.8 * 1036 years
256 AES 2256 ≈ 1.2 * 1077 2255 ns = 1.8 * 1060 years 1.8 * 1056 years
26 characters
(permutation)
Monoalphabetic 2! = 4 * 1026 2 * 1026 ns = 6.3 * 109 years 6.3 * 106 years
Table 4.5 Average Time Required for Exhaustive Key Search
the weaknesses in the S-boxes. This assertion is tantalizing, and over the years a
number of regularities and unexpected behaviors of the S-boxes have been discov-
ered. Despite this, no one has so far succeeded in discovering the supposed fatal
weaknesses in the S-boxes.9
Timing Attacks
We discuss timing attacks in more detail in Part Two, as they relate to public-key
algorithms. However, the issue may also be relevant for symmetric ciphers. In
essence, a timing attack is one in which information about the key or the plaintext is
obtained by observing how long it takes a given implementation to perform decryp-
tions on various ciphertexts. A timing attack exploits the fact that an encryption
or decryption algorithm often takes slightly different amounts of time on different
inputs. [HEVI99] reports on an approach that yields the Hamming weight (number
of bits equal to one) of the secret key. This is a long way from knowing the actual
key, but it is an intriguing first step. The authors conclude that DES appears to be
fairly resistant to a successful timing attack but suggest some avenues to explore.
Although this is an interesting line of attack, it so far appears unlikely that this tech-
nique will ever be successful against DES or more powerful symmetric ciphers such
as triple DES and AES.
4.5 BLOCK CIPHER DESIGN PRINCIPLES
Although much progress has been made in designing block ciphers that are cryp-
tographically strong, the basic principles have not changed all that much since the
work of Feistel and the DES design team in the early 1970s. In this section we look
at three critical aspects of block cipher design: the number of rounds, design of the
function F, and key scheduling.
9At least, no one has publicly acknowledged such a discovery.
136 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD
Number of Rounds
The cryptographic strength of a Feistel cipher derives from three aspects of the
design: the number of rounds, the function F, and the key schedule algorithm. Let
us look first at the choice of the number of rounds.
The greater the number of rounds, the more difficult it is to perform crypt-
analysis, even for a relatively weak F. In general, the criterion should be that the
number of rounds is chosen so that known cryptanalytic efforts require greater
effort than a simple brute-force key search attack. This criterion was certainly used
in the design of DES. Schneier [SCHN96] observes that for 16-round DES, a dif-
ferential cryptanalysis attack is slightly less efficient than brute force: The differen-
tial cryptanalysis attack requires 255.1 operations,10 whereas brute force requires 255.
If DES had 15 or fewer rounds, differential cryptanalysis would require less effort
than a brute-force key search.
This criterion is attractive, because it makes it easy to judge the strength of
an algorithm and to compare different algorithms. In the absence of a cryptana-
lytic breakthrough, the strength of any algorithm that satisfies the criterion can be
judged solely on key length.
Design of Function F
The heart of a Feistel block cipher is the function F, which provides the element of
confusion in a Feistel cipher. Thus, it must be difficult to “unscramble” the substitu-
tion performed by F. One obvious criterion is that F be nonlinear, as we discussed
previously. The more nonlinear F, the more difficult any type of cryptanalysis will be.
There are several measures of nonlinearity, which are beyond the scope of this
book. In rough terms, the more difficult it is to approximate F by a set of linear
equations, the more nonlinear F is.
Several other criteria should be considered in designing F. We would like the
algorithm to have good avalanche properties. Recall that, in general, this means that
a change in one bit of the input should produce a change in many bits of the output.
A more stringent version of this is the strict avalanche criterion (SAC) [WEBS86], which states that any output bit j of an S-box (see Appendix S for a discussion of S-boxes) should change with probability 1/2 when any single input bit i is inverted for all i, j. Although SAC is expressed in terms of S-boxes, a similar criterion could be applied to F as a whole. This is important when considering designs that do not
include S-boxes.
Another criterion proposed in [WEBS86] is the bit independence criterion (BIC), which states that output bits j and k should change independently when any single input bit i is inverted for all i, j, and k. The SAC and BIC criteria appear to strengthen the effectiveness of the confusion function.
10Differential cryptanalysis of DES requires 247 chosen plaintext. If all you have to work with is known plaintext, then you must sort through a large quantity of known plaintext–ciphertext pairs looking for the useful ones. This brings the level of effort up to 255.1.
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4.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 137
Key Schedule Algorithm
With any Feistel block cipher, the key is used to generate one subkey for each round.
In general, we would like to select subkeys to maximize the difficulty of deducing
individual subkeys and the difficulty of working back to the main key. No general
principles for this have yet been promulgated.
Adams suggests [ADAM94] that, at minimum, the key schedule should guar-
antee key/ciphertext Strict Avalanche Criterion and Bit Independence Criterion.
4.6 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
Key Terms
avalanche effect
block cipher
confusion
Data Encryption Standard
(DES)
diffusion
Feistel cipher
irreversible mapping
key
permutation
product cipher
reversible mapping
round
round function
subkey
substitution
Review Questions 4.1 Briefly define a nonsingular transformation. 4.2 What is the difference between a block cipher and a stream cipher? 4.3 Why is it not practical to use an arbitrary reversible substitution cipher of the kind
shown in Table 4.1?
4.4 Briefly define the terms substitution and permutation. 4.5 What is the difference between diffusion and confusion? 4.6 Which parameters and design choices determine the actual algorithm of a Feistel
cipher?
4.7 What are the critical aspects of Feistel cipher design?
Problems
4.1 a. In Section 4.1, under the subsection on the motivation for the Feistel cipher struc- ture, it was stated that, for a block of n bits, the number of different reversible mappings for the ideal block cipher is 2n!. Justify.
b. In that same discussion, it was stated that for the ideal block cipher, which allows all possible reversible mappings, the size of the key is n * 2n bits. But, if there are 2n! possible mappings, it should take log2 2
n! bits to discriminate among the different mappings, and so the key length should be log2 2
n!. However, log2 2 n! 6 n * 2n.
Explain the discrepancy.
138 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD
4.2 Consider a Feistel cipher composed of sixteen rounds with a block length of 128 bits and a key length of 128 bits. Suppose that, for a given k, the key scheduling algorithm determines values for the first eight round keys, k1, k2, c k8, and then sets
k9 = k8, k10 = k7, k11 = k6, c , k16 = k1
Suppose you have a ciphertext c. Explain how, with access to an encryption oracle, you can decrypt c and determine m using just a single oracle query. This shows that such a cipher is vulnerable to a chosen plaintext attack. (An encryption oracle can be thought of as a device that, when given a plaintext, returns the corresponding cipher- text. The internal details of the device are not known to you and you cannot break open the device. You can only gain information from the oracle by making queries to it and observing its responses.)
4.3 Let p be a permutation of the integers 0, 1, 2, c , (2n - 1), such that p(m) gives the permuted value of m, 0 … m 6 2n. Put another way, p maps the set of n-bit integers into itself and no two integers map into the same integer. DES is such a permutation for 64-bit integers. We say that p has a fixed point at m if p(m) = m. That is, if p is an encryption mapping, then a fixed point corresponds to a message that encrypts to itself. We are interested in the number of fixed points in a randomly chosen permuta- tion p. Show the somewhat unexpected result that the number of fixed points for p is 1 on an average, and this number is independent of the size of the permutation.
4.4 Consider a block encryption algorithm that encrypts blocks of length n, and let N = 2n. Say we have t plaintext–ciphertext pairs Pi, Ci = E(K, Pi), where we assume that the key K selects one of the N! possible mappings. Imagine that we wish to find K by exhaustive search. We could generate key K′ and test whether Ci = E(K′, Pi) for 1 … i … t. If K′ encrypts each Pi to its proper Ci, then we have evidence that K = K′. However, it may be the case that the mappings E(K, # ) and E(K′, # ) exactly agree on the t plaintext–cipher text pairs Pi, Ci and agree on no other pairs. a. What is the probability that E(K, # ) and E(K′, # ) are in fact distinct mappings? b. What is the probability that E(K, # ) and E(K′, # ) agree on another t′ plaintext–
ciphertext pairs where 0 … t′ … N - t? 4.5 For any block cipher, the fact that it is a nonlinear function is crucial to its security. To
see this, suppose that we have a linear block cipher EL that encrypts 256-bit blocks of plaintext into 256-bit blocks of ciphertext. Let EL(k, m) denote the encryption of a 256-bit message m under a key k (the actual bit length of k is irrelevant). Thus,
EL(k, [m1 ⊕ m2]) = EL(k, m1) ⊕ EL(k, m2) for all 128@bit patterns m1, m2.
Describe how, with 256 chosen ciphertexts, an adversary can decrypt any ciphertext without knowledge of the secret key k. (A “chosen ciphertext” means that an adver- sary has the ability to choose a ciphertext and then obtain its decryption. Here, you have 256 plaintext/ciphertext pairs to work with and you have the ability to choose the value of the ciphertexts.)
4.6 Suppose the DES F function mapped every 32-bit input R, regardless of the value of the input K, to; a. 32-bit string of zero b. R
Then 1. What function would DES then compute? 2. What would the decryption look like?
Hint: Use the following properties of the XOR operation:
(A ⊕ B) ⊕ C = A ⊕ (B ⊕ C) (A ⊕ A) = 0 (A⊕ 0 ) = A
A ⊕ 1 = bitwise complement of A
4.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 139
where
A,B,C are n-bit strings of bits 0 is an n-bit string of zeros 1 is an n-bit string of one
4.7 Show that DES decryption is, in fact, the inverse of DES encryption. 4.8 The 32-bit swap after the sixteenth iteration of the DES algorithm is needed to make
the encryption process invertible by simply running the ciphertext back through the algorithm with the key order reversed. This was demonstrated in the preceding prob- lem. However, it still may not be entirely clear why the 32-bit swap is needed. To demonstrate why, solve the following exercises. First, some notation:
A ‘ B = the concatenation of the bit strings A and B Ti(R ‘ L) = the transformation defined by the ith iteration of the encryption
algorithm for 1 … I … 16 TDi(R ‘ L) = the transformation defined by the ith iteration of the decryption
algorithm for 1 … I … 16 T17(R ‘ L) = L ‘ R, where this transformation occurs after the sixteenth iteration
of the encryption algorithm
a. Show that the composition TD1(IP(IP -1(T17(T16(L15 ‘ R15))))) is equivalent to the
transformation that interchanges the 32-bit halves, L15 and R15. That is, show that
TD1(IP(IP -1(T17(T16(L15 ‘ R15))))) = R15 ‘ L15
b. Now suppose that we did away with the final 32-bit swap in the encryption algo- rithm. Then we would want the following equality to hold:
TD1(IP(IP -1(T16(L15 ‘ R15)))) = L15 ‘ R15
Does it?
Note: The following problems refer to details of DES that are described in Appendix S.
4.9 Consider the substitution defined by row 1 of S-box S1 in Table S.2. Show a block diagram similar to Figure 4.2 that corresponds to this substitution.
4.10 Compute the bits number 4, 17, 41, and 45 at the output of the first round of the DES decryption, assuming that the ciphertext block is composed of all ones and the exter- nal key is composed of all ones.
4.11 This problem provides a numerical example of encryption using a one-round version of DES. We start with the same bit pattern for the key K and the plaintext, namely:
Hexadecimal notation: 0 1 2 3 4 5 6 7 8 9 A B C D E F
Binary notation: 0000 0001 0010 0011 0100 0101 0110 0111
1000 1001 1010 1011 1100 1101 1110 1111
a. Derive K1, the first-round subkey. b. Derive L0, R0. c. Expand R0 to get E[R0], where E[ # ] is the expansion function of Table S.1. d. Calculate A = E[R0] ⊕ K1. e. Group the 48-bit result of (d) into sets of 6 bits and evaluate the corresponding
S-box substitutions. f. Concatenate the results of (e) to get a 32-bit result, B.
140 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD
g. Apply the permutation to get P(B). h. Calculate R1 = P(B) ⊕ L0. i. Write down the ciphertext.
4.12 Analyze the amount of left shifts in the DES key schedule by studying Table S.3 (d). Is there a pattern? What could be the reason for the choice of these constants?
4.13 When using the DES algorithm for decryption, the 16 keys (K1, K2, c , K16) are used in reverse order. Therefore, the right-hand side of Figure S.1 is not valid for decryption. Design a key-generation scheme with the appropriate shift schedule (analogous to Table S.3d) for the decryption process.
4.14 a. Let X′ be the bitwise complement of X. Prove that if the complement of the plaintext block is taken and the complement of an encryption key is taken, then the result of DES encryption with these values is the complement of the original ciphertext. That is,
If Y = E(K, X)
Then Y′ = E(K′, X′)
Hint: Begin by showing that for any two bit strings of equal length, A and B, (A ⊕ B)′ = A′ ⊕ B.
b. It has been said that a brute-force attack on DES requires searching a key space of 256 keys. Does the result of part (a) change that?
4.15 a. We say that a DES key K is weak if DESK is an involution. Exhibit four weak keys for DES.
b. We say that a DES key K is semi-weak if it is not weak and if there exists a key K′ such that DESK
- 1 = DESK′. Exhibit four semi-weak keys for DES.
Note: The following problems refer to simplified DES, described in Appendix G. 4.16 Refer to Figure G.3, which explains encryption function for S-DES.
a. How important is the initial permutation IP? b. How important is the SW function in the middle?
4.17 The equations for the variables q and r for S-DES are defined in the section on S-DES analysis. Provide the equations for s and t.
4.18 Using S-DES, decrypt the string 01000110 using the key 1010000010 by hand. Show intermediate results after each function (IP, FK, SW, FK, IP
-1). Then decode the first 4 bits of the plaintext string to a letter and the second 4 bits to another letter where we encode A through P in base 2 (i.e., A = 0000, B = 0001, c , P = 1111). Hint: As a midway check, after the xoring with K2, the string should be 11000001.
Programming Problems
4.19 Create software that can encrypt and decrypt using a general substitution block cipher.
4.20 Create software that can encrypt and decrypt using S-DES. Test data: use plaintext, ciphertext, and key of Problem 4.18.
141
5.1 Groups
Groups
Abelian Group
Cyclic Group
5.2 Rings
5.3 Fields
5.4 Finite Fields of the Form GF(p)
Finite Fields of Order p Finding the Multiplicative Inverse in GF(p) Summary
5.5 Polynomial Arithmetic
Ordinary Polynomial Arithmetic
Polynomial Arithmetic with Coefficients in Zp Finding the Greatest Common Divisor
Summary
5.6 Finite Fields of the form GF(2n)
Motivation
Modular Polynomial Arithmetic
Finding the Multiplicative Inverse
Computational Considerations
Using a Generator
Summary
5.7 Key Terms, Review Questions, and Problems
CHAPTER
Finite Fields
142 CHAPTER 5 / FINITE FIELDS
Finite fields have become increasingly important in cryptography. A number of
cryptographic algorithms rely heavily on properties of finite fields, notably the
Advanced Encryption Standard (AES) and elliptic curve cryptography. Other exam-
ples include the message authentication code CMAC and the authenticated encryption
scheme GCM.
This chapter provides the reader with sufficient background on the concepts of
finite fields to be able to understand the design of AES and other cryptographic algo-
rithms that use finite fields. Because students unfamiliar with abstract algebra may find
the concepts behind finite fields somewhat difficult to grasp, we approach the topic in a
way designed to enhance understanding. Our plan of attack is as follows:
1. Fields are a subset of a larger class of algebraic structures called rings, which are in turn a subset of the larger class of groups. In fact, as shown in Figure 5.1,
both groups and rings can be further differentiated. Groups are defined by
a simple set of properties and are easily understood. Each successive subset
(abelian group, ring, commutative ring, and so on) adds additional properties
and is thus more complex. Sections 5.1 through 5.3 will examine groups, rings,
and fields, successively.
2. Finite fields are a subset of fields, consisting of those fields with a finite num- ber of elements. These are the class of fields that are found in cryptographic
algorithms. With the concepts of fields in hand, we turn in Section 5.4 to a
specific class of finite fields, namely those with p elements, where p is prime. Certain asymmetric cryptographic algorithms make use of such fields.
3. A more important class of finite fields, for cryptography, comprises those with 2n elements depicted as fields of the form GF(2n). These are used in a wide
variety of cryptographic algorithms. However, before discussing these fields, we
need to analyze the topic of polynomial arithmetic, which is done in Section 5.5.
4. With all of this preliminary work done, we are able at last, in Section 5.6, to discuss finite fields of the form GF(2n).
Before proceeding, the reader may wish to review Sections 2.1 through 2.3, which
cover relevant topics in number theory.
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Distinguish among groups, rings, and fields.
◆ Define finite fields of the form GF(p).
◆ Explain the differences among ordinary polynomial arithmetic, polynomial arithmetic with coefficients in Zp, and modular polynomial arithmetic in
GF(2n).
◆ Define finite fields of the form GF(2n).
◆ Explain the two different uses of the mod operator.
5.1 / GROUPS 143
5.1 GROUPS
Groups, rings, and fields are the fundamental elements of a branch of mathematics
known as abstract algebra, or modern algebra. In abstract algebra, we are concerned
with sets on whose elements we can operate algebraically; that is, we can combine
two elements of the set, perhaps in several ways, to obtain a third element of the set.
These operations are subject to specific rules, which define the nature of the set. By
convention, the notation for the two principal classes of operations on set elements is
usually the same as the notation for addition and multiplication on ordinary numbers.
However, it is important to note that, in abstract algebra, we are not limited to ordi-
nary arithmetical operations. All this should become clear as we proceed.
Groups
A group G, sometimes denoted by {G, # }, is a set of elements with a binary opera- tion denoted by # that associates to each ordered pair (a, b) of elements in G an element (a # b) in G, such that the following axioms are obeyed:1
(A1) Closure: If a and b belong to G, then a # b is also in G. (A2) Associative: a # (b # c) = (a # b) # c for all a, b, c in G.
1 The operator # is generic and can refer to addition, multiplication, or some other mathematical operation.
Figure 5.1 Groups, Rings, and Fields
Groups
Abelian groups
Rings
Commutative rings
Integral domains
Fields
Finite fields
144 CHAPTER 5 / FINITE FIELDS
(A3) Identity element: There is an element e in G such that a # e = e # a = a for all a in G.
(A4) Inverse element: For each a in G, there is an element a′ in G such that a # a′ = a′ # a = e.
Let Nn denote a set of n distinct symbols that, for convenience, we represent as {1, 2, c , n}. A permutation of n distinct symbols is a one-to-one mapping from Nn to Nn.
2 Define Sn to be the set of all permutations of n distinct symbols. Each element of Sn is represented by a permutation p of the integers in 1, 2, . . . , n. It is easy to demonstrate that Sn is a group:
A1: If (p, r∈ Sn), then the composite mapping p # r is formed by per- muting the elements of r according to the permutation p. For
example, {3, 2, 1} # {1, 3, 2} = {2, 3, 1}. The notation for this map- ping is explained as follows: The value of the first element of p
indicates which element of r is to be in the first position in p # r; the value of the second element of p indicates which element of r is
to be in the second position in p # r; and so on. Clearly, p # r∈ Sn. A2: The composition of mappings is also easily seen to be associative.
A3: The identity mapping is the permutation that does not alter the order of the n elements. For Sn, the identity element is {1, 2, c , n}.
A4: For any p∈ Sn, the mapping that undoes the permutation defined by p is the inverse element for p. There will always be such an
inverse. For example {2, 3, 1} # {3, 1, 2} = {1, 2, 3}.
2This is equivalent to the definition of permutation in Chapter 2, which stated that a permutation of a finite set of elements S is an ordered sequence of all the elements of S, with each element appearing exactly once.
The set of integers (positive, negative, and 0) under addition is an abelian group.
The set of nonzero real numbers under multiplication is an abelian group. The
set Sn from the preceding example is a group but not an abelian group for n 7 2.
If a group has a finite number of elements, it is referred to as a finite group, and the order of the group is equal to the number of elements in the group. Otherwise, the group is an infinite group.
Abelian Group
A group is said to be abelian if it satisfies the following additional condition:
(A5) Commutative: a # b = b # a for all a, b in G.
5.2 / RINGS 145
When the group operation is addition, the identity element is 0; the in-
verse element of a is - a; and subtraction is defined with the following rule: a - b = a + ( - b).
Cyclic Group
We define exponentiation within a group as a repeated application of the group
operator, so that a3 = a # a # a. Furthermore, we define a0 = e as the identity ele- ment, and a-n = (a′)n, where a′ is the inverse element of a within the group. A group G is cyclic if every element of G is a power ak (k is an integer) of a fixed element a ∈ G. The element a is said to generate the group G or to be a generator of G. A cyclic group is always abelian and may be finite or infinite.
The additive group of integers is an infinite cyclic group generated by the element
1. In this case, powers are interpreted additively, so that n is the nth power of 1.
5.2 RINGS
A ring R, sometimes denoted by {R, + , * }, is a set of elements with two binary operations, called addition and multiplication,3 such that for all a, b, c in R the fol- lowing axioms are obeyed.
(A1–A5) R is an abelian group with respect to addition; that is, R satisfies axioms A1 through A5. For the case of an additive group, we denote the identity element
as 0 and the inverse of a as - a. (M1) Closure under multiplication: If a and b belong to R, then ab is also in R.
(M2) Associativity of multiplication: a(bc) = (ab)c for all a, b, c in R.
(M3) Distributive laws: a(b + c) = ab + ac for all a, b, c in R. (a + b)c = ac + bc for all a, b, c in R.
In essence, a ring is a set of elements in which we can do addition, subtraction
[a - b = a + ( - b)], and multiplication without leaving the set.
3Generally, we do not use the multiplication symbol, * , but denote multiplication by the concatenation of two elements.
With respect to addition and multiplication, the set of all n-square matrices over the real numbers is a ring.
A ring is said to be commutative if it satisfies the following additional condition:
(M4) Commutativity of multiplication: ab = ba for all a, b in R.
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146 CHAPTER 5 / FINITE FIELDS
Next, we define an integral domain, which is a commutative ring that obeys the following axioms.
(M5) Multiplicative identity: There is an element 1 in R such that a1 = 1a = a for all a in R.
(M6) No zero divisors: If a, b in R and ab = 0, then either a = 0 or b = 0.
Let S be the set of even integers (positive, negative, and 0) under the usual operations of addition and multiplication. S is a commutative ring. The set of all n-square matrices defined in the preceding example is not a commutative ring.
The set Zn of integers {0, 1, c , n - 1}, together with the arithmetic oper- ations modulo n, is a commutative ring (Table 4.3).
Let S be the set of integers (positive, negative, and 0) under the usual operations of addition and multiplication. S is an integral domain.
Familiar examples of fields are the rational numbers, the real numbers, and the
complex numbers. Note that the set of all integers is not a field, because not every
element of the set has a multiplicative inverse; in fact, only the elements 1 and - 1 have multiplicative inverses in the integers.
5.3 FIELDS
A field F, sometimes denoted by {F, + , * }, is a set of elements with two binary operations, called addition and multiplication, such that for all a, b, c in F the follow- ing axioms are obeyed.
(A1–M6) F is an integral domain; that is, F satisfies axioms A1 through A5 and M1 through M6.
(M7) Multiplicative inverse: For each a in F, except 0, there is an element a-1 in F such that aa-1 = (a-1)a = 1.
In essence, a field is a set of elements in which we can do addition, subtraction,
multiplication, and division without leaving the set. Division is defined with the fol-
lowing rule: a/b = a(b-1).
In gaining insight into fields, the following alternate characterization may be
useful. A field F, denoted by {F, + }, is a set of elements with two binary operations, called addition and multiplication, such that the following conditions hold:
1. F forms an abelian group with respect to addition.
2. The nonzero elements of F form an abelian group with respect to multiplication.
5.4 / FINITE FIELDS OF THE FORM GF(p) 147
3. The distributive law holds. That is, for all a, b, c in F,
a(b + c) = ab + ac.
(a + b)c = ac + bc
4. Figure 5.2 summarizes the axioms that define groups, rings, and fields.
5.4 FINITE FIELDS OF THE FORM GF(p)
In Section 5.3, we defined a field as a set that obeys all of the axioms of Figure 5.2
and gave some examples of infinite fields. Infinite fields are not of particular inter-
est in the context of cryptography. However, in addition to infinite fields, there are
two types of finite fields, as illustrated in Figure 5.3. Finite fields play a crucial role
in many cryptographic algorithms.
It can be shown that the order of a finite field (number of elements in the
field) must be a power of a prime pn, where n is a positive integer. The finite field of order pn is generally written GF(pn); GF stands for Galois field, in honor of the mathematician who first studied finite fields. Two special cases are of interest for
our purposes. For n = 1, we have the finite field GF(p); this finite field has a differ- ent structure than that for finite fields with n 7 1 and is studied in this section. For finite fields of the form GF(pn), GF(2n) fields are of particular cryptographic inter- est, and these are covered in Section 5.6.
Finite Fields of Order p
For a given prime, p, we define the finite field of order p, GF(p), as the set Zp of integers {0, 1, c , p - 1} together with the arithmetic operations modulo p. Note therefore that we are using ordinary modular arithmetic to define the operations over these fields.
Figure 5.2 Properties of Groups, Rings, and Fields
(A1) Closure under addition: If a and b belong to S, then a + b is also in S (A2) Associativity of addition: a + (b + c) = (a + b) + c for all a, b, c in S (A3) Additive identity: There is an element 0 in R such that
a + 0 = 0 + a = a for all a in S (A4) Additive inverse: For each a in S there is an element –a in S
such that a + (–a) = (–a) + a = 0
(A5) Commutativity of addition: a + b = b + a for all a, b in S
(M1) Closure under multiplication: If a and b belong to S, then ab is also in S (M2) Associativity of multiplication: a(bc) = (ab)c for all a, b, c in S (M3) Distributive laws: a(b + c) = ab + ac for all a, b, c in S
(a + b)c = ac + bc for all a, b, c in S
(M4) Commutativity of multiplication: ab = ba for all a, b in S
(M5) Multiplicative identity: There is an element 1 in S such that a1 = 1a = a for all a in S
(M6) No zero divisors: If a, b in S and ab = 0, then either a = 0 or b = 0
(M7) Multiplicative inverse: If a belongs to S and a ≠ 0, there is an element a –1 in S such that aa –1 = a –1a = 1
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148 CHAPTER 5 / FINITE FIELDS
Recall that we showed in Section 5.2 that the set Zn of integers {0, 1, c , n - 1}, together with the arithmetic operations modulo n, is a commutative ring (Table 2.5). We further observed that any integer in Zn has a multiplicative inverse if and only if
that integer is relatively prime to n [see discussion of Equation (2.5)].4 If n is prime, then all of the nonzero integers in Zn are relatively prime to n, and therefore there exists a multiplicative inverse for all of the nonzero integers in Zn. Thus, for Zp we
can add the following properties to those listed in Table 5.2:
Multiplicative
inverse (w -1) For each w ∈ Zp, w ≠ 0, there exists a z ∈ Zp such that w * z K 1 (mod p)
Because w is relatively prime to p, if we multiply all the elements of Zp by w, the resulting residues are all of the elements of Zp permuted. Thus, exactly one of the residues has the value 1. Therefore, there is some integer in Zp that, when
multiplied by w, yields the residue 1. That integer is the multiplicative inverse of w, designated w -1. Therefore, Zp is in fact a finite field. Furthermore, Equation (2.5) is consistent with the existence of a multiplicative inverse and can be rewritten with-
out the condition:
if (a * b) K (a * c)(mod p) then b K c(mod p) (5.1)
Multiplying both sides of Equation (5.1) by the multiplicative inverse of a, we have
((a-1) * a * b) K ((a-1) * a * c)(mod p) b K c (mod p)
4As stated in the discussion of Equation (2.5), two integers are relatively prime if their only common positive integer factor is 1.
Figure 5.3 Types of Fields
Fields
Fields with an infinite number
of elements
Finite fields
GF(p) Finite fields
with p elements
GF(pn) Finite fields
with pn elements
The simplest finite field is GF(2). Its arithmetic operations are easily summarized:
+ 0 1 0 0 1
1 1 0
Addition
* 0 1 0 0 0
1 0 1
Multiplication
w - w w -1
0 0 - 1 1 1
Inverses
In this case, addition is equivalent to the exclusive-OR (XOR) operation, and
multiplication is equivalent to the logical AND operation.
5.4 / FINITE FIELDS OF THE FORM GF(p) 149
The right-hand side of Table 5.1 shows arithmetic operations in GF(7). This is a
field of order 7 using modular arithmetic modulo 7. As can be seen, it satisfies all
of the properties required of a field (Figure 5.2). Compare with the left-hand side
of Table 5.1, which reproduces Table 2.2. In the latter case, we see that the set Z8,
using modular arithmetic modulo 8, is not a field. Later in this chapter, we show
how to define addition and multiplication operations on Z8 in such a way as to
form a finite field.
Finding the Multiplicative Inverse in GF(p)
It is easy to find the multiplicative inverse of an element in GF(p) for small values
of p. You simply construct a multiplication table, such as shown in Table 5.1e, and the desired result can be read directly. However, for large values of p, this approach is not practical.
If a and b are relatively prime, then b has a multiplicative inverse modulo a. That is, if gcd(a, b) = 1, then b has a multiplicative inverse modulo a. That is, for positive integer b 6 a, there exists a b-1 6 a such that bb-1 = 1 mod a. If a is a prime number and b 6 a, then clearly a and b are relatively prime and have a great- est common divisor of 1. We now show that we can easily compute b-1 using the extended Euclidean algorithm.
We repeat here Equation (2.7), which we showed can be solved with the ex-
tended Euclidean algorithm:
ax + by = d = gcd(a, b)
Now, if gcd(a, b) = 1, then we have ax + by = 1. Using the basic equalities of modular arithmetic, defined in Section 2.3, we can say
[(ax mod a) + (by mod a)] mod a = 1 mod a 0 + (by mod a) = 1
But if by mod a = 1, then y = b-1. Thus, applying the extended Euclidean algorithm to Equation (2.7) yields the value of the multiplicative inverse of b if gcd(a, b) = 1.
Consider the example that was shown in Table 2.4. Here we have a = 1759, which is a prime number, and b = 550. The solution of the equation 1759x + 550y = d yields a value of y = 355. Thus, b-1 = 355. To verify, we cal- culate 550 * 355 mod 1759 = 195250 mod 1759 = 1.
More generally, the extended Euclidean algorithm can be used to find a
multiplicative inverse in Zn for any n. If we apply the extended Euclidean algorithm to the equation nx + by = d, and the algorithm yields d = 1, then y = b-1 in Zn.
150 CHAPTER 5 / FINITE FIELDS
+ 0 1 2 3 4 5 6 7
0 0 1 2 3 4 5 6 7
1 1 2 3 4 5 6 7 0
2 2 3 4 5 6 7 0 1
3 3 4 5 6 7 0 1 2
4 4 5 6 7 0 1 2 3
5 5 6 7 0 1 2 3 4
6 6 7 0 1 2 3 4 5
7 7 0 1 2 3 4 5 6
(a) Addition modulo 8
* 0 1 2 3 4 5 6 7
0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7
2 0 2 4 6 0 2 4 6
3 0 3 6 1 4 7 2 5
4 0 4 0 4 0 4 0 4
5 0 5 2 7 4 1 6 3
6 0 6 4 2 0 6 4 2
7 0 7 6 5 4 3 2 1
(b) Multiplication modulo 8
w 0 1 2 3 4 5 6 7
- w 0 7 6 5 4 3 2 1
w -1 — 1 — 3 — 5 — 7
(c) Additive and multiplicative
inverses modulo 8
+ 0 1 2 3 4 5 6
0 0 1 2 3 4 5 6
1 1 2 3 4 5 6 0
2 2 3 4 5 6 0 1
3 3 4 5 6 0 1 2
4 4 5 6 0 1 2 3
5 5 6 0 1 2 3 4
6 6 0 1 2 3 4 5
(d) Addition modulo 7
* 0 1 2 3 4 5 6
0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6
2 0 2 4 6 1 3 5
3 0 3 6 2 5 1 4
4 0 4 1 5 2 6 3
5 0 5 3 1 6 4 2
6 0 6 5 4 3 2 1
(e) Multiplication modulo 7
w 0 1 2 3 4 5 6
- w 0 6 5 4 3 2 1
w -1 — 1 4 5 2 3 6
(f) Additive and multiplicative
inverses modulo 7
Table 5.1 Arithmetic Modulo 8 and Modulo 7
Summary
In this section, we have shown how to construct a finite field of order p, where p is prime. Specifically, we defined GF(p) with the following properties.
1. GF(p) consists of p elements.
2. The binary operations + and * are defined over the set. The operations of addition, subtraction, multiplication, and division can be performed without
leaving the set. Each element of the set other than 0 has a multiplicative in-
verse, and division is performed by multiplication by the multiplicative inverse.
We have shown that the elements of GF(p) are the integers {0, 1, c , p - 1} and that the arithmetic operations are addition and multiplication mod p.
5.5 / POLYNOMIAL ARITHMETIC 151
5.5 POLYNOMIAL ARITHMETIC
Before continuing our discussion of finite fields, we need to introduce the interest-
ing subject of polynomial arithmetic. We are concerned with polynomials in a single
variable x, and we can distinguish three classes of polynomial arithmetic (Figure 5.4).
■ Ordinary polynomial arithmetic, using the basic rules of algebra.
■ Polynomial arithmetic in which the arithmetic on the coefficients is performed
modulo p; that is, the coefficients are in GF(p).
■ Polynomial arithmetic in which the coefficients are in GF(p), and the poly- nomials are defined modulo a polynomial m(x) whose highest power is some integer n.
This section examines the first two classes, and the next section covers the
last class.
Ordinary Polynomial Arithmetic
A polynomial of degree n (integer n Ú 0) is an expression of the form
f(x) = anx n + an - 1xn - 1 + g + a1x + a0 = a
n
i = 0 aix
i
where the ai are elements of some designated set of numbers S, called the coefficient set, and an ≠ 0. We say that such polynomials are defined over the coefficient set S.
A zero-degree polynomial is called a constant polynomial and is simply an element of the set of coefficients. An nth-degree polynomial is said to be a monic polynomial if an = 1.
In the context of abstract algebra, we are usually not interested in evaluating a
polynomial for a particular value of x [e.g., f(7)]. To emphasize this point, the vari- able x is sometimes referred to as the indeterminate.
Polynomial arithmetic includes the operations of addition, subtraction, and
multiplication. These operations are defined in a natural way as though the variable
Figure 5.4 Treatment of Polynomials
Polynomial f(x)
x treated as a variable, and evaluated for
a particular value of x
x treated as an indeterminate
Ordinary polynomial arithmetic
Arithmetic on coefficients is
performed modulo p
Arithmetic on coefficients is performed modulo p
and polynomials are defined modulo a polynomial m(x)
152 CHAPTER 5 / FINITE FIELDS
x was an element of S. Division is similarly defined, but requires that S be a field. Examples of fields include the real numbers, rational numbers, and Zp for p prime. Note that the set of all integers is not a field and does not support polynomial
division.
Addition and subtraction are performed by adding or subtracting correspond-
ing coefficients. Thus, if
f(x) = a n
i = 0 aix
i; g(x) = a m
i = 0 bix
i; n Ú m
then addition is defined as
f(x) + g(x) = a m
i = 0 (ai + bi)xi + a
n
i = m + 1 aix
i
and multiplication is defined as
f(x) * g(x) = a n + m
i = 0 cix
i
where
ck = a0bk + a1bk - 1 + g + ak - 1b1 + akb0
In the last formula, we treat ai as zero for i 7 n and bi as zero for i 7 m. Note that the degree of the product is equal to the sum of the degrees of the two polynomials.
As an example, let f(x) = x3 + x2 + 2 and g(x) = x2 - x + 1, where S is the set of integers. Then
f(x) + g(x) = x3 + 2x2 - x + 3 f(x) - g(x) = x3 + x + 1 f(x) * g(x) = x5 + 3x2 - 2x + 2
Figures 5.5a through 5.5c show the manual calculations. We comment on division
subsequently.
Polynomial Arithmetic with Coefficients in Zp Let us now consider polynomials in which the coefficients are elements of some
field F; we refer to this as a polynomial over the field F. In this case, it is easy to
show that the set of such polynomials is a ring, referred to as a polynomial ring. That is, if we consider each distinct polynomial to be an element of the set, then that set
is a ring.5
When polynomial arithmetic is performed on polynomials over a field, then
division is possible. Note that this does not mean that exact division is possible. Let
5In fact, the set of polynomials whose coefficients are elements of a commutative ring forms a polynomial ring, but that is of no interest in the present context.
5.5 / POLYNOMIAL ARITHMETIC 153
us clarify this distinction. Within a field, given two elements a and b, the quotient a/b is also an element of the field. However, given a ring R that is not a field, in gen- eral, division will result in both a quotient and a remainder; this is not exact division.
Figure 5.5 Examples of Polynomial Arithmetic
x3
x3
+ +x2
+2x2 x2 x
2
+–+ ( )
× ( )
– ( )
x–
1
+ 3
(a) Addition
(d) Division(c) Multiplication
x3
x3
+ +x2
+ x2 x2 x
2
x3 x 2
+
+
+x2
x3 – x2
2x2 + x
– x
x
2
+ 2
2x2 – 2x + 2
x4 –– –x3 2x
– 2x
x5 + +x4 2x2
x5 +3x2
+– 1 x2 x +– 1
+ 2
+ 2
x3
x3
+ +x2
x2 x
2
+–
x+
1
+ 1
(b) Subtraction
Consider the division 5/3 within a set S. If S is the set of rational numbers, which is a field, then the result is simply expressed as 5/3 and is an element of S. Now suppose that S is the field Z7. In this case, we calculate (using Table 5.1f)
5/3 = (5 * 3-1) mod 7 = (5 * 5) mod 7 = 4 which is an exact solution. Finally, suppose that S is the set of integers, which is a ring but not a field. Then 5/3 produces a quotient of 1 and a remainder of 2:
5/3 = 1 + 2/3 5 = 1 * 3 + 2
Thus, division is not exact over the set of integers.
Now, if we attempt to perform polynomial division over a coefficient set that
is not a field, we find that division is not always defined.
If the coefficient set is the integers, then (5x2)/(3x) does not have a solution, because it would require a coefficient with a value of 5/3, which is not in the coef-
ficient set. Suppose that we perform the same polynomial division over Z7. Then
we have (5x2)/(3x) = 4x, which is a valid polynomial over Z7.
However, as we demonstrate presently, even if the coefficient set is a field,
polynomial division is not necessarily exact. In general, division will produce a quo-
tient and a remainder. We can restate the division algorithm of Equation (2.1) for
polynomials over a field as follows. Given polynomials f(x) of degree n and g(x)
154 CHAPTER 5 / FINITE FIELDS
of degree (m), (n Ú m), if we divide f(x) by g(x), we get a quotient q(x) and a remainder r(x) that obey the relationship
f(x) = q(x)g(x) + r(x) (5.2)
with polynomial degrees:
Degree f(x) = n Degree g(x) = m Degree q(x) = n - m Degree r(x) … m - 1
With the understanding that remainders are allowed, we can say that poly-
nomial division is possible if the coefficient set is a field. One common technique
used for polynomial division is polynomial long division, similar to long division for
integers. Examples of this are shown subsequently.
In an analogy to integer arithmetic, we can write f(x) mod g(x) for the remain- der r(x) in Equation (5.2). That is, r(x) = f(x) mod g(x). If there is no remainder [i.e., r(x) = 0], then we can say g(x) divides f(x), written as g(x)�f(x). Equivalently, we can say that g(x) is a factor of f(x) or g(x) is a divisor of f(x).
For the preceding example [f(x) = x3 + x2 + 2 and g(x) = x2 - x + 1], f(x)/g(x) produces a quotient of q(x) = x + 2 and a remainder r(x) = x, as shown in Figure 5.5d. This is easily verified by noting that
q(x)g(x) + r(x) = (x + 2)(x2 - x + 1) + x = (x3 + x2 - x + 2) + x = x3 + x2 + 2 = f(x)
For our purposes, polynomials over GF(2) are of most interest. Recall from
Section 5.4 that in GF(2), addition is equivalent to the XOR operation, and multi-
plication is equivalent to the logical AND operation. Further, addition and subtrac-
tion are equivalent mod 2:
1 + 1 = 1 - 1 = 0 1 + 0 = 1 - 0 = 1 0 + 1 = 0 - 1 = 1
Figure 5.6 shows an example of polynomial arithmetic over GF(2). For
f(x) = (x7 + x5 + x4 + x3 + x + 1) and g(x) = (x3 + x + 1), the figure shows f(x) + g(x); f(x) - g(x); f(x) * g(x); and f(x)/g(x). Note that g(x)�f(x).
A polynomial f(x) over a field F is called irreducible if and only if f(x) can- not be expressed as a product of two polynomials, both over F, and both of degree lower than that of f(x). By analogy to integers, an irreducible polynomial is also called a prime polynomial.
The polynomial6 f(x) = x4 + 1 over GF(2) is reducible, because x4 + 1 = (x + 1)(x3 + x2 + x + 1).
6In the reminder of this chapter, unless otherwise noted, all examples are of polynomials over GF(2).
Hiva-Network.Com
5.5 / POLYNOMIAL ARITHMETIC 155
Consider the polynomial f(x) = x3 + x + 1. It is clear by inspection that x is not a factor of f(x). We easily show that x + 1 is not a factor of f(x):
x2 + x x + 1�x3 + x + 1
x3 + x2
x2 + x x2 + x
1
Thus, f(x) has no factors of degree 1. But it is clear by inspection that if f(x) is reducible, it must have one factor of degree 2 and one factor of degree 1. There-
fore, f(x) is irreducible.
Figure 5.6 Examples of Polynomial Arithmetic over GF(2)
(a) Addition
(c) Multiplication
(d) Division
x4x5 ++x7 xx3
x3x4 ++x5 ++x7 +x 1
+++ ( )1
x3x4 ++x5 ++x7 +x 1
x4x5 ++x7 x3 x
x3 ++ +x 1
+ 1
x5x6 ++x8 x4 ++ +x2
+ x2
x
x7x8 ++x10 x6 ++ +x4
x10 + x4 x3
++× ( )1
x3x4 ++x5 ++x7
x4x5 ++x7
+x
x3 x
1
++– ( )1
(b) Subtraction
x3x4 ++x5 ++
++
x7
x4x5x7 +x 1
x3 + +x 1
x3 + +x 1
x4 1+
x3 x ++ 1
156 CHAPTER 5 / FINITE FIELDS
Finding the Greatest Common Divisor
We can extend the analogy between polynomial arithmetic over a field and integer
arithmetic by defining the greatest common divisor as follows. The polynomial c(x) is said to be the greatest common divisor of a(x) and b(x) if the following are true.
1. c(x) divides both a(x) and b(x).
2. Any divisor of a(x) and b(x) is a divisor of c(x).
An equivalent definition is the following: gcd[a(x), b(x)] is the polynomial of maximum degree that divides both a(x) and b(x).
We can adapt the Euclidean algorithm to compute the greatest common divisor
of two polynomials. Recall Equation (2.6), from Chapter 2, which is the basis of the
Euclidean algorithm: gcd(a, b) = gcd(b, a mod b). This equality can be rewritten as the following equation:
gcd[a(x), b(x)] = gcd[b(x), a(x) mod b(x)] (5.3)
Equation (5.3) can be used repetitively to determine the greatest common divisor.
Compare the following scheme to the definition of the Euclidean algorithm for integers.
Euclidean Algorithm for Polynomials
Calculate Which satisfies
r1(x) = a(x) mod b(x) a(x) = q1(x)b(x) + r1(x) r2(x) = b(x) mod r1(x) b(x) = q2(x)r1(x) + r2(x) r3(x) = r1(x) mod r2(x) r1(x) = q3(x)r2(x) + r3(x)
rn(x) = rn - 2(x) mod rn - 1(x) rn - 2(x) = qn(x)rn - 1(x) + rn(x)
rn + 1(x) = rn - 1(x) mod rn(x) = 0 rn - 1(x) = qn + 1(x)rn(x) + 0
d(x) = gcd(a(x), b(x)) = rn(x)
At each iteration, we have d(x) = gcd(ri + 1(x), ri(x)) until finally d(x) = gcd(rn(x), 0) = rn(x). Thus, we can find the greatest common divisor of two integers by repetitive application of the division algorithm. This is the Euclidean
algorithm for polynomials. The algorithm assumes that the degree of a(x) is greater than the degree of b(x).
Find gcd[a(x), b(x)] for a(x) = x6 + x5 + x4 + x3 + x2 + x + 1 and b(x) = x4 + x2 + x + 1. First, we divide a(x) by b(x):
x2 + x x4 + x2 + x + 1�x6 + x5 + x4 + x3 + x2 + x + 1
x6 + x4 + x3 + x2
x5 + x + 1 x5 + x3 + x2 + x
x3 + x2 + 1
5.6 / FINITE FIELDS OF THE FORM GF(2n) 157
Summary
We began this section with a discussion of arithmetic with ordinary polynomials. In
ordinary polynomial arithmetic, the variable is not evaluated; that is, we do not plug
a value in for the variable of the polynomials. Instead, arithmetic operations are
performed on polynomials (addition, subtraction, multiplication, division) using the
ordinary rules of algebra. Polynomial division is not allowed unless the coefficients
are elements of a field.
Next, we discussed polynomial arithmetic in which the coefficients are ele-
ments of GF(p). In this case, polynomial addition, subtraction, multiplication, and division are allowed. However, division is not exact; that is, in general division re-
sults in a quotient and a remainder.
Finally, we showed that the Euclidean algorithm can be extended to find the
greatest common divisor of two polynomials whose coefficients are elements of a
field.
All of the material in this section provides a foundation for the following sec-
tion, in which polynomials are used to define finite fields of order pn.
5.6 FINITE FIELDS OF THE FORM GF(2n)
Earlier in this chapter, we mentioned that the order of a finite field must be of the
form pn, where p is a prime and n is a positive integer. In Section 5.4, we looked at the special case of finite fields with order p. We found that, using modular arith- metic in Zp, all of the axioms for a field (Figure 5.2) are satisfied. For polynomials
over pn, with n 7 1, operations modulo pn do not produce a field. In this section, we show what structure satisfies the axioms for a field in a set with pn elements and concentrate on GF(2n).
Motivation
Virtually all encryption algorithms, both symmetric and asymmetric, involve arith-
metic operations on integers. If one of the operations that is used in the algorithm is
division, then we need to work in arithmetic defined over a field. For convenience
This yields r1(x) = x 3 + x2 + 1 and q1 (x) = x2 + x.
Then, we divide b(x) by r1(x).
x + 1 x3 + x2 + 1�x4 + x2 + x + 1
x4 + x3 + x x3 + x2 + 1 x3 + x2 + 1
This yields r2(x) = 0 and q2(x) = x + 1. Therefore, gcd[a(x), b(x)] = r1(x) = x
3 + x2 + 1.
158 CHAPTER 5 / FINITE FIELDS
and for implementation efficiency, we would also like to work with integers that fit
exactly into a given number of bits with no wasted bit patterns. That is, we wish to
work with integers in the range 0 through 2n - 1, which fit into an n-bit word.
Suppose we wish to define a conventional encryption algorithm that operates on
data 8 bits at a time, and we wish to perform division. With 8 bits, we can repre-
sent integers in the range 0 through 255. However, 256 is not a prime number, so
that if arithmetic is performed in Z256 (arithmetic modulo 256), this set of inte-
gers will not be a field. The closest prime number less than 256 is 251. Thus, the
set Z251, using arithmetic modulo 251, is a field. However, in this case the 8-bit
patterns representing the integers 251 through 255 would not be used, resulting
in inefficient use of storage.
As the preceding example points out, if all arithmetic operations are to be
used and we wish to represent a full range of integers in n bits, then arithmetic modulo 2n will not work. Equivalently, the set of integers modulo 2n for n 7 1, is not a field. Furthermore, even if the encryption algorithm uses only addition and
multiplication, but not division, the use of the set Z2n is questionable, as the follow-
ing example illustrates.
Suppose we wish to use 3-bit blocks in our encryption algorithm and use only the
operations of addition and multiplication. Then arithmetic modulo 8 is well defined,
as shown in Table 5.1. However, note that in the multiplication table, the nonzero
integers do not appear an equal number of times. For example, there are only four
occurrences of 3, but twelve occurrences of 4. On the other hand, as was mentioned,
there are finite fields of the form GF(2n), so there is in particular a finite field of
order 23 = 8. Arithmetic for this field is shown in Table 5.2. In this case, the number of occurrences of the nonzero integers is uniform for multiplication. To summarize,
Integer 1 2 3 4 5 6 7
Occurrences in Z8 4 8 4 12 4 8 4
Occurrences in GF(23) 7 7 7 7 7 7 7
For the moment, let us set aside the question of how the matrices of Table 5.2
were constructed and instead make some observations.
1. The addition and multiplication tables are symmetric about the main diago- nal, in conformance to the commutative property of addition and multiplica-
tion. This property is also exhibited in Table 5.1, which uses mod 8 arithmetic.
2. All the nonzero elements defined by Table 5.2 have a multiplicative inverse, unlike the case with Table 5.1.
3. The scheme defined by Table 5.2 satisfies all the requirements for a finite field. Thus, we can refer to this scheme as GF(23).
4. For convenience, we show the 3-bit assignment used for each of the elements of GF(23).
5.6 / FINITE FIELDS OF THE FORM GF(2n) 159
Intuitively, it would seem that an algorithm that maps the integers unevenly
onto themselves might be cryptographically weaker than one that provides a uni-
form mapping. That is, a cryptanalytic technique might be able to exploit the fact
that some integers occur more frequently and some less frequently in the ciphertext.
Thus, the finite fields of the form GF(2n) are attractive for cryptographic algorithms.
To summarize, we are looking for a set consisting of 2n elements, together
with a definition of addition and multiplication over the set that define a field. We
can assign a unique integer in the range 0 through 2n - 1 to each element of the set. Keep in mind that we will not use modular arithmetic, as we have seen that this
does not result in a field. Instead, we will show how polynomial arithmetic provides
a means for constructing the desired field.
Modular Polynomial Arithmetic
Consider the set S of all polynomials of degree n - 1 or less over the field Zp. Thus, each polynomial has the form
f(x) = an - 1x n - 1 + an - 2xn - 2 + g + a1x + a0 = a
n - 1
i = 0 aix
i
000 001 010 011 100 101 110 111
+ 0 1 2 3 4 5 6 7
000 0 0 1 2 3 4 5 6 7
001 1 1 0 3 2 5 4 7 6
010 2 2 3 0 1 6 7 4 5
011 3 3 2 1 0 7 6 5 4
100 4 4 5 6 7 0 1 2 3
101 5 5 4 7 6 1 0 3 2
110 6 6 7 4 5 2 3 0 1
111 7 7 6 5 4 3 2 1 0
(a) Addition
000 001 010 011 100 101 110 111
* 0 1 2 3 4 5 6 7
000 0 0 0 0 0 0 0 0 0
001 1 0 1 2 3 4 5 6 7
010 2 0 2 4 6 3 1 7 5
011 3 0 3 6 5 7 4 1 2
100 4 0 4 3 7 6 2 5 1
101 5 0 5 1 4 2 7 3 6
110 6 0 6 7 1 5 3 2 4
111 7 0 7 5 2 1 6 4 3
(b) Multiplication
w - w w -1
0 0 -
1 1 1
2 2 5
3 3 6
4 4 7
5 5 2
6 6 3
7 7 4
(c) Additive and multiplicative
inverses
Table 5.2 Arithmetic in GF(23)
160 CHAPTER 5 / FINITE FIELDS
where each ai takes on a value in the set {0, 1, c , p - 1}. There are a total of pn different polynomials in S.
For p = 3 and n = 2, the 32 = 9 polynomials in the set are 0, 1, 2, x, x + 1, x + 2, 2x, 2x + 1, 2x + 2
For p = 2 and n = 3, the 23 = 8 polynomials in the set are 0, 1, x, x + 1, x2, x2 + 1, x2 + x, x2 + x + 1
With the appropriate definition of arithmetic operations, each such set S is a finite field. The definition consists of the following elements.
1. Arithmetic follows the ordinary rules of polynomial arithmetic using the basic rules of algebra, with the following two refinements.
2. Arithmetic on the coefficients is performed modulo p. That is, we use the rules of arithmetic for the finite field Zp.
3. If multiplication results in a polynomial of degree greater than n - 1, then the polynomial is reduced modulo some irreducible polynomial m(x) of degree n. That is, we divide by m(x) and keep the remainder. For a polynomial f(x), the remainder is expressed as r(x) = f(x) mod m(x).
The Advanced Encryption Standard (AES) uses arithmetic in the finite field
GF(28), with the irreducible polynomial m(x) = x8 + x4 + x3 + x + 1. Consider the two polynomials f(x) = x6 + x4 + x2 + x + 1 and g(x) = x7 + x + 1. Then
f(x) + g(x) = x6 + x4 + x2 + x + 1 + x7 + x + 1 = x7 + x6 + x4 + x2
f(x) * g(x) = x13 + x11 + x9 + x8 + x7
+ x7 + x5 + x3 + x2 + x + x6 + x4 + x2 + x + 1
= x13 + x11 + x9 + x8 + x6 + x5 + x4 + x3 + 1
x5 + x3
x8 + x4 + x3 + x + 1>x13 + x11 + x9 + x8 + x6 + x5 + x4 + x3 + 1 x13 + x9 + x8 + x6 + x5
x11 + x4 + x3
x11 + x7 + x6 + x4 + x3
x7 + x6 + 1
Therefore, f(x) * g(x) mod m(x) = x7 + x6 + 1.
5.6 / FINITE FIELDS OF THE FORM GF(2n) 161
As with ordinary modular arithmetic, we have the notion of a set of residues
in modular polynomial arithmetic. The set of residues modulo m(x), an nth-degree polynomial, consists of pn elements. Each of these elements is represented by one of the pn polynomials of degree m 6 n.
The residue class [x + 1], (mod m(x)), consists of all polynomials a(x) such that a(x) K (x + 1)(mod m(x)). Equivalently, the residue class [x + 1] consists of all polynomials a(x) that satisfy the equality a(x) mod m(x) = x + 1.
It can be shown that the set of all polynomials modulo an irreducible nth- degree polynomial m(x) satisfies the axioms in Figure 5.2, and thus forms a finite field. Furthermore, all finite fields of a given order are isomorphic; that is, any two
finite-field structures of a given order have the same structure, but the representa-
tion or labels of the elements may be different.
To construct the finite field GF(23), we need to choose an irreducible poly-
nomial of degree 3. There are only two such polynomials: (x3 + x2 + 1) and (x3 + x + 1). Using the latter, Table 5.3 shows the addition and multiplication tables for GF(23). Note that this set of tables has the identical structure to those
of Table 5.2. Thus, we have succeeded in finding a way to define a field of order 23.
We can now read additions and multiplications from the table easily. For exam-
ple, consider binary 100 + 010 = 110. This is equivalent to x2 + x. Also consider 100 * 010 = 011, which is equivalent to x2 * x = x3 and reduces to x + 1. That is, x3 mod (x3 + x + 1) = x + 1, which is equivalent to 011.
Finding the Multiplicative Inverse
Just as the Euclidean algorithm can be adapted to find the greatest common divisor
of two polynomials, the extended Euclidean algorithm can be adapted to find the
multiplicative inverse of a polynomial. Specifically, the algorithm will find the mul-
tiplicative inverse of b(x) modulo a(x) if the degree of b(x) is less than the degree of a(x) and gcd[a(x), b(x)] = 1. If a(x) is an irreducible polynomial, then it has no fac- tor other than itself or 1, so that gcd[a(x), b(x)] = 1. The algorithm can be charac- terized in the same way as we did for the extended Euclidean algorithm for integers.
Given polynomials a(x) and b(x) with the degree of a(x) greater than the degree of b(x), we wish to solve the following equation for the values v(x), w(x), and d(x), where d(x) = gcd[a(x), b(x)]:
a(x)v(x) + b(x)w(x) = d(x)
If d(x) = 1, then w(x) is the multiplicative inverse of b(x) modulo a(x). The calcula- tions are as follows.
162 CHAPTER 5 / FINITE FIELDS
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
+ 0
1 x
x +
1 x
2 x
2 +
1 x
2 +
x x
2 +
x +
1
0 0 0
0 0
1 x
x +
1 x
2 x
2 +
1 x
2 +
x x
2 +
x +
1
0 0 1
1 1
0 x
+ 1
x x
2 +
1 x
2 x
2 +
x +
1 x
2 +
x
0 1 0
x x
x +
1 0
1 x
2 +
x x
2 +
x +
1 x
2 x
2 +
1
0 1 1
x +
1 x
+ 1
x 1
0 x
2 +
x +
1 x
2 +
x x
2 +
1 x
2
1 0 0
x 2
x 2
x 2
+ 1
x 2
+ x
x 2
+ x
+ 1
0 1
x x
+ 1
1 0 1
x 2
+ 1
x 2
+ 1
x 2
x 2
+ x
+ 1
x 2
+ x
1 0
x +
1 x
1 1 0
x 2
+ x
x 2
+ x
x 2
+ x
+ 1
x 2
x 2
+ 1
x x
+ 1
0 1
1 1 1
x 2
+ x
+ 1
x 2
+ x
+ 1
x 2
+ x
x 2
+ 1
x 2
x +
1 x
1 0
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
* 0
1 x
x +
1 x
2 x
2 +
1 x
2 +
x x
2 +
x +
1
0 0 0
0 0
0 0
0 0
0 0
0
0 0 1
1 0
1 x
x +
1 x
2 x
2 +
1 x
2 +
x x
2 +
x +
1
0 1 0
x 0
x x
2 x
2 +
x x
+ 1
1 x
2 +
x +
1 x
2 +
1
0 1 1
x +
1 0
x +
1 x
2 +
x x
2 +
1 x
2 +
x +
1 x
2 1
x
1 0 0
x 2
0 x
2 x
+ 1
x 2
+ x
+ 1
x 2
+ x
x x
2 +
1 1
1 0 1
x 2
+ 1
0 x
2 +
1 1
x 2
x x
2 +
x +
1 x
+ 1
x 2
+ x
1 1 0
x 2
+ x
0 x
2 +
x x
2 +
x +
1 1
x 2
+ 1
x +
1 x
x 2
1 1 1
x 2
+ x
+ 1
0 x
2 +
x +
1 x
2 +
1 x
1 x
2 +
1 x
2 x
+ 1
(a )
A d
d it
io n
(b )
M u
lt ip
li c a ti
o n
T ab
le 5
.3
P o
ly n
o m
ia l A
ri th
m e ti
c M
o d
u lo
( x3
+ x
+ 1 )
5.6 / FINITE FIELDS OF THE FORM GF(2n) 163
Extended Euclidean Algorithm for Polynomials
Calculate Which satisfies Calculate Which satisfies
r-1(x) = a(x) v-1(x) = 1; w-1(x) = 0 a(x) = a(x)v-1(x) + bw-1(x)
r0(x) = b(x) v0(x) = 0; w0(x) = 1 b(x) = a(x)v0(x) + b(x)w0(x)
r1(x) = a(x) mod b(x) q1(x) = quotient of a(x)/b(x)
a(x) = q1(x)b(x) + r1(x)
v1(x) = v-1(x) - q1(x)v0(x) = 1 w1(x) = w-1(x) - q1(x)w0(x) = - q1(x)
r1(x) = a(x)v1(x) + b(x)w1(x)
r2(x) = b(x) mod r1(x) q2(x) = quotient of b(x)/r1(x)
b(x) = q2(x)r1(x) + r2(x)
v2(x) = v0(x) - q2(x)v1(x) w2(x) = w0(x) - q2(x)w1(x)
r2(x) = a(x)v2(x) + b(x)w2(x)
r3(x) = r1(x) mod r2(x) q3(x) = quotient of r1(x)/r2(x)
r1(x) = q3(x)r2(x) + r3(x)
v3(x) = v1(x) - q3(x)v2(x) w3(x) = w1(x) - q3(x)w2(x)
r3(x) = a(x)v3(x) + b(x)w3(x)
f
rn(x) = rn - 2(x) mod rn - 1(x) qn(x) = quotient of rn - 2(x)/rn - 2(x)
rn - 2(x) = qn(x)rn - 1(x) + rn(x)
vn(x) = vn - 2(x) - qn(x)vn - 1(x) wn(x) = wn - 2(x) - qn(x)wn - 1(x)
rn(x) = a(x)vn(x) + b(x)wn(x)
rn + 1(x) = rn - 1(x) mod rn(x) = 0 qn + 1(x) = quotient of rn - 1(x)/rn(x)
rn - 1(x) = qn + 1(x)rn(x) + 0
d(x) = gcd(a(x), b(x)) = rn(x) v(x) = vn(x); w(x) = wn(x)
Table 5.4 shows the calculation of the multiplicative inverse of (x7 + x + 1) mod (x8 + x4 + x3 + x + 1). The result is that (x7 + x + 1)-1 = (x7). That is, (x7 + x + 1)(x7) K 1(mod (x8 + x4 + x3 + x + 1)).
Computational Considerations
A polynomial f(x) in GF(2n)
f(x) = an - 1x n - 1 + an - 2xn - 2 + g + a1x + a0 = a
n - 1
i = 0 aix
i
can be uniquely represented by the sequence of its n binary coefficients (an - 1, an - 2, c , a0). Thus, every polynomial in GF(2
n) can be represented by an
n-bit number.
Hiva-Network.Com
164 CHAPTER 5 / FINITE FIELDS
ADDITION We have seen that addition of polynomials is performed by adding cor- responding coefficients, and, in the case of polynomials over Z2, addition is just the
XOR operation. So, addition of two polynomials in GF(2n) corresponds to a bitwise
XOR operation.
Initialization a(x) = x8 + x4 + x3 + x + 1; v-1(x) = 1; w-1(x) = 0 b(x) = x7 + x + 1; v0(x) = 0; w0(x) = 1
Iteration 1 q1(x) = x; r1(x) = x 4 + x3 + x2 + 1
v1(x) = 1; w1(x) = x
Iteration 2 q2(x) = x 3 + x2 + 1; r2(x) = x
v2(x) = x 3 + x2 + 1; w2(x) = x4 + x3 + x + 1
Iteration 3 q3(x) = x 3 + x2 + x; r3(x) = 1
v3(x) = x 6 + x2 + x + 1; w3(x) = x7
Iteration 4 q4(x) = x; r4(x) = 0 v4(x) = x
7 + x + 1; w4(x) = x8 + x4 + x3 + x + 1
Result d(x) = r3(x) = gcd(a(x), b(x)) = 1 w(x) = w3(x) = (x
7 + x + 1)-1 mod (x8 + x4 + x3 + x + 1) = x7
Table 5.4 Extended Euclid [(x8 + x4 + x3 + x + 1), (x7 + x + 1)]
Tables 5.2 and 5.3 show the addition and multiplication tables for GF(23) modulo
m(x) = (x3 + x + 1). Table 5.2 uses the binary representation, and Table 5.3 uses the polynomial representation.
Consider the two polynomials in GF(28) from our earlier example:
f(x) = x6 + x4 + x2 + x + 1 and g(x) = x7 + x + 1.
(x6 + x4 + x2 + x + 1) + (x7 + x + 1) = x7 + x6 + x4 + x2 (polynomial notation) (01010111) ⊕ (10000011) = (11010100) (binary notation) {57} ⊕ {83} = {D4} (hexadecimal notation)7
7A basic refresher on number systems (decimal, binary, hexadecimal) can be found at the Computer Science Student Resource Site at WilliamStallings.com/StudentSupport.html. Here each of two groups of 4 bits in a byte is denoted by a single hexadecimal character, and the two characters are enclosed in brackets.
MULTIPLICATION There is no simple XOR operation that will accomplish multi- plication in GF(2n). However, a reasonably straightforward, easily implemented
technique is available. We will discuss the technique with reference to GF(28) using
m(x) = x8 + x4 + x3 + x + 1, which is the finite field used in AES. The technique readily generalizes to GF(2n).
The technique is based on the observation that
x8 mod m(x) = [m(x) - x8] = (x4 + x3 + x + 1) (5.4)
5.6 / FINITE FIELDS OF THE FORM GF(2n) 165
A moment’s thought should convince you that Equation (5.4) is true; if you
are not sure, divide it out. In general, in GF(2n) with an nth-degree polynomial p(x), we have xn mod p(x) = [p(x) - xn].
Now, consider a polynomial in GF(28), which has the form
f(x) = b7x 7 + b6x6 + b5x5 + b4x4 + b3x3 + b2x2 + b1x + b0. If we multiply by x,
we have
x * f(x) = (b7x8 + b6x7 + b5x6 + b4x5 + b3x4
+ b2x3 + b1x2 + b0x) mod m(x) (5.5)
If b7 = 0, then the result is a polynomial of degree less than 8, which is already in reduced form, and no further computation is necessary. If b7 = 1, then reduction modulo m(x) is achieved using Equation (5.4):
x * f(x) = (b6x7 + b5x6 + b4x5 + b3x4 + b2x3 + b1x2 + b0x) + (x4 + x3 + x + 1)
It follows that multiplication by x (i.e., 00000010) can be implemented as a 1-bit left shift followed by a conditional bitwise XOR with (00011011), which represents
(x4 + x3 + x + 1). To summarize,
x * f(x) = b (b6b5b4b3b2b1b00) if b7 = 0 (b6b5b4b3b2b1b00) ⊕ (00011011) if b7 = 1
(5.6)
Multiplication by a higher power of x can be achieved by repeated application of Equation (5.6). By adding intermediate results, multiplication by any constant in
GF(28) can be achieved.
In an earlier example, we showed that for f(x) = x6 + x4 + x2 + x + 1, g(x) = x7 + x + 1, and m(x) = x8 + x4 + x3 + x + 1, we have f(x) * g(x) mod m(x) = x7 + x6 + 1. Redoing this in binary arithmetic, we need to compute (01010111) * (10000011). First, we determine the results of multiplication by powers of x:
(01010111) * (00000010) = (10101110) (01010111) * (00000100) = (01011100) ⊕ (00011011) = (01000111) (01010111) * (00001000) = (10001110) (01010111) * (00010000) = (00011100) ⊕ (00011011) = (00000111) (01010111) * (00100000) = (00001110) (01010111) * (01000000) = (00011100) (01010111) * (10000000) = (00111000)
So,
(01010111) * (10000011) = (01010111) * [(00000001) ⊕ (00000010) ⊕ (10000000)] = (01010111) ⊕ (10101110) ⊕ (00111000) = (11000001)
which is equivalent to x7 + x6 + 1.
166 CHAPTER 5 / FINITE FIELDS
Using a Generator
An equivalent technique for defining a finite field of the form GF(2n), using the
same irreducible polynomial, is sometimes more convenient. To begin, we need two
definitions: A generator g of a finite field F of order q (contains q elements) is an element whose first q - 1 powers generate all the nonzero elements of F. That is, the elements of F consist of 0, g0, g1, c , gq - 2. Consider a field F defined by a polynomial f(x). An element b contained in F is called a root of the polynomial if f(b) = 0. Finally, it can be shown that a root g of an irreducible polynomial is a gen- erator of the finite field defined on that polynomial.
Power Representation
Polynomial Representation
Binary Representation
Decimal (Hex) Representation
0 0 000 0
g0(= g7) 1 001 1
g1 g 010 2
g2 g2 100 4
g3 g + 1 011 3
g4 g2 + g 110 6
g5 g2 + g + 1 111 7
g6 g2 + 1 101 5
Table 5.5 Generator for GF(23) using x3 + x + 1
Let us consider the finite field GF(23), defined over the irreducible poly-
nomial x3 + x + 1, discussed previously. Thus, the generator g must satisfy f(g) = g3 + g + 1 = 0. Keep in mind, as discussed previously, that we need not find a numerical solution to this equality. Rather, we deal with polynomial arith-
metic in which arithmetic on the coefficients is performed modulo 2. Therefore,
the solution to the preceding equality is g3 = - g - 1 = g + 1. We now show that g in fact generates all of the polynomials of degree less than 3. We have the
following.
g4 = g(g3) = g(g + 1) = g2 + g g5 = g(g4) = g(g2 + g) = g3 + g2 = g2 + g + 1 g6 = g(g5) = g(g2 + g + 1) = g3 + g2 + g = g2 + g + g + 1 = g2 + 1 g7 = g(g6) = g(g2 + 1) = g3 + g = g + g + 1 = 1 = g0
We see that the powers of g generate all the nonzero polynomials in GF(23). Also, it should be clear that gk = gk mod7 for any integer k. Table 5.5 shows the power representation, as well as the polynomial and binary representations.
5.6 / FINITE FIELDS OF THE FORM GF(2n) 167
In general, for GF(2n) with irreducible polynomial f(x), determine gn = f(g) - gn. Then calculate all of the powers of g from gn + 1 through g2
n - 2.
The elements of the field correspond to the powers of g from g0 through g2 n - 2
plus the value 0. For multiplication of two elements in the field, use the equality
gk = gk mod(2 n - 1) for any integer k.
Summary
In this section, we have shown how to construct a finite field of order 2n. Specifically,
we defined GF(2n) with the following properties.
1. GF(2n) consists of 2n elements.
2. The binary operations + and * are defined over the set. The operations of addition, subtraction, multiplication, and division can be performed with-
out leaving the set. Each element of the set other than 0 has a multiplicative
inverse.
We have shown that the elements of GF(2n) can be defined as the set of all
polynomials of degree n - 1 or less with binary coefficients. Each such polynomial can be represented by a unique n-bit value. Arithmetic is defined as polynomial arithmetic modulo some irreducible polynomial of degree n. We have also seen that an equivalent definition of a finite field GF(2n) makes use of a generator and that
arithmetic is defined using powers of the generator.
This power representation makes multiplication easy. To multiply in the
power notation, add exponents modulo 7. For example, g4 * g6 = g(10 mod 7) = g3 = g + 1. The same result is achieved using polynomial arithmetic: We have g4 = g2 + g and g6 = g2 + 1. Then, (g2 + g) * (g2 + 1) = g4 + g3 + g2 + g. Next, we need to determine (g4 + g3 + g2 + 1) mod (g3 + g + 1) by division:
g + 1 g3 + g + 1�g4 + g3 + g2 + g
g4 + g2 + g g3
g3 + g + 1 g + 1
We get a result of g + 1, which agrees with the result obtained using the power representation.
Table 5.6 shows the addition and multiplication tables for GF(23) using
the power representation. Note that this yields the identical results to the
polynomial representation (Table 5.3) with some of the rows and columns
i nterchanged.
168 CHAPTER 5 / FINITE FIELDS
0 0 0
0 0 1
0 1 0
1 0 0
0 1 1
1 1 0
1 1 1
1 0 1
+ 0
1 G
g 2
g 3
g 4
g 5
g 6
0 0 0
0 0
1 G
g 2
g +
1 g
2 +
g g
2 +
g +
1 g
2 +
1
0 0 1
1 1
0 g
+ 1
g 2
+ 1
g g
2 +
g +
1 g
2 +
g g
2
0 1 0
g g
g +
1 0
g 2
+ g
1 g
2 g
2 +
1 g
2 +
g +
1
1 0 0
g 2
g 2
g 2
+ 1
g 2
+ g
0 g
2 +
g +
1 g
g +
1 1
0 1 1
g 3
g +
1 g
1 g
2 +
g +
1 0
g 2
+ 1
g 2
g 2
+ g
1 1 0
g 4
g 2
+ g
g 2
+ g
+ 1
g 2
g g
2 +
1 0
1 g
+ 1
1 1 1
g 5
g 2
+ g
+ 1
g 2
+ g
g 2
+ 1
g +
1 g
2 1
0 g
1 0 1
g 6
g 2
+ 1
g 2
g 2
+ g
+ 1
1 g
2 +
g g
+ 1
g 0
(a )
A d
d it
io n
0 0 0
0 0 1
0 1 0
1 0 0
0 1 1
1 1 0
1 1 1
1 0 1
* 0
1 G
g 2
g 3
g 4
g 5
g 6
0 0 0
0 0
0 0
0 0
0 0
0
0 0 1
1 0
1 G
g 2
g +
1 g
2 +
g g
2 +
g +
1 g
2 +
1
0 1 0
g 0
g g
2 g
+ 1
g 2
+ g
g 2
+ g
+ 1
g 2
+ 1
1
1 0 0
g 2
0 g
2 g
+ 1
g 2
+ g
g 2
+ g
+ 1
g 2
+ 1
1 g
0 1 1
g 3
0 g
+ 1
g 2
+ g
g 2
+ g
+ 1
g 2
+ 1
1 g
g 2
1 1 0
g 4
0 g
2 +
g g
2 +
g +
1 g
2 +
1 1
g g
2 g
+ 1
1 1 1
g 5
0 g
2 +
g +
1 g
2 +
1 1
g g
2 g
+ 1
g 2
+ g
1 0 1
g 6
0 g
2 +
1 1
g g
2 g
+ 1
g 2
+ g
g 2
+ g
+ 1
(b )
M u
lt ip
li c a ti
o n
T ab
le 5
.6
G F
(2 3 )
A ri
th m
e ti
c U
si n
g G
e n
e ra
to r
fo r
th e P
o ly
n o
m ia
l (x
3 +
x +
1 )
5.7 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 169
5.7 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
Key Terms
abelian group
associative
coefficient set
commutative
commutative ring
cyclic group
divisor
Euclidean algorithm
field
finite field
finite group
generator
greatest common divisor
group
identity element
infinite field
infinite group
integral domain
inverse element
irreducible polynomial
modular arithmetic
modular polynomial
arithmetic
monic polynomial
order
polynomial
polynomial arithmetic
polynomial ring
prime number
prime polynomial
relatively prime
residue
ring
Review Questions
5.1 Briefly define a group. 5.2 Briefly define a ring. 5.3 Briefly define a field. 5.4 List three classes of polynomial arithmetic.
Problems
5.1 For the group Sn of all permutations of n distinct symbols, a. what is the number of elements in Sn? b. show that Sn is not abelian for n 7 2.
5.2 Does the set of residue classes (mod3) form a group a. with respect to modular addition? b. with respect to modular multiplication?
5.3 Let S = {0, a, b, c}. The addition and multiplication on the set S is defined in the following tables:
+ 0 a B C 0 0 a B C
A a 0 c B
B b c 0 A
C c b a 0
* 0 a b c 0 0 0 0 0
a 0 a b c
b 0 a b c
c 0 0 0 0
Is S a noncommutative ring? Justify your answer.
5.4 Develop a set of tables similar to Table 5.1 for GF(5). 5.5 Demonstrate that the set of polynomials whose coefficients form a field is a ring. 5.6 Demonstrate whether each of these statements is true or false for polynomials over a
field.
170 CHAPTER 5 / FINITE FIELDS
a. The product of monic polynomials is monic. b. The product of polynomials of degrees m and n has degree m + n. c. The sum of polynomials of degrees m and n has degree max [m, n].
5.7 For polynomial arithmetic with coefficients in Z1 1 , perform the following calculations. a. (x 2 + 2 x + 9 )(x 3 + 1 1 x 2 + x + 7 ) b. (8 x 2 + 3 x + 2 )(5 x 2 + 6 )
5.8 Determine which of the following polynomials are reducible over GF(2). a. x 2 + 1 b. x 2 + x + 1 c. x 4 + x + 1
5.9 Determine the gcd of the following pairs of polynomials. a. (x3 + 1) and (x2 + x + 1) over GF(2) b. (x3 + x + 1) and (x2 + 1) over GF(3) c. (x3 - 2x + 1) and (x2 - x - 2) over GF(5) d. (x4 + 8x3 + 7x + 8) and (2x3 + 9x2 + 10x + 1) over GF(11)
5.10 Develop a set of tables similar to Table 5.3 for GF(3) with m(x) = x2 + x + 1. 5.11 Determine the multiplicative inverse of x 2 + 1 in GF(2 3 ) with m (x ) = x 3 + x - 1 . 5.12 Develop a table similar to Table 5.5 for GF(2 5 ) with m (x ) = x 5 + x 4 + x 3 + x + 1 .
Programming Problems
5.13 Write a simple four-function calculator in GF(24). You may use table lookups for the multiplicative inverses.
5.14 Write a simple four-function calculator in GF(28). You should compute the multiplica- tive inverses on the fly.
171
6.1 Finite Field Arithmetic
6.2 AES Structure
General Structure
Detailed Structure
6.3 AES Transformation Functions
Substitute Bytes Transformation
ShiftRows Transformation
MixColumns Transformation
AddRoundKey Transformation
6.4 AES Key Expansion
Key Expansion Algorithm
Rationale
6.5 An AES Example
Results
Avalanche Effect
6.6 AES Implementation
Equivalent Inverse Cipher
Implementation Aspects
6.7 Key Terms, Review Questions, and Problems
Appendix 6A Polynomials with Coefficients in GF(28)
CHAPTER
Advanced Encryption Standard
172 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD
The Advanced Encryption Standard (AES) was published by the National Institute of
Standards and Technology (NIST) in 2001. AES is a symmetric block cipher that is
intended to replace DES as the approved standard for a wide range of applications.
Compared to public-key ciphers such as RSA, the structure of AES and most symmet-
ric ciphers is quite complex and cannot be explained as easily as many other
cryptographic algorithms. Accordingly, the reader may wish to begin with a simplified
version of AES, which is described in Appendix I. This version allows the reader to
perform encryption and decryption by hand and gain a good understanding of the
working of the algorithm details. Classroom experience indicates that a study of this
simplified version enhances understanding of AES.1 One possible approach is to read
the chapter first, then carefully read Appendix I, and then re-read the main body
of the chapter.
Appendix H looks at the evaluation criteria used by NIST to select from among
the candidates for AES, plus the rationale for picking Rijndael, which was the winning
candidate. This material is useful in understanding not just the AES design but also the
criteria by which to judge any symmetric encryption algorithm.
6.1 FINITE FIELD ARITHMETIC
In AES, all operations are performed on 8-bit bytes. In particular, the arithmetic
operations of addition, multiplication, and division are performed over the finite
field GF(28). Section 5.6 discusses such operations in some detail. For the reader
who has not studied Chapter 5, and as a quick review for those who have, this sec-
tion summarizes the important concepts.
In essence, a field is a set in which we can do addition, subtraction, multiplica-
tion, and division without leaving the set. Division is defined with the following rule:
a/b = a(b-1). An example of a finite field (one with a finite number of elements) is the set Zp consisting of all the integers {0, 1, c , p - 1}, where p is a prime num- ber and in which arithmetic is carried out modulo p.
1However, you may safely skip Appendix I, at least on a first reading. If you get lost or bogged down in the details of AES, then you can go back and start with simplified AES.
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Present an overview of the general structure of Advanced Encryption Standard (AES).
◆ Understand the four transformations used in AES.
◆ Explain the AES key expansion algorithm.
◆ Understand the use of polynomials with coefficients in GF(28).
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6.1 / FINITE FIELD ARITHMETIC 173
Virtually all encryption algorithms, both conventional and public-key, involve
arithmetic operations on integers. If one of the operations used in the algorithm
is division, then we need to work in arithmetic defined over a field; this is because
division requires that each nonzero element have a multiplicative inverse. For con-
venience and for implementation efficiency, we would also like to work with inte-
gers that fit exactly into a given number of bits, with no wasted bit patterns. That is,
we wish to work with integers in the range 0 through 2n - 1, which fit into an n-bit word. Unfortunately, the set of such integers, Z2n, using modular arithmetic, is not a field. For example, the integer 2 has no multiplicative inverse in Z2n, that is, there is no integer b, such that 2b mod 2n = 1.
There is a way of defining a finite field containing 2n elements; such a field is
referred to as GF(2n). Consider the set, S, of all polynomials of degree n - 1 or less with binary coefficients. Thus, each polynomial has the form
f(x) = an - 1x n - 1 + an - 2xn - 2 + g + a1x + a0 = a
n - 1
i = 0 aix
i
where each ai takes on the value 0 or 1. There are a total of 2 n different polynomials
in S. For n = 3, the 23 = 8 polynomials in the set are
0 x x2 x2 + x 1 x + 1 x2 + 1 x2 + x + 1
With the appropriate definition of arithmetic operations, each such set S is a finite field. The definition consists of the following elements.
1. Arithmetic follows the ordinary rules of polynomial arithmetic using the basic rules of algebra with the following two refinements.
2. Arithmetic on the coefficients is performed modulo 2. This is the same as the XOR operation.
3. If multiplication results in a polynomial of degree greater than n - 1, then the polynomial is reduced modulo some irreducible polynomial m(x) of degree n. That is, we divide by m(x) and keep the remainder. For a polynomial f(x), the remainder is expressed as r(x) = f(x) mod m(x). A polynomial m(x) is called irreducible if and only if m(x) cannot be expressed as a product of two polynomials, both of degree lower than that of m(x).
For example, to construct the finite field GF(23), we need to choose an irre-
ducible polynomial of degree 3. There are only two such polynomials: (x3 + x2 + 1) and (x3 + x + 1). Addition is equivalent to taking the XOR of like terms. Thus, (x + 1) + x = 1.
A polynomial in GF(2n) can be uniquely represented by its n binary coeffi cients (an - 1an - 2 c a0). Therefore, every polynomial in GF(2
n) can be represented by
an n-bit number. Addition is performed by taking the bitwise XOR of the two n-bit elements. There is no simple XOR operation that will accomplish multiplication in
GF(2n). However, a reasonably straightforward, easily implemented, technique is
available. In essence, it can be shown that multiplication of a number in GF(2n) by
174 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD
2 consists of a left shift followed by a conditional XOR with a constant. Multiplication
by larger numbers can be achieved by repeated application of this rule.
For example, AES uses arithmetic in the finite field GF(28) with the irreducible
polynomial m(x) = x8 + x4 + x3 + x + 1. Consider two elements A = (a7a6 c a1a0) and B = (b7b6 c b1b0). The sum A + B = (c7c6 c c1c0), where ci = ai ⊕ bi. The multiplication {02} # A equals (a6 c a1a00) if a7 = 0 and equals (a6 c a1a00) ⊕ (00011011) if a7 = 1.
2
To summarize, AES operates on 8-bit bytes. Addition of two bytes is defined
as the bitwise XOR operation. Multiplication of two bytes is defined as multiplica-
tion in the finite field GF(28), with the irreducible polynomial3 m(x) = x8 + x4 + x3 + x + 1. The developers of Rijndael give as their motivation for selecting this one of the 30 possible irreducible polynomials of degree 8 that it is the first one on the list
given in [LIDL94].
6.2 AES STRUCTURE
General Structure
Figure 6.1 shows the overall structure of the AES encryption process. The cipher
takes a plaintext block size of 128 bits, or 16 bytes. The key length can be 16, 24, or
32 bytes (128, 192, or 256 bits). The algorithm is referred to as AES-128, AES-192,
or AES-256, depending on the key length.
The input to the encryption and decryption algorithms is a single 128-bit block.
In FIPS PUB 197, this block is depicted as a 4 * 4 square matrix of bytes. This block is copied into the State array, which is modified at each stage of encryption or decryption. After the final stage, State is copied to an output matrix. These opera- tions are depicted in Figure 6.2a. Similarly, the key is depicted as a square matrix of
bytes. This key is then expanded into an array of key schedule words. Figure 6.2b
shows the expansion for the 128-bit key. Each word is four bytes, and the total key
schedule is 44 words for the 128-bit key. Note that the ordering of bytes within a ma-
trix is by column. So, for example, the first four bytes of a 128-bit plaintext input to
the encryption cipher occupy the first column of the in matrix, the second four bytes occupy the second column, and so on. Similarly, the first four bytes of the expanded
key, which form a word, occupy the first column of the w matrix. The cipher consists of N rounds, where the number of rounds depends on the
key length: 10 rounds for a 16-byte key, 12 rounds for a 24-byte key, and 14 rounds
for a 32-byte key (Table 6.1). The first N - 1 rounds consist of four distinct trans- formation functions: SubBytes, ShiftRows, MixColumns, and AddRoundKey,
which are described subsequently. The final round contains only three transforma-
tions, and there is a initial single transformation (AddRoundKey) before the first
round, which can be considered Round 0. Each transformation takes one or more
2In FIPS PUB 197, a hexadecimal number is indicated by enclosing it in curly brackets. We use that convention in this chapter. 3In the remainder of this discussion, references to GF(28) refer to the finite field defined with this polynomial.
6.2 / AES STRUCTURE 175
Figure 6.1 AES Encryption Process
Initial transformation
K ey
e xp
an si
on
Plaintext—16 bytes (128 bits) Key—M bytes
Key (M bytes)Round 0 key
(16 bytes)
Round 1 key (16 bytes)
Round N – 1 key (16 bytes)
Round N key (16 bytes)
Cipehertext—16 bytes (128 bits)
No. of rounds
10 16
Key Length (bytes)
Input state (16 bytes)
State after initial
transformation (16 bytes)
Final state (16 bytes)
Round N – 1 output state (16 bytes)
Round 1 output state (16 bytes)
Round 1 (4 transformations)
Round N – 1 (4 transformations)
Round N (3 transformations)
12 24
14 32
4 * 4 matrices as input and produces a 4 * 4 matrix as output. Figure 6.1 shows that the output of each round is a 4 * 4 matrix, with the output of the final round being the ciphertext. Also, the key expansion function generates N + 1 round keys, each of which is a distinct 4 * 4 matrix. Each round key serves as one of the inputs to the AddRoundKey transformation in each round.
F ig
u re
6 .2
A
E S
D a ta
S tr
u c tu
re s
in 0
in 4
in 8
in 12
in 1
in 5
in 9
in 13
in 2
in 6
in 10
in 14
in 3
in 7
in 11
in 15
k 0
w 0
w 1
w 2
w 43
w 42
k 4 k 8
k 1 2
k 1 k 5
k 9 k 1
3
k 2 k 6
k 1 0
k 1 4
k 3 k 7
k 1 1
k 1 5
ou t 0
ou t 4
ou t 8
ou t 1
2
ou t 1
ou t 5
ou t 9
ou t 1
3
ou t 2
ou t 6
ou t 1
0 ou
t 1 4
ou t 3
ou t 7
ou t 1
1 ou
t 1 5
s 0 ,0
s 1 ,0
s 2 ,0
s 3 ,0
s 0 ,1
s 1 ,1
s 2 ,1
s 3 ,1
s 0 ,2
s 1 ,2
s 2 ,2
s 3 ,2
s 0 ,3
s 1 ,3
s 2 ,3
s 3 ,3
s 0 ,0
s 1 ,0
s 2 ,0
s 3 ,0
s 0 ,1
s 1 ,1
s 2 ,1
s 3 ,1
s 0 ,2
s 1 ,2
s 2 ,2
s 3 ,2
s 0 ,3
s 1 ,3
s 2 ,3
s 3 ,3
(a )
In pu
t, s
ta te
a rr
ay , a
nd o
ut pu
t
(b )
K ey
a nd
e xp
an de
d ke
y
176
6.2 / AES STRUCTURE 177
Key Size (words/bytes/bits) 4/16/128 6/24/192 8/32/256 Plaintext Block Size (words/bytes/bits) 4/16/128 4/16/128 4/16/128 Number of Rounds 10 12 14 Round Key Size (words/bytes/bits) 4/16/128 4/16/128 4/16/128 Expanded Key Size (words/bytes) 44/176 52/208 60/240
Table 6.1 AES Parameters
Detailed Structure
Figure 6.3 shows the AES cipher in more detail, indicating the sequence of transfor-
mations in each round and showing the corresponding decryption function. As was
done in Chapter 4, we show encryption proceeding down the page and decryption
proceeding up the page.
Before delving into details, we can make several comments about the overall
AES structure.
1. One noteworthy feature of this structure is that it is not a Feistel structure. Recall that, in the classic Feistel structure, half of the data block is used to
modify the other half of the data block and then the halves are swapped. AES
instead processes the entire data block as a single matrix during each round
using substitutions and permutation.
2. The key that is provided as input is expanded into an array of forty-four 32-bit words, w[i]. Four distinct words (128 bits) serve as a round key for each round; these are indicated in Figure 6.3.
3. Four different stages are used, one of permutation and three of substitution:
■ Substitute bytes: Uses an S-box to perform a byte-by-byte substitution of the block.
■ ShiftRows: A simple permutation.
■ MixColumns: A substitution that makes use of arithmetic over GF(28).
■ AddRoundKey: A simple bitwise XOR of the current block with a portion of the expanded key.
4. The structure is quite simple. For both encryption and decryption, the cipher begins with an AddRoundKey stage, followed by nine rounds that each in-
cludes all four stages, followed by a tenth round of three stages. Figure 6.4
depicts the structure of a full encryption round.
5. Only the AddRoundKey stage makes use of the key. For this reason, the cipher begins and ends with an AddRoundKey stage. Any other stage, applied at the
beginning or end, is reversible without knowledge of the key and so would add
no security.
6. The AddRoundKey stage is, in effect, a form of Vernam cipher and by itself would not be formidable. The other three stages together provide confusion,
diffusion, and nonlinearity, but by themselves would provide no security be-
cause they do not use the key. We can view the cipher as alternating operations
of XOR encryption (AddRoundKey) of a block, followed by scrambling of the
178 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD
Figure 6.3 AES Encryption and Decryption
Add round key
w[4, 7]
Plaintext (16 bytes)
Plaintext (16 bytes)
Substitute bytes
Expand key
Shift rows
Mix columnsR ou
nd 1
R ou
nd 9
R ou
nd 1
0
Add round key
Substitute bytes
Shift rows
Mix columns
Add round key
Substitute bytes
Shift rows
Add round key
Ciphertext (16 bytes)
(a) Encryption
Key (16 bytes)
Add round key
Inverse sub bytes
Inverse shift rows
Inverse mix cols
R ou
nd 1
0 R
ou nd
9 R
ou nd
1
Add round key
Inverse sub bytes
Inverse shift rows
Inverse mix cols
Add round key
Inverse sub bytes
Inverse shift rows
Add round key
Ciphertext (16 bytes)
(b) Decryption
w[36, 39]
w[40, 43]
w[0, 3]
block (the other three stages), followed by XOR encryption, and so on. This
scheme is both efficient and highly secure.
7. Each stage is easily reversible. For the Substitute Byte, ShiftRows, and MixColumns stages, an inverse function is used in the decryption algorithm.
For the AddRoundKey stage, the inverse is achieved by XORing the same
round key to the block, using the result that A ⊕ B ⊕ B = A. 8. As with most block ciphers, the decryption algorithm makes use of the
expanded key in reverse order. However, the decryption algorithm is not
6.3 / AES TRANSFORMATION FUNCTIONS 179
Figure 6.4 AES Encryption Round
SSubBytes
State
State
State
State
State
ShiftRows
MixColumns
AddRoundKey
S S S S S S S S S S S S S S S
M M M M
r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15
identical to the encryption algorithm. This is a consequence of the particular
structure of AES.
9. Once it is established that all four stages are reversible, it is easy to verify that decryption does recover the plaintext. Figure 6.3 lays out encryption
and decryption going in opposite vertical directions. At each horizontal point
(e.g., the dashed line in the figure), State is the same for both encryption and decryption.
10. The final round of both encryption and decryption consists of only three stages. Again, this is a consequence of the particular structure of AES and is required
to make the cipher reversible.
6.3 AES TRANSFORMATION FUNCTIONS
We now turn to a discussion of each of the four transformations used in AES. For
each stage, we describe the forward (encryption) algorithm, the inverse ( decryption)
algorithm, and the rationale for the stage.
180 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD
Substitute Bytes Transformation
FORWARD AND INVERSE TRANSFORMATIONS The forward substitute byte transformation, called SubBytes, is a simple table lookup (Figure 6.5a). AES defines a 16 * 16 matrix of byte values, called an S-box (Table 6.2a), that con- tains a permutation of all possible 256 8-bit values. Each individual byte of State is mapped into a new byte in the following way: The leftmost 4 bits of the byte are
used as a row value and the rightmost 4 bits are used as a column value. These row
and column values serve as indexes into the S-box to select a unique 8-bit output
value. For example, the hexadecimal value {95} references row 9, column 5 of the
S-box, which contains the value {2A}. Accordingly, the value {95} is mapped into
the value {2A}.
Figure 6.5 AES Byte-Level Operations
s0,0 s0,1 s0,2 s0,3
s1,0 s1,2 s1,3
s2,0 s2,1 s2,2 s2,3
s3,0 s3,1 s3,2 s3,3
s0,0 s0,1 s0,2 s0,3
s1,0 s1,2 s1,3
s2,0 s2,1 s2,2 s2,3
s3,0 s3,1 s3,2 s3,3
(b) Add round key transformation
(a) Substitute byte transformation
S-box
x
y
¿ ¿ ¿ ¿
¿ ¿¿¿
s1,1
s0,0
wi wi+2 wi+3
s0,2 s0,3
s1,0 s1,2 s1,3 =
s2,0 s2,2 s2,3
s3,0 s3,2 s3,3
s1,1
s0,0 s0,2 s0,3
s1,0 s1,2 s1,3
s2,0 s2,2 s2,3
s3,0 s3,2 s3,3
s1,1
s0,1
s2,1
s3,1
wi+1
s0,1
s2,1
s3,1
s1,1
¿¿¿
¿ ¿ ¿ ¿
¿
¿
¿
¿ ¿
¿ ¿
¿ ¿ ¿ ¿
¿ ¿ ¿
¿ ¿ ¿
6.3 / AES TRANSFORMATION FUNCTIONS 181
y 0 1 2 3 4 5 6 7 8 9 A B C D E F
0 63 7C 77 7B F2 6B 6F C5 30 01 67 2B FE D7 AB 76
1 CA 82 C9 7D FA 59 47 F0 AD D4 A2 AF 9C A4 72 C0
2 B7 FD 93 26 36 3F F7 CC 34 A5 E5 F1 71 D8 31 15
3 04 C7 23 C3 18 96 05 9A 07 12 80 E2 EB 27 B2 75
4 09 83 2C 1A 1B 6E 5A A0 52 3B D6 B3 29 E3 2F 84
5 53 D1 00 ED 20 FC B1 5B 6A CB BE 39 4A 4C 58 CF
6 D0 EF AA FB 43 4D 33 85 45 F9 02 7F 50 3C 9F A8
x 7 51 A3 40 8F 92 9D 38 F5 BC B6 DA 21 10 FF F3 D2
8 CD 0C 13 EC 5F 97 44 17 C4 A7 7E 3D 64 5D 19 73
9 60 81 4F DC 22 2A 90 88 46 EE B8 14 DE 5E 0B DB
A E0 32 3A 0A 49 06 24 5C C2 D3 AC 62 91 95 E4 79
B E7 C8 37 6D 8D D5 4E A9 6C 56 F4 EA 65 7A AE 08
C BA 78 25 2E 1C A6 B4 C6 E8 DD 74 1F 4B BD 8B 8A
D 70 3E B5 66 48 03 F6 0E 61 35 57 B9 86 C1 1D 9E
E E1 F8 98 11 69 D9 8E 94 9B 1E 87 E9 CE 55 28 DF
F 8C A1 89 0D BF E6 42 68 41 99 2D 0F B0 54 BB 16
(a) S-box
y 0 1 2 3 4 5 6 7 8 9 A B C D E F
0 52 09 6A D5 30 36 A5 38 BF 40 A3 9E 81 F3 D7 FB
1 7C E3 39 82 9B 2F FF 87 34 8E 43 44 C4 DE E9 CB
2 54 7B 94 32 A6 C2 23 3D EE 4C 95 0B 42 FA C3 4E
3 08 2E A1 66 28 D9 24 B2 76 5B A2 49 6D 8B D1 25
4 72 F8 F6 64 86 68 98 16 D4 A4 5C CC 5D 65 B6 92
5 6C 70 48 50 FD ED B9 DA 5E 15 46 57 A7 8D 9D 84
6 90 D8 AB 00 8C BC D3 0A F7 E4 58 05 B8 B3 45 06
x 7 D0 2C 1E 8F CA 3F 0F 02 C1 AF BD 03 01 13 8A 6B
8 3A 91 11 41 4F 67 DC EA 97 F2 CF CE F0 B4 E6 73
9 96 AC 74 22 E7 AD 35 85 E2 F9 37 E8 1C 75 DF 6E
A 47 F1 1A 71 1D 29 C5 89 6F B7 62 0E AA 18 BE 1B
B FC 56 3E 4B C6 D2 79 20 9A DB C0 FE 78 CD 5A F4
C 1F DD A8 33 88 07 C7 31 B1 12 10 59 27 80 EC 5F
D 60 51 7F A9 19 B5 4A 0D 2D E5 7A 9F 93 C9 9C EF
E A0 E0 3B 4D AE 2A F5 B0 C8 EB BB 3C 83 53 99 61
F 17 2B 04 7E BA 77 D6 26 E1 69 14 63 55 21 0C 7D
(b) Inverse S-box
Table 6.2 AES S-Boxes
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182 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD
Here is an example of the SubBytes transformation:
EA 04 65 85 87 F2 4D 97
83 45 5D 96 EC 6E 4C 90
5C 33 98 B0 S 4A C3 46 E7 F0 2D AD C5 8C D8 95 A6
The S-box is constructed in the following fashion (Figure 6.6a).
Figure 6.6 Constuction of S-Box and IS-Box
b0 b1 b2 b3 b4 b5 b6 b7
=
1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1
b0 b1 b2 b3 b4 b5 b6 b7
+
1 1 0 0 0 1 1 0
Inverse in GF(28)
Byte to bit column vector
Bit column vector to byte
Byte at row y, column x
initialized to yx yx
S(yx)
(a) Calculation of byte at row y, column x of S-box
(a) Calculation of byte at row y, column x of IS-box
Inverse in GF(28)
Byte to bit column vector
Bit column vector to byte
Byte at row y, column x
initialized to yx yx
b0¿
b¿
b¿ b ¿
1
2
3
b4 b5 b6 b7
=
0 0 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0
b0 b1 b2 b3 b4 b5 b6 b7
+
1 0 1 0 0 0 0 0
IS(yx)
¿
¿ ¿
¿
¿
¿
¿
¿
¿
¿
¿ ¿
6.3 / AES TRANSFORMATION FUNCTIONS 183
1. Initialize the S-box with the byte values in ascending sequence row by row. The first row contains {00}, {01}, {02}, c , {0F}; the second row contains {10}, {11}, etc.; and so on. Thus, the value of the byte at row y, column x is {yx}.
2. Map each byte in the S-box to its multiplicative inverse in the finite field GF(28); the value {00} is mapped to itself.
3. Consider that each byte in the S-box consists of 8 bits labeled (b7, b6, b5, b4, b3, b2, b1, b0). Apply the following transformation to each bit of each byte in the S-box:
bi = = bi ⊕ b(i + 4) mod 8 ⊕ b(i + 5) mod 8 ⊕ b(i + 6) mod 8 ⊕ b(i + 7) mod 8 ⊕ ci (6.1)
where ci is the ith bit of byte c with the value {63}; that is, (c7c6c5c4c3c2c1c0) = (01100011). The prime (′) indicates that the variable is to be updated by the value on the right. The AES standard depicts this transfor-
mation in matrix form as follows.
H b0 =
b1 =
b2 =
b3 =
b4 =
b5 =
b6 =
b7 =
X = H 1 0 0 0 1 1 1 1
1 1 0 0 0 1 1 1
1 1 1 0 0 0 1 1
1 1 1 1 0 0 0 1
1 1 1 1 1 0 0 0
0 1 1 1 1 1 0 0
0 0 1 1 1 1 1 0
0 0 0 1 1 1 1 1
X H b0 b1 b2 b3 b4 b5 b6 b7
X + H 1
1
0
0
0
1
1
0
X (6.2) Equation (6.2) has to be interpreted carefully. In ordinary matrix multiplica-
tion,4 each element in the product matrix is the sum of products of the elements of
one row and one column. In this case, each element in the product matrix is the
bitwise XOR of products of elements of one row and one column. Furthermore, the
final addition shown in Equation (6.2) is a bitwise XOR. Recall from Section 5.6
that the bitwise XOR is addition in GF(28).
As an example, consider the input value {95}. The multiplicative inverse in
GF(28) is {95}-1 = {8A}, which is 10001010 in binary. Using Equation (6.2),
H 1 0 0 0 1 1 1 1
1 1 0 0 0 1 1 1
1 1 1 0 0 0 1 1
1 1 1 1 0 0 0 1
1 1 1 1 1 0 0 0
0 1 1 1 1 1 0 0
0 0 1 1 1 1 1 0
0 0 0 1 1 1 1 1
X H 0
1
0
1
0
0
0
1
X ⊕ H 1
1
0
0
0
1
1
0
X = H 1
0
0
1
0
0
1
0
X ⊕ H 1
1
0
0
0
1
1
0
X = H 0
1
0
1
0
1
0
0
X
4For a brief review of the rules of matrix and vector multiplication, refer to Appendix E.
184 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD
The result is {2A}, which should appear in row {09} column {05} of the S-box.
This is verified by checking Table 6.2a.
The inverse substitute byte transformation, called InvSubBytes, makes use of the inverse S-box shown in Table 6.2b. Note, for example, that the input {2A}
produces the output {95}, and the input {95} to the S-box produces {2A}. The inverse
S-box is constructed (Figure 6.6b) by applying the inverse of the transformation in
Equation (6.1) followed by taking the multiplicative inverse in GF(28). The inverse
transformation is
bi = = b(i + 2) mod 8 ⊕ b(i + 5) mod 8 ⊕ b(i + 7) mod 8 ⊕ di
where byte d = {05}, or 00000101. We can depict this transformation as follows.
H
b0 =
b1 =
b2 =
b3 =
b4 =
b5 =
b6 =
b7 =
X = H 0 0 1 0 0 1 0 1
1 0 0 1 0 0 1 0
0 1 0 0 1 0 0 1
1 0 1 0 0 1 0 0
0 1 0 1 0 0 1 0
0 0 1 0 1 0 0 1
1 0 0 1 0 1 0 0
0 1 0 0 1 0 1 0
X H b0 b1 b2 b3 b4 b5 b6 b7
X + H 1
0
1
0
0
0
0
0
X
To see that InvSubBytes is the inverse of SubBytes, label the matrices in
SubBytes and InvSubBytes as X and Y, respectively, and the vector versions of con- stants c and d as C and D, respectively. For some 8-bit vector B, Equation (6.2) becomes B= = XB ⊕ C. We need to show that Y(XB ⊕ C) ⊕ D = B. To multiply out, we must show YXB ⊕ YC ⊕ D = B. This becomes
H
0 0 1 0 0 1 0 1
1 0 0 1 0 0 1 0
0 1 0 0 1 0 0 1
1 0 1 0 0 1 0 0
0 1 0 1 0 0 1 0
0 0 1 0 1 0 0 1
1 0 0 1 0 1 0 0
0 1 0 0 1 0 1 0
X H 1 0 0 0 1 1 1 1
1 1 0 0 0 1 1 1
1 1 1 0 0 0 1 1
1 1 1 1 0 0 0 1
1 1 1 1 1 0 0 0
0 1 1 1 1 1 0 0
0 0 1 1 1 1 1 0
0 0 0 1 1 1 1 1
X H b0 b1 b2 b3 b4 b5 b6 b7
X ⊕
H 0 0 1 0 0 1 0 1
1 0 0 1 0 0 1 0
0 1 0 0 1 0 0 1
1 0 1 0 0 1 0 0
0 1 0 1 0 0 1 0
0 0 1 0 1 0 0 1
1 0 0 1 0 1 0 0
0 1 0 0 1 0 1 0
X H 1
1
0
0
0
1
1
0
X ⊕ H 1
0
1
0
0
0
0
0
X =
6.3 / AES TRANSFORMATION FUNCTIONS 185
H 1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
X H b0 b1 b2 b3 b4 b5 b6 b7
X ⊕ H 1
0
1
0
0
0
0
0
X ⊕ H 1
0
1
0
0
0
0
0
X = H b0 b1 b2 b3 b4 b5 b6 b7
X We have demonstrated that YX equals the identity matrix, and the YC = D,
so that YC ⊕ D equals the null vector.
RATIONALE The S-box is designed to be resistant to known cryptanalytic attacks. Specifically, the Rijndael developers sought a design that has a low correlation
between input bits and output bits and the property that the output is not a linear
mathematical function of the input [DAEM01]. The nonlinearity is due to the use
of the multiplicative inverse. In addition, the constant in Equation (6.1) was chosen
so that the S-box has no fixed points [S@box(a) = a] and no “opposite fixed points” [S@box(a) = a], where a is the bitwise complement of a.
Of course, the S-box must be invertible, that is, IS@box[S@box(a)] = a. However, the S-box does not self-inverse in the sense that it is not true that
S@box(a) = IS@box(a). For example, S@box({95}) = {2A}, but IS@box({95}) = {AD}.
ShiftRows Transformation
FORWARD AND INVERSE TRANSFORMATIONS The forward shift row transformation, called ShiftRows, is depicted in Figure 6.7a. The first row of State is not altered. For the second row, a 1-byte circular left shift is performed. For the third row, a 2-byte
circular left shift is performed. For the fourth row, a 3-byte circular left shift is per-
formed. The following is an example of ShiftRows.
87 F2 4D 97 87 F2 4D 97
EC 6E 4C 90 6E 4C 90 EC
4A C3 46 E7 S 46 E7 4A C3 8C D8 95 A6 A6 8C D8 95
The inverse shift row transformation, called InvShiftRows, performs the cir- cular shifts in the opposite direction for each of the last three rows, with a 1-byte
circular right shift for the second row, and so on.
RATIONALE The shift row transformation is more substantial than it may first appear. This is because the State, as well as the cipher input and output, is treated as an array of four 4-byte columns. Thus, on encryption, the first 4 bytes
of the plaintext are copied to the first column of State, and so on. Furthermore, as will be seen, the round key is applied to State column by column. Thus, a row shift moves an individual byte from one column to another, which is a linear
186 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD
5We follow the convention of FIPS PUB 197 and use the symbol # to indicate multiplication over the finite field GF(28) and ⊕ to indicate bitwise XOR, which corresponds to addition in GF(28).
distance of a multiple of 4 bytes. Also note that the transformation ensures that
the 4 bytes of one column are spread out to four different columns. Figure 6.4
illustrates the effect.
MixColumns Transformation
FORWARD AND INVERSE TRANSFORMATIONS The forward mix column transformation, called MixColumns, operates on each column individually. Each byte of a column
is mapped into a new value that is a function of all four bytes in that column. The
transformation can be defined by the following matrix multiplication on State (Figure 6.7b):
D 02 03 01 0101 02 03 01 01 01 02 03
03 01 01 02
T D s0,0 s0,1 s0,2 s0,3s1,0 s1,1 s1,2 s1,3 s2,0 s2,1 s2,2 s2,3 s3,0 s3,1 s3,2 s3,3
T = D s0,0= s0,1= s0,2= s0,3=s1,0= s1,1= s1,2= s1,3= s2,0 = s2,1
= s2,2 = s2,3
=
s3,0 = s3,1
= s3,2 = s3,3
=
T (6.3) Each element in the product matrix is the sum of products of elements of one row
and one column. In this case, the individual additions and multiplications5 are
Figure 6.7 AES Row and Column Operations
s0,0 s0,1 s0,2 s0,3
s1,0 s1,1 s1,2 s1,3
s2,0 s2,1 s2,2 s2,3
s3,0 s3,1 s3,2 s3,3
s0,0 s0,1 s0,2 s0,3
s1,0 s1,1 s1,2 s1,3
s2,0 s2,1 s2,2 s2,3
s3,0 s3,1 s3,2 s3,3
s0,0 s0,1 s0,2 s0,3
s1,0 s1,1 s1,2 s1,3
s2,0 s2,1 s2,2 s2,3
s3,0 s3,1 s3,2 s3,3
s0,0 s0,1 s0,2 s0,3
s1,1 s1,2 s1,3 s1,0
s2,2 s2,3 s2,0 s2,1
s3,3 s3,0 s3,1 s3,2
(a) Shift row transformation
(b) Mix column transformation
2 3 1 1 1 2 3 1 1 1 2 3 3 1 1 2
=*
¿ ¿ ¿ ¿
¿¿¿¿
¿ ¿ ¿ ¿
¿¿¿¿
6.3 / AES TRANSFORMATION FUNCTIONS 187
performed in GF(28). The MixColumns transformation on a single column of State can be expressed as
s0, j = = (2 # s0, j) ⊕ (3 # s1, j) ⊕ s2, j ⊕ s3, j
s1, j = = s0, j ⊕ (2 # s1, j) ⊕ (3 # s2, j) ⊕ s3, j
s2, j = = s0, j ⊕ s1, j ⊕ (2 # s2, j) ⊕ (3 # s3, j)
s3, j = = (3 # s0, j) ⊕ s1, j ⊕ s2, j ⊕ (2 # s3, j)
(6.4)
The following is an example of MixColumns:
87 F2 4D 97 47 40 A3 4C
6E 4C 90 EC 37 D4 70 9F
46 E7 4A C3 S 94 E4 3A 42 A6 8C D8 95 ED A5 A6 BC
Let us verify the first column of this example. Recall from Section 5.6 that, in
GF(28), addition is the bitwise XOR operation and that multiplication can be per-
formed according to the rule established in Equation (4.14). In particular, multipli-
cation of a value by x (i.e., by {02}) can be implemented as a 1-bit left shift followed by a conditional bitwise XOR with (0001 1011) if the leftmost bit of the original
value (prior to the shift) is 1. Thus, to verify the MixColumns transformation on the
first column, we need to show that
({02} # {87}) ⊕ ({03} # {6E}) ⊕ {46} ⊕ {A6} = {47} {87} ⊕ ({02} # {6E}) ⊕ ({03} # {46}) ⊕ {A6} = {37} {87} ⊕ {6E} ⊕ ({02} # {46}) ⊕ ({03} # {A6}) = {94} ({03} # {87}) ⊕ {6E} ⊕ {46} ⊕ ({02} # {A6}) = {ED} For the first equation, we have {02} # {87} = (0000 1110) ⊕ (0001 1011) =
(0001 0101) and {03} # {6E} = {6E} ⊕ ({02} # {6E}) = (0110 1110) ⊕ (1101 1100) = (1011 0010). Then,
{02} # {87} = 0001 0101 {03} # {6E} = 1011 0010 {46} = 0100 0110 {A6} = 1010 0110
0100 0111 = {47}
The other equations can be similarly verified.
The inverse mix column transformation, called InvMixColumns, is defined by the following matrix multiplication:
D 0E 0B 0D 0909 0E 0B 0D 0D 09 0E 0B
0B 0D 09 0E
T D s0,0 s0,1 s0,2 s0,3s1,0 s1,1 s1,2 s1,3 s2,0 s2,1 s2,2 s2,3 s3,0 s3,1 s3,2 s3,3
T = D s0,0= s0,1= s0,2= s0,3=s1,0= s1,1= s1,2= s1,3= s2,0 = s2,1
= s2,2 = s2,3
=
s3,0 = s3,1
= s3,2 = s3,3
=
T (6.5)
188 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD
It is not immediately clear that Equation (6.5) is the inverse of Equation (6.3). We need to show
D 0E 0B 0D 0909 0E 0B 0D 0D 09 0E 0B
0B 0D 09 0E
T D 02 03 01 0101 02 03 01 01 01 02 03
03 01 01 02
T D s0,0 s0,1 s0,2 s0,3s1,0 s1,1 s1,2 s1,3 s2,0 s2,1 s2,2 s2,3 s3,0 s3,1 s3,2 s3,3
T = D s0,0 s0,1 s0,2 s0,3s1,0 s1,1 s1,2 s1,3 s2,0 s2,1 s2,2 s2,3 s0,3 s3,1 s3,2 s3,3
T which is equivalent to showing
D 0E 0B 0D 0909 0E 0B 0D 0D 09 0E 0B
0B 0D 09 0E
T D 02 03 01 0101 02 03 01 01 01 02 03
03 01 01 02
T = D 1 0 0 00 1 0 0 0 0 1 0
0 0 0 1
T (6.6) That is, the inverse transformation matrix times the forward transformation matrix
equals the identity matrix. To verify the first column of Equation (6.6), we need
to show
({0E} # {02}) ⊕ {0B} ⊕ {0D} ⊕ ({09} # {03}) = {01} ({09} # {02}) ⊕ {0E} ⊕ {0B} ⊕ ({0D} # {03}) = {00} ({0D} # {02}) ⊕ {09} ⊕ {0E} ⊕ ({0B} # {03}) = {00}
({0B} # {02}) ⊕ {0D} ⊕ {09} ⊕ ({0E} # {03}) = {00} For the first equation, we have {0E} # {02} = 00011100 and {09} # {03} =
{09} ⊕ ({09} # {02}) = 00001001 ⊕ 00010010 = 00011011. Then
{0E} # {02} = 00011100 {0B} = 00001011 {0D} = 00001101 {09} # {03} = 00011011
00000001
The other equations can be similarly verified.
The AES document describes another way of characterizing the MixColumns
transformation, which is in terms of polynomial arithmetic. In the standard,
MixColumns is defined by considering each column of State to be a four-term poly- nomial with coefficients in GF(28). Each column is multiplied modulo (x4 + 1) by the fixed polynomial a(x), given by
a(x) = {03}x3 + {01}x2 + {01}x + {02} (6.7)
Appendix 5A demonstrates that multiplication of each column of State by a(x) can be written as the matrix multiplication of Equation (6.3). Similarly, it can be seen that the transformation in Equation (6.5) corresponds to treating
6.3 / AES TRANSFORMATION FUNCTIONS 189
each column as a four-term polynomial and multiplying each column by b(x), given by
b(x) = {0B}x3 + {0D}x2 + {09}x + {0E} (6.8)
It readily can be shown that b(x) = a-1(x) mod (x4 + 1).
RATIONALE The coefficients of the matrix in Equation (6.3) are based on a linear code with maximal distance between code words, which ensures a good mixing
among the bytes of each column. The mix column transformation combined with
the shift row transformation ensures that after a few rounds all output bits depend
on all input bits. See [DAEM99] for a discussion.
In addition, the choice of coefficients in MixColumns, which are all {01}, {02},
or {03}, was influenced by implementation considerations. As was discussed, multi-
plication by these coefficients involves at most a shift and an XOR. The coefficients
in InvMixColumns are more formidable to implement. However, encryption was
deemed more important than decryption for two reasons:
1. For the CFB and OFB cipher modes (Figures 7.5 and 7.6; described in Chapter 7), only encryption is used.
2. As with any block cipher, AES can be used to construct a message authentica- tion code (Chapter 13), and for this, only encryption is used.
AddRoundKey Transformation
FORWARD AND INVERSE TRANSFORMATIONS In the forward add round key transfor- mation, called AddRoundKey, the 128 bits of State are bitwise XORed with the 128 bits of the round key. As shown in Figure 6.5b, the operation is viewed as a
columnwise operation between the 4 bytes of a State column and one word of the round key; it can also be viewed as a byte-level operation. The following is an
example of AddRoundKey:
47 40 A3 4C AC 19 28 57 EB 59 8B 1B
37 D4 70 9F 77 FA D1 5C 40 2E A1 C3
94 E4 3A 42 ⊕ 66 DC 29 00 = F2 38 13 42
ED A5 A6 BC F3 21 41 6A 1E 84 E7 D6
The first matrix is State, and the second matrix is the round key. The inverse add round key transformation is identical to the forward add
round key transformation, because the XOR operation is its own inverse.
RATIONALE The add round key transformation is as simple as possible and affects every bit of State. The complexity of the round key expansion, plus the complexity of the other stages of AES, ensure security.
Figure 6.8 is another view of a single round of AES, emphasizing the mecha-
nisms and inputs of each transformation.
190 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD
6.4 AES KEY EXPANSION
Key Expansion Algorithm
The AES key expansion algorithm takes as input a four-word (16-byte) key and
produces a linear array of 44 words (176 bytes). This is sufficient to provide a four-
word round key for the initial AddRoundKey stage and each of the 10 rounds of the
cipher. The pseudocode on the next page describes the expansion.
The key is copied into the first four words of the expanded key. The remain-
der of the expanded key is filled in four words at a time. Each added word w[i] depends on the immediately preceding word, w[i - 1], and the word four positions back, w[i - 4]. In three out of four cases, a simple XOR is used. For a word whose position in the w array is a multiple of 4, a more complex function is used. Figure 6.9 illustrates the generation of the expanded key, using the symbol g to represent that
complex function. The function g consists of the following subfunctions.
Figure 6.8 Inputs for Single AES Round
SubBytes
State matrix at beginning
of round
State matrix at end
of round
MixColumns matrix Round
key
Variable inputConstant inputs
ShiftRows
MixColumns
AddRoundKey
S-box
02 03 01 01 01 02 03 01 01 01 02 03 03 01 01 02
Hiva-Network.Com
6.4 / AES KEY EXPANSION 191
KeyExpansion (byte key[16], word w[44]) { word temp for (i = 0; i < 4; i++) w[i] = (key[4*i], key[4*i+1], key[4*i+2], key[4*i+3]); for (i = 4; i < 44; i++) { temp = w[i − 1]; if (i mod 4 = 0) temp = SubWord (RotWord (temp)) ⊕ Rcon[i/4]; w[i] = w[i−4] ⊕ temp } }
Figure 6.9 AES Key Expansion
k3
(a) Overall algorithm
(b) Function g
k7 k11 k15
k2 k6 k10 k14
k1 k5 k9 k13
k0 k4 k8 k12
w0 w1 w2 w3 g
w4 w5 w6 w7
w40 w41 w42 w43
g
B0 B1 B2 B3
w
w
B1 B2 B3 B0
0 0 0
B1
S S
B2' ' B3
S S
B0' '
RCj
œ
192 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD
1. RotWord performs a one-byte circular left shift on a word. This means that an input word [B0, B1, B2, B3] is transformed into [B1, B2, B3, B0].
2. SubWord performs a byte substitution on each byte of its input word, using the S-box (Table 6.2a).
3. The result of steps 1 and 2 is XORed with a round constant, Rcon[j].
The round constant is a word in which the three rightmost bytes are always 0.
Thus, the effect of an XOR of a word with Rcon is to only perform an XOR on the
leftmost byte of the word. The round constant is different for each round and is de-
fined as Rcon[j] = (RC[j], 0, 0, 0), with RC[1] = 1, RC[j] = 2 # RC[j - 1] and with multiplication defined over the field GF(28). The values of RC[j] in hexadecimal are
j 1 2 3 4 5 6 7 8 9 10
RC[j] 01 02 04 08 10 20 40 80 1B 36
For example, suppose that the round key for round 8 is
EA D2 73 21 B5 8D BA D2 31 2B F5 60 7F 8D 29 2F
Then the first 4 bytes (first column) of the round key for round 9 are calculated as
follows:
i (decimal) temp After
RotWord
After
SubWord Rcon (9)
After XOR
with Rcon w[i - 4]
w[i] = temp ⊕ w[i - 4]
36 7F8D292F 8D292F7F 5DA515D2 1B000000 46A515D2 EAD27321 AC7766F3
Rationale
The Rijndael developers designed the expansion key algorithm to be resistant to
known cryptanalytic attacks. The inclusion of a round-dependent round constant
eliminates the symmetry, or similarity, between the ways in which round keys are
generated in different rounds. The specific criteria that were used are [DAEM99]
■ Knowledge of a part of the cipher key or round key does not enable calcula-
tion of many other round-key bits.
■ An invertible transformation [i.e., knowledge of any Nk consecutive words of the expanded key enables regeneration of the entire expanded key (Nk = key size in words)].
■ Speed on a wide range of processors.
■ Usage of round constants to eliminate symmetries.
■ Diffusion of cipher key differences into the round keys; that is, each key bit
affects many round key bits.
■ Enough nonlinearity to prohibit the full determination of round key differ-
ences from cipher key differences only.
■ Simplicity of description.
6.5 / AN AES EXAMPLE 193
The authors do not quantify the first point on the preceding list, but the idea
is that if you know less than Nk consecutive words of either the cipher key or one of the round keys, then it is difficult to reconstruct the remaining unknown bits. The
fewer bits one knows, the more difficult it is to do the reconstruction or to deter-
mine other bits in the key expansion.
6.5 AN AES EXAMPLE
We now work through an example and consider some of its implications. Although
you are not expected to duplicate the example by hand, you will find it informative
to study the hex patterns that occur from one step to the next.
For this example, the plaintext is a hexadecimal palindrome. The plaintext,
key, and resulting ciphertext are
Plaintext: 0123456789abcdeffedcba9876543210
Key: 0f1571c947d9e8590cb7add6af7f6798
Ciphertext: ff0b844a0853bf7c6934ab4364148fb9
Results
Table 6.3 shows the expansion of the 16-byte key into 10 round keys. As previ-
ously explained, this process is performed word by word, with each four-byte word
occupying one column of the word round-key matrix. The left-hand column shows
Key Words Auxiliary Function
w0 = 0f 15 71 c9 w1 = 47 d9 e8 59 w2 = 0c b7 ad d6 w3 = af 7f 67 98
RotWord (w3) = 7f 67 98 af = x1 SubWord (x1) = d2 85 46 79 = y1 Rcon (1) = 01 00 00 00 y1 ⊕ Rcon (1) = d3 85 46 79 = z1
w4 = w0 ⊕ z1 = dc 90 37 b0 w5 = w4 ⊕ w1 = 9b 49 df e9 w6 = w5 ⊕ w2 = 97 fe 72 3f w7 = w6 ⊕ w3 = 38 81 15 a7
RotWord (w7) = 81 15 a7 38 = x2 SubWord (x2) = 0c 59 5c 07 = y2 Rcon (2) = 02 00 00 00 y2 ⊕ Rcon (2) = 0e 59 5c 07 = z2
w8 = w4 ⊕ z2 = d2 c9 6b b7 w9 = w8 ⊕ w5 = 49 80 b4 5e w10 = w9 ⊕ w6 = de 7e c6 61 w11 = w10 ⊕ w7 = e6 ff d3 c6
RotWord (w11) = ff d3 c6 e6 = x3 SubWord (x3) = 16 66 b4 83 = y3 Rcon (3) = 04 00 00 00 y3 ⊕ Rcon (3) = 12 66 b4 8e = z3
w12 = w8 ⊕ z3 = c0 af df 39 w13 = w12 ⊕ w9 = 89 2f 6b 67 w14 = w13 ⊕ w10 = 57 51 ad 06 w15 = w14 ⊕ w11 = b1 ae 7e c0
RotWord (w15) = ae 7e c0 b1 = x4 SubWord (x4) = e4 f3 ba c8 = y4 Rcon (4) = 08 00 00 00 y4 ⊕ Rcon (4) = ec f3 ba c8 = 4
Table 6.3 Key Expansion for AES Example
(Continued)
194 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD
Key Words Auxiliary Function
w16 = w12 ⊕ z4 = 2c 5c 65 f1 w17 = w16 ⊕ w13 = a5 73 0e 96 w18 = w17 ⊕ w14 = f2 22 a3 90 w19 = w18 ⊕ w15 = 43 8c dd 50
RotWord (w19) = 8c dd 50 43 = x5 SubWord (x5) = 64 c1 53 1a = y5 Rcon(5) = 10 00 00 00 y5 ⊕ Rcon (5) = 74 c1 53 1a = z5
w20 = w16 ⊕ z5 = 58 9d 36 eb w21 = w20 ⊕ w17 = fd ee 38 7d w22 = w21 ⊕ w18 = 0f cc 9b ed w23 = w22 ⊕ w19 = 4c 40 46 bd
RotWord (w23) = 40 46 bd 4c = x6 SubWord (x6) = 09 5a 7a 29 = y6 Rcon(6) = 20 00 00 00 y6 ⊕ Rcon(6) = 29 5a 7a 29 = z6
w24 = w20 ⊕ z6 = 71 c7 4c c2 w25 = w24 ⊕ w21 = 8c 29 74 bf w26 = w25 ⊕ w22 = 83 e5 ef 52 w27 = w26 ⊕ w23 = cf a5 a9 ef
RotWord (w27) = a5 a9 ef cf = x7 SubWord (x7) = 06 d3 bf 8a = y7 Rcon (7) = 40 00 00 00 y7 ⊕ Rcon(7) = 46 d3 df 8a = z7
w28 = w24 ⊕ z7 = 37 14 93 48 w29 = w28 ⊕ w25 = bb 3d e7 f7 w30 = w29 ⊕ w26 = 38 d8 08 a5 w31 = w30 ⊕ w27 = f7 7d a1 4a
RotWord (w31) = 7d a1 4a f7 = x8 SubWord (x8) = ff 32 d6 68 = y8 Rcon (8) = 80 00 00 00 y8 ⊕ Rcon(8) = 7f 32 d6 68 = z8
w32 = w28 ⊕ z8 = 48 26 45 20 w33 = w32 ⊕ w29 = f3 1b a2 d7 w34 = w33 ⊕ w30 = cb c3 aa 72 w35 = w34 ⊕ w32 = 3c be 0b 3
RotWord (w35) = be 0b 38 3c = x9 SubWord (x9) = ae 2b 07 eb = y9 Rcon (9) = 1B 00 00 00 y9 ⊕ Rcon (9) = b5 2b 07 eb = z9
w36 = w32 ⊕ z9 = fd 0d 42 cb w37 = w36 ⊕ w33 = 0e 16 e0 1c w38 = w37 ⊕ w34 = c5 d5 4a 6e w39 = w38 ⊕ w35 = f9 6b 41 56
RotWord (w39) = 6b 41 56 f9 = x10 SubWord (x10) = 7f 83 b1 99 = y10 Rcon (10) = 36 00 00 00 y10 ⊕ Rcon (10) = 49 83 b1 99 = z10
w40 = w36 ⊕ z10 = b4 8e f3 52 w41 = w40 ⊕ w37 = ba 98 13 4e w42 = w41 ⊕ w38 = 7f 4d 59 20 w43 = w42 ⊕ w39 = 86 26 18 76
Table 6.3 Continued
the four round-key words generated for each round. The right-hand column shows
the steps used to generate the auxiliary word used in key expansion. We begin, of
course, with the key itself serving as the round key for round 0.
Next, Table 6.4 shows the progression of State through the AES encryption process. The first column shows the value of State at the start of a round. For the first row, State is just the matrix arrangement of the plaintext. The second, third, and fourth columns show the value of State for that round after the SubBytes, ShiftRows, and MixColumns transformations, respectively. The fifth column shows the round
key. You can verify that these round keys equate with those shown in Table 6.3. The
first column shows the value of State resulting from the bitwise XOR of State after the preceding MixColumns with the round key for the preceding round.
Avalanche Effect
If a small change in the key or plaintext were to produce a corresponding small
change in the ciphertext, this might be used to effectively reduce the size of the
6.5 / AN AES EXAMPLE 195
Start of Round After SubBytes After ShiftRows After MixColumns Round Key
01 89 fe 76 23 ab dc 54 45 cd ba 32 67 ef 98 10
0f 47 0c af 15 d9 b7 7f 71 e8 ad 67 c9 59 d6 98
0e ce f2 d9 36 72 6b 2b 34 25 17 55 ae b6 4e 88
ab 8b 89 35 05 40 7f f1 18 3f f0 fc e4 4e 2f c4
ab 8b 89 35 40 7f f1 05 f0 fc 18 3f c4 e4 4e 2f
b9 94 57 75 e4 8e 16 51 47 20 9a 3f c5 d6 f5 3b
dc 9b 97 38 90 49 fe 81 37 df 72 15 b0 e9 3f a7
65 0f c0 4d 74 c7 e8 d0 70 ff e8 2a 75 3f ca 9c
4d 76 ba e3 92 c6 9b 70 51 16 9b e5 9d 75 74 de
4d 76 ba e3 c6 9b 70 92 9b e5 51 16 de 9d 75 74
8e 22 db 12 b2 f2 dc 92 df 80 f7 c1 2d c5 1e 52
d2 49 de e6 c9 80 7e ff 6b b4 c6 d3 b7 5e 61 c6
5c 6b 05 f4 7b 72 a2 6d b4 34 31 12 9a 9b 7f 94
4a 7f 6b bf 21 40 3a 3c 8d 18 c7 c9 b8 14 d2 22
4a 7f 6b bf 40 3a 3c 21 c7 c9 8d 18 22 b8 14 d2
b1 c1 0b cc ba f3 8b 07 f9 1f 6a c3 1d 19 24 5c
c0 89 57 b1 af 2f 51 ae df 6b ad 7e 39 67 06 c0
71 48 5c 7d 15 dc da a9 26 74 c7 bd 24 7e 22 9c
a3 52 4a ff 59 86 57 d3 f7 92 c6 7a 36 f3 93 de
a3 52 4a ff 86 57 d3 59 c6 7a f7 92 de 36 f3 93
d4 11 fe 0f 3b 44 06 73 cb ab 62 37 19 b7 07 ec
2c a5 f2 43 5c 73 22 8c 65 0e a3 dd f1 96 90 50
f8 b4 0c 4c 67 37 24 ff ae a5 c1 ea e8 21 97 bc
41 8d fe 29 85 9a 36 16 e4 06 78 87 9b fd 88 65
41 8d fe 29 9a 36 16 85 78 87 e4 06 65 9b fd 88
2a 47 c4 48 83 e8 18 ba 84 18 27 23 eb 10 0a f3
58 fd 0f 4c 9d ee cc 40 36 38 9b 46 eb 7d ed bd
72 ba cb 04 1e 06 d4 fa b2 20 bc 65 00 6d e7 4e
40 f4 1f f2 72 6f 48 2d 37 b7 65 4d 63 3c 94 2f
40 f4 1f f2 6f 48 2d 72 65 4d 37 b7 2f 63 3c 94
7b 05 42 4a 1e d0 20 40 94 83 18 52 94 c4 43 fb
71 8c 83 cf c7 29 e5 a5 4c 74 ef a9 c2 bf 52 ef
0a 89 c1 85 d9 f9 c5 e5 d8 f7 f7 fb 56 7b 11 14
67 a7 78 97 35 99 a6 d9 61 68 68 0f b1 21 82 fa
67 a7 78 97 99 a6 d9 35 68 0f 61 68 fa b1 21 82
ec 1a c0 80 0c 50 53 c7 3b d7 00 ef b7 22 72 e0
37 bb 38 f7 14 3d d8 7d 93 e7 08 a1 48 f7 a5 4a
db a1 f8 77 18 6d 8b ba a8 30 08 4e ff d5 d7 aa
b9 32 41 f5 ad 3c 3d f4 c2 04 30 2f 16 03 0e ac
b9 32 41 f5 3c 3d f4 ad 30 2f c2 04 ac 16 03 0e
b1 1a 44 17 3d 2f ec b6 0a 6b 2f 42 9f 68 f3 b1
48 f3 cb 3c 26 1b c3 be 45 a2 aa 0b 20 d7 72 38
f9 e9 8f 2b 1b 34 2f 08 4f c9 85 49 bf bf 81 89
99 1e 73 f1 af 18 15 30 84 dd 97 3b 08 08 0c a7
99 1e 73 f1 18 15 30 af 97 3b 84 dd a7 08 08 0c
31 30 3a c2 ac 71 8c c4 46 65 48 eb 6a 1c 31 62
fd 0e c5 f9 0d 16 d5 6b 42 e0 4a 41 cb 1c 6e 56
cc 3e ff 3b a1 67 59 af 04 85 02 aa a1 00 5f 34
4b b2 16 e2 32 85 cb 79 f2 97 77 ac 32 63 cf 18
4b b2 16 e2 85 cb 79 32 77 ac f2 97 18 32 63 cf
b4 ba 7f 86 8e 98 4d 26 f3 13 59 18 52 4e 20 76
ff 08 69 64 0b 53 34 14 84 bf ab 8f 4a 7c 43 b9
Table 6.4 AES Example
196 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD
Round Number of Bits
that Differ
0123456789abcdeffedcba9876543210 0023456789abcdeffedcba9876543210
1
0 0e3634aece7225b6f26b174ed92b5588 0f3634aece7225b6f26b174ed92b5588
1
1 657470750fc7ff3fc0e8e8ca4dd02a9c c4a9ad090fc7ff3fc0e8e8ca4dd02a9c
20
2 5c7bb49a6b72349b05a2317ff46d1294 fe2ae569f7ee8bb8c1f5a2bb37ef53d5
58
3 7115262448dc747e5cdac7227da9bd9c ec093dfb7c45343d689017507d485e62
59
4 f867aee8b437a5210c24c1974cffeabc 43efdb697244df808e8d9364ee0ae6f5
61
5 721eb200ba06206dcbd4bce704fa654e 7b28a5d5ed643287e006c099bb375302
68
6 0ad9d85689f9f77bc1c5f71185e5fb14 3bc2d8b6798d8ac4fe36a1d891ac181a
64
7 db18a8ffa16d30d5f88b08d777ba4eaa 9fb8b5452023c70280e5c4bb9e555a4b
67
8 f91b4fbfe934c9bf8f2f85812b084989 20264e1126b219aef7feb3f9b2d6de40
65
9 cca104a13e678500ff59025f3bafaa34 b56a0341b2290ba7dfdfbddcd8578205
61
10 ff0b844a0853bf7c6934ab4364148fb9 612b89398d0600cde116227ce72433f0
58
Table 6.5 Avalanche Effect in AES: Change in Plaintext
plaintext (or key) space to be searched. What is desired is the avalanche effect, in
which a small change in plaintext or key produces a large change in the ciphertext.
Using the example from Table 6.4, Table 6.5 shows the result when the
eighth bit of the plaintext is changed. The second column of the table shows the
value of the State matrix at the end of each round for the two plaintexts. Note that after just one round, 20 bits of the State vector differ. After two rounds, close to half the bits differ. This magnitude of difference propagates through
the remaining rounds. A bit difference in approximately half the positions in the
most desirable outcome. Clearly, if almost all the bits are changed, this would be
logically equivalent to almost none of the bits being changed. Put another way, if
we select two plaintexts at random, we would expect the two plaintexts to differ
in about half of the bit positions and the two ciphertexts to also differ in about
half the positions.
Table 6.6 shows the change in State matrix values when the same plaintext is used and the two keys differ in the eighth bit. That is, for the second case, the
key is 0e1571c947d9e8590cb7add6af7f6798. Again, one round produces a significant change, and the magnitude of change after all subsequent rounds
is roughly half the bits. Thus, based on this example, AES exhibits a very strong
avalanche effect.
6.6 / AES IMPLEMENTATION 197
Round Number of Bits
that Differ
0123456789abcdeffedcba9876543210 0123456789abcdeffedcba9876543210
0
0 0e3634aece7225b6f26b174ed92b5588 0f3634aece7225b6f26b174ed92b5588
1
1 657470750fc7ff3fc0e8e8ca4dd02a9c c5a9ad090ec7ff3fc1e8e8ca4cd02a9c
22
2 5c7bb49a6b72349b05a2317ff46d1294 90905fa9563356d15f3760f3b8259985
58
3 7115262448dc747e5cdac7227da9bd9c 18aeb7aa794b3b66629448d575c7cebf
67
4 f867aee8b437a5210c24c1974cffeabc f81015f993c978a876ae017cb49e7eec
63
5 721eb200ba06206dcbd4bce704fa654e 5955c91b4e769f3cb4a94768e98d5267
81
6 0ad9d85689f9f77bc1c5f71185e5fb14 dc60a24d137662181e45b8d3726b2920
70
7 db18a8ffa16d30d5f88b08d777ba4eaa fe8343b8f88bef66cab7e977d005a03c
74
8 f91b4fbfe934c9bf8f2f85812b084989 da7dad581d1725c5b72fa0f9d9d1366a
67
9 cca104a13e678500ff59025f3bafaa34 0ccb4c66bbfd912f4b511d72996345e0
59
10 ff0b844a0853bf7c6934ab4364148fb9 fc8923ee501a7d207ab670686839996b
53
Table 6.6 Avalanche Effect in AES: Change in Key
Note that this avalanche effect is stronger than that for DES (Table 4.2),
which requires three rounds to reach a point at which approximately half the bits
are changed, both for a bit change in the plaintext and a bit change in the key.
6.6 AES IMPLEMENTATION
Equivalent Inverse Cipher
As was mentioned, the AES decryption cipher is not identical to the encryption
cipher (Figure 6.3). That is, the sequence of transformations for decryption differs
from that for encryption, although the form of the key schedules for encryption
and decryption is the same. This has the disadvantage that two separate software
or firmware modules are needed for applications that require both encryption and
decryption. There is, however, an equivalent version of the decryption algorithm
that has the same structure as the encryption algorithm. The equivalent version has
the same sequence of transformations as the encryption algorithm (with transfor-
mations replaced by their inverses). To achieve this equivalence, a change in key
schedule is needed.
198 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD
Two separate changes are needed to bring the decryption structure in line
with the encryption structure. As illustrated in Figure 6.3, an encryption round has
the structure SubBytes, ShiftRows, MixColumns, AddRoundKey. The standard
decryption round has the structure InvShiftRows, InvSubBytes, AddRoundKey,
InvMixColumns. Thus, the first two stages of the decryption round need to be inter-
changed, and the second two stages of the decryption round need to be interchanged.
INTERCHANGING INVSHIFTROWS AND INVSUBBYTES InvShiftRows affects the se- quence of bytes in State but does not alter byte contents and does not depend on byte contents to perform its transformation. InvSubBytes affects the contents of
bytes in State but does not alter byte sequence and does not depend on byte se- quence to perform its transformation. Thus, these two operations commute and can
be interchanged. For a given State Si,
InvShiftRows [InvSubBytes (Si)] = InvSubBytes [InvShiftRows (Si)]
INTERCHANGING ADDROUNDKEY AND INVMIXCOLUMNS The transformations AddRoundKey and InvMixColumns do not alter the sequence of bytes in State. If we view the key as a sequence of words, then both AddRoundKey and InvMixColumns
operate on State one column at a time. These two operations are linear with respect to the column input. That is, for a given State Si and a given round key wj,
InvMixColumns (Si ⊕ wj) = [InvMixColumns (Si)] ⊕ [InvMixColumns (wj)]
To see this, suppose that the first column of State Si is the sequence (y0, y1, y2, y3) and the first column of the round key wj is (k0, k1, k2, k3). Then we need to show
D 0E 0B 0D 0909 0E 0B 0D 0D 09 0E 0B
0B 0D 09 0E
T D y0 ⊕ k0y1 ⊕ k1 y2 ⊕ k2 y3 ⊕ k3
T = D 0E 0B 0D 0909 0E 0B 0D 0D 09 0E 0B
0B 0D 09 0E
T D y0y1 y2 y3
T ⊕ D 0E 0B 0D 0909 0E 0B 0D 0D 09 0E 0B
0B 0D 09 0E
T D k0k1 k2 k3
T Let us demonstrate that for the first column entry. We need to show
[{0E} # (y0 ⊕ k0)] ⊕ [{0B} # (y1 ⊕ k1)] ⊕ [{0D} # (y2 ⊕ k2)] ⊕ [{09} # (y3 ⊕ k3)] = [{0E} # y0] ⊕ [{0B} # y1] ⊕ [{0D} # y2] ⊕ [{09} # y3] ⊕
[{0E} # k0] ⊕ [{0B} # k1] ⊕ [{0D} # k2] ⊕ [{09} # k3] This equation is valid by inspection. Thus, we can interchange AddRoundKey
and InvMixColumns, provided that we first apply InvMixColumns to the round
key. Note that we do not need to apply InvMixColumns to the round key for the
input to the first AddRoundKey transformation (preceding the first round) nor to
the last AddRoundKey transformation (in round 10). This is because these two
AddRoundKey transformations are not interchanged with InvMixColumns to pro-
duce the equivalent decryption algorithm.
Figure 6.10 illustrates the equivalent decryption algorithm.
6.6 / AES IMPLEMENTATION 199
Figure 6.10 Equivalent Inverse Cipher
Add round key
w[36, 39]
w[40, 43]
Ciphertext
Inverse sub bytes
Inverse shift rows
Inverse mix cols R ou
nd 1
R ou
nd 9
R ou
nd 1
0
Add round keyInverse mix cols
Inverse sub bytes
Inverse shift rows
Inverse mix cols
Add round keyInverse mix cols
Inverse sub bytes
Inverse shift rowsExpand key
Add round key
PlaintextKey
w[4, 7]
w[0, 3]
Implementation Aspects
The Rijndael proposal [DAEM99] provides some suggestions for efficient im-
plementation on 8-bit processors, typical for current smart cards, and on 32-bit
processors, typical for PCs.
8-BIT PROCESSOR AES can be implemented very efficiently on an 8-bit proces- sor. AddRoundKey is a bytewise XOR operation. ShiftRows is a simple byte-
shifting operation. SubBytes operates at the byte level and only requires a table
of 256 bytes.
The transformation MixColumns requires matrix multiplication in the field
GF(28), which means that all operations are carried out on bytes. MixColumns only
requires multiplication by {02} and {03}, which, as we have seen, involved simple
shifts, conditional XORs, and XORs. This can be implemented in a more efficient
Hiva-Network.Com
200 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD
way that eliminates the shifts and conditional XORs. Equation set (6.4) shows the
equations for the MixColumns transformation on a single column. Using the iden-
tity {03} # x = ({02} # x) ⊕ x, we can rewrite Equation set (6.4) as follows. Tmp = s0, j ⊕ s1, j ⊕ s2, j ⊕ s3, j
s0, j = = s0, j ⊕ Tmp ⊕ [2 # (s0, j ⊕ s1, j)]
s1, j = = s1, j ⊕ Tmp ⊕ [2 # (s1, j ⊕ s2, j)] (6.9)
s2, j = = s2, j ⊕ Tmp ⊕ [2 # (s2, j ⊕ s3, j)]
s3, j = = s3, j ⊕ Tmp ⊕ [2 # (s3, j ⊕ s0, j)]
Equation set (6.9) is verified by expanding and eliminating terms.
The multiplication by {02} involves a shift and a conditional XOR. Such
an implementation may be vulnerable to a timing attack of the sort described in
Section 4.4. To counter this attack and to increase processing efficiency at the
cost of some storage, the multiplication can be replaced by a table lookup. Define
the 256-byte table X2, such that X2[i] = {02} # i. Then Equation set (6.9) can be rewritten as
Tmp = s0, j ⊕ s1, j ⊕ s2, j ⊕ s3, j s0, j = = s0, j ⊕ Tmp ⊕ X2[s0, j ⊕ s1, j]
s1, c = = s1, j ⊕ Tmp ⊕ X2[s1, j ⊕ s2, j]
s2, c = = s2, j ⊕ Tmp ⊕ X2[s2, j ⊕ s3, j]
s3, j = = s3, j ⊕ Tmp ⊕ X2[s3, j ⊕ s0, j]
32-BIT PROCESSOR The implementation described in the preceding subsection uses only 8-bit operations. For a 32-bit processor, a more efficient implementation can be
achieved if operations are defined on 32-bit words. To show this, we first define the
four transformations of a round in algebraic form. Suppose we begin with a State matrix consisting of elements ai, j and a round-key matrix consisting of elements ki, j. Then the transformations can be expressed as follows.
SubBytes bi, j = S[ai, j]
ShiftRows D c0, jc1, j c2, j c3, j
T = D b0, jb1, j - 1 b2, j - 2 b3, j - 3
T MixColumns D d0, jd1, j
d2, j d3, j
T = D 02 03 01 0101 02 03 01 01 01 02 03
03 01 01 02
T D c0, jc1, j c2, j c3, j
T AddRoundKey D e0, je1, j
e2, j e3, j
T = D d0, jd1, j d2, j d3, j
T ⊕ D k0, jk1, j k2, j k3, j
T
6.6 / AES IMPLEMENTATION 201
In the ShiftRows equation, the column indices are taken mod 4. We can
combine all of these expressions into a single equation:
D e0, je1, j e2, j e3, j
T = D 02 03 01 0101 02 03 01 01 01 02 03
03 01 01 02
T D S[a0, j]S[a1, j - 1] S[a2, j - 2] S[a3, j - 3]
T ⊕ D k0, jk1, j k2, j k3, j
T = § D 0201
01
03
T # S[a0, j] ¥ ⊕ § D 0302 01
01
T # S[a1, j - 1] ¥ ⊕ § D 0103 02
01
T # S[a2, j - 2] ¥ ⊕ § D 0101
03
02
T # S[a3, j - 3] ¥ ⊕ D k0, jk1, jk2, j k3, j
T In the second equation, we are expressing the matrix multiplication as a linear com-
bination of vectors. We define four 256-word (1024-byte) tables as follows.
T0[x] = § D 0201 01
03
T # S[x] ¥ T1[x] = § D 0302 01
01
T # S[x] ¥ T2[x] = § D 0103 02
01
T # S[x] ¥ T3[x] = § D 0101 03
02
T # S[x] ¥ Thus, each table takes as input a byte value and produces a column vector (a 32-bit
word) that is a function of the S-box entry for that byte value. These tables can be
calculated in advance.
We can define a round function operating on a column in the following fashion.
D s0, j=s1, j= s2, j =
s3, j =
T = T0[s0, j] ⊕ T1[s1, j - 1] ⊕ T2[s2, j - 2] ⊕ T3[s3, j - 3] ⊕ D k0, jk1, jk2, j k3, j
T As a result, an implementation based on the preceding equation requires only
four table lookups and four XORs per column per round, plus 4 Kbytes to store the
table. The developers of Rijndael believe that this compact, efficient implementa-
tion was probably one of the most important factors in the selection of Rijndael
for AES.
202 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD
6.7 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
Advanced Encryption
Standard (AES)
avalanche effect
field
finite field
irreducible
polynomial
key expansion
National Institute of Standards
and Technology (NIST)
Rijndael
S-box
Key Terms
Review Questions
6.1 What was the original set of criteria used by NIST to evaluate candidate AES ciphers? 6.2 What was the final set of criteria used by NIST to evaluate candidate AES ciphers? 6.3 What is the difference between Rijndael and AES? 6.4 What is the purpose of the State array? 6.5 How is the S-box constructed? 6.6 Briefly describe SubBytes. 6.7 Briefly describe ShiftRows. 6.8 How many bytes in State are affected by ShiftRows? 6.9 Briefly describe MixColumns. 6.10 Briefly describe AddRoundKey. 6.11 Briefly describe the key expansion algorithm. 6.12 What is the difference between SubBytes and SubWord? 6.13 What is the difference between ShiftRows and RotWord? 6.14 What is the difference between the AES decryption algorithm and the equivalent
inverse cipher?
Problems
6.1 In the discussion of MixColumns and InvMixColumns, it was stated that
b(x) = a-1(x) mod(x4 + 1)
where a(x) = {03}x3 + {01}x2 + {01}x + {02} and b(x) = {0B}x3 + {0D}x2 + {09}x + {0E.} Show that this is true.
6.2 a. What is {0 2 }-1 in GF(28)? b. Verify the entry for {0 2 } in the S-box.
6.3 Show the first eight words of the key expansion for a 128-bit key of all ones. 6.4 Given the plaintext {0F0E0D0C0B0A09080706050403020100} and the key
{02020202020202020202020202020202}: a. Show the original contents of State, displayed as a 4 * 4 matrix. b. Show the value of State after initial AddRoundKey. c. Show the value of State after SubBytes. d. Show the value of State after ShiftRows. e. Show the value of State after MixColumns.
6.5 Verify Equation (6.11) in Appendix 6A. That is, show that xi mod (x4 + 1) = xi mod 4.
APPENDIX 6A / POLYNOMIALS WITH COEFFICIENTS IN GF(28) 203
6.6 Compare AES to DES. For each of the following elements of DES, indicate the com- parable element in AES or explain why it is not needed in AES. a. XOR of subkey material with the input to the f function b. XOR of the f function output with the left half of the block c. f function d. permutation P e. swapping of halves of the block
6.7 In the subsection on implementation aspects, it is mentioned that the use of tables helps thwart timing attacks. Suggest an alternative technique.
6.8 In the subsection on implementation aspects, a single algebraic equation is developed that describes the four stages of a typical round of the encryption algorithm. Provide the equivalent equation for the tenth round.
6.9 Compute the output of the MixColumns transformation for the following sequence of input bytes “A1 B2 C3 D4.” Apply the InvMixColumns transformation to the ob- tained result to verify your calculations. Change the first byte of the input from “A1” to “A3” perform the MixColumns transformation again for the new input, and deter- mine how many bits have changed in the output.
Note: You can perform all calculations by hand or write a program supporting these computations. If you choose to write a program, it should be written entirely by you; no use of libraries or public domain source code is allowed in this assignment.
6.10 Use the key 1010 1001 1100 0011 to encrypt the plaintext “hi” as expressed in ASCII as 0110 1000 0110 1001. The designers of S-AES got the ciphertext 0011 1110 1111 1011. Do you?
6.11 Show that the matrix given here, with entries in GF(24), is the inverse of the matrix used in the MixColumns step of S-AES.
¢x3 + 1 x x x3 + 1
≤ 6.12 Carefully write up a complete decryption of the ciphertext 0011 1110 1111 1011 using
the key 1010 1001 1100 0011 and the S-AES algorithm. You should get the plaintext we started with in Problem 6.10. Note that the inverse of the S-boxes can be done with a reverse table lookup. The inverse of the MixColumns step is given by the ma- trix in the previous problem.
6.13 Demonstrate that Equation (6.9) is equivalent to Equation (6.4).
Programming Problems
6.14 Create software that can encrypt and decrypt using S-AES. Test data: A binary plaintext of 0110 1111 0110 1011 encrypted with a binary key of 1010 0111 0011 1011 should give a binary ciphertext of 0000 0111 0011 1000. Decryption should work correspondingly.
6.15 Implement a differential cryptanalysis attack on 1-round S-AES.
APPENDIX 6A POLYNOMIALS WITH COEFFICIENTS IN GF(28)
In Section 5.5, we discussed polynomial arithmetic in which the coefficients are in Zp
and the polynomials are defined modulo a polynomial m(x) whose highest power is some integer n. In this case, addition and multiplication of coefficients occurred within the field Zp; that is, addition and multiplication were performed modulo p.
204 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD
The AES document defines polynomial arithmetic for polynomials of degree 3
or less with coefficients in GF(28). The following rules apply.
1. Addition is performed by adding corresponding coefficients in GF(28). As was pointed out Section 5.4, if we treat the elements of GF(28) as 8-bit strings, then
addition is equivalent to the XOR operation. So, if we have
a(x) = a3x 3 + a2x2 + a1x + a0 (6.10)
and
b(x) = b3x 3 + b2x2 + b1x + b0 (6.11)
then
a(x) + b(x) = (a3 ⊕ b3)x3 + (a2 ⊕ b2)x2 + (a1 ⊕ b1)x + (a0 ⊕ b0)
2. Multiplication is performed as in ordinary polynomial multiplication with two refinements:
a. Coefficients are multiplied in GF(28). b. The resulting polynomial is reduced mod (x4 + 1).
We need to keep straight which polynomial we are talking about. Recall from
Section 5.6 that each element of GF(28) is a polynomial of degree 7 or less with bi-
nary coefficients, and multiplication is carried out modulo a polynomial of degree
8. Equivalently, each element of GF(28) can be viewed as an 8-bit byte whose bit
values correspond to the binary coefficients of the corresponding polynomial. For
the sets defined in this section, we are defining a polynomial ring in which each ele-
ment of this ring is a polynomial of degree 3 or less with coefficients in GF(28), and
multiplication is carried out modulo a polynomial of degree 4. Equivalently, each
element of this ring can be viewed as a 4-byte word whose byte values are elements
of GF(28) that correspond to the 8-bit coefficients of the corresponding polynomial.
We denote the modular product of a(x) and b(x) by a(x) ⊕ b(x). To com- pute d(x) = a(x) ⊕ b(x), the first step is to perform a multiplication without the modulo operation and to collect coefficients of like powers. Let us express this as
c(x) = a(x) * b(x). Then
c(x) = c6x 6 + c5x5 + c4x4 + c3x3 + c2x2 + c1x + c0 (6.12)
where
c0 = a0 # b0 c4 = (a3 # b1) ⊕ (a2 # b2) ⊕ (a1 # b3) c1 = (a1 # b0) ⊕ (a0 # b1) c5 = (a3 # b2) ⊕ (a2 # b3) c2 = (a2 # b0) ⊕ (a1 # b1) ⊕ (a0 # b2) c6 = a3 # b3 c3 = (a3 # b0) ⊕ (a2 # b1) ⊕ (a1 # b2) ⊕ (a0 # b3)
The final step is to perform the modulo operation
d(x) = c(x) mod (x4 + 1)
That is, d(x) must satisfy the equation
c(x) = [(x4 + 1) * q(x)] ⊕ d(x)
such that the degree of d(x) is 3 or less. A practical technique for performing multiplication over this polynomial ring
is based on the observation that
xi mod (x4 + 1) = xi mod 4 (6.13)
If we now combine Equations (6.12) and (6.13), we end up with
d(x) = c(x) mod (x4 + 1) = [c6x
6 + c5x5 + c4x4 + c3x3 + c2x2 + c1x + c0] mod (x4 + 1) = c3x
3 + (c2 ⊕ c6)x2 + (c1 ⊕ c5)x + (c0 ⊕ c4)
Expanding the ci coefficients, we have the following equations for the coef- ficients of d(x).
d0 = (a0 # b0) ⊕ (a3 # b1) ⊕ (a2 # b2) ⊕ (a1 # b3) d1 = (a1 # b0) ⊕ (a0 # b1) ⊕ (a3 # b2) ⊕ (a2 # b3) d2 = (a2 # b0) ⊕ (a1 # b1) ⊕ (a0 # b2) ⊕ (a3 # b3) d3 = (a3 # b0) ⊕ (a2 # b1) ⊕ (a1 # b2) ⊕ (a0 # b3)
This can be written in matrix form:
D d0d1 d2 d3
T = D a0 a3 a2 a1a1 a0 a3 a2 a2 a1 a0 a3 a3 a2 a1 a0
T D b0b1 b2 b3
T (6.14) MixColumns Transformation
In the discussion of MixColumns, it was stated that there were two equivalent
ways of defining the transformation. The first is the matrix multiplication shown in
Equation (6.3), which is repeated here:
D 02 03 01 0101 02 03 01 01 01 02 03
03 01 01 02
T D s0, 0 s0, 1 s0, 2 s0, 3s1, 0 s1, 1 s1, 2 s1, 3 s2, 0 s2, 1 s2, 2 s2, 3 s3, 0 s3, 1 s3, 2 s3, 3
T = D s0, 0= s0, 1= s0, 2= s0, 3=s1, 0= s1, 1= s1, 2= s1, 3= s2, 0 = s2, 1
= s2, 2 = s2, 3
=
s3, 0 = s3, 1
= s3, 2 = s3, 3
=
T The second method is to treat each column of State as a four-term polynomial
with coefficients in GF(28). Each column is multiplied modulo (x4 + 1) by the fixed polynomial a(x), given by
a(x) = {03}x3 + {01}x2 + {01}x + {02}
APPENDIX 6A / POLYNOMIALS WITH COEFFICIENTS IN GF(28) 205
206 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD
From Equation (6.10), we have a3 = {03}; a2 = {01}; a1 = {01}; and a0 = {02}. For the jth column of State, we have the polynomial colj(x) = s3,jx
3 + s2,jx
2 + s1,jx + s0, j. Substituting into Equation (6.14), we can express d(x) = a(x) * colj(x) as
D d0d1 d2 d3
T = D a0 a3 a2 a1a1 a0 a3 a2 a2 a1 a0 a3 a3 a2 a1 a0
T D s0,js1,j s2,j s3,j
T = D 02 03 01 0101 02 03 01 01 01 02 03
03 01 01 02
T D s0,js1,j s2,j s3,j
T which is equivalent to Equation (6.3).
Multiplication by x
Consider the multiplication of a polynomial in the ring by x: c(x) = x ⊕ b(x). We have
c(x) = x ⊕ b(x) = [x * (b3x3 + b2x2 + b1x + b0] mod (x4 + 1) = (b3x
4 + b2x3 + b1x2 + b0x) mod (x4 + 1) = b2x
3 + b1x2 + b0x + b3
Thus, multiplication by x corresponds to a 1-byte circular left shift of the 4 bytes in the word representing the polynomial. If we represent the polynomial as
a 4-byte column vector, then we have
D c0c1 c2 c3
T = D 00 00 00 0101 00 00 00 00 01 00 00
00 00 01 00
T D b0b1 b2 b3
T
207
Block Cipher Operation 7.1 Multiple Encryption and Triple DES
Double DES
Triple DES with Two Keys
Triple DES with Three Keys
7.2 Electronic Codebook
7.3 Cipher Block Chaining Mode
7.4 Cipher Feedback Mode
7.5 Output Feedback Mode
7.6 Counter Mode
7.7 XTS-AES Mode for Block-Oriented Storage Devices
Tweakable Block Ciphers
Storage Encryption Requirements
Operation on a Single Block
Operation on a Sector
7.8 Format-Preserving Encryption
Motivation
Difficulties in Designing an FPE
Feistel Structure for Format-Preserving Encryption
NIST Methods for Format-Preserving Encryption
7.9 Key Terms, Review Questions, and Problems
CHAPTER
208 CHAPTER 7 / BLOCK CIPHER OPERATION
This chapter continues our discussion of symmetric ciphers. We begin with the topic of
multiple encryption, looking in particular at the most widely used multiple-encryption
scheme: triple DES.
The chapter next turns to the subject of block cipher modes of operation. We
find that there are a number of different ways to apply a block cipher to plaintext, each
with its own advantages and particular applications.
7.1 MULTIPLE ENCRYPTION AND TRIPLE DES
Because of its vulnerability to brute-force attack, DES, once the most widely used
symmetric cipher, has been largely replaced by stronger encryption schemes. Two
approaches have been taken. One approach is to design a completely new algo-
rithm that is resistant to both cryptanalytic and brute-force attacks, of which AES
is a prime example. Another alternative, which preserves the existing investment in
software and equipment, is to use multiple encryption with DES and multiple keys.
We begin by examining the simplest example of this second alternative. We then
look at the widely accepted triple DES (3DES) algorithm.
Double DES
The simplest form of multiple encryption has two encryption stages and two keys
(Figure 7.1a). Given a plaintext P and two encryption keys K1 and K2, ciphertext C is generated as
C = E(K2, E(K1, P))
Decryption requires that the keys be applied in reverse order:
P = D(K1, D(K2, C))
For DES, this scheme apparently involves a key length of 56 * 2 = 112 bits, and should result in a dramatic increase in cryptographic strength. But we need to exam-
ine the algorithm more closely.
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Analyze the security of multiple encryption schemes.
◆ Explain the meet-in-the-middle attack.
◆ Compare and contrast ECB, CBC, CFB, OFB, and counter modes of operation.
◆ Present an overview of the XTS-AES mode of operation.
Hiva-Network.Com
7.1 / MULTIPLE ENCRYPTION AND TRIPLE DES 209
REDUCTION TO A SINGLE STAGE Suppose it were true for DES, for all 56-bit key val- ues, that given any two keys K1 and K2, it would be possible to find a key K3 such that
E(K2, E(K1, P)) = E(K3, P) (7.1)
If this were the case, then double encryption, and indeed any number of stages of
multiple encryption with DES, would be useless because the result would be equiv-
alent to a single encryption with a single 56-bit key.
On the face of it, it does not appear that Equation (7.1) is likely to hold.
Consider that encryption with DES is a mapping of 64-bit blocks to 64-bit blocks.
In fact, the mapping can be viewed as a permutation. That is, if we consider all 264
possible input blocks, DES encryption with a specific key will map each block into a
unique 64-bit block. Otherwise, if, say, two given input blocks mapped to the same
output block, then decryption to recover the original plaintext would be impossible.
Figure 7.1 Multiple Encryption
(3-key)
(2-key)K1
K3 or
(3-key)
(2-key)K1
K3 or
E E
K1
P
K2
C X
Encryption
D D
K1
C
K2
P X
Decryption (a) Double encryption
E D E
K1
P
K2
C A B
Encryption
D E D
K1
C
K2
P
Decryption (b) Triple encryption
B A
210 CHAPTER 7 / BLOCK CIPHER OPERATION
With 264 possible inputs, how many different mappings are there that generate a
permutation of the input blocks? The value is easily seen to be
(264)! = 10347380000000000000000 7 (1010 20
)
On the other hand, DES defines one mapping for each different key, for a total
number of mappings:
256 6 1017
Therefore, it is reasonable to assume that if DES is used twice with different keys, it
will produce one of the many mappings that are not defined by a single application
of DES. Although there was much supporting evidence for this assumption, it was
not until 1992 that the assumption was proven [CAMP92].
MEET-IN-THE-MIDDLE ATTACK Thus, the use of double DES results in a mapping that is not equivalent to a single DES encryption. But there is a way to attack this
scheme, one that does not depend on any particular property of DES but that will
work against any block encryption cipher.
The algorithm, known as a meet-in-the-middle attack, was first described in [DIFF77]. It is based on the observation that, if we have
C = E(K2, E(K1, P))
then (see Figure 7.1a)
X = E(K1, P) = D(K2, C)
Given a known pair, (P, C), the attack proceeds as follows. First, encrypt P for all 256 possible values of K1. Store these results in a table and then sort the table by the values of X. Next, decrypt C using all 256 possible values of K2. As each decryption is produced, check the result against the table for a match. If a match occurs, then
test the two resulting keys against a new known plaintext–ciphertext pair. If the two
keys produce the correct ciphertext, accept them as the correct keys.
For any given plaintext P, there are 264 possible ciphertext values that could be produced by double DES. Double DES uses, in effect, a 112-bit key, so that there
are 2112 possible keys. Therefore, for a given plaintext P, the maximum number of different 112-bit keys that could produce a given ciphertext C is 2112/264 = 248. Thus, the foregoing procedure can produce about 248 false alarms on the first (P, C) pair. A similar argument indicates that with an additional 64 bits of known plaintext
and ciphertext, the false alarm rate is reduced to 248 - 64 = 2-16. Put another way, if the meet-in-the-middle attack is performed on two blocks of known plaintext–
ciphertext, the probability that the correct keys are determined is 1 - 2-16. The result is that a known plaintext attack will succeed against double DES, which has a
key size of 112 bits, with an effort on the order of 256, which is not much more than
the 255 required for single DES.
Triple DES with Two Keys
An obvious counter to the meet-in-the-middle attack is to use three stages of
encryption with three different keys. Using DES as the underlying algorithm,
this approach is commonly referred to as 3DES, or Triple Data Encryption
7.1 / MULTIPLE ENCRYPTION AND TRIPLE DES 211
Algorithm (TDEA). As shown in Figure 7.1b, there are two versions of 3DES;
one using two keys and one using three keys. NIST SP 800-67 (Recommendation for the Triple Data Encryption Block Cipher, January 2012) defines the two-key and three-key versions. We look first at the strength of the two-key version and
then examine the three-key version.
Two-key triple encryption was first proposed by Tuchman [TUCH79]. The
function follows an encrypt-decrypt-encrypt (EDE) sequence (Figure 7.1b):
C = E(K1, D(K2, E(K1, P))) P = D(K1, E(K2, D(K1, C)))
There is no cryptographic significance to the use of decryption for the second
stage. Its only advantage is that it allows users of 3DES to decrypt data encrypted by
users of the older single DES:
C = E(K1, D(K1, E(K1, P))) = E(K1, P) P = D(K1, E(K1, D(K1, C))) = D(K1, C)
3DES with two keys is a relatively popular alternative to DES and has been
adopted for use in the key management standards ANSI X9.17 and ISO 8732.1
Currently, there are no practical cryptanalytic attacks on 3DES. Coppersmith
[COPP94] notes that the cost of a brute-force key search on 3DES is on the order of
2112 ≈ (5 * 1033) and estimates that the cost of differential cryptanalysis suffers an exponential growth, compared to single DES, exceeding 1052.
It is worth looking at several proposed attacks on 3DES that, although not
practical, give a flavor for the types of attacks that have been considered and that
could form the basis for more successful future attacks.
The first serious proposal came from Merkle and Hellman [MERK81]. Their
plan involves finding plaintext values that produce a first intermediate value of
A = 0 (Figure 7.1b) and then using the meet-in-the-middle attack to determine the two keys. The level of effort is 256, but the technique requires 256 chosen plain-
text–ciphertext pairs, which is a number unlikely to be provided by the holder of
the keys.
A known-plaintext attack is outlined in [VANO90]. This method is an im-
provement over the chosen-plaintext approach but requires more effort. The attack
is based on the observation that if we know A and C (Figure 7.1b), then the problem reduces to that of an attack on double DES. Of course, the attacker does not know
A, even if P and C are known, as long as the two keys are unknown. However, the attacker can choose a potential value of A and then try to find a known (P, C) pair that produces A. The attack proceeds as follows.
1. Obtain n (P, C) pairs. This is the known plaintext. Place these in a table (Table 1) sorted on the values of P (Figure 7.2b).
1American National Standards Institute (ANSI): Financial Institution Key Management (Wholesale). From its title, X9.17 appears to be a somewhat obscure standard. Yet a number of techniques specified in this standard have been adopted for use in other standards and applications, as we shall see throughout this book.
212 CHAPTER 7 / BLOCK CIPHER OPERATION
2. Pick an arbitrary value a for A, and create a second table (Figure 7.2c) with en- tries defined in the following fashion. For each of the 256 possible keys K1 = i, calculate the plaintext value Pi such that
Pi = D(i, a)
For each Pi that matches an entry in Table 1, create an entry in Table 2 consist- ing of the K1 value and the value of B that is produced for the (P, C) pair from Table 1, assuming that value of K1:
B = D(i, C)
At the end of this step, sort Table 2 on the values of B.
3. We now have a number of candidate values of K1 in Table 2 and are in a position to search for a value of K2. For each of the 2
56 possible keys K2 = j, calculate the second intermediate value for our chosen value of a:
Bj = D(j, a)
At each step, look up Bj in Table 2. If there is a match, then the corresponding key i from Table 2 plus this value of j are candidate values for the unknown keys (K1, K2). Why? Because we have found a pair of keys (i, j) that produce a known (P, C) pair (Figure 7.2a).
4. Test each candidate pair of keys (i, j) on a few other plaintext–ciphertext pairs. If a pair of keys produces the desired ciphertext, the task is complete. If no pair
succeeds, repeat from step 1 with a new value of a.
Figure 7.2 Known-Plaintext Attack on Triple DES
E D E
i j i
Ci a Bj
(a) Two-key triple encryption with candidate pair of keys
Pi
Pi Ci
(b) Table of n known plaintext–ciphertext
pairs, sorted on P
Bj Key i
(c) Table of intermediate values and candidate
keys
7.2 / ELECTRONIC CODEBOOK 213
For a given known (P, C), the probability of selecting the unique value of a that leads to success is 1/264. Thus, given n (P, C) pairs, the probability of success for a single selected value of a is n/264. A basic result from probability theory is that the expected number of draws required to draw one red ball out of a bin containing n red balls and N - n green balls is (N + 1)/(n + 1) if the balls are not replaced. So the expected number of values of a that must be tried is, for large n,
264 + 1 n + 1
≈ 264
n
Thus, the expected running time of the attack is on the order of
(256) 264
n = 2120 - log2 n
Triple DES with Three Keys
Although the attacks just described appear impractical, anyone using two-key 3DES
may feel some concern. Thus, many researchers now feel that three-key 3DES is the
preferred alternative (e.g., [KALI96a]). In SP 800-57, Part 1 (Recommendation for Key Management—Part 1: General, July 2012) NIST recommends that 2-key 3DES be retired as soon as practical and replaced with 3-key 3DES.
Three-key 3DES is defined as
C = E(K3, D(K2, E(K1, P)))
Backward compatibility with DES is provided by putting K3 = K2 or K1 = K2. One might expect that 3TDEA would provide 56 # 3 = 168 bits of strength. However, there is an attack on 3TDEA that reduces the strength to the work that would be
involved in exhausting a 112-bit key [MERK81].
A number of Internet-based applications have adopted three-key 3DES, in-
cluding PGP and S/MIME, both discussed in Chapter 19.
7.2 ELECTRONIC CODEBOOK
A block cipher takes a fixed-length block of text of length b bits and a key as input and produces a b-bit block of ciphertext. If the amount of plaintext to be encrypted is greater than b bits, then the block cipher can still be used by breaking the plain- text up into b-bit blocks. When multiple blocks of plaintext are encrypted using the same key, a number of security issues arise. To apply a block cipher in a variety of
applications, five modes of operation have been defined by NIST (SP 800-38A). In essence, a mode of operation is a technique for enhancing the effect of a cryp-
tographic algorithm or adapting the algorithm for an application, such as applying
a block cipher to a sequence of data blocks or a data stream. The five modes are
intended to cover a wide variety of applications of encryption for which a block
cipher could be used. These modes are intended for use with any symmetric block
cipher, including triple DES and AES. The modes are summarized in Table 7.1 and
described in this and the following sections.
214 CHAPTER 7 / BLOCK CIPHER OPERATION
The simplest mode is the electronic codebook (ECB) mode, in which plaintext is handled one block at a time and each block of plaintext is encrypted using the
same key (Figure 7.3). The term codebook is used because, for a given key, there is a unique ciphertext for every b-bit block of plaintext. Therefore, we can imagine a gigantic codebook in which there is an entry for every possible b-bit plaintext pat- tern showing its corresponding ciphertext.
For a message longer than b bits, the procedure is simply to break the message into b-bit blocks, padding the last block if necessary. Decryption is performed one block at a time, always using the same key. In Figure 7.3, the plaintext (padded as
necessary) consists of a sequence of b-bit blocks, P1, P2, c , PN; the correspond- ing sequence of ciphertext blocks is C1, C2, c , CN. We can define ECB mode as follows.
ECB C j = E(K, Pj) j = 1, c , N Pj = D(K, Cj) j = 1, c , N
The ECB mode should be used only to secure messages shorter than a single
block of underlying cipher (i.e., 64 bits for 3DES and 128 bits for AES), such as to
encrypt a secret key. Because in most of the cases messages are longer than the en-
cryption block mode, this mode has a minimum practical value.
The most significant characteristic of ECB is that if the same b-bit block of plaintext appears more than once in the message, it always produces the same
ciphertext.
Mode Description Typical Application
Electronic Codebook (ECB) Each block of plaintext bits is
encoded independently using the
same key.
Secure transmission of
single values (e.g., an
encryption key)
Cipher Block Chaining (CBC) The input to the encryption algo-
rithm is the XOR of the next block
of plaintext and the preceding
block of ciphertext.
General-purpose block-
oriented transmission
Authentication
Cipher Feedback (CFB) Input is processed s bits at a time. Preceding ciphertext is used as
input to the encryption algorithm
to produce pseudorandom output,
which is XORed with plaintext to
produce next unit of ciphertext.
General-purpose
stream-oriented
transmission
Authentication
Output Feedback (OFB) Similar to CFB, except that the
input to the encryption algorithm
is the preceding encryption output,
and full blocks are used.
Stream-oriented
transmission over noisy
channel (e.g., satellite
communication)
Counter (CTR) Each block of plaintext is XORed
with an encrypted counter. The
counter is incremented for each
subsequent block.
General-purpose block-
oriented transmission
Useful for high-speed
requirements
Table 7.1 Block Cipher Modes of Operation
7.2 / ELECTRONIC CODEBOOK 215
For lengthy messages, the ECB mode may not be secure. If the message is
highly structured, it may be possible for a cryptanalyst to exploit these regularities.
For example, if it is known that the message always starts out with certain predefined
fields, then the cryptanalyst may have a number of known plaintext–ciphertext pairs
to work with. If the message has repetitive elements with a period of repetition a
multiple of b bits, then these elements can be identified by the analyst. This may help in the analysis or may provide an opportunity for substituting or rearranging blocks.
We now turn to more complex modes of operation. [KNUD00] lists the fol-
lowing criteria and properties for evaluating and constructing block cipher modes of
operation that are superior to ECB:
■ Overhead: The additional operations for the encryption and decryption opera- tion when compared to encrypting and decrypting in the ECB mode.
■ Error recovery: The property that an error in the ith ciphertext block is inher- ited by only a few plaintext blocks after which the mode resynchronizes.
■ Error propagation: The property that an error in the ith ciphertext block is inherited by the ith and all subsequent plaintext blocks. What is meant here is a bit error that occurs in the transmission of a ciphertext block, not a computa-
tional error in the encryption of a plaintext block.
Figure 7.3 Electronic Codebook (ECB) Mode
C1
P1
Encrypt
K
P2
C2
Encrypt
K
P N
CN
Encrypt
K
(a) Encryption
P1
C1
Decrypt
K
C2
P2
Decrypt
K
CN
PN
Decrypt
K
(b) Decryption
216 CHAPTER 7 / BLOCK CIPHER OPERATION
■ Diffusion: How the plaintext statistics are reflected in the ciphertext. Low en- tropy plaintext blocks should not be reflected in the ciphertext blocks. Roughly,
low entropy equates to predictability or lack of randomness (see Appendix F).
■ Security: Whether or not the ciphertext blocks leak information about the plaintext blocks.
7.3 CIPHER BLOCK CHAINING MODE
To overcome the security deficiencies of ECB, we would like a technique in which
the same plaintext block, if repeated, produces different ciphertext blocks. A
simple way to satisfy this requirement is the cipher block chaining (CBC) mode (Figure 7.4). In this scheme, the input to the encryption algorithm is the XOR of the
current plaintext block and the preceding ciphertext block; the same key is used for
each block. In effect, we have chained together the processing of the sequence of
plaintext blocks. The input to the encryption function for each plaintext block bears
no fixed relationship to the plaintext block. Therefore, repeating patterns of b bits are not exposed. As with the ECB mode, the CBC mode requires that the last block
be padded to a full b bits if it is a partial block.
Figure 7.4 Cipher Block Chaining (CBC) Mode
C1
P1
Encrypt
IV
K
P2
C2
Encrypt
K
PN
CN
CN–1
Encrypt
K
(a) Encryption
P1
C1
Decrypt
IV
K
C2
P2
Decrypt
K
CN
PN
CN–1
Decrypt
K
(b) Decryption
7.3 / CIPHER BLOCK CHAINING MODE 217
For decryption, each cipher block is passed through the decryption algorithm.
The result is XORed with the preceding ciphertext block to produce the plaintext
block. To see that this works, we can write
Cj = E(K, [Cj - 1 ⊕ Pj])
Then
D(K, Cj) = D(K, E(K, [Cj - 1 ⊕ Pj])) D(K, Cj) = Cj - 1 ⊕ Pj
Cj - 1 ⊕ D(K, Cj) = Cj - 1 ⊕ Cj - 1 ⊕ Pj = Pj
To produce the first block of ciphertext, an initialization vector (IV) is XORed
with the first block of plaintext. On decryption, the IV is XORed with the output
of the decryption algorithm to recover the first block of plaintext. The IV is a data
block that is the same size as the cipher block. We can define CBC mode as
CBC C1 = E(K, [P1 ⊕ IV])
Cj = E(K, [Pj ⊕ Cj - 1])j = 2, c , N
P1 = D(K, C1) ⊕ IV
Pj = D(K, Cj) ⊕ Cj - 1 j = 2, c , N
The IV must be known to both the sender and receiver but be unpredictable
by a third party. In particular, for any given plaintext, it must not be possible to
predict the IV that will be associated to the plaintext in advance of the generation
of the IV. For maximum security, the IV should be protected against unauthorized
changes. This could be done by sending the IV using ECB encryption. One reason
for protecting the IV is as follows: If an opponent is able to fool the receiver into
using a different value for IV, then the opponent is able to invert selected bits in the
first block of plaintext. To see this, consider
C1 = E(K, [IV ⊕ P1]) P1 = IV ⊕ D(K, C1)
Now use the notation that X[i] denotes the ith bit of the b-bit quantity X. Then
P1[i] = IV[i] ⊕ D(K, C1)[i]
Then, using the properties of XOR, we can state
P1[i]′ = IV[i]′ ⊕ D(K, C1)[i]
where the prime notation denotes bit complementation. This means that if an oppo-
nent can predictably change bits in IV, the corresponding bits of the received value
of P1 can be changed. For other possible attacks based on prior knowledge of IV, see [VOYD83].
So long as it is unpredictable, the specific choice of IV is unimportant.
SP 800-38A recommends two possible methods: The first method is to apply
the encryption function, under the same key that is used for the encryption of the
plaintext, to a nonce.2 The nonce must be a data block that is unique to each
2NIST SP 800-90 (Recommendation for Random Number Generation Using Deterministic Random Bit Generators) defines nonce as follows: A time-varying value that has at most a negligible chance of repeat- ing, for example, a random value that is generated anew for each use, a timestamp, a sequence number, or some combination of these.
Hiva-Network.Com
218 CHAPTER 7 / BLOCK CIPHER OPERATION
execution of the encryption operation. For example, the nonce may be a counter,
a timestamp, or a message number. The second method is to generate a random
data block using a random number generator.
In conclusion, because of the chaining mechanism of CBC, it is an appropriate
mode for encrypting messages of length greater than b bits. In addition to its use to achieve confidentiality, the CBC mode can be used for
authentication. This use is described in Chapter 12.
7.4 CIPHER FEEDBACK MODE
For AES, DES, or any block cipher, encryption is performed on a block of b bits. In the case of DES, b = 64 and in the case of AES, b = 128. However, it is pos- sible to convert a block cipher into a stream cipher, using one of the three modes
to be discussed in this and the next two sections: cipher feedback (CFB) mode, output feedback (OFB) mode, and counter (CTR) mode. A stream cipher elimi- nates the need to pad a message to be an integral number of blocks. It also can
operate in real time. Thus, if a character stream is being transmitted, each char-
acter can be encrypted and transmitted immediately using a character-oriented
stream cipher.
One desirable property of a stream cipher is that the ciphertext be of the same
length as the plaintext. Thus, if 8-bit characters are being transmitted, each charac-
ter should be encrypted to produce a ciphertext output of 8 bits. If more than 8 bits
are produced, transmission capacity is wasted.
Figure 7.5 depicts the CFB scheme. In the figure, it is assumed that the unit of
transmission is s bits; a common value is s = 8. As with CBC, the units of plaintext are chained together, so that the ciphertext of any plaintext unit is a function of all
the preceding plaintext. In this case, rather than blocks of b bits, the plaintext is divided into segments of s bits.
First, consider encryption. The input to the encryption function is a b-bit shift register that is initially set to some initialization vector (IV). The leftmost (most
significant) s bits of the output of the encryption function are XORed with the first segment of plaintext P1 to produce the first unit of ciphertext C1, which is then transmitted. In addition, the contents of the shift register are shifted left by s bits, and C1 is placed in the rightmost (least significant) s bits of the shift register. This process continues until all plaintext units have been encrypted.
For decryption, the same scheme is used, except that the received ciphertext
unit is XORed with the output of the encryption function to produce the plaintext
unit. Note that it is the encryption function that is used, not the decryption function. This is easily explained. Let MSBs(X) be defined as the most significant s bits of X. Then
C1 = P1 ⊕ MSBs[E(K, IV)]
Therefore, by rearranging terms:
P1 = C1 ⊕ MSBs[E(K, IV)]
The same reasoning holds for subsequent steps in the process.
7.4 / CIPHER FEEDBACK MODE 219
We can define CFB mode as follows.
CFB
I1 = IV
Ij = LSBb - s(Ij - 1) } Cj - 1 j = 2, c, N
Oj = E(K, Ij) j = 1, c, N
Cj = Pj ⊕ MSBs(Oj) j = 1, c, N
I1 = IV
Ij = LSBb - s(Ij - 1) }Cj - 1 j = 2, c, N
Oj = E(K, Ij) j = 1, c, N
Pj = Cj ⊕ MSBs(Oj) j = 1, c, N
Although CFB can be viewed as a stream cipher, it does not conform to the
typical construction of a stream cipher. In a typical stream cipher, the cipher takes
Figure 7.5 s-bit Cipher Feedback (CFB) Mode
C1
IV I1
O1
I1
O1
I2
O2
I2
O2
IN
ON
IN
ON
P1
Encrypt
Select s bits
Discard b – s bits
K
(a) Encryption
CN–1
(b) Decryption
s bits
s bits s bits
C2
P2
Encrypt
Select s bits
Discard b – s bits
K
s bits
s bitsb – s bits Shift register
s bits
CN
PN
Encrypt
Select s bits
Discard b – s bits
K
s bits
s bitsb – s bits Shift register
P1
IV
C1
Encrypt
Select s bits
Discard b – s bits
K
CN–1
s bits C2
s bits CN
s bits
s bits s bits
P2
Encrypt
Select s bits
Discard b – s bits
K s bitsb – s bits
Shift register s bitsb – s bits
Shift register
s bits
PN
Encrypt
Select s bits
Discard b – s bits
K
220 CHAPTER 7 / BLOCK CIPHER OPERATION
as input some initial value and a key and generates a stream of bits, which is then
XORed with the plaintext bits (see Figure 4.1). In the case of CFB, the stream of
bits that is XORed with the plaintext also depends on the plaintext.
In CFB encryption, like CBC encryption, the input block to each forward
cipher function (except the first) depends on the result of the previous forward
cipher function; therefore, multiple forward cipher operations cannot be performed
in parallel. In CFB decryption, the required forward cipher operations can be per-
formed in parallel if the input blocks are first constructed (in series) from the IV
and the ciphertext.
7.5 OUTPUT FEEDBACK MODE
The output feedback (OFB) mode is similar in structure to that of CFB. For OFB, the output of the encryption function is fed back to become the input for encrypting
the next block of plaintext (Figure 7.6). In CFB, the output of the XOR unit is fed
back to become input for encrypting the next block. The other difference is that the
OFB mode operates on full blocks of plaintext and ciphertext, whereas CFB oper-
ates on an s-bit subset. OFB encryption can be expressed as
Cj = Pj ⊕ E(K, Oj - 1)
where
Oj - 1 = E(K, Oj - 2)
Some thought should convince you that we can rewrite the encryption expres-
sion as:
Cj = Pj ⊕ E(K, [Cj - 1 ⊕ Pj - 1])
By rearranging terms, we can demonstrate that decryption works.
Pj = Cj ⊕ E(K, [Cj - 1 ⊕ Pj - 1])
We can define OFB mode as follows.
OFB
I1 = Nonce
Ij = Oj - 1 j = 2, c , N
Oj = E(K, Ij) j = 1, c , N
Cj = Pj ⊕ Oj j = 1, c , N - 1 C N * = PN
* ⊕ MSBu(ON)
I1 = Nonce
Ij = Oj - 1 j = 2, c , N
Oj = E(K, Ij) j = 1, c , N
Pj = Cj ⊕ Oj j = 1, c , N - 1 PN * = C N
* ⊕ MSBu(ON)
Let the size of a block be b. If the last block of plaintext contains u bits (indi- cated by *), with u 6 b, the most significant u bits of the last output block ON are used for the XOR operation; the remaining b - u bits of the last output block are discarded.
As with CBC and CFB, the OFB mode requires an initialization vector. In
the case of OFB, the IV must be a nonce; that is, the IV must be unique to each
execution of the encryption operation. The reason for this is that the sequence of
7.5 / OUTPUT FEEDBACK MODE 221
encryption output blocks, Oi, depends only on the key and the IV and does not de- pend on the plaintext. Therefore, for a given key and IV, the stream of output bits
used to XOR with the stream of plaintext bits is fixed. If two different messages had
an identical block of plaintext in the identical position, then an attacker would be
able to determine that portion of the Oi stream. One advantage of the OFB method is that bit errors in transmission do not
propagate. For example, if a bit error occurs in C1, only the recovered value of P1 is affected; subsequent plaintext units are not corrupted. With CFB, C1 also serves as input to the shift register and therefore causes additional corruption downstream.
The disadvantage of OFB is that it is more vulnerable to a message stream
modification attack than is CFB. Consider that complementing a bit in the cipher-
text complements the corresponding bit in the recovered plaintext. Thus, controlled
Figure 7.6 Output Feedback (OFB) Mode
(a) Encryption
P1
C1
Nonce
Encrypt
K
P2 PN
C2
Encrypt
K
CN
Encrypt
K
(b) Decryption
C1
I1 I2 IN
I1 I2 IN
O1 O2 ON
O1 O2 ON
P1
Nonce
Encrypt
K
C2 CN
P2
Encrypt
K
PN
Encrypt
K
222 CHAPTER 7 / BLOCK CIPHER OPERATION
changes to the recovered plaintext can be made. This may make it possible for an
opponent, by making the necessary changes to the checksum portion of the message
as well as to the data portion, to alter the ciphertext in such a way that it is not de-
tected by an error-correcting code. For a further discussion, see [VOYD83].
OFB has the structure of a typical stream cipher, because the cipher gener-
ates a stream of bits as a function of an initial value and a key, and that stream of
bits is XORed with the plaintext bits (see Figure 4.1). The generated stream that is
XORed with the plaintext is itself independent of the plaintext; this is highlighted
by dashed boxes in Figure 7.6. One distinction from the stream ciphers we discuss
in Chapter 8 is that OFB encrypts plaintext a full block at a time, where typically a
block is 64 or 128 bits. Many stream ciphers encrypt one byte at a time.
7.6 COUNTER MODE
Although interest in the counter (CTR) mode has increased recently with appli- cations to ATM (asynchronous transfer mode) network security and IPsec
(IP security), this mode was proposed in 1979 (e.g., [DIFF79]).
Figure 7.7 depicts the CTR mode. A counter equal to the plaintext block size
is used. The only requirement stated in SP 800-38A is that the counter value must be
different for each plaintext block that is encrypted. Typically, the counter is initial-
ized to some value and then incremented by 1 for each subsequent block (modulo 2b,
where b is the block size). For encryption, the counter is encrypted and then XORed with the plaintext block to produce the ciphertext block; there is no chaining. For
decryption, the same sequence of counter values is used, with each encrypted coun-
ter XORed with a ciphertext block to recover the corresponding plaintext block.
Thus, the initial counter value must be made available for decryption. Given a
sequence of counters T1, T2, c , TN, we can define CTR mode as follows.
CTR Cj = Pj ⊕ E(K, Tj) j = 1, c , N - 1
C N * = PN
* ⊕ MSBu[E(K, TN)]
Pj = Cj ⊕ E(K, Tj) j = 1, c , N - 1
PN * = C N
* ⊕ MSBu[E(K, TN)]
For the last plaintext block, which may be a partial block of u bits, the most significant u bits of the last output block are used for the XOR operation; the re- maining b - u bits are discarded. Unlike the ECB, CBC, and CFB modes, we do not need to use padding because of the structure of the CTR mode.
As with the OFB mode, the initial counter value must be a nonce; that is, T1 must be different for all of the messages encrypted using the same key. Further,
all Ti values across all messages must be unique. If, contrary to this requirement, a counter value is used multiple times, then the confidentiality of all of the plaintext
blocks corresponding to that counter value may be compromised. In particular, if
any plaintext block that is encrypted using a given counter value is known, then
the output of the encryption function can be determined easily from the associated
ciphertext block. This output allows any other plaintext blocks that are encrypted
using the same counter value to be easily recovered from their associated ciphertext
blocks.
7.6 / COUNTER MODE 223
One way to ensure the uniqueness of counter values is to continue to incre-
ment the counter value by 1 across messages. That is, the first counter value of the
each message is one more than the last counter value of the preceding message.
[LIPM00] lists the following advantages of CTR mode.
■ Hardware efficiency: Unlike the three chaining modes, encryption (or decryp- tion) in CTR mode can be done in parallel on multiple blocks of plaintext or
ciphertext. For the chaining modes, the algorithm must complete the computa-
tion on one block before beginning on the next block. This limits the maximum
throughput of the algorithm to the reciprocal of the time for one execution of
block encryption or decryption. In CTR mode, the throughput is only limited
by the amount of parallelism that is achieved.
Figure 7.7 Counter (CTR) Mode
(a) Encryption
P1
C1
Counter 1
Encrypt
K
Counter 2 Counter N
P2 PN
C2
Encrypt
K
CN
Encrypt
K
(b) Decryption
C1
P1
Counter 1
Encrypt
K
Counter 2 Counter N
C2 CN
P2
Encrypt
K
PN
Encrypt
K
224 CHAPTER 7 / BLOCK CIPHER OPERATION
■ Software efficiency: Similarly, because of the opportunities for parallel execu- tion in CTR mode, processors that support parallel features, such as aggressive
pipelining, multiple instruction dispatch per clock cycle, a large number of reg-
isters, and SIMD instructions, can be effectively utilized.
■ Preprocessing: The execution of the underlying encryption algorithm does not depend on input of the plaintext or ciphertext. Therefore, if sufficient
memory is available and security is maintained, preprocessing can be used to
prepare the output of the encryption boxes that feed into the XOR functions,
as in Figure 7.7. When the plaintext or ciphertext input is presented, then
the only computation is a series of XORs. Such a strategy greatly enhances
throughput.
■ Random access: The ith block of plaintext or ciphertext can be processed in random-access fashion. With the chaining modes, block Ci cannot be com- puted until the i - 1 prior blocks are computed. There may be applications in which a ciphertext is stored and it is desired to decrypt just one block; for such
applications, the random access feature is attractive.
■ Provable security: It can be shown that CTR is at least as secure as the other modes discussed in this chapter.
■ Simplicity: Unlike ECB and CBC modes, CTR mode requires only the imple- mentation of the encryption algorithm and not the decryption algorithm. This
matters most when the decryption algorithm differs substantially from the en-
cryption algorithm, as it does for AES. In addition, the decryption key schedul-
ing need not be implemented.
Note that, with the exception of ECB, all of the NIST-approved block ci-
pher modes of operation involve feedback. This is clearly seen in Figure 7.8. To
highlight the feedback mechanism, it is useful to think of the encryption function
as taking input from an input register whose length equals the encryption block
length and with output stored in an output register. The input register is updated
one block at a time by the feedback mechanism. After each update, the encryp-
tion algorithm is executed, producing a result in the output register. Meanwhile,
a block of plaintext is accessed. Note that both OFB and CTR produce output
that is independent of both the plaintext and the ciphertext. Thus, they are natu-
ral candidates for stream ciphers that encrypt plaintext by XOR one full block at
a time.
7.7 XTS-AES MODE FOR BLOCK-ORIENTED STORAGE DEVICES
In 2010, NIST approved an additional block cipher mode of operation, XTS-AES.
This mode is also an IEEE standard, IEEE Std 1619-2007, which was developed
by the IEEE Security in Storage Working Group (P1619). The standard describes
a method of encryption for data stored in sector-based devices where the threat
model includes possible access to stored data by the adversary. The standard has
received widespread industry support.
7.7 / XTS-AES MODE FOR BLOCK-ORIENTED STORAGE DEVICES 225
Tweakable Block Ciphers
The XTS-AES mode is based on the concept of a tweakable block cipher, intro- duced in [LISK02], which functions in much the same manner as a salt used with
passwords, as described in Chapter 22. The form of this concept used in XTS-AES
was first described in [ROGA04].
Before examining XTS-AES, let us consider the general structure of a tweak-
able block cipher. A tweakable block cipher is one that has three inputs: a plain-
text P, a symmetric key K, and a tweak T; and produces a ciphertext output C. We can write this as C = E(K, T, P). The tweak need not be kept secret. Whereas the
Figure 7.8 Feedback Characteristic of Modes of Operation
Plaintext block
Plaintext block
Encrypt
Input register
Output register
Ciphertext Ciphertext
(a) Cipher block chaining (CBC) mode
Key
Encrypt
Input register
Output register
Key
(b) Cipher feedback (CFB) mode
Plaintext block
Ciphertext
Key
Encrypt
Input register
Output register
(c) Output feedback (OFB) mode
Plaintext block
Ciphertext
Key
Encrypt
Input register
Output register
Counter
(d) Counter (CTR) mode
226 CHAPTER 7 / BLOCK CIPHER OPERATION
purpose of the key is to provide security, the purpose of the tweak is to provide
variability. That is, the use of different tweaks with the same plaintext and same key
produces different outputs. The basic structure of several tweakable clock ciphers
that have been implemented is shown in Figure 7.9. Encryption can be expressed as:
C = H(T) ⊕ E(K, H(T) ⊕ P)
where H is a hash function. For decryption, the same structure is used with the
plaintext as input and decryption as the function instead of encryption. To see that
this works, we can write
H(T) ⊕ C = E(K, H(T) ⊕ P) D[K, H(T) ⊕ C] = H(T) ⊕ P H(T) ⊕ D(K, H(T) ⊕ C) = P
It is now easy to construct a block cipher mode of operation by using a differ-
ent tweak value on each block. In essence, the ECB mode is used but for each block
the tweak is changed. This overcomes the principal security weakness of ECB,
which is that two encryptions of the same block yield the same ciphertext.
Storage Encryption Requirements
The requirements for encrypting stored data, also referred to as “data at rest” dif-
fer somewhat from those for transmitted data. The P1619 standard was designed to
have the following characteristics:
1. The ciphertext is freely available for an attacker. Among the circumstances that lead to this situation:
a. A group of users has authorized access to a database. Some of the records in the database are encrypted so that only specific users can successfully read/
Figure 7.9 Tweakable Block Cipher
K
Hash function
Tj
H(Tj)
Pj
Cj
Encrypt
(a) Encryption
K
Hash function
Tj Cj
Pj
Decrypt
(b) Decryption
Hiva-Network.Com
7.7 / XTS-AES MODE FOR BLOCK-ORIENTED STORAGE DEVICES 227
write them. Other users can retrieve an encrypted record but are unable to
read it without the key.
b. An unauthorized user manages to gain access to encrypted records. c. A data disk or laptop is stolen, giving the adversary access to the encrypted
data.
2. The data layout is not changed on the storage medium and in transit. The en- crypted data must be the same size as the plaintext data.
3. Data are accessed in fixed sized blocks, independently from each other. That is, an authorized user may access one or more blocks in any order.
4. Encryption is performed in 16-byte blocks, independently from other blocks (except the last two plaintext blocks of a sector, if its size is not a multiple of
16 bytes).
5. There are no other metadata used, except the location of the data blocks within the whole data set.
6. The same plaintext is encrypted to different ciphertexts at different locations, but always to the same ciphertext when written to the same location again.
7. A standard conformant device can be constructed for decryption of data en- crypted by another standard conformant device.
The P1619 group considered some of the existing modes of operation for use with
stored data. For CTR mode, an adversary with write access to the encrypted media can
flip any bit of the plaintext simply by flipping the corresponding ciphertext bit.
Next, consider requirement 6 and the use of CBC. To enforce the requirement
that the same plaintext encrypts to different ciphertext in different locations, the IV
could be derived from the sector number. Each sector contains multiple blocks. An
adversary with read/write access to the encrypted disk can copy a ciphertext sec-
tor from one position to another, and an application reading the sector off the new
location will still get the same plaintext sector (except perhaps the first 128 bits).
For example, this means that an adversary that is allowed to read a sector from the
second position but not the first can find the content of the sector in the first posi-
tion by manipulating the ciphertext. Another weakness is that an adversary can flip
any bit of the plaintext by flipping the corresponding ciphertext bit of the previous
block, with the side-effect of “randomizing” the previous block.
Operation on a Single Block
Figure 7.10 shows the encryption and decryption of a single block. The operation in-
volves two instances of the AES algorithm with two keys. The following parameters
are associated with the algorithm.
Key The 256 or 512 bit XTS-AES key; this is parsed as a concatenation of two fields of equal size called Key1 and Key2, such that Ke y = Ke y1 }Ke y2 .
Pj The jth block of plaintext. All blocks except possibly the final block have a length of 128 bits. A plaintext data unit, typically a disk sector, consists of a
sequence of plaintext blocks P1, P2, c , Pm. Cj The jth block of ciphertext. All blocks except possibly the final block have a
length of 128 bits.
228 CHAPTER 7 / BLOCK CIPHER OPERATION
j The sequential number of the 128-bit block inside the data unit.
i The value of the 128-bit tweak. Each data unit (sector) is assigned a tweak value that is a nonnegative integer. The tweak values are assigned
consecutively, starting from an arbitrary nonnegative integer.
a A primitive element of GF(2128) that corresponds to polynomial x (i.e., 0000 c 0102).
aj a multiplied by itself j times, in GF(2128).
⊕ Bitwise XOR.
⊗ Modular multiplication of two polynomials with binary coefficients modulo x128 + x7 + x2 + x + 1. Thus, this is multiplication in GF(2128).
Figure 7.10 XTS-AES Operation on Single Block
Key2
Key1
AES Encrypt
i
T
CC
PP
Pj
Cj
AES Encrypt
(a) Encryption
(b) Decryption
j
Key2
Key1
AES Encrypt
i
T
CC
PP
Cj
Pj
AES Decrypt
j
7.7 / XTS-AES MODE FOR BLOCK-ORIENTED STORAGE DEVICES 229
In essence, the parameter j functions much like the counter in CTR mode. It assures that if the same plaintext block appears at two different positions within a
data unit, it will encrypt to two different ciphertext blocks. The parameter i functions much like a nonce at the data unit level. It assures that, if the same plaintext block
appears at the same position in two different data units, it will encrypt to two differ-
ent ciphertext blocks. More generally, it assures that the same plaintext data unit will
encrypt to two different ciphertext data units for two different data unit positions.
The encryption and decryption of a single block can be described as
XTS-AES block
operation
T = E(K2, i) ⊗a j
PP = P ⊕ T CC = E(K1, PP)
C = CC ⊕ T
T = E(K2, i) ⊗a j
CC = C ⊕ T PP = D(K1, CC)
P = PP ⊕ T
To see that decryption recovers the plaintext, let us expand the last line of both en-
cryption and decryption. For encryption, we have
C = CC ⊕ T = E(K1, PP) ⊕ T = E(K1, P ⊕ T) ⊕ T
and for decryption, we have
P = PP ⊕ T = D(K1, CC) ⊕ T = D(K1, C ⊕ T) ⊕ T
Now, we substitute for C:
P = D(K1, C ⊕ T) ⊕ T = D(K1, [E(K1, P ⊕ T) ⊕ T] ⊕ T) ⊕ T = D(K1, E(K1, P ⊕ T)) ⊕ T = (P ⊕ T) ⊕ T = P
Operation on a Sector
The plaintext of a sector or data unit is organized into blocks of 128 bits. Blocks are
labeled P0, P1, c , Pm. The last block my be null or may contain from 1 to 127 bits. In other words, the input to the XTS-AES algorithm consists of m 128-bit blocks and possibly a final partial block.
For encryption and decryption, each block is treated independently and en-
crypted/decrypted as shown in Figure 7.10. The only exception occurs when the
last block has less than 128 bits. In that case, the last two blocks are encrypted/de-
crypted using a ciphertext-stealing technique instead of padding. Figure 7.11 shows the scheme. Pm - 1 is the last full plaintext block, and Pm is the final plaintext block, which contains s bits with 1 … s … 127. Cm - 1 is the last full ciphertext block, and Cm is the final ciphertext block, which contains s bits. This technique is commonly called ciphertext stealing because the processing of the last block “steals” a tempo-
rary ciphertext of the penultimate block to complete the cipher block.
Let us label the block encryption and decryption algorithms of Figure 7.10 as
Block encryption: XTS-AES-blockEnc(K, Pj, i, j) Block decryption: XTS-AES-blockDec(K, Cj, i, j)
230 CHAPTER 7 / BLOCK CIPHER OPERATION
Then, XTS-AES mode is defined as follows:
XTS-AES mode with null
final block Cj = XTS@AES@blockEnc(K, Pj, i, j) j = 0, c , m - 1
Pj = XTS@AES@blockEnc(K, Cj, i, j) j = 0, c , m - 1
XTS-AES mode with final
block containing s bits
Cj = XTS@AES@blockEnc(K, Pj, i, j) j = 0, c , m - 2 XX = XTS@AES@blockEnc(K, Pm - 1, i, m - 1) CP = LSB128 - s(XX) YY = Pm }CP
Cm - 1 = XTS@AES@blockEnc(K, YY, i, m) Cm = MSBs(XX)
Pj = XTS@AES@blockDec(K, Cj, i, j) j = 0, c , m - 2 YY = XTS@AES@blockDec(K, Cm - 1, i, m - 1) CP = LSB128 - s(YY) XX = Cm }CP
Pm - 1 = XTS@AES@blockDec(K, XX, i, m) Pm = MSBs(YY)
Figure 7.11 XTS-AES Mode
C0
P0
XTS-AES block
encryption
Key
i, 0
C1
P1
XTS-AES block
encryption
Key
i, 1
CP
XX
XX
YY
YY
Cm
CPPmPm–1
XTS-AES block
encryption
Key
i, m–1
Cm–1
Cm–1
XTS-AES block
encryption
Key
i, m
Cm (a) Encryption
(b) Decryption
P0
C0
XTS-AES block
decryption
Key
i, 0
P1
C1
XTS-AES block
decryption
Key
i, 1
CPPm
CPCmCm–1
XTS-AES block
decryption
Key
i, m
Pm–1
Pm–1
XTS-AES block
decryption
Key
i, m–1
Pm
7.8 / FORMAT-PRESERVING ENCRYPTION 231
As can be seen, XTS-AES mode, like CTR mode, is suitable for parallel oper-
ation. Because there is no chaining, multiple blocks can be encrypted or decrypted
simultaneously. Unlike CTR mode, XTS-AES mode includes a nonce (the param-
eter i) as well as a counter (parameter j).
7.8 FORMAT-PRESERVING ENCRYPTION
Format-preserving encryption (FPE) refers to any encryption technique that takes
a plaintext in a given format and produces a ciphertext in the same format. For
example, credit cards consist of 16 decimal digits. An FPE that can accept this type of
input would produce a ciphertext output of 16 decimal digits. Note that the ciphertext
need not be, and in fact is unlikely to be, a valid credit card number. But it will have
the same format and can be stored in the same way as credit card number plaintext.
A simple encryption algorithm is not format preserving, with the exception
that it preserves the format of binary strings. For example, Table 7.2 shows three
types of plaintext for which it might be desired to perform FPE. The third row
shows examples of what might be generated by an FPE algorithm. The fourth row
shows (in hexadecimal) what is produced by AES with a given key.
Motivation
FPE facilitates the retrofitting of encryption technology to legacy applications,
where a conventional encryption mode might not be feasible because it would dis-
rupt data fields/pathways. FPE has emerged as a useful cryptographic tool, whose
applications include financial-information security, data sanitization, and transpar-
ent encryption of fields in legacy databases.
The principal benefit of FPE is that it enables protection of particular data
elements in a legacy database that did not provide encryption of those data ele-
ments, while still enabling workflows that were in place before FPE was in use. With
FPE, as opposed to ordinary AES encryption or TDEA encryption, no database
schema changes and minimal application changes are required. Only applications
that need to see the plaintext of a data element need to be modified and generally
these modifications will be minimal.
Some examples of legacy applications where FPE is desirable:
■ COBOL data-processing applications: Any changes in the structure of a re-
cord requires corresponding changes in all code that references that record
structure. Typical code sizes involve hundreds of modules, each containing
around 5,000–10,000 lines on average.
Credit Card Tax ID Bank Account Number
Plaintext 8123 4512 3456 6780 219-09-9999 800N2982K-22
FPE 8123 4521 7292 6780 078-05-1120 709G9242H-35
AES (hex) af411326466add24 c86abd8aa525db7a
7b9af4f3f218ab25
07c7376869313afa
9720ec7f793096ff
d37141242e1c51bd
Table 7.2 Comparison of Format-Preserving Encryption and AES
232 CHAPTER 7 / BLOCK CIPHER OPERATION
■ Database applications: Fields that are specified to take only character strings
cannot be used to store conventionally encrypted binary ciphertext. Base64
encoding of such binary ciphertext is not always feasible without increase in
data lengths, requiring augmentation of corresponding field lengths.
■ FPE-encrypted characters can be significantly compressed for efficient trans-
mission. This cannot be said about AES-encrypted binary ciphertext.
Difficulties in Designing an FPE
A general-purpose standardized FPE should meet a number of requirements:
1. The ciphertext is of the same length and format as the plaintext.
2. It should be adaptable to work with a variety of character and number types. Examples include decimal digits, lowercase alphabetic characters, and the full
character set of a standard keyboard or international keyboard.
3. It should work with variable plaintext lengths.
4. Security strength should be comparable to that achieved with AES.
5. Security should be strong even for very small plaintext lengths.
Meeting the first requirement is not at all straightforward. As illustrated in
Table 7.2, a straightforward encryption with AES yields a 128-bit binary block that
does not resemble the required format. Also, a standard symmetric block cipher is
not easily adaptable to produce an FPE.
Consider a simple example. Assume that we want an algorithm that can en-
crypt decimal digit strings of maximum length of 32 digits. The input to the algo-
rithm can be stored in 16 bytes (128 bits) by encoding each digit as four bits and
using the corresponding binary value for each digit (e.g., 6 is encoded as 0101).
Next, we use AES to encrypt the 128-bit block, in the following fashion:
1. The plaintext input X is represented by the string of 4-bit decimal digits X[1] . . . X[16]. If the plaintext is less than 16 digits long, it is padded out to the left (most significant) with zeros.
2. Treating X as a 128-bit binary string and using key K, form ciphertext Y = AESK(X).
3. Treat Y as a string of length 16 of 4-bit elements.
4. Some of the entries in Y may have values greater than 9 (e.g., 1100). To gener- ate ciphertext Z in the required format, calculate
Z[i] = Y[i] mod 10, 1 … i … 16
This generates a ciphertext of 16 decimal digits, which conforms to the de-
sired format. However, this algorithm does not meet the basic requirement of
any encryption algorithm of reversibility. It is impossible to decrypt Z to recover the original plaintext X because the operation is one-way; that is, it is a many- to-one function. For example, 12 mod 10 = 2 mod 10 = 2. Thus, we need to de- sign a reversible function that is both a secure encryption algorithm and format
preserving.
7.8 / FORMAT-PRESERVING ENCRYPTION 233
A second difficulty in designing an FPE is that some of the input strings are
quite short. For example, consider the 16-digit credit card number (CCN). The first
six digits provide the issuer identification number (IIN), which identifies the insti-
tution that issued the card. The final digit is a check digit to catch typographical
errors or other mistakes. The remaining nine digits are the user’s account number.
However, a number of applications require that the last four digits be in the clear
(the check digit plus three account digits) for applications such as credit card re-
ceipts, which leaves only six digits for encryption. Now suppose that an adversary
is able to obtain a number of plaintext/ciphertext pairs. Each such pair corresponds
to not just one CCN, but multiple CCNs that have the same middle six digits. In a
large database of credit card numbers, there may be multiple card numbers with
the same middle six digits. An adversary may be able to assemble a large diction-
ary mapping known as six-digit plaintexts to their corresponding ciphertexts. This
could be used to decrypt unknown ciphertexts from the database. As pointed out
in [BELL10a], in a database of 100 million entries, on average about 100 CCNs
will share any given middle-six digits. Thus, if the adversary has learned k CCNs and gains access to such a database, the adversary can decrypt approximately
100k CCNs. The solution to this second difficulty is to use a tweakable block cipher; this
concept is described in Section 7.7. For example, the tweak for CCNs could be the
first two and last four digits of the CCN. Prior to encryption, the tweak is added,
digit-by-digit mod 10, to the middle six-digit plaintext, and the result is then en-
crypted. Two different CCNs with identical middle six digits will yield different
tweaked inputs and therefore different ciphertexts. Consider the following:
CCN Tweak Plaintext Plaintext + Tweak
4012 8812 3456 1884 401884 123456 524230
5105 1012 3456 6782 516782 123456 639138
Two CCNs with the same middle six digits have different tweaks and there-
fore different values to the middle six digits after the tweak is added.
Feistel Structure for Format-Preserving Encryption
As the preceding discussion shows, the challenge with FPE is to design an algo-
rithm for scrambling the plaintext that is secure, preserves format, and is reversible.
A number of approaches have been proposed in recent years [ROGA10, BELL09]
for FPE algorithms. The majority of these proposals use a Feistel structure.
Although IBM introduced this structure with their Lucifer cipher [SMIT71] almost
half a century ago, it remains a powerful basis for implementing ciphers.
This section provides a general description of how the Feistel structure can
be used to implement an FPE. In the following section, we look at three specific
Feistel-based algorithms that are in the process of receiving NIST approval.
ENCRYPTION AND DECRYPTION Figure 7.12 shows the Feistel structure used in all of the NIST algorithms, with encryption shown on the left-hand side and decryption
on the right-hand side. The structure in Figure 7.12 is the same as that shown in
234 CHAPTER 7 / BLOCK CIPHER OPERATION
Figure 4.3 but, to simplify the presentation, it is untwisted, not illustrating the swap
that occurs at the end of each round.
The input to the encryption algorithm is a plaintext character string of
n = u + v characters. If n is even, then u = v, otherwise u and v differ by 1. The two parts of the string pass through an even number of rounds of processing to
produce a ciphertext block of n characters and the same format as the plaintext. Each round i has inputs Ai and Bi, derived from the preceding round (or plaintext for round 0).
All rounds have the same structure. On even-numbered rounds, a substitution
is performed on the left part (length u) of the data, Ai. This is done by applying the round function FK to the right part (length v) of the data, Bi, and then performing
Figure 7.12 Feistel Structure for Format-Preserving Encryption
Input (plaintext)
Output (ciphertext)
(a) Encryption (b) Decryption
R ou
nd 0
R ou
nd 1
A0
C0
C1
u characters v characters B0
n, T, 0
n, T, 1
A2 B1 B2 C1
+ FK
+
B1 C0 A1 B0
FK
R ou
nd r
–2 R
ou nd
r –1
Ar–2
Cr–2
Br–2
n, T, r–2
n, T, r–1
Ar Br–1 Br Cr–1
+ FK
+
Br–1 Cr–2 Ar–1 Br–2
FK
Output (plaintext)
Input (ciphertext)
R ou
nd r
–1 R
ou nd
r –2
A0 C0
C0
C1
u characters v characters B0 A1
n, T, 0
n, T, 1
A2 C2 B2 A3
– FK
–
B1 A2 A1 C1
FK
R ou
nd 1
R ou
nd 0
Ci–2
Cr–1
Cr–1
n, T, i–2
n, T, r–1
Ar Br
– FK
–
Br–1 Ar Ar–1 Cr–1
Ar–2 Cr–2 Br–2 Ar–1
FK
7.8 / FORMAT-PRESERVING ENCRYPTION 235
a modular addition of the output of FK with Ai. The modular addition function and the selection of modulus are described subsequently. On odd-numbered rounds,
the substitution is done on the right part of the data. FK is a one-way function that
converts the input into a binary string, performs a scrambling transformation on the
string, and then converts the result back into a character string of suitable format
and length. The function has as parameters the secret key K, the plaintext length n, a tweak T, and the round number i.
Note that on even-numbered rounds, FK has an input of v characters, and that the modular addition produces a result of u characters, whereas on odd-numbered rounds, FK has an input of u characters, and that the modular addition produces a result of v characters. The total number of rounds is even, so that the output consists of an A portion of length u concatenated with a B portion of length v, matching the partition of the plaintext.
The process of decryption is essentially the same as the encryption process.
The differences are: (1) the addition function is replaced by a subtraction function
that is its inverse; and (2) the order of the round indices is reversed.
To demonstrate that the decryption produces the correct result, Figure 7.12b
shows the encryption process going down the left-hand side and the decryption pro-
cess going up the right-hand side. The diagram indicates that, at every round, the
intermediate value of the decryption process is equal to the corresponding value of
the encryption process. We can walk through the figure to validate this, starting at
the bottom. The ciphertext is produced at the end of round r - 1 as a string of the form A
r }B r, with Ar and Br having string lengths u and v, respectively. Encryption round r - 1 can be described with the following equations:
Ar = Br - 1 Br = Ar - 1 + FK[Br - 1]
Because we define the subtraction function to be the inverse of the addition
function, these equations can be rewritten:
Br - 1 = Ar Ar - 1 = Br - FK[Br - 1]
It can be seen that the last two equations describe the action of round 0 of the
decryption function, so that the output of round 0 of decryption equals the input
of round r - 1 of encryption. This correspondence holds all the way through the r iterations, as is easily shown.
Note that the derivation does not require that F be a reversible function. To
see this, take a limiting case in which F produces a constant output (e.g., all ones)
regardless of the values of its input. The equations still hold.
CHARACTER STRINGS The NIST algorithms, and the other FPE algorithms that have been proposed, are used with plaintext consisting of a string of elements, called
characters. Specifically, a finite set of two or more symbols is called an alphabet, and the elements of an alphabet are called characters. A character string is a finite sequence of characters from an alphabet. Individual characters may repeat in the
string. The number of different characters in an alphabet is called the base, also
Hiva-Network.Com
236 CHAPTER 7 / BLOCK CIPHER OPERATION
referred to as the radix of the alphabet. For example, the lowercase English alpha- bet a, b, c, . . . has a radix, or base, of 26. For purposes of encryption and decryption,
the plaintext alphabet must be converted to numerals, where a numeral is a non- negative integer that is less than the base. For example, for the lowercase alphabet,
the assignment could be characters a, b, c, . . . , z map into 0, 1, 2, . . . , 25.
A limitation of this approach is that all of the elements in a plaintext format
must have the same radix. So, for example, an identification number that consists
of an alphabetic character followed by nine numeric digits cannot be handled in
format-preserving fashion by the FPEs that have been implemented so far.
The NIST document defines notation for specifying these conversions
(Table 7.3a). To begin, it is assumed that the character string is represented by
a numeral string. To convert a numeral string X into a number x, the function NUMradix(X) is used. Viewing X as the string X[1] . . . X [m] with the most signifi- cant numeral first, the function is defined as
NUMradix(X) = a m
i = 1 X[i] radixm - i = a
m - 1
i = 0 X[m - i] radixi
Observe that 0 … NUMradix(X) 6 radixm and that 0 … X[i] 6 radix.
[x]s Converts an integer into a byte string; it is the string of s bytes that encodes the number x, with 0 … x 6 28s. The equivalent notation is STR28s(x).
LEN(X) Length of the character string X.
NUMradix(X) Converts strings to numbers. The number that the numeral string X represents in base radix, with the most significant character first. In other words, it is the nonnegative integer less than radixLEN(X) whose most-significant-character-first representation in base radix is X.
PRFK(X) A pseudorandom function that produces a 128-bit output with X as the input, using encryption key K.
STRradix m (x) Given a nonnegative integer x less than radixm, this function produces a repre-
sentation of x as a string of m characters in base radix, with the most significant character first.
[i .. j] The set of integers between two integers i and j, including i and j.
X[i .. j] The substring of characters of a string X from X[i] to X[j], including X[i] and X[j].
REV(X) Given a bit string, X, the string that consists of the bits of X in reverse order.
(a) Notation
radix The base, or number of characters, in a given plaintext alphabet.
tweak Input parameter to the encryption and decryption functions whose confidentiality is not protected by the mode.
tweakradix The base for tweak strings
minlen Minimum message length, in characters.
maxlen Maximum message length, in characters.
maxTlen Maximum tweak length
(b) Parameters
Table 7.3 Notation and Parameters Used in FPE Algorithms
7.8 / FORMAT-PRESERVING ENCRYPTION 237
For example, consider the string zaby in radix 26, which converts to the
numeral string 25 0 1 24. This converts to the number x = (25 * 263) + (1 * 261) + 2 4 = 4 3 9 4 5 0 . To go in the opposite direction and convert from a number x 6 radixm to a numeral string X of length m, the function STRradixm (x) is used:
STRradix m (x) = X[1] c X[m], where
X[i] = j x radixm - i
kmod radix, i = 1, c, m With the mapping of characters to numerals and the use of the NUM func-
tion, a plaintext character string can be mapped to a number and stored as an
unsigned integer. We would like to treat this unsigned integer as a bit string that
can be input to a bit-scrambling algorithm in FK. However, different platforms store
unsigned integers differently, some in little-endian and some in big-endian fashion.
So one more step is needed. By the definition of the STR function, STR2 8s(x) will
generate a bit string of length 8s, equivalently a byte string of length s, which is a binary integer with the most significant bit first, regardless of how x is stored as an unsigned integer. For convenience the following notation is used: [x]s = STR2
8s(x). Thus, [NUMradix(X)]
s will convert the character string X into an unsigned integer and then convert that to a byte string of length s bytes with the most significant bit first.
Continuing, the preceding example should help clarify the issues involved.
Character string “zaby”
Numeral string X representation of character string
25 0 1 24
Convert X to number x = NUM26(X)
decimal: 439450 hex: 6B49A binary: 1101011010010011010
x stored on big-endian byte order machine as a 32-bit unsigned
integer
hex: 00 06 B4 9A binary: 00000000000001101011010010011010
x stored on little-endian byte order machine as a 32-bit unsigned
integer
hex: 9A B4 06 00 binary: 10011010101101000000011000000000
Convert x, regardless of endian format, to a bit string of length
32 bits (4 bytes), expressed as [x]4
00000000000001101011010010011010
THE FUNCTION FK We can now define in general terms the function FK. The core of FK is some type of randomizing function whose input and output are bit
strings. For convenience, the strings should be multiples of 8 bits, forming byte
strings. Define m to be u for even rounds and v for odd rounds; this specifies the desired output character string length. Define b to be the number of bytes needed to store the number representing a character string of m bytes. Then the
238 CHAPTER 7 / BLOCK CIPHER OPERATION
round, including FK, consists of the following general steps (A and B refer to Ai and Bi for round i):
1. Q d [NUMradix(B)]b Converts numeral string X into byte string Q of length b bytes.
2. Y d RAN[Q] A pseudorandom function PRNF that produces a pseudorandom byte string Y as a function of the bits of Q.
3. y d NUM2(Y) Converts Y into unsigned integer.
4. c d (NUMradix(A) + y) mod radixm Converts numeral string A into an integer and adds to y, modulo radixm.
5. C d STRradixm (c) Converts c into a numeral string C of length m. 6. A d B;
B d C Completes the round by placing the unchanged
value of B from the preceding round into A, and placing C into B.
Steps 1 through 3 constitute the round function FK. Step 3 is presented with Y, which is an unstructured bit string. Because different platforms may store unsigned
integers using different word lengths and endian conventions, it is necessary to per-
form NUM2(Y) to get an unsigned integer y. The stored bit sequence for y may or may not be identical to the bit sequence for Y.
As mentioned, the pseudorandom function in step 2 need not be reversible. Its
purpose is to provide a randomized, scrambled bit string. For DES, this is achieved
by using fixed S-boxes, as described in Appendix S. Virtually all FPE schemes that
use the Feistel structure use AES as the basis for the scrambling function to achieve
stronger security.
RELATIONSHIP BETWEEN RADIX, MESSAGE LENGTH, AND BIT LENGTH Consider a numeral string X of length len and base radix. If we convert this to a number x = NUMradix(X), then the maximum value of x is radix
len - 1. The number of bits needed to encode x is
bitlen = <LOG2(radixlen)= = <lenLOG2(radix)= Observe that an increase in either radix or len increases bitlen. Often, we want
to limit the value of bitlen to some fixed upper limit, for example, 128 bits, which is the size of the input to AES encryption. We also want the FPE to handle a variety of
radix values. The typical FPE, and all of those discussed subsequently, allow a given
range of radix values and then define a maximum character string length in order to
provide the algorithm with a fixed value of bitlen. Let the range of radix values be from 2 to maxradix, and the maximum allowable character string value be maxlen. Then the following relationship holds:
maxlen … :bitlen/LOG2(radix);, or equivalently maxlen … :bitlen * LOGradix(2);
For example, for a radix of 10, maxlen … :0.3 * bitlen;; for a radix of 26, maxlen … :0.21 * bitlen;. The larger the radix, the smaller the maximum charac- ter length for a given bit length.
7.8 / FORMAT-PRESERVING ENCRYPTION 239
NIST Methods for Format-Preserving Encryption
In 2013, NIST issued SP 800-38G: Recommendation for Block Cipher Modes of Operation: Methods for Format-Preserving Encryption. This Recommendation specifies three methods for format-preserving encryption, called FF1, FF2, and FF3.
The three methods all use the Feistel structure shown in Figure 7.12. They employ
somewhat different round functions FK, which are built using AES. Important dif- ferences are the following:
■ FF1 supports the greatest range of lengths for the plaintext character string
and the tweak. To achieve this, the round function uses a cipher-block-chaining
(CBC) style of encryption, whereas FF2 and FF3 employ simple electronic
codebook (ECB) encryption.
■ FF2 uses a subkey generated from the encryption key and the tweak, whereas
FF1 and FF3 use the encryption key directly. The use of a subkey may help
protect the original key from side-channel analysis, which is an attack based
on information gained from the physical implementation of a cryptosystem,
rather than brute force or cryptanalysis. Examples of such attacks are attempts
to deduce key bits based on power consumption or execution time.
■ FF3 offers the lowest round count, eight, compared to ten for FF1 and FF2,
and is the least flexible in the tweaks that it supports.
ALGORITHM FF1 Algorithm FF1 was submitted to NIST as a proposed FPE mode [BELL10a, BELL10b] with the name FFX[Radix]. FF1 uses a pseudorandom func-
tion PRFK(X) that produces a 128-bit output with inputs X that is a multiple of 128 bits and encryption key K (Figure 7.13). In essence, PRFK(X) use CBC encryption (Figure 7.4) with X as the plaintext input, encryption key K, and an initial vector (IV) of all zeros. The output is the last block of ciphertext produced. This is also
Figure 7.13 Algorithm PRF(X)
Prerequisites:
Approved, 128-bit block cipher, CIPH;
Key, K, for the block cipher;
Input:
Nonempty bit string, X, such that LEN(X) is a multiple of 128. Output: 128-bit block, Y
Steps:
1. Let m = LEN(X)/128. 2. Partition X into m 128-bit blocks X1, c , Xm, so that X = X1 } c }Xm 3. Let Y0 = [0]
16
4. For j from 1 to m:
4.i let Yj = CIPHK(Yj - 1 ⊕ Xj). 6. Return Ym.
240 CHAPTER 7 / BLOCK CIPHER OPERATION
equivalent to the message authentication code known as CBC-MAC, or CMAC,
described in Chapter 12.
The FF1 encryption algorithm is illustrated in Figure 7.14. The shaded lines
correspond to the function FK. The algorithm has 10 rounds and the following parameters (Table 7.3b):
■ radix ∈ [2 .. 216] ■ radixminlen Ú 100 ■ minlen Ú 2 ■ maxlen 6 232. For the maximum radix value of 216, the maximum bit length to
store the integer value of X is 16 * 232 bits; for the minimum radix value of 2, the maximum bit length to store the integer value of X is 232 bits.
■ maxTlen 6 232
The inputs to the encryption algorithm are a character string X of length n and a tweak T of length t. The tweak is optional in that it may be the empty string.
Prerequisites:
Approved, 128-bit block cipher, CIPH;
Key, K, for the block cipher; Base, radix, for the character alphabet; Range of supported message lengths, [minlen .. maxlen];
Maximum byte length for tweaks, maxTlen.
Inputs:
Character string, X, in base radix of length n such that n ∈ [minlen .. maxlen]; Tweak T, a byte string of byte length t, such that t ∈ [0 .. maxTlen].
Output:
Character string, Y, such that LEN(Y) = n.
Steps:
1. Let u = :n/2;; v = n - u. 2. Let A = X[1 .. u]; B = X[u + 1 .. n]. 3. Let b = <<v LOG2(radix)=/8=; d = 4<b/4= + 4 4. Let P = [1]1 }[2]1 }[1]1 }[radix]3 }[10]1 }[u mod 256]1 }[n]4 }[t]4. 5. For i from 0 to 9:
i. Let Q = T}[0](-t - b - 1) mod 16 }[i]1 }[NUMradix(B)] b.
ii. Let R = PRFK(P}Q). iii. Let S be the first d bytes of the following string of [d/16] 128-bit blocks:
R}CIPHK(R ⊕ [1] 16) }CIPHK(R ⊕ [2]
16) } c }CIPHK(R ⊕ [<d/16= - 1]16). iv. Let y = NUM2(S). v. If i is even, let m = u; else, let m = v.
vi. Let c = (NUMradix(A) + y) mod radixm. vii. Let C = STRradix
m (c). viii. Let A = B.
ix. Let B = C. 6. Return Y = A}B.
Figure 7.14 Algorithm FF1 (FFX[Radix])
7.8 / FORMAT-PRESERVING ENCRYPTION 241
The output is the encrypted character string Y of length n. What follows is a step-by- step description of the algorithm.
1., 2. The input X is split into two substrings A and B. If n is even, A and B are of equal length. Otherwise, B is one character longer than A.
3. The expression <v LOG2(radix)= equals the number of bits needed to encode B, which is v characters long. Encoding B as a byte string, b is the number of bytes in the encoding. The definition of d ensures that the output of the Feistel round function is at least 4 bytes longer than this
encoding of B, which minimizes any bias in the modular reduction in step 5.vi, as explained subsequently.
4. P is a 128-bit (16-byte) block that is a function of radix, u, n, and t. It serves as the first block of plaintext input to the CBC encryption mode
used in 5.ii, and is intended to increase security.
5. The loop through the 10 rounds of encryption.
5.i The tweak, T, the substring, B, and the round number, i, are encoded as a binary string, Q, which is one or more 128-bit blocks in length. To understand this step, first note that the value NUM radix(B) produces a numeral string that represents B in base radix. How this numeral string is formatted and stored is outside the scope of the standard. Then, the value
[NUM radix(B)] b produces the representation of the numerical value of B
as a binary number in a string of b bytes. We also have the length of T is t bytes, and the round number is stored in a single byte. This yields a length of (t + b + 1) bytes. This is padded out with z = ( - t - b - 1) mod 16 bytes. From the rules of modular arithmetic, we know that
(z + t + b + 1) mod 16 = 0. Thus the length of Q is one or more 128- bit blocks.
5.ii The concatenation of P and Q is input to the pseudorandom func- tion PRF to produce a 128-bit output R. This function is the pseudo- random core of the Feistel round function. It scrambles the bits of Bi (Figure 7.12).
5.iii This step either truncates or expands R to a byte string S of length d bytes. That is, if d … 16 bytes, then R is the first d bytes of R. Otherwise the 16-byte R is concatenated with successive encryptions of R XORed with successive constants to produce the shortest string of 16-byte blocks
whose length is greater than or equal to d bytes.
5.iv This step begins the process of converting the results of the scrambling of Bi into a form suitable for combining with Ai. In this step, the d-byte string S is converted into a numeral string in base 2 that represents S. That is, S is represented as a binary string y.
5.v This step determines the length m of the character string output that is required to match the length of the B portion of the round output. For
even-numbered rounds, the length is u characters, and for odd- numbered rounds it is v characters, as shown in Figure 7.12.
5.vi The numerical values of A and y are added modulo radixm. This trun- cates the value of the sum to a value c that can be stored in m characters.
242 CHAPTER 7 / BLOCK CIPHER OPERATION
5.vii This step converts the c into the proper representation C as a string of m characters.
5.viii, 5.ix These steps complete the round by placing the unchanged value of B from the preceding round into A, and placing C into B.
6. After the final round, the result is returned as the concatenation of A and B.
It may be worthwhile to clarify the various uses of the NUM function in FF1.
NUM converts strings with a given radix into integers. In step 5.i, B is a character string in base radix, so NUMradix(B) converts this into an integer, which is stored as a byte string, suitable for encryption in step 5.ii. For step 5.iv, S is a byte string output of an encryption function, which can be viewed a bit string, so NUM 2(S) converts this into an integer.
Finally, a brief explanation of the variable d is in order, which is best ex- plained by example. Let radix = 26 and v = 30 characters. Then b = 18 bytes, and d = 24 bytes. Step 5.ii produces an output R of 16 bytes. We desire a scram- bled output of b bytes to match the input, and so R needs to be padded out. Rather than padding with a constant value such as all zeros, step 5.iii pads out with random
bits. The result, in step 5.iv is a number greater than radixm of fully randomized bits. The use of randomized padding avoids a potential security risk of using a fixed
padding.
ALGORITHM FF2 Algorithm FF2 was submitted to NIST as a proposed FPE mode with the name VAES3 [VANC11]. The encryption algorithm is defined in
Figure 7.15. The shaded lines correspond to the function FK. The algorithm has the
following parameters:
■ radix ∈ [2 .. 28] ■ tweakradix ∈ [2 .. 28] ■ radixminlen Ú 100 ■ minlen Ú 2 ■ maxlen … 2:120/LOG2(radix); if radix is a power of 2. For the maximum radix
value of 28, maxlen … 30; for the minimum radix value of 2, maxlen … 240. In both cases, the maximum bit length to store the integer value of X is 240 bits, or 30 bytes.
■ maxlen … 2:98/LOG2(radix); if radix is a not a power of 2. For the maxi- mum radix value of 255, maxlen … 24; for the minimum radix value of 3, maxlen … 124.
■ maxTlen … :104/LOG2(tweakradix);. For the maximum tweakradix value of 28, maxTlen … 13.
For FF2, the plaintext character alphabet and that of the tweak may be different.
The first two steps of FF2 are the same as FF1, setting values for v, u, A, and B. FF2 proceeds with the following steps:
3. P is a 128-bit (16-byte) block. If there is a tweak, then P is a function of radix, t, n, and the 13-byte numerical value of the tweak. If there is no tweak (t = 0), then P is a function of radix and n. P is used to form an encryption key in step 4.
7.8 / FORMAT-PRESERVING ENCRYPTION 243
4. J is the encryption of P using the input key K.
5. The loop through the 10 rounds of encryption.
5.i B is converted into a 15-byte number, prepended by the round number to form a 16-byte block Q.
5.ii Q is encrypted using the encryption key J to yield Y.
The remaining steps are the same as for FF1. The essential difference is in the
way in which all of the parameters are incorporated into the encryption that takes
place in the block FK. In both cases, the encryption is not simply an encryption of B using key K. For FF1, B is combined with the tweak, the round number, t, n, u, and radix to form a string of multiple 16-byte blocks. Then CBC encryption is used with K to produce a 16-byte output. For FF2, all of the parameters besides B are com- bined to form a 16-byte block, which is then encrypted with K to form the key value J. J is then used as the key for the one-block encryption of B.
The structure of FF2 explains the maximum length restrictions. In step 3, P incorporates the radix, tweak length, the numeral string length, and the tweak into
the calculation. As input to AES, P is restricted to 16 bytes. With a maximum radix value of 28, the radix value can be stored in one byte (byte value 0 corresponds
to 256). The string length n and tweak length t each easily fits into one byte. This leaves a restriction that the value of the tweak should be stored in at most 13 bytes,
Approved, 128-bit block cipher, CIPH;
Key, K, for the block cipher; Base, tweakradix, for the tweak character alphabet; Range of supported message lengths, [minlen .. maxlen]; Maximum supported tweak length, maxTlen.
Inputs:
Numeral string, X, in base radix, of length n such that n ∈ [minlen .. maxlen]; Tweak numeral string, T, in base tweakradix, of length t such that t ∈ [0 .. maxTlen].
Output: Numeral string, Y, such that LEN(Y) = n.
Steps:
1. Let u = :n/2;; v = n - u. 2. Let A = X[1 .. u]; B = X[u + 1 .. n]. 3. If t 7 0, P = [radix]1 } [t]1 } [n]1 } [NUMtweakradix(T)]13; else P = [radix]1 } [0]1 } [n]1 } [0]13. 4. Let J = CIPHK(P). 5. For i from 0 to 9:
i. Let Q d [i]1 } [NUMradix(B)]15 ii. Let Y d CIPHJ(Q).
iii. Let y d NUM2(Y). iv. If i is even, let m = u; else, let m = v.
v. Let c = (NUMradix(A) + y) mod radixm. vi. Let C = STRradix
m (c). vii. Let A = B.
viii. Let B = C. 6. Return Y = A}B.
Figure 7.15 Algorithm FF2 (VAES3)
244 CHAPTER 7 / BLOCK CIPHER OPERATION
or 104 bits. The number of bits to store the tweak is LOG2(tweakradix Tlen). This
leads to the restriction maxTlen Ú :104/LOG2(tweakradix);. Similarly step 5i incorporates B and the round number into a 16-byte input to AES, leaving 15 bytes to encode B, or 120 bits, so that the length must be less than or equal to :120/LOG2(radix);. The parameter maxlen refers to the entire block, consisting of partitions A and B, thus maxlen Ú 2:120/LOG2(radix);.
There is a further restriction on maxlen for a radix that is not a power of 2. As explained in [VANC11], when the radix is not a power of 2, modular arithme-
tic causes the value (y mod radixm) to not have uniform distribution in the output space, which can result in a cryptographic weakness.
ALGORITHM FF3 Algorithm FF3 was submitted to NIST as a proposed FPE mode with the name BPS-BC [BRIE10]. The encryption algorithm is illustrated in
Figure 7.16. The shaded lines correspond to the function FK. The algorithm has the
following parameters:
■ radix ∈ [2 .. 216] ■ radixminlen Ú 100 ■ minlen Ú 2
Approved, 128-bit block cipher, CIPH;
Key, K, for the block cipher; Base, radix, for the character alphabet such that radix ∈ [2..216]; Range of supported message lengths, [minlen .. maxlen], such that minlen Ú 2 and maxlen … 2:log radix(296);. Inputs:
Numeral string, X, in base radix of length n such that n ∈ [minlen .. maxlen]; Tweak bit string, T, such that LEN(T) = 64.
Output: Numeral string, Y, such that LEN(Y) = n.
Steps:
1. Let u = <n/2=; v = n - u. 2. Let A = X[1 .. u]; B = X[u + 1 .. n]. 3. Let TL = T[0 .. 31] and TR = T[32 .. 63]. 4. For i from 0 to 7:
i. If i is even, let m = u and W = TR, else let m = v and W = TL. ii. Let P = REV([NUMradix(REV(B))]
12) }[W ⊕ REV([i]4]). iii. Let Y = CIPHK(P). iv. Let y = NUM2(REV(Y)).
v. Let c = (NUMradix(REV(A)) + y) mod radixm. vi. Let C = REV(STRradix
m (c)). vii. Let A = B.
viii. Let B = C. 5. Return A}B.
Figure 7.16 Algorithm FF3 (BPS-BC)
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7.9 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 245
■ maxlen … 2:LOGradix(296);. For the maximum radix value of 216, maxlen … 12; for the minimum radix value of 2, maxlen … 192. In both cases, the maximum bit length to store the integer value of X is 192 bits, or 24 bytes.
■ Tweak length = 64 bits
FF3 proceeds with the following steps:
1., 2. The input X is split into two substrings A and B. If n is even, A and B are of equal length. Otherwise, A is one character longer than B, in contrast to FF1 and FF2, where B is one character longer than A.
3. The tweak is partitioned into a 32-bit left tweak TL and a 32-bit right tweak TR.
4. The loop through the 8 rounds of encryption.
4.i As in FF1 and FF2, this step determines the length m of the character string output that is required to match the length of the B portion of the round output. The step also determines whether TL or TR will be used as W in step 4ii.
4.ii The bits of B are reversed, then NUM radix(B) produces a 12-byte numeral string in base radix; the results are again reversed. A 32-bit encoding of the round number i is stored in a 4-byte unit, which is reversed and then XORed with W. P is formed by concatenating these two results to form a 16-byte block.
4.iii P is encrypted using the encryption key K to yield Y.
4.iv This is similar to step 5.iv in FF1, except that Y is reversed before convert- ing it into a numeral string in base 2.
4.v The numerical values of the reverse of A and y are added modulo radixm. This truncates the value of the sum to a value c that can be stored in m characters.
4.vi This step converts c to a numeral string C.
The remaining steps are the same as for FF1.
7.9 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
Key Terms
block cipher modes of
operation
cipher block chaining mode
(CBC)
cipher feedback mode
(CFB)
ciphertext stealing
counter mode (CTR)
electronic codebook mode
(ECB)
meet-in-the-middle attack
nonce
output feedback mode
(OFB)
Triple DES (3DES)
tweakable block cipher
XTS-AES mode
246 CHAPTER 7 / BLOCK CIPHER OPERATION
Review Questions
7.1 What is triple encryption? 7.2 What is a meet-in-the-middle attack? 7.3 How many keys are used in triple encryption? 7.4 List and briefly define the block cipher modes of operation. 7.5 Why do some block cipher modes of operation only use encryption while others use
both encryption and decryption?
Problems
7.1 You want to build a hardware device to do block encryption in the cipher block chain- ing (CBC) mode using an algorithm stronger than DES. 3DES is a good candidate. Figure 7.17 shows two possibilities, both of which follow from the definition of CBC. Which of the two would you choose: a. For security? b. For performance?
7.2 Can you suggest a security improvement to either option in Figure 7.17, using only three DES chips and some number of XOR functions? Assume you are still limited to two keys.
Figure 7.17 Use of Triple DES in CBC Mode
Pn
K1 E
An-1
An
K2 D
K1 E
Bn-1
Bn Cn-1
Cn
(b) Three-loop CBC
Pn
K1, K2 EDE
Cn-1
Cn
(a) One-loop CBC
7.9 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 247
7.3 The Merkle–Hellman attack on 3DES begins by assuming a value of A = 0 (Figure 7.1b). Then, for each of the 256 possible values of K1, the plaintext P that produces A = 0 is determined. Describe the rest of the algorithm.
7.4 With the ECB mode, if there is an error in a block of the transmitted ciphertext, only the corresponding plaintext block is affected. However, in the CBC mode, this error propagates. For example, an error in the transmitted C1 (Figure 7.4) obviously cor- rupts P1 and P2. a. Are any blocks beyond P2 affected? b. Suppose that there is a bit error in the source version of P1. Through how many
ciphertext blocks is this error propagated? What is the effect at the receiver?
7.5 Is it possible to perform encryption operations in parallel on multiple blocks of plain- text in CBC mode? How about decryption?
7.6 CBC-Pad is a block cipher mode of operation used in the RC5 block cipher, but it could be used in any block cipher. CBC-Pad handles plaintext of any length. The ciphertext is longer then the plaintext by at most the size of a single block. Padding is used to assure that the plaintext input is a multiple of the block length. It is assumed that the original plaintext is an integer number of bytes. This plaintext is padded at the end by from 1 to bb bytes, where bb equals the block size in bytes. The pad bytes are all the same and set to a byte that represents the number of bytes of padding. For example, if there are 8 bytes of padding, each byte has the bit pattern 00001000. Why not allow zero bytes of padding? That is, if the original plaintext is an integer multiple of the block size, why not refrain from padding?
7.7 For the ECB, CBC, and CFB modes, the plaintext must be a sequence of one or more complete data blocks (or, for CFB mode, data segments). In other words, for these three modes, the total number of bits in the plaintext must be a positive multiple of the block (or segment) size. One common method of padding, if needed, consists of a 1 bit followed by as few zero bits, possibly none, as are necessary to complete the final block. It is considered good practice for the sender to pad every message, including messages in which the final message block is already complete. What is the motiva- tion for including a padding block when padding is not needed?
7.8 If a bit error occurs in the transmission of a ciphertext character in 8-bit CFB mode, how far does the error propagate?
7.9 In discussing OFB, it was mentioned that if it was known that two different messages had an identical block of plaintext in the identical position, it is possible to recover the corresponding Oi block. Show the calculation.
7.10 In discussing the CTR mode, it was mentioned that if any plaintext block that is encrypted using a given counter value is known, then the output of the encryption function can be determined easily from the associated ciphertext block. Show the calculation.
7.11 Padding may not always be appropriate. For example, one might wish to store the encrypted data in the same memory buffer that originally contained the plaintext. In that case, the ciphertext must be the same length as the original plaintext. We saw the use of ciphertext stealing in the case of XTS-AES to deal with partial blocks. Figure 7.18a shows the use of ciphertext stealing to modify CBC mode, called CBC-CTS. a. Explain how it works. b. Describe how to decrypt Cn - 1 and Cn.
7.12 Figure 7.18b shows an alternative to CBC-CTS for producing ciphertext of equal length to the plaintext when the plaintext is not an integer multiple of the block size. a. Explain the algorithm. b. Explain why CBC-CTS is preferable to this approach illustrated in Figure 7.18b.
7.13 Draw a figure similar to those of Figure 7.8 for XTS-AES mode. 7.14 Work out the following problems from first principles without converting to binary
and counting the bits. Then compare with the formulae presented for encoding a
248 CHAPTER 7 / BLOCK CIPHER OPERATION
character string into an integer, and vice-versa, in the specified radix. (Hint: Consider the next-lower and next-higher power of two for each integer.) a. How many bits are exactly required to encode the following integers? (The num-
ber shown as an integer’s subscript refers to the radix of that integer.) i. 2 0 4 7 1 0
ii. 2 0 4 8 1 0 iii. 3 2 7 6 7 1 0 iv. 3 2 7 6 8 1 0 v. 3 2 7 6 7 1 6
vi. 3 2 7 6 8 1 6 vii. 5 3 7 F1 6
viii. 2 9 4 3 1 1 0 b. Exactly how many bytes are required to represent the numbers in (a) above?
7.15 a. In radix-26, write down the numeral string X for each of the following character strings, followed by the number of “digits” (i.e., the length of the numeral string) in each case. i. “hex”
ii. “cipher” iii. “not” iv. “symbol”
Figure 7.18 Block Cipher Modes for Plaintext not a Multiple of Block Size
IV P1
C1
K K K K
+
PN-2
CN-2
CN-3 +
Encrypt Encrypt Encrypt Encrypt
Encrypt Encrypt
(a) Ciphertext stealing mode
(b) Alternative method
Encrypt Encrypt
+
CN X
PN-1
+
CN-1
PN 00…0
IV P1
(bb bits)
C1 (bb bits)
K K K K
+
PN-2 (bb bits)
CN-2 (bb bits)
CN-3 +
select leftmost
j bits
PN-1 (bb bits)
CN-1 (bb bits)
+
PN (j bits)
CN (j bits)
+
7.9 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 249
b. For each case of problem (a), determine the number x = NUM26(X) c. Determine the byte form [x] for each number x computed in problem (b). d. What is the smallest power of the radix (26) that is greater than each of the nu-
merical strings determined in (b)? e. Is it related to the length of the numeral string in each case, in problem (a)? If so,
what is this relationship?
7.16 Refer to algorithms FF1 and FF2. a. For step 1, for each algorithm, u d :n/2; and v d <n - u=. Show that for any
three integers x, y, and n: if x = :n/2; and y = <n - x=, then:
i. Either x = n/2, or x = (n - 1)/2. ii. Either y = n/2, or y = (n + 1)/2.
iii. x … y. (Under what condition is x = y?) b. What is the significance of result in the previous sub-problem (iii), in terms of the
lengths u and v of the left and right half-strings, respectively? 7.17 In step 3 of Algorithm FF1, what do b and d represent? What is the unit of measure-
ment (bits, bytes, digits, characters) of each of these quantities?
7.18 In the inputs to algorithms FF1, FF2, and FF3, why are the specified radix ranges important? For example, why should radix ∈ [0..28] for Algorithm FF2, or radix ∈ [2..216] in the case of Algorithm FF3?
Programming Problems
7.19 Create software that can encrypt and decrypt in cipher block chaining mode using one of the following ciphers: affine modulo 256, Hill modulo 256, S-DES, DES.
Test data for S-DES using a binary initialization vector of 1010 1010. A binary plain- text of 0000 0001 0010 0011 encrypted with a binary key of 01111 11101 should give a binary plaintext of 1111 0100 0000 1011. Decryption should work correspondingly.
7.20 Create software that can encrypt and decrypt in 4-bit cipher feedback mode using one of the following ciphers: additive modulo 256, affine modulo 256, S-DES;
or 8-bit cipher feedback mode using one of the following ciphers: 2 * 2 Hill modulo 256.
Test data for S-DES using a binary initialization vector of 1010 1011. A binary plain- text of 0001 0010 0011 0100 encrypted with a binary key of 01111 11101 should give a binary plaintext of 1110 1100 1111 1010. Decryption should work correspondingly.
7.21 Create software that can encrypt and decrypt in counter mode using one of the follow- ing ciphers: affine modulo 256, Hill modulo 256, S-DES.
Test data for S-DES using a counter starting at 0000 0000. A binary plaintext of 0000 0001 0000 0010 0000 0100 encrypted with a binary key of 01111 11101 should give a binary plaintext of 0011 1000 0100 1111 0011 0010. Decryption should work cor- respondingly.
7.22 Implement a differential cryptanalysis attack on 3-round S-DES.
250250
8.1 Principles of Pseudorandom Number Generation
The Use of Random Numbers
TRNGs, PRNGs, and PRFs
PRNG Requirements
Algorithm Design
8.2 Pseudorandom Number Generators
Linear Congruential Generators
Blum Blum Shub Generator
8.3 Pseudorandom Number Generation Using a Block Cipher
PRNG Using Block Cipher Modes of Operation
ANSI X9.17 PRNG
NIST CTR_DRBG
8.4 Stream Ciphers
8.5 RC4
Initialization of S
Stream Generation
Strength of RC4
8.6 True Random Number Generators
Entropy Sources
Comparison of PRNGs and TRNGs
Conditioning
Health Testing
Intel Digital Random Number Generator
8.7 Key Terms, Review Questions, and Problems
CHAPTER
Random Bit Generation and Stream Ciphers
RANDOM BIT GENERATION AND STREAM CIPHERS 251
An important cryptographic function is the generation of random bit streams. Random
bits streams are used in a wide variety of contexts, including key generation and
encryption. In essence, there are two fundamentally different strategies for generating
random bits or random numbers. One strategy, which until recently dominated in cryp-
tographic applications, computes bits deterministically using an algorithm. This class of
random bit generators is known as pseudorandom number generators (PRNGs) or
deterministic random bit generators (DRBGs). The other strategy is to produce bits
non-deterministically using some physical source that produces some sort of random
output. This latter class of random bit generators is known as true random number
generators (TRNGs) or non-deterministic random bit generators (NRBGs).
The chapter begins with an analysis of the basic principles of PRNGs. Next, we
look at some common PRNGs, including PRNGs based on the use of a symmetric
block cipher. The chapter then moves on to the topic of symmetric stream ciphers,
which are based on the use of a PRNG. The chapter next examines the most important
stream cipher, RC4.
The remainder of the chapter is devoted to TRNGs. We look first at the basic
principles and structure of TRNGs, and then examine a specific product, the Intel
Digital Random Number Generator.
Throughout this chapter, reference is made to four important NIST documents:
■ SP 80 0-90A (Recommendation for Random Number Generation Using Deterministic Random Bit Generators, January 2012) covers DRNGs.
■ SP 800-90B (Recommendation for the Entropy Sources Used for Random Bit Generation, August 2012) covers criteria for entropy sources (ES), the devices from which we get unpredictable randomness and NRNGs.
■ SP 80 0-90C (Recommendation for Random Bit Generator (RBG) Constructions, August 2012) discusses how to combine the entropy sources in 90B with the DRNG’s from 90A to provide large quantities of unpredictable
bits for cryptographic applications.
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Explain the concepts of randomness and unpredictability with respect to random numbers.
◆ Understand the differences among true random number generators, pseudorandom number generators, and pseudorandom functions.
◆ Present an overview of requirements for pseudorandom number generators.
◆ Explain how a block cipher can be used to construct a pseudorandom number generator.
◆ Present an overview of stream ciphers and RC4.
◆ Explain the significance of skew.
252 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS
■ SP 800-22 (A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications, April 2010) discusses the selection and testing of NRBGs and DRBGs.
These specifications have heavily influenced the implementation of random bit
generators in industry both in the U.S. and worldwide.
8.1 PRINCIPLES OF PSEUDORANDOM NUMBER GENERATION
Random numbers play an important role in the use of encryption for various net-
work security applications. In this section, we provide a brief overview of the use
of random numbers in cryptography and network security and then focus on the
principles of pseudorandom number generation.
The Use of Random Numbers
A number of network security algorithms and protocols based on cryptography
make use of random binary numbers. For example,
■ Key distribution and reciprocal (mutual) authentication schemes, such as
those discussed in Chapters 14 and 15. In such schemes, two communicating
parties cooperate by exchanging messages to distribute keys and/or authen-
ticate each other. In many cases, nonces are used for handshaking to prevent
replay attacks. The use of random numbers for the nonces frustrates an oppo-
nent’s efforts to determine or guess the nonce, in order to repeat an obsolete
transaction.
■ Session key generation. We will see a number of protocols in this book where a
secret key for symmetric encryption is generated for use for a particular trans-
action (or session) and is valid for a short period of time. This key is generally
called a session key.
■ Generation of keys for the RSA public-key encryption algorithm (described
in Chapter 9).
■ Generation of a bit stream for symmetric stream encryption (described in this
chapter).
These applications give rise to two distinct and not necessarily compatible
requirements for a sequence of random numbers: randomness and unpredictability.
RANDOMNESS Traditionally, the concern in the generation of a sequence of alleg- edly random numbers has been that the sequence of numbers be random in some
well-defined statistical sense. The following two criteria are used to validate that a
sequence of numbers is random:
■ Uniform distribution: The distribution of bits in the sequence should be uniform; that is, the frequency of occurrence of ones and zeros should be
approximately equal.
■ Independence: No one subsequence in the sequence can be inferred from the others.
8.1 / PRINCIPLES OF PSEUDORANDOM NUMBER GENERATION 253
Although there are well-defined tests for determining that a sequence of bits
matches a particular distribution, such as the uniform distribution, there is no such
test to “prove” independence. Rather, a number of tests can be applied to demon-
strate if a sequence does not exhibit independence. The general strategy is to apply
a number of such tests until the confidence that independence exists is sufficiently
strong. That is, if each of a number of tests fails to show that a sequence of bits is
not independent, then we can have a high level of confidence that the sequence is in
fact independent.
In the context of our discussion, the use of a sequence of numbers that appear
statistically random often occurs in the design of algorithms related to cryptography.
For example, a fundamental requirement of the RSA public-key encryption scheme
discussed in Chapter 9 is the ability to generate prime numbers. In general, it is
difficult to determine if a given large number N is prime. A brute-force approach would be to divide N by every odd integer less than 2N. If N is on the order, say, of 10150, which is a not uncommon occurrence in public-key cryptography, such a
brute-force approach is beyond the reach of human analysts and their computers.
However, a number of effective algorithms exist that test the primality of a num-
ber by using a sequence of randomly chosen integers as input to relatively simple
computations. If the sequence is sufficiently long (but far, far less than 210150), the primality of a number can be determined with near certainty. This type of approach,
known as randomization, crops up frequently in the design of algorithms. In es-
sence, if a problem is too hard or time-consuming to solve exactly, a simpler, shorter
approach based on randomization is used to provide an answer with any desired
level of confidence.
UNPREDICTABILITY In applications such as reciprocal authentication, session key generation, and stream ciphers, the requirement is not just that the sequence of
numbers be statistically random but that the successive members of the sequence
are unpredictable. With “true” random sequences, each number is statistically inde-
pendent of other numbers in the sequence and therefore unpredictable. Although
true random numbers are used in some applications, they have their limitations,
such as inefficiency, as is discussed shortly. Thus, it is more common to imple-
ment algorithms that generate sequences of numbers that appear to be random. In
this latter case, care must be taken that an opponent not be able to predict future
elements of the sequence on the basis of earlier elements.
TRNGs, PRNGs, and PRFs
Cryptographic applications typically make use of algorithmic techniques for ran-
dom number generation. These algorithms are deterministic and therefore produce
sequences of numbers that are not statistically random. However, if the algorithm is
good, the resulting sequences will pass many tests of randomness. Such numbers are
referred to as pseudorandom numbers. You may be somewhat uneasy about the concept of using numbers generated
by a deterministic algorithm as if they were random numbers. Despite what might be
called philosophical objections to such a practice, it generally works. That is, under
most circumstances, pseudorandom numbers will perform as well as if they were
random for a given use. The phrase “as well as” is unfortunately subjective, but the
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254 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS
use of pseudorandom numbers is widely accepted. The same principle applies in
statistical applications, in which a statistician takes a sample of a population and
assumes that the results will be approximately the same as if the whole population
were measured.
Figure 8.1 contrasts a true random number generator (TRNG) with two forms of pseudorandom number generators. A TRNG takes as input a source that is
effectively random; the source is often referred to as an entropy source. We discuss such sources in Section 8.6. In essence, the entropy source is drawn from the physi-
cal environment of the computer and could include things such as keystroke timing
patterns, disk electrical activity, mouse movements, and instantaneous values of the
system clock. The source, or combination of sources, serve as input to an algorithm
that produces random binary output. The TRNG may simply involve conversion of
an analog source to a binary output. The TRNG may involve additional processing
to overcome any bias in the source; this is discussed in Section 8.6.
In contrast, a PRNG takes as input a fixed value, called the seed, and produces a sequence of output bits using a deterministic algorithm. Quite often, the seed is
generated by a TRNG. Typically, as shown, there is some feedback path by which
some of the results of the algorithm are fed back as input as additional output bits
are produced. The important thing to note is that the output bit stream is deter-
mined solely by the input value or values, so that an adversary who knows the algo-
rithm and the seed can reproduce the entire bit stream.
Figure 8.1 shows two different forms of PRNGs, based on application.
■ Pseudorandom number generator: An algorithm that is used to produce an open-ended sequence of bits is referred to as a PRNG. A common application
for an open-ended sequence of bits is as input to a symmetric stream cipher,
as discussed in Section 8.4. Also, see Figure 4.1a.
Figure 8.1 Random and Pseudorandom Number Generators
Conversion to binary
Source of true
randomness
Random bit stream
(a) TRNG
TRNG = true random number generator PRNG = pseudorandom number generator PRF = pseudorandom function
Deterministic algorithm
Seed
Pseudorandom bit stream
(b) PRNG
Deterministic algorithm
Seed
Pseudorandom value
(c) PRF
Context- specific values
8.1 / PRINCIPLES OF PSEUDORANDOM NUMBER GENERATION 255
■ Pseudorandom function (PRF): A PRF is used to produce a pseudorandom string of bits of some fixed length. Examples are symmetric encryption keys
and nonces. Typically, the PRF takes as input a seed plus some context specific
values, such as a user ID or an application ID. A number of examples of PRFs
will be seen throughout this book, notably in Chapters 17 and 18.
Other than the number of bits produced, there is no difference between a
PRNG and a PRF. The same algorithms can be used in both applications. Both
require a seed and both must exhibit randomness and unpredictability. Further,
a PRNG application may also employ context-specific input. In what follows, we
make no distinction between these two applications.
PRNG Requirements
When a PRNG or PRF is used for a cryptographic application, then the basic
requirement is that an adversary who does not know the seed is unable to determine
the pseudorandom string. For example, if the pseudorandom bit stream is used in
a stream cipher, then knowledge of the pseudorandom bit stream would enable the
adversary to recover the plaintext from the ciphertext. Similarly, we wish to pro-
tect the output value of a PRF. In this latter case, consider the following scenario.
A 128-bit seed, together with some context-specific values, are used to generate a
128-bit secret key that is subsequently used for symmetric encryption. Under nor-
mal circumstances, a 128-bit key is safe from a brute-force attack. However, if the
PRF does not generate effectively random 128-bit output values, it may be possible
for an adversary to narrow the possibilities and successfully use a brute force attack.
This general requirement for secrecy of the output of a PRNG or PRF leads
to specific requirements in the areas of randomness, unpredictability, and the char-
acteristics of the seed. We now look at these in turn.
RANDOMNESS In terms of randomness, the requirement for a PRNG is that the gen- erated bit stream appear random even though it is deterministic. There is no single
test that can determine if a PRNG generates numbers that have the characteristic
of randomness. The best that can be done is to apply a sequence of tests to the
PRNG. If the PRNG exhibits randomness on the basis of multiple tests, then it can
be assumed to satisfy the randomness requirement. NIST SP 800-22 specifies that
the tests should seek to establish the following three characteristics.
■ Uniformity: At any point in the generation of a sequence of random or pseu- dorandom bits, the occurrence of a zero or one is equally likely, that is, the
probability of each is exactly 1/2. The expected number of zeros (or ones) is
n/2, where n = the sequence length. ■ Scalability: Any test applicable to a sequence can also be applied to subse-
quences extracted at random. If a sequence is random, then any such extracted
subsequence should also be random. Hence, any extracted subsequence should
pass any test for randomness.
■ Consistency: The behavior of a generator must be consistent across starting values (seeds). It is inadequate to test a PRNG based on the output from
a single seed or a TRNG on the basis of an output produced from a single
physical output.
256 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS
SP 800-22 lists 15 separate tests of randomness. An understanding of these
tests requires a basic knowledge of statistical analysis, so we don’t attempt a techni-
cal description here. Instead, to give some flavor for the tests, we list three of the
tests and the purpose of each test, as follows.
■ Frequency test: This is the most basic test and must be included in any test suite. The purpose of this test is to determine whether the number of ones and
zeros in a sequence is approximately the same as would be expected for a truly
random sequence.
■ Runs test: The focus of this test is the total number of runs in the sequence, where a run is an uninterrupted sequence of identical bits bounded before
and after with a bit of the opposite value. The purpose of the runs test is to
determine whether the number of runs of ones and zeros of various lengths is
as expected for a random sequence.
■ Maurer’s universal statistical test: The focus of this test is the number of bits between matching patterns (a measure that is related to the length of a
compressed sequence). The purpose of the test is to detect whether or not
the sequence can be significantly compressed without loss of information.
A significantly compressible sequence is considered to be non-random.
UNPREDICTABILITY A stream of pseudorandom numbers should exhibit two forms of unpredictability:
■ Forward unpredictability: If the seed is unknown, the next output bit in the sequence should be unpredictable in spite of any knowledge of previous bits
in the sequence.
■ Backward unpredictability: It should also not be feasible to determine the seed from knowledge of any generated values. No correlation between a seed
and any value generated from that seed should be evident; each element of the
sequence should appear to be the outcome of an independent random event
whose probability is 1/2.
The same set of tests for randomness also provide a test of unpredictability. If
the generated bit stream appears random, then it is not possible to predict some bit
or bit sequence from knowledge of any previous bits. Similarly, if the bit sequence
appears random, then there is no feasible way to deduce the seed based on the bit
sequence. That is, a random sequence will have no correlation with a fixed value
(the seed).
SEED REQUIREMENTS For cryptographic applications, the seed that serves as input to the PRNG must be secure. Because the PRNG is a deterministic algorithm, if the
adversary can deduce the seed, then the output can also be determined. Therefore,
the seed must be unpredictable. In fact, the seed itself must be a random or pseudo-
random number.
Typically, the seed is generated by a TRNG, as shown in Figure 8.2. This is
the scheme recommended by SP 800-90A. The reader may wonder, if a TRNG is
available, why it is necessary to use a PRNG. If the application is a stream cipher,
then a TRNG is not practical. The sender would need to generate a keystream of
8.1 / PRINCIPLES OF PSEUDORANDOM NUMBER GENERATION 257
bits as long as the plaintext and then transmit the keystream and the ciphertext
securely to the receiver. If a PRNG is used, the sender need only find a way to
deliver the stream cipher key, which is typically 54 or 128 bits, to the receiver in a
secure fashion.
Even in the case of a PRF application, in which only a limited number of bits
is generated, it is generally desirable to use a TRNG to provide the seed to the
PRF and use the PRF output rather than use the TRNG directly. As is explained in
Section 8.6, a TRNG may produce a binary string with some bias. The PRF would
have the effect of conditioning the output of the TRNG so as to eliminate that bias.
Finally, the mechanism used to generate true random numbers may not be
able to generate bits at a rate sufficient to keep up with the application requiring
the random bits.
Algorithm Design
Cryptographic PRNGs have been the subject of much research over the years,
and a wide variety of algorithms have been developed. These fall roughly into two
categories.
■ Purpose-built algorithms: These are algorithms designed specifically and solely for the purpose of generating pseudorandom bit streams. Some of these
algorithms are used for a variety of PRNG applications; several of these are
described in the next section. Others are designed specifically for use in a
stream cipher. The most important example of the latter is RC4, described in
Section 8.5.
■ Algorithms based on existing cryptographic algorithms: Cryptographic algorithms have the effect of randomizing input data. Indeed, this is a require-
ment of such algorithms. For example, if a symmetric block cipher produced
ciphertext that had certain regular patterns in it, it would aid in the process of
Figure 8.2 Generation of Seed Input to PRNG
Entropy source
Pseudorandom number generator
(PRNG)
Seed
Pseudorandom bit stream
True random number generator
(TRNG)
258 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS
cryptanalysis. Thus, cryptographic algorithms can serve as the core of PRNGs.
Three broad categories of cryptographic algorithms are commonly used to
create PRNGs:
–Symmetric block ciphers: This approach is discussed in Section 8.3.
–Asymmetric ciphers: The number theoretic concepts used for an asymmet- ric cipher can also be adapted for a PRNG; this approach is examined in
Chapter 10.
–Hash functions and message authentication codes: This approach is exam- ined in Chapter 12.
Any of these approaches can yield a cryptographically strong PRNG.
A purpose-built algorithm may be provided by an operating system for general use.
For applications that already use certain cryptographic algorithms for encryption or
authentication, it makes sense to reuse the same code for the PRNG. Thus, all of
these approaches are in common use.
8.2 PSEUDORANDOM NUMBER GENERATORS
In this section, we look at two types of algorithms for PRNGs.
Linear Congruential Generators
A widely used technique for pseudorandom number generation is an algorithm first
proposed by Lehmer [LEHM51], which is known as the linear congruential method.
The algorithm is parameterized with four numbers, as follows:
m the modulus m 7 0 a the multiplier 0 6 a 6 m c the increment 0 … c 6 m X0 the starting value, or seed 0 … X0 6 m
The sequence of random numbers {Xn} is obtained via the following iterative equation:
Xn + 1 = (aXn + c) mod m
If m, a, c, and X0 are integers, then this technique will produce a sequence of inte- gers with each integer in the range 0 … Xn 6 m.
The selection of values for a, c, and m is critical in developing a good ran- dom number generator. For example, consider a = c = 1. The sequence produced is obviously not satisfactory. Now consider the values a = 7, c = 0, m = 32, and X0 = 1. This generates the sequence {7, 17, 23, 1, 7, etc.}, which is also clearly unsatisfactory. Of the 32 possible values, only four are used; thus, the sequence
is said to have a period of 4. If, instead, we change the value of a to 5, then the sequence is {5, 25, 29, 17, 21, 9, 13, 1, 5, etc. }, which increases the period to 8.
We would like m to be very large, so that there is the potential for producing a long series of distinct random numbers. A common criterion is that m be nearly
8.2 / PSEUDORANDOM NUMBER GENERATORS 259
equal to the maximum representable nonnegative integer for a given computer.
Thus, a value of m near to or equal to 231 is typically chosen. [PARK88] proposes three tests to be used in evaluating a random number
generator:
T1: The function should be a full-period generating function. That is, the function
should generate all the numbers from 0 through m - 1 before repeating. T2: The generated sequence should appear random.
T3: The function should implement efficiently with 32-bit arithmetic.
With appropriate values of a, c, and m, these three tests can be passed. With respect to T1, it can be shown that if m is prime and c = 0, then for certain values of a the period of the generating function is m - 1, with only the value 0 missing. For 32-bit arithmetic, a convenient prime value of m is 231 - 1. Thus, the generating function becomes
Xn + 1 = (aXn) mod (2 31 - 1)
Of the more than 2 billion possible choices for a, only a handful of multipliers pass all three tests. One such value is a = 75 = 16807, which was originally selected for use in the IBM 360 family of computers [LEWI69]. This generator is widely
used and has been subjected to a more thorough testing than any other PRNG. It is
frequently recommended for statistical and simulation work (e.g., [JAIN91]).
The strength of the linear congruential algorithm is that if the multiplier and
modulus are properly chosen, the resulting sequence of numbers will be statistically
indistinguishable from a sequence drawn at random (but without replacement) from
the set 1, 2, c , m - 1. But there is nothing random at all about the algorithm, apart from the choice of the initial value X0. Once that value is chosen, the remaining num- bers in the sequence follow deterministically. This has implications for cryptanalysis.
If an opponent knows that the linear congruential algorithm is being used and
if the parameters are known (e.g., a = 75, c = 0, m = 231 - 1), then once a single number is discovered, all subsequent numbers are known. Even if the opponent
knows only that a linear congruential algorithm is being used, knowledge of a small
part of the sequence is sufficient to determine the parameters of the algorithm.
Suppose that the opponent is able to determine values for X0, X1, X2, and X3. Then
X1 = (aX0 + c) mod m X2 = (aX1 + c) mod m X3 = (aX2 + c) mod m
These equations can be solved for a, c, and m. Thus, although it is nice to be able to use a good PRNG, it is desirable to make
the actual sequence used nonreproducible, so that knowledge of part of the se-
quence on the part of an opponent is insufficient to determine future elements of the
sequence. This goal can be achieved in a number of ways. For example, [BRIG79]
suggests using an internal system clock to modify the random number stream. One
way to use the clock would be to restart the sequence after every N numbers using the current clock value (mod m) as the new seed. Another way would be simply to add the current clock value to each random number (mod m).
260 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS
Blum Blum Shub Generator
A popular approach to generating secure pseudorandom numbers is known as
the Blum Blum Shub (BBS) generator (see Figure 8.3), named for its developers
[BLUM86]. It has perhaps the strongest public proof of its cryptographic strength
of any purpose-built algorithm. The procedure is as follows. First, choose two large
prime numbers, p and q, that both have a remainder of 3 when divided by 4. That is,
p K q K 3(mod 4)
This notation, explained more fully in Chapter 4, simply means that (p mod 4) = (q mod 4) = 3. For example, the prime numbers 7 and 11 satisfy 7 K 11 K 3(mod 4). Let n = p * q. Next, choose a random number s, such that s is relatively prime to n; this is equivalent to saying that neither p nor q is a factor of s. Then the BBS genera- tor produces a sequence of bits Bi according to the following algorithm:
X0 = s 2 mod n
for i = 1 to ∞ Xi = (Xi−1)
2 mod n Bi = Xi mod 2
Thus, the least significant bit is taken at each iteration. Table 8.1 shows an example
of BBS operation. Here, n = 192649 = 383 * 503, and the seed s = 101355. The BBS is referred to as a cryptographically secure pseudorandom bit
generator (CSPRBG). A CSPRBG is defined as one that passes the next-bit test, which, in turn, is defined as follows [MENE97]: A pseudorandom bit generator is
said to pass the next-bit test if there is not a polynomial-time algorithm1 that, on
input of the first k bits of an output sequence, can predict the (k + 1)st bit with probability significantly greater than 1/2. In other words, given the first k bits of the
1A polynomial-time algorithm of order k is one whose running time is bounded by a polynomial of order k.
Figure 8.3 Blum Blum Shub Block Diagram
Generate x2 mod n
Select least significant bit
Initialize with seed s
[0, 1]
8.3 / PSEUDORANDOM NUMBER GENERATION USING A BLOCK CIPHER 261
sequence, there is not a practical algorithm that can even allow you to state that the
next bit will be 1 (or 0) with probability greater than 1/2. For all practical purposes,
the sequence is unpredictable. The security of BBS is based on the difficulty of
factoring n. That is, given n, we need to determine its two prime factors p and q.
8.3 PSEUDORANDOM NUMBER GENERATION USING A BLOCK CIPHER
A popular approach to PRNG construction is to use a symmetric block cipher as
the heart of the PRNG mechanism. For any block of plaintext, a symmetric block
cipher produces an output block that is apparently random. That is, there are no
patterns or regularities in the ciphertext that provide information that can be used
to deduce the plaintext. Thus, a symmetric block cipher is a good candidate for
building a pseudorandom number generator.
If an established, standardized block cipher is used, such as DES or AES, then
the security characteristics of the PRNG can be established. Further, many applica-
tions already make use of DES or AES, so the inclusion of the block cipher as part
of the PRNG algorithm is straightforward.
PRNG Using Block Cipher Modes of Operation
Two approaches that use a block cipher to build a PNRG have gained widespread
acceptance: the CTR mode and the OFB mode. The CTR mode is recommended in
NIST SP 800-90A, in the ANSI standard X9.82 (Random Number Generation), and in RFC 4086 (Randomness Requirements for Security, June 2005). The OFB mode is recommended in X9.82 and RFC 4086.
Figure 8.4 illustrates the two methods. In each case, the seed consists of two
parts: the encryption key value and a value V that will be updated after each block of pseudorandom numbers is generated. Thus, for AES-128, the seed consists of a
128-bit key and a 128-bit V value. In the CTR case, the value of V is incremented
Table 8.1 Example Operation of BBS Generator
i Xi Bi 0 20749
1 143135 1
2 177671 1
3 97048 0
4 89992 0
5 174051 1
6 80649 1
7 45663 1
8 69442 0
9 186894 0
10 177046 0
i Xi Bi 11 137922 0
12 123175 1
13 8630 0
14 114386 0
15 14863 1
16 133015 1
17 106065 1
18 45870 0
19 137171 1
20 48060 0
262 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS
by 1 after each encryption. In the case of OFB, the value of V is updated to equal the value of the preceding PRNG block. In both cases, pseudorandom bits are produced
one block at a time (e.g., for AES, PRNG bits are generated 128 bits at a time).
The CTR algorithm for PRNG, called CTR_DRBG, can be summarized
as follows.
while (len (temp) < requested_number_of_bits) do V = (V + 1) mod 2128
output_block = E(Key, V) temp = temp || output_block
The OFB algorithm can be summarized as follows.
while (len (temp) < requested_number_of_bits) do V = E(Key, V) temp = temp || V
To get some idea of the performance of these two PRNGs, consider the fol-
lowing short experiment. A random bit sequence of 256 bits was obtained from
random.org, which uses three radios tuned between stations to pick up atmospheric
noise. These 256 bits form the seed, allocated as
Key: cfb0ef3108d49cc4562d5810b0a9af60
V: 4c89af496176b728ed1e2ea8ba27f5a4
The total number of one bits in the 256-bit seed is 124, or a fraction of 0.48,
which is reassuringly close to the ideal of 0.5.
For the OFB PRNG, Table 8.2 shows the first eight output blocks (1024 bits)
with two rough measures of security. The second column shows the fraction of one
bits in each 128-bit block. This corresponds to one of the NIST tests. The results
indicate that the output is split roughly equally between zero and one bits. The
third column shows the fraction of bits that match between adjacent blocks. If this
Figure 8.4 PRNG Mechanisms Based on Block Ciphers
(a) CTR mode
V
Encrypt
Pseudorandom bits
K
1
+
(b) OFB mode
V
Encrypt
Pseudorandom bits
K
Hiva-Network.Com
8.3 / PSEUDORANDOM NUMBER GENERATION USING A BLOCK CIPHER 263
Output Block Fraction of One Bits
Fraction of Bits that Match with Preceding Block
1786f4c7ff6e291dbdfdd90ec3453176 0.57 —
5e17b22b14677a4d66890f87565eae64 0.51 0.52
fd18284ac82251dfb3aa62c326cd46cc 0.47 0.54
c8e545198a758ef5dd86b41946389bd5 0.50 0.44
fe7bae0e23019542962e2c52d215a2e3 0.47 0.48
14fdf5ec99469598ae0379472803accd 0.49 0.52
6aeca972e5a3ef17bd1a1b775fc8b929 0.57 0.48
f7e97badf359d128f00d9b4ae323db64 0.55 0.45
Table 8.2 Example Results for PRNG Using OFB
Output Block Fraction of One Bits
Fraction of Bits that Match with Preceding Block
1786f4c7ff6e291dbdfdd90ec3453176 0.57 —
60809669a3e092a01b463472fdcae420 0.41 0.41
d4e6e170b46b0573eedf88ee39bff33d 0.59 0.45
5f8fcfc5deca18ea246785d7fadc76f8 0.59 0.52
90e63ed27bb07868c753545bdd57ee28 0.53 0.52
0125856fdf4a17f747c7833695c52235 0.50 0.47
f4be2d179b0f2548fd748c8fc7c81990 0.51 0.48
1151fc48f90eebac658a3911515c3c66 0.47 0.45
Table 8.3 Example Results for PRNG Using CTR
number differs substantially from 0.5, that suggests a correlation between blocks,
which could be a security weakness. The results suggest no correlation.
Table 8.3 shows the results using the same key and V values for CTR mode. Again, the results are favorable.
ANSI X9.17 PRNG
One of the strongest (cryptographically speaking) PRNGs is specified in ANSI
X9.17. A number of applications employ this technique, including financial security
applications and PGP (the latter described in Chapter 19).
Figure 8.5 illustrates the algorithm, which makes use of triple DES for encryp-
tion. The ingredients are as follows.
■ Input: Two pseudorandom inputs drive the generator. One is a 64-bit represen- tation of the current date and time, which is updated on each number genera-
tion. The other is a 64-bit seed value; this is initialized to some arbitrary value
and is updated during the generation process.
■ Keys: The generator makes use of three triple DES encryption modules. All three make use of the same pair of 56-bit keys, which must be kept secret and
are used only for pseudorandom number generation.
264 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS
■ Output: The output consists of a 64-bit pseudorandom number and a 64-bit seed value.
Let us define the following quantities.
DTi Date/time value at the beginning of ith generation stage Vi Seed value at the beginning of ith generation stage Ri Pseudorandom number produced by the ith generation stage K1, K2 DES keys used for each stage
Then
Ri = EDE([K1, K2], [Vi ⊕ EDE([K1, K2], DTi)]) Vi + 1 = EDE([K1, K2], [Ri ⊕ EDE([K1, K2], DTi)])
where EDE([K1, K2], X) refers to the sequence encrypt-decrypt-encrypt using two- key triple DES to encrypt X.
Several factors contribute to the cryptographic strength of this method. The
technique involves a 112-bit key and three EDE encryptions for a total of nine DES
encryptions. The scheme is driven by two independent inputs, the date and time
value, and a seed produced by the generator that is distinct from the pseudorandom
number produced by the generator. Thus, the amount of material that must be com-
promised by an opponent appears to be overwhelming. Even if a pseudorandom
number Ri were compromised, it would be impossible to deduce the Vi + 1 from the Ri, because an additional EDE operation is used to produce the Vi + 1.
NIST CTR_DRBG
We now look more closely at the details of the PRNG defined in NIST SP 800-90A
based on the CTR mode of operation. The PRNG is referred to as CTRDRBG
(counter mode–deterministic random bit generator). CTR_DRBG is widely imple-
mented and is part of the hardware random number generator implemented on all
recent Intel processor chips (discussed in Section 8.6).
Figure 8.5 ANSI X9.17 Pseudorandom Number Generator
EDE
EDE
EDE
K1, K2
DTi
Vi
Vi+1
Ri
8.3 / PSEUDORANDOM NUMBER GENERATION USING A BLOCK CIPHER 265
The DRBG assumes that an entropy source is available to provide random
bits. Typically, the entropy source will be a TRNG based on some physical source.
Other sources are possible if they meet the required entropy measure of the appli-
cation. Entropy is an information theoretic concept that measures unpredictability,
or randomness; see Appendix F for details. The encryption algorithm used in the
DRBG may be 3DES with three keys or AES with a key size of 128, 192, or 256 bits.
Four parameters are associated with the algorithm:
■ Output block length (outlen): Length of the output block of the encryption algorithm.
■ Key length (keylen): Length of the encryption key.
■ Seed length (seedlen): The seed is a string of bits that is used as input to a DRBG mechanism. The seed will determine a portion of the internal state of
the DRBG, and its entropy must be sufficient to support the security strength
of the DRBG. seedlen = outlen + keylen. ■ Reseed interval (reseed_interval): Length of the encryption key. It is the maxi-
mum number of output blocks generated before updating the algorithm with
a new seed.
Table 8.4 lists the values specified in SP 800-90A for these parameters.
INITIALIZE Figure 8.6 shows the two principal functions that comprise CTR_DRBG. We first consider how CTR_DRBG is initialized, using the initialize and update
function (Figure 8.6a). Recall that the CTR block cipher mode requires both an
encryption key K and an initial counter value, referred to in SP 800-90A as the counter V. The combination of K and V is referred to as the seed. To start the DRGB operation, initial values for K and V are needed, and can be chosen arbi- trarily. As an example, the Intel Digital Random Number Generator, discussed in
Section 8.6, uses the values K = 0 and V = 0. These values are used as param- eters for the CTR mode of operation to produce at least seedlen bits. In addition, exactly seedlen bits must be supplied from what is referred to as an entropy source. Typically, the entropy source would be some form of TRNG.
With these inputs, the CTR mode of encryption is iterated to produce a
sequence of output blocks, with V incremented by 1 after each encryption. The pro- cess continues until at least seedlen bits have been generated. The leftmost seedlen bits of output are then XORed with the seedlen entropy bits to produce a new seed. In turn, the leftmost keylen bits of the seed form the new key and the rightmost outlen bits of the seed form the new counter value V.
3DES AES-128 AES-192 AES-256
outlen 64 128 128 128
keylen 168 128 192 256
seedlen 232 256 320 384
reseed_interval … 232 … 248 … 248 … 248
Table 8.4 CTR_DRBG Parameters
266 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS
GENERATE Once values of Key and V are obtained, the DRBG enters the generate phase and is able to generate pseudorandom bits, one output block at a time
(Figure 8.6b). The encryption function is iterated to generate the number of pseu-
dorandom bits desired. Each iteration uses the same encryption key. The counter
value V is incremented by 1 for each iteration.
UPDATE To enhance security, the number of bits generated by any PRNG should be limited. CTR_DRGB uses the parameter reseed_interval to set that limit. During the generate phase, a reseed counter is initialized to 1 and then incremented with each
Figure 8.6 CTR_DRBG Functions
Encrypt
Iterate1
V
Entropy source
1st time
+
Key
B0
Key
(a) Initialize and update function
(b) Generate function
Bi
V
Key V
Encrypt
Iterate1
+
8.4 / STREAM CIPHERS 267
iteration (each production of an output block). When the reseed counter reaches
reseed_interval, the update function is invoked (Figure 8.6a). The update function is the same as the initialize function. In the update case, the Key and V values last used by the generate function serve as the input parameters to the update function.
The update function takes seedlen new bits from an entropy source and produces a new seed (Key, V). The generate function can then resume production of pseudo- random bits. Note that the result of the update function is to change both the Key
and V values used by the generate function.
8.4 STREAM CIPHERS
A typical stream cipher encrypts plaintext one byte at a time, although a stream
cipher may be designed to operate on one bit at a time or on units larger than a byte
at a time. Figure 8.7 is a representative diagram of stream cipher structure. In this
structure, a key is input to a pseudorandom bit generator that produces a stream
of 8-bit numbers that are apparently random. The output of the generator, called
a keystream, is combined one byte at a time with the plaintext stream using the bitwise exclusive-OR (XOR) operation. For example, if the next byte generated by
the generator is 01101100 and the next plaintext byte is 11001100, then the resulting
ciphertext byte is
11001100 plaintext ⊕ 01101100 key stream
10100000 ciphertext
Decryption requires the use of the same pseudorandom sequence:
10100000 ciphertext ⊕ 01101100 key stream
11001100 plaintext
Figure 8.7 Stream Cipher Diagram
Pseudorandom byte generator
(key stream generator)
Plaintext byte stream
M
Key K
Key K
k Plaintext
byte stream M
Ciphertext byte stream
CENCRYPTION
Pseudorandom byte generator
(key stream generator)
DECRYPTION
k
268 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS
The stream cipher is similar to the one-time pad discussed in Chapter 3. The
difference is that a one-time pad uses a genuine random number stream, whereas a
stream cipher uses a pseudorandom number stream.
[KUMA97] lists the following important design considerations for a stream cipher.
1. The encryption sequence should have a large period. A pseudorandom num- ber generator uses a function that produces a deterministic stream of bits that
eventually repeats. The longer the period of repeat the more difficult it will be
to do cryptanalysis. This is essentially the same consideration that was discussed
with reference to the Vigenère cipher, namely that the longer the keyword
the more difficult the cryptanalysis.
2. The keystream should approximate the properties of a true random number stream as close as possible. For example, there should be an approximately
equal number of 1s and 0s. If the keystream is treated as a stream of bytes,
then all of the 256 possible byte values should appear approximately equally
often. The more random-appearing the keystream is, the more randomized the
ciphertext is, making cryptanalysis more difficult.
3. Note from Figure 8.7 that the output of the pseudorandom number genera- tor is conditioned on the value of the input key. To guard against brute-force
attacks, the key needs to be sufficiently long. The same considerations that
apply to block ciphers are valid here. Thus, with current technology, a key
length of at least 128 bits is desirable.
With a properly designed pseudorandom number generator, a stream cipher
can be as secure as a block cipher of comparable key length. A potential advantage
of a stream cipher is that stream ciphers that do not use block ciphers as a building
block are typically faster and use far less code than do block ciphers. The example
in this chapter, RC4, can be implemented in just a few lines of code. In recent years,
this advantage has diminished with the introduction of AES, which is quite efficient
in software. Furthermore, hardware acceleration techniques are now available for
AES. For example, the Intel AES Instruction Set has machine instructions for one
round of encryption and decryption and key generation. Using the hardware in-
structions results in speedups of about an order of magnitude compared to pure
software implementations [XU10].
One advantage of a block cipher is that you can reuse keys. In contrast, if two
plaintexts are encrypted with the same key using a stream cipher, then cryptanalysis
is often quite simple [DAWS96]. If the two ciphertext streams are XORed together,
the result is the XOR of the original plaintexts. If the plaintexts are text strings,
credit card numbers, or other byte streams with known properties, then cryptanaly-
sis may be successful.
For applications that require encryption/decryption of a stream of data, such as
over a data communications channel or a browser/Web link, a stream cipher might
be the better alternative. For applications that deal with blocks of data, such as file
transfer, email, and database, block ciphers may be more appropriate. However,
either type of cipher can be used in virtually any application.
A stream cipher can be constructed with any cryptographically strong PRNG,
such as the ones discussed in Sections 8.2 and 8.3. In the next section, we look at a
stream cipher that uses a PRNG designed specifically for the stream cipher.
8.5 / RC4 269
8.5 RC4
RC4 is a stream cipher designed in 1987 by Ron Rivest for RSA Security. It is a
variable key size stream cipher with byte-oriented operations. The algorithm is
based on the use of a random permutation. Analysis shows that the period of the
cipher is overwhelmingly likely to be greater than 10100 [ROBS95a]. Eight to sixteen
machine operations are required per output byte, and the cipher can be expected
to run very quickly in software. RC4 is used in the WiFi Protected Access (WPA)
protocol that are part of the IEEE 802.11 wireless LAN standard. It is optional for
use in Secure Shell (SSH) and Kerberos. RC4 was kept as a trade secret by RSA
Security. In September 1994, the RC4 algorithm was anonymously posted on the
Internet on the Cypherpunks anonymous remailers list.
The RC4 algorithm is remarkably simple and quite easy to explain.
A variable-length key of from 1 to 256 bytes (8 to 2048 bits) is used to initialize a
256-byte state vector S, with elements S[0],S[1], . . . ,S[255]. At all times, S contains
a permutation of all 8-bit numbers from 0 through 255. For encryption and decryp-
tion, a byte k (see Figure 8.7) is generated from S by selecting one of the 255 entries in a systematic fashion. As each value of k is generated, the entries in S are once again permuted.
Initialization of S
To begin, the entries of S are set equal to the values from 0 through 255 in ascending
order; that is, S[0] = 0, S[1] = 1, c , S[255] = 255. A temporary vector, T, is also created. If the length of the key K is 256 bytes, then K is transferred to T. Otherwise,
for a key of length keylen bytes, the first keylen elements of T are copied from K, and then K is repeated as many times as necessary to fill out T. These preliminary
operations can be summarized as
/* Initialization */ for i = 0 to 255 do S[i] = i; T[i] = K[i mod keylen];
Next we use T to produce the initial permutation of S. This involves starting
with S[0] and going through to S[255], and for each S[i], swapping S[i] with another
byte in S according to a scheme dictated by T[i]:
/* Initial Permutation of S */ j = 0; for i = 0 to 255 do
j = (j + S[i] + T[i]) mod 256; Swap (S[i], S[j]);
Because the only operation on S is a swap, the only effect is a permutation.
S still contains all the numbers from 0 through 255.
270 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS
Stream Generation
Once the S vector is initialized, the input key is no longer used. Stream generation
involves cycling through all the elements of S[i], and for each S[i], swapping S[i]
with another byte in S according to a scheme dictated by the current configuration
of S. After S[255] is reached, the process continues, starting over again at S[0]:
/* Stream Generation */ i, j = 0; while (true) i = (i + 1) mod 256; j = (j + S[i]) mod 256; Swap (S[i], S[j]); t = (S[i] + S[j]) mod 256; k = S[t];
To encrypt, XOR the value k with the next byte of plaintext. To decrypt, XOR
the value k with the next byte of ciphertext. Figure 8.8 illustrates the RC4 logic.
Figure 8.8 RC4
25525425343210S
T
S
(a) Initial state of S and T
(b) Initial permutation of S
Swap
T
K
T[i]
j = j + S[i] + T[i]
t = S[i] + S[j]
]j[S]i[S
Keylen
i
S
(c) Stream generation
Swap
j = j + S[i]
]t[S]j[S]i[S
k
i
8.6 / TRUE RANDOM NUMBER GENERATORS 271
Strength of RC4
A number of papers have been published analyzing methods of attacking RC4 (e.g.,
[KNUD98], [FLUH00], [MANT01]). None of these approaches is practical against
RC4 with a reasonable key length, such as 128 bits. A more serious problem is reported
in [FLUH01]. The authors demonstrate that the WEP protocol, intended to provide
confidentiality on 802.11 wireless LAN networks, is vulnerable to a particular attack
approach. In essence, the problem is not with RC4 itself but the way in which keys are
generated for use as input to RC4. This particular problem does not appear to be rele-
vant to other applications using RC4 and can be remedied in WEP by changing the way
in which keys are generated. This problem points out the difficulty in designing a secure
system that involves both cryptographic functions and protocols that make use of them.
More recently, [PAUL07] revealed a more fundamental vulnerability in the
RC4 key scheduling algorithm that reduces the amount of effort to discover the
key. Recent cryptanalysis results [ALFA13] exploit biases in the RC4 keystream to
recover repeatedly encrypted plaintexts. As a result of the discovered weaknesses,
particularly those reported in [ALFA13], the IETF issued RFC 7465 prohibiting the
use of RC4 in TLS (Prohibiting RC4 Cipher Suites, February 2015). In its latest TLS guidelines, NIST also prohibited the use of RC4 for government use (SP 800-52,
Guidelines for the Selection, Configuration, and Use of Transport Layer Security (TLS) Implementations, September 2013).
8.6 TRUE RANDOM NUMBER GENERATORS
Entropy Sources
A true random number generator (TRNG) uses a nondeterministic source to pro-
duce randomness. Most operate by measuring unpredictable natural processes, such
as pulse detectors of ionizing radiation events, gas discharge tubes, and leaky capac-
itors. Intel has developed a commercially available chip that samples thermal noise
by sampling the output of a coupled pair of inverters. LavaRnd is an open source
project for creating truly random numbers using inexpensive cameras, open source
code, and inexpensive hardware. The system uses a saturated CCD in a light-tight
can as a chaotic source to produce the seed. Software processes the result into truly
random numbers in a variety of formats.
RFC 4086 lists the following possible sources of randomness that, with care,
easily can be used on a computer to generate true random sequences.
■ Sound/video input: Many computers are built with inputs that digitize some real-world analog source, such as sound from a microphone or video input
from a camera. The “input” from a sound digitizer with no source plugged in or
from a camera with the lens cap on is essentially thermal noise. If the system
has enough gain to detect anything, such input can provide reasonably high
quality random bits.
■ Disk drives: Disk drives have small random fluctuations in their rotational speed due to chaotic air turbulence [JAKO98]. The addition of low-level disk
seek-time instrumentation produces a series of measurements that contain this
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272 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS
randomness. Such data is usually highly correlated, so significant processing is
needed. Nevertheless, experimentation a decade ago showed that, with such
processing, even slow disk drives on the slower computers of that day could
easily produce 100 bits a minute or more of excellent random data.
There is also an online service (random.org), which can deliver random
sequences securely over the Internet.
Comparison of PRNGs and TRNGs
Table 8.5 summarizes the principal differences between PRNGs and TRNGs.
PRNGs are efficient, meaning they can produce many numbers in a short time, and
deterministic, meaning that a given sequence of numbers can be reproduced at a
later date if the starting point in the sequence is known. Efficiency is a nice char-
acteristic if your application needs many numbers, and determinism is handy if you
need to replay the same sequence of numbers again at a later stage. PRNGs are
typically also periodic, which means that the sequence will eventually repeat itself.
While periodicity is hardly ever a desirable characteristic, modern PRNGs have a
period that is so long that it can be ignored for most practical purposes.
TRNGs are generally rather inefficient compared to PRNGs, taking consid-
erably longer time to produce numbers. This presents a difficulty in many applica-
tions. For example, cryptography system in banking or national security might need
to generate millions of random bits per second. TRNGs are also nondeterministic,
meaning that a given sequence of numbers cannot be reproduced, although the same
sequence may of course occur several times by chance. TRNGs have no period.
Conditioning2
A TRNG may produce an output that is biased in some way, such as having more
ones than zeros or vice versa. More generally, NIST SP 800-90B defines a random
process as biased with respect to an assumed discrete set of potential outcomes (i.e., possible output values) if some of those outcomes have a greater probability
of occurring than do others. For example, a physical source such as electronic noise
may contain a superposition of regular structures, such as waves or other periodic
phenomena, which may appear to be random, yet are determined to be non-random
using statistical tests.
2 The reader unfamiliar with the concepts of entropy and min-entropy should read Appendix F before proceeding.
Pseudorandom Number Generators
True Random Number Generators
Efficiency Very efficient Generally inefficient Determinism Deterministic Nondeterministic Periodicity Periodic Aperiodic
Table 8.5 Comparison of PRNGs and TRNGs
8.6 / TRUE RANDOM NUMBER GENERATORS 273
In addition to bias, another concept used by SP 800-98B is that of entropy rate. SP 800-90B defines entropy rate as the rate at which a digitized noise source (or
entropy source) provides entropy; it is computed as the assessed amount of entropy
provided by a bit string output from the source, divided by the total number of
bits in the bit string (yielding assessed bits of entropy per output bit). This will be
a value between 0 (no entropy) and 1 (full entropy). Entropy rate is a measure
of the randomness or unpredictability of a bit string. Another way of express-
ing it is that the entropy rate is k/n for a random source of length n bits and min- entropy k. Min-entropy is a measure of the number of random bits and is explained in Appendix F. In essence, a block of bits or a bit stream that is unbiased, and in
which each bit and each group of bits is independent of all other bits and groups of
bits will have an entropy rate of 1.
For hardware sources of random bits, the recommended approach is to assume
that there may be bias and/or an entropy rate of less than 1 and to apply techniques
to further “randomize” the bits. Various methods of modifying a bit stream for this
purpose have been developed. These are referred to as conditioning algorithms or deskewing algorithms.
Typically, conditioning is done by using a cryptographic algorithm to “ scramble”
the random bits so as to eliminate bias and increase entropy. The two most common
approaches are the use of a hash function or a symmetric block cipher.
HASH FUNCTION As we describe in Chapter 11, a hash function produces an n-bit output from an input of arbitrary length. A simple way to use a hash function for
conditioning is as follows. Blocks of m input bits, with m Ú n, are passed through the hash function and the n output bits are used as random bits. To generate a stream of random bits, successive input blocks pass through the hash function to
produce successive hashed output blocks.
Operating systems typically provide a built-in mechanism for generating ran-
dom numbers. For example, Linux uses four entropy sources: mouse and keyboard
activity, disk I/O operations, and specific interrupts. Bits are generated from these
four sources and combined in a pooled buffer. When random bits are needed, the
appropriate number of bits are read from the buffer and passed through the SHA-1
hash function [GUTT06].
A more complex approach is the hash derivation function specified in
SP800-90A. Hash_df can be defined as follows:
Parameters:
input_string: The string to be hashed.
outlen: Output length.
no_of_bits_to_return: The number of bits to be returned by Hash_df. The maxi- mum length (max_number_of_bits) is implementation dependent, but shall be less than or equal to (255 * outlen). no_of_bits_to_return is represented as a 32-bit integer.
requested_bits: The result of performing the Hash_df.
274 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS
Hash_df Process:
1. temp = the Null string
2. len = l no_of_bits_to_return outlen
m 3. counter = 0x01 Comment: An 8-bit binary value representing the integer “1”. 4. For i = 1 to len do Comment: In 4.1, no_of_bits_to_return is used as a 32-bit
string.
4.1. temp = temp } Hash (counter } no_of_bits_to_return } input_string). 4.2. counter = counter + 1.
5. requested_bits = leftmost (temp, no_of_bits_to_return). 6. Return (SUCCESS, requested_bits).
This algorithm takes an input block of bits of arbitrary length and returns the
requested number of bits, which may be up to 255 times as long as the hash output
length.
The reader may be uneasy that the output consists of hashed blocks in which
the input to the hash function for each block is the same input string and differs
only by the value of the counter. However, cryptographically strong hash functions,
such as the SHA family, provide excellent diffusion (as defined in Chapter 4) so that
change in the counter value results in dramatically different outputs.
BLOCK CIPHER Instead of a hash function, a block cipher such as AES can be used to scramble the TRNG bits. Using AES, a simple approach would be to take
128-bit blocks of TRNG bits and encrypt each block with AES and some arbitrary
key. SP 800-90B outlines an approach similar to the hash_df function described pre-
viously. The Intel implementation discussed subsequently provides an example of
using AES for conditioning.
Health Testing
Figure 8.9 provides a general model for a nondeterministic random bit generator.
A hardware noise source produces a true random output. This is digitized to pro-
duce true, or nondeterministic, source of bits. This bit source then passes through a
conditioning module to mitigate bias and maximize entropy.
Figure 8.9 also shows a health-testing module, which is used on the outputs
of both the digitizer and conditioner. In essence, health testing is used to validate
that the noise source is working as expected and that the conditioning module is
produced output with the desired characteristics. Both forms of health testing are
recommended by SP 800-90B.
HEALTH TESTS ON THE NOISE SOURCE The nature of the health testing of the noise source depends strongly on the technology used to produce noise. In general, we
can assume that the digitized output of the noise source will exhibit some bias. Thus,
the traditional statistical tests, such as those defined in SP 800-22 and discussed in
Section 8.1, are not useful for monitoring the noise source, because the noise source
8.6 / TRUE RANDOM NUMBER GENERATORS 275
Figure 8.9 NRBG Model
Nondetermistic bit source
Noise source
Digitization
Conditioning
A
B
Output
Health testing
NONDETERMINISTIC RANDOM
BIT GENERATOR
is likely to always fail. Rather, the tests on the noise source need to be tailored to
the expected statistical behavior of the correctly operating noise source. The goal
is not to determine if the source is unbiased, which it isn’t, but if it is operating
as expected.
SP 800-90B specifies that continuous tests be done on digitized samples
obtained from the noise source (point A in Figure 8.9). The purpose is to test for
variability. More specifically, the purpose is to determine if the noise source is pro-
ducing at the expected entropy rate. SP 800-909B mandates the use of two tests: the
Repetition Count Test and the Adaptive Proportion Test.
The Repetition Count Test is designed to quickly detect a catastrophic failure that causes the noise source to become “stuck” on a single output value for a long
time. For this test, it is assumed that a given noise source is assessed to have a given
min-entropy value of H. The entropy is expressed as the amount of entropy per sam- ple, where a sample could be a single bit or some block of bits of length n. With an assessed value of H, it is straightforward to calculate the probability that a sequence of C consecutive samples will yield identical sample values. For example, a noise source with one bit of min-entropy per sample has no more than a 1/2 probability
of repeating some sample value twice in a row, no more than 1/4 probability of
repeating some sample value three times in a row, and in general, no more than
(1/2)C - 1 probability of repeating some sample value C times in a row. To generalize, for a noise source with H bits of min-entropy per sample, we have:
Pr[C identical samples in a row] … (2-H)(C - 1)
The Repetition Count Test involves looking for consecutive identical sam-
ples. If the count reaches some cutoff value C, then an error condition is raised. To determine the value of C used in the test, the test must be configured with
276 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS
a parameter W, which is the acceptable false-positive probability associated with an alarm triggered by C repeated sample values. To avoid false positives, W should be set at some very small number greater than 0. Given W, we can now determine the value of C. Specifically, we want C to be the smallest number that satisfies the equation W … (2-H)(C - 1). Reworking terms, this gives us a value of:
C = l1 + - log(W) H
m For example, for W = 2-30, an entropy source with H = 7.3 bits per sample
would have a cutoff value C of l1 + 30 7.3 m = 6.
The Repetition Count Test starts by recording a sample value and then count-
ing the number of repetitions of the same value. If the counter reaches the cutoff
value C, an error is reported. If a sample value is encountered that differs from the
preceding sample, then the counter is reset to 1 and the algorithm starts over.
The Adaptive Proportion Test is designed to detect a large loss of entropy, such as might occur as a result of some physical failure or environmental change
affecting the noise source. The test continuously measures the local frequency of
occurrence of some sample value in a sequence of noise source samples to determine
if the sample occurs too frequently.
The test starts by recording a sample value and then observes N successive sample values. If the initial sample value is observed at least C times, then an error condition is reported. SP 800-90B recommends that a probability of a false positive
of W = 2-30 be used for the test and provides guidance on the selection of values for N and C.
HEALTH TESTS ON THE CONDITIONING FUNCTION SP 800-90B specifies that health tests should also be applied to the output of the conditioning component (point B
in Figure 8.9), but does not indicate which tests to use. The purpose of the health
tests on the conditioning component is to assure that the output behaves as a true
random bit stream. Thus, it is reasonable to use the tests for randomness defined in
SP 800-22, and described in Section 8.1.
Intel Digital Random Number Generator
As was mentioned, TRNGs have traditionally been used only for key generation
and other applications where only a small number of random bits were required.
This is because TRNGs have generally been inefficient, with a low bit rate of
random bit production.
The first commercially available TRNG that achieves bit production rates
comparable with that of PRNGs is the Intel digital random number generator
(DRNG) [TAYL11, MECH14], offered on new multicore chips since May 2012.3
3It is unfortunate that Intel chose the acronym DRNG for an NRBG. It confuses with DRBG, which is a pseudorandom number bit generator.
8.6 / TRUE RANDOM NUMBER GENERATORS 277
Two notable aspects of the DRNG:
1. It is implemented entirely in hardware. This provides greater security than a facility that includes a software component. A hardware-only implementa-
tion should also be able to achieve greater computation speed than a software
module.
2. The entire DRNG is on the same multicore chip as the processors. This elimi- nates the I/O delays found in other hardware random number generators.
DRNG HARDWARE ARCHITECTURE Figure 8.10 shows the overall structure of the DRNG. The first stage of the DRNG generates random numbers from thermal
noise. The heart of the stage consists of two inverters (NOT gates), with the output
of each inverter connected to the input of the other. Such an arrangement has two
stable states, with one inverter having an output of logical 1 and the other having an
output of logical 0. The circuit is then configured so that both inverters are forced
to have the same indeterminate state (both inputs and both outputs at logical 1) by
clock pulses. Random thermal noise within the inverters soon jostles the two invert-
ers into a mutually stable state. Additional circuitry is intended to compensate for
any biases or correlations. This stage is capable, with current hardware, of generat-
ing random bits at a rate of 4 Gbps.
Figure 8.10 Intel Processor Chip with Random Number Generator
Hardware AES-CBC- MAC based conditioner
Digital Random Number Generator
Processor chip
Hardware SP 800-90A AES-CTR
based DRBGHardware
entropy source
RDSEED instruction
Core 0
Core N–1 RDSEED instruction
RDRAND instruction
RDRAND instruction
Hardware SP 800- 90B & C ENRNG
278 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS
The output of the first stage is generated 512 bits at a time. To assure that
the bit stream does not have skew or bias, a conditioner randomizes its input using
a cryptographic function. In this case, the function is referred to as CBC-MAC or
CMAC, as specified in NIST SP 800-38B. In essence, CMAC encrypts its input using
the cipher block chaining (CBC) mode (Figure 8.4) and outputs the final block.
We examine CMAC in detail in Chapter 12. The output of this stage is generated
256 bits at a time and is intended to exhibit true randomness with no skew or bias.
While the hardware’s circuitry generates random numbers from thermal noise
much more quickly than its predecessors, it is still not fast enough for some of to-
day’s computing requirements. To enable the DRNG to generate random numbers
as quickly as a software DRBG, and also maintain the high quality of the random
numbers, a third stage is added. This stage uses the 256-bit random numbers to
seed a cryptographically secure DRBG that creates 128-bit numbers. From one
256-bit seed, the DRBG can output many pseudorandom numbers, exceeding the
3-Gbps rate of the entropy source. An upper bound of 511 128-bit samples can
be generated per seed. The algorithm used for this stage is CTR_DRBG, described
in Section 8.3.
The output of the PRNG stage is available to each of the cores on the chip via
the RDRAND instruction. RDRAND retrieves a 16-, 32-, or 64-bit random value
and makes it available in a software-accessible register.
Preliminary data from a pre-production sample on a system with a third
generation Intel® Core™ family processor produced the following performance
[INTE12]: up to 70 million RDRAND invocations per second, and a random data
production rate of over 4 Gbps.
The output of the conditioner is also made available to another module,
known as an enhanced nondeterministic random number generator (ENRNG) that
provides random numbers that can be used as seeds for various cryptographic algo-
rithms. The ENRNG is compliant with specifications in SP 800-90B and 900-90C.
The output of the ENRNG stage is available to each of the cores on the chip via
the RDSEED instruction. RDSEED retrieves a hardware-generated random seed
value from the ENRNG and stores it in the destination register given as an argu-
ment to the instruction.
DRNG LOGICAL STRUCTURE Figure 8.11 provides a simplified view of the logical flow of the Intel DRBG. As was described, the heart of the hardware entropy source
is a pair of inverters that feed each other. Two transistors, driven by the same clock,
force the inputs and outputs of both inverters to the logical 1 state. Because this is
an unstable state, thermal noise will cause the configuration to settle randomly into
a stable state with either Node A at logical 1 and Node B at logical 0, or the reverse.
Thus the module generates random bits at the clock rate.
The output of the entropy source is collected 512 bits at a time and used to
feed to two CBC hardware implementations using AES encryption. Each imple-
mentation takes two blocks of 128 bits of “plaintext” and encrypts using the CBC
mode. The output of the second encryption is retained. For both CBC modules, an
all-zeros key is used initially. Subsequently, the output of the PRNG stage is fed
back to become the key for the conditioner stage.
8.6 / TRUE RANDOM NUMBER GENERATORS 279
Figure 8.11 Intel DRNG Logical Structure
EncryptEncrypt
128 bits 128 bits
Clock
Transistor 1 Transistor 2
Inverters
Node A Node B
128 bits 128 bits
128 bits
Key V
K K K K
Encrypt Encrypt
Pseudorandom bits
128 bits
101st time
+
256 bits
EncryptEncrypt
128 bits
1
+
Hardware entropy source
AES CBC Mac-based conditioner
AES-CTR- based PRNG
K = 0
The output of the conditioner stage consists of 256 bits. This block is provided
as input to the update function of the DRGB stage. The update function is initial-
ized with the all-zeros key and the counter value 0. The function is iterated twice
to produce a 256-block, which is then XORed with the input from the conditioner
stage. The results are used as the 128-bit key and the 128-bit seed for the generate
function. The generate function produces pseudorandom bits in 128-bit blocks.
280 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS
8.7 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
Key Terms
backward unpredictability
Blum Blum Shub generator
deskewing
entropy source
forward unpredictability
keystream
linear congruential generator
pseudorandom function
(PRF)
pseudorandom number
generator (PRNG)
randomness
RC4
seed
stream cipher
skew
true random number
generator (TRNG)
unpredictability
Review Questions
8.1 List two criteria to validate the randomness of a sequence of numbers. 8.2 What is ANSI X9.17 PRNG? 8.3 What is the difference between a one-time pad and a stream cipher? 8.4 List a few applications of stream ciphers and block ciphers.
Problems
8.1 If we take the linear congruential algorithm with an additive component of 0,
Xn + 1 = (aXn) mod m
Then it can be shown that if m is prime and if a given value of a produces the maxi- mum period of m - 1, then ak will also produce the maximum period, provided that k is less than m and that k and m - 1 are relatively prime. Demonstrate this by using X0 = 1 and m = 31 and producing the sequences for a
k = 3, 32, 33, and 34. 8.2 a. What is the maximum period obtainable from the following generator?
Xn + 1 = (aXn) mod 2 4
b. What should be the value of a? c. What restrictions are required on the seed?
8.3 You may wonder why the modulus m = 231 - 1 was chosen for the linear congruen- tial method instead of simply 231, because this latter number can be represented with no additional bits and the mod operation should be easier to perform. In general, the modulus 2k - 1 is preferable to 2k. Why is this so?
8.4 With the linear congruential algorithm, a choice of parameters that provides a full period does not necessarily provide a good randomization. For example, consider the following two generators:
Xn + 1 = (11Xn) mod 13
Xn + 1 = (2Xn) mod 13
Write out the two sequences to show that both are full period. Which one appears more random to you?
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8.7 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 281
8.5 In any use of pseudorandom numbers, whether for encryption, simulation, or statisti- cal design, it is dangerous to trust blindly the random number generator that happens to be available in your computer’s system library. [PARK88] found that many con- temporary textbooks and programming packages make use of flawed algorithms for pseudorandom number generation. This exercise will enable you to test your system.
The test is based on a theorem attributed to Ernesto Cesaro (see [KNUT98] for a proof), which states the following: Given two randomly chosen integers, x and y, the probability that gcd(x, y) = 1 is 6/p2. Use this theorem in a program to determine statistically the value of p. The main program should call three subprograms: the ran- dom number generator from the system library to generate the random integers; a subprogram to calculate the greatest common divisor of two integers using Euclid’s Algorithm; and a subprogram that calculates square roots. If these latter two pro- grams are not available, you will have to write them as well. The main program should loop through a large number of random numbers to give an estimate of the afore- mentioned probability. From this, it is a simple matter to solve for your estimate of p.
If the result is close to 3.14, congratulations! If not, then the result is probably low, usually a value of around 2.7. Why would such an inferior result be obtained?
8.6 What RC4 key value will leave S unchanged during initialization? That is, after the initial permutation of S, the entries of S will be equal to the values from 0 through 255 in ascending order.
8.7 RC4 has a secret internal state which is a permutation of all the possible values of the vector S and the two indices i and j. a. Using a straightforward scheme to store the internal state, how many bits are used? b. Suppose we think of it from the point of view of how much information is repre-
sented by the state. In that case, we need to determine how may different states there are, then take the log to base 2 to find out how many bits of information this represents. Using this approach, how many bits would be needed to represent the state?
8.8 Alice and Bob agree to communicate privately via email using a scheme based on RC4, but they want to avoid using a new secret key for each transmission. Alice and Bob privately agree on a 128-bit key k. To encrypt a message m, consisting of a string of bits, the following procedure is used. 1. Choose a random 64-bit value v 2. Generate the ciphertext c = RC4(v} k) ⊕ m 3. Send the bit string (v} c)
a. Suppose Alice uses this procedure to send a message m to Bob. Describe how Bob can recover the message m from (v} c) using k.
b. If an adversary observes several values (v1 } c1), (v2 } c2), c transmitted between Alice and Bob, how can he/she determine when the same key stream has been used to encrypt two messages?
c. Approximately how many messages can Alice expect to send before the same key stream will be used twice? Use the result from the birthday paradox described in Appendix U.
d. What does this imply about the lifetime of the key k (i.e., the number of mes- sages that can be encrypted using k)?
8.9 Suppose you have a true random bit generator where each bit in the generated stream has the same probability of being a 0 or 1 as any other bit in the stream and that the bits are not correlated; that is the bits are generated from identical independent dis- tribution. However, the bit stream is biased. The probability of a 1 is 0.5 + 0 and the probability of a 0 is 0.5 - 0, where 0 6 0 6 0.5. A simple conditioning algorithm is as follows: Examine the bit stream as a sequence of nonoverlapping pairs. Discard all 00 and 11 pairs. Replace each 01 pair with 0 and each 10 pair with 1.
282 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS
a. What is the probability of occurrence of each pair in the original sequence? b. What is the probability of occurrence of 0 and 1 in the modified sequence? c. What is the expected number of input bits to produce x output bits? d. Suppose that the algorithm uses overlapping successive bit pairs instead of non-
overlapping successive bit pairs. That is, the first output bit is based on input bits 1 and 2, the second output bit is based on input bits 2 and 3, and so on. What can you say about the output bit stream?
8.10 Another approach to conditioning is to consider the bit stream as a sequence of non- overlapping groups of n bits each and output the parity of each group. That is, if a group contains an odd number of ones, the output is 1; otherwise the output is 0. a. Express this operation in terms of a basic Boolean function. b. Assume, as in the preceding problem, that the probability of a 1 is 0.5 + 0. If each
group consists of 2 bits, what is the probability of an output of 1? c. If each group consists of 4 bits, what is the probability of an output of 1? d. Generalize the result to find the probability of an output of 1 for input groups of
n bits. 8.11 It is important to note that the Repetition Count Test described in Section 8.6 is not a
very powerful health test. It is able to detect only catastrophic failures of an entropy source. For example, a noise source evaluated at 8 bits of min-entropy per sample has a cutoff value of 5 repetitions to ensure a false-positive rate of approximately once per four billion samples generated. If that noise source somehow failed to the point that it was providing only 6 bits of min-entropy per sample, how many samples would be expected to be needed before the Repetition Count Test would notice the problem?
283
Public-Key Cryptography and RSA
9.1 Principles of Public-Key Cryptosystems
Public-Key Cryptosystems
Applications for Public-Key Cryptosystems
Requirements for Public-Key Cryptography
Public-Key Cryptanalysis
9.2 The RSA Algorithm
Description of the Algorithm
Computational Aspects
The Security of RSA
9.3 Key Terms, Review Questions, and Problems
CHAPTER
PART THREE: ASYMMETRIC CIPHERS
284 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA
The development of public-key, or asymmetric, cryptography is the greatest and per-
haps the only true revolution in the entire history of cryptography. From its earliest
beginnings to modern times, virtually all cryptographic systems have been based on
the elementary tools of substitution and permutation. After millennia of working with
algorithms that could be calculated by hand, a major advance in symmetric cryptogra-
phy occurred with the development of the rotor encryption/decryption machine. The
electromechanical rotor enabled the development of fiendishly complex cipher sys-
tems. With the availability of computers, even more complex systems were devised,
the most prominent of which was the Lucifer effort at IBM that culminated in the Data
Encryption Standard (DES). But both rotor machines and DES, although represent-
ing significant advances, still relied on the bread-and-butter tools of substitution and
permutation.
Public-key cryptography provides a radical departure from all that has gone be-
fore. For one thing, public-key algorithms are based on mathematical functions rather
than on substitution and permutation. More important, public-key cryptography is
asymmetric, involving the use of two separate keys, in contrast to symmetric encryp-
tion, which uses only one key. The use of two keys has profound consequences in the
areas of confidentiality, key distribution, and authentication, as we shall see.
Before proceeding, we should mention several common misconceptions con-
cerning public-key encryption. One such misconception is that public-key encryption
is more secure from cryptanalysis than is symmetric encryption. In fact, the security of
any encryption scheme depends on the length of the key and the computational work
involved in breaking a cipher. There is nothing in principle about either symmetric or
public-key encryption that makes one superior to another from the point of view of
resisting cryptanalysis.
A second misconception is that public-key encryption is a general-purpose tech-
nique that has made symmetric encryption obsolete. On the contrary, because of the
computational overhead of current public-key encryption schemes, there seems no
foreseeable likelihood that symmetric encryption will be abandoned. As one of the
inventors of public-key encryption has put it [DIFF88], “the restriction of public-key
cryptography to key management and signature applications is almost universally
accepted.”
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Present an overview of the basic principles of public-key cryptosystems.
◆ Explain the two distinct uses of public-key cryptosystems.
◆ List and explain the requirements for a public-key cryptosystem.
◆ Present an overview of the RSA algorithm.
◆ Understand the timing attack.
◆ Summarize the relevant issues related to the complexity of algorithms.
9.1 / PRINCIPLES OF PUBLIC-KEY CRYPTOSYSTEMS 285
Finally, there is a feeling that key distribution is trivial when using public-key
encryption, compared to the rather cumbersome handshaking involved with key dis-
tribution centers for symmetric encryption. In fact, some form of protocol is needed,
generally involving a central agent, and the procedures involved are not simpler nor
any more efficient than those required for symmetric encryption (e.g., see analysis in
[NEED78]).
This chapter and the next provide an overview of public-key cryptography. First,
we look at its conceptual framework. Interestingly, the concept for this technique was
developed and published before it was shown to be practical to adopt it. Next, we ex-
amine the RSA algorithm, which is the most important encryption/decryption algo-
rithm that has been shown to be feasible for public-key encryption. Other important
public-key cryptographic algorithms are covered in Chapter 10.
Much of the theory of public-key cryptosystems is based on number theory. If
one is prepared to accept the results given in this chapter, an understanding of number
theory is not strictly necessary. However, to gain a full appreciation of public-key
algorithms, some understanding of number theory is required. Chapter 2 provides the
necessary background in number theory.
Table 9.1 defines some key terms.
9.1 PRINCIPLES OF PUBLIC-KEY CRYPTOSYSTEMS
The concept of public-key cryptography evolved from an attempt to attack two of
the most difficult problems associated with symmetric encryption. The first problem
is that of key distribution, which is examined in some detail in Chapter 14.
As Chapter 14 discusses, key distribution under symmetric encryption requires
either (1) that two communicants already share a key, which somehow has been dis-
tributed to them; or (2) the use of a key distribution center. Whitfield Diffie, one
Asymmetric Keys Two related keys, a public key and a private key, that are used to perform complementary operations, such as
encryption and decryption or signature generation and signature verification.
Public Key Certificate A digital document issued and digitally signed by the private key of a Certification Authority that binds the
name of a subscriber to a public key. The certificate indicates that the subscriber identified in the certificate
has sole control and access to the corresponding private key.
Public Key (Asymmetric) Cryptographic Algorithm A cryptographic algorithm that uses two related keys, a public key and a private key. The two keys have the
property that deriving the private key from the public key is computationally infeasible.
Public Key Infrastructure (PKI) A set of policies, processes, server platforms, software and workstations used for the purpose of administer-
ing certificates and public-private key pairs, including the ability to issue, maintain, and revoke public key
certificates.
Source: Glossary of Key Information Security Terms, NIST IR 7298 [KISS06].
Table 9.1 Terminology Related to Asymmetric Encryption
286 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA
of the discoverers of public-key encryption (along with Martin Hellman, both at
Stanford University at the time), reasoned that this second requirement negated the
very essence of cryptography: the ability to maintain total secrecy over your own
communication. As Diffie put it [DIFF88], “what good would it do after all to de-
velop impenetrable cryptosystems, if their users were forced to share their keys with
a KDC that could be compromised by either burglary or subpoena?”
The second problem that Diffie pondered, and one that was apparently un-
related to the first, was that of digital signatures. If the use of cryptography was to become widespread, not just in military situations but for commercial and private
purposes, then electronic messages and documents would need the equivalent of
signatures used in paper documents. That is, could a method be devised that would
stipulate, to the satisfaction of all parties, that a digital message had been sent by a
particular person? This is a somewhat broader requirement than that of authentica-
tion, and its characteristics and ramifications are explored in Chapter 13.
Diffie and Hellman achieved an astounding breakthrough in 1976 [DIFF76 a, b]
by coming up with a method that addressed both problems and was radically different
from all previous approaches to cryptography, going back over four millennia.1
In the next subsection, we look at the overall framework for public-key cryp-
tography. Then we examine the requirements for the encryption/decryption algo-
rithm that is at the heart of the scheme.
Public-Key Cryptosystems
Asymmetric algorithms rely on one key for encryption and a different but related
key for decryption. These algorithms have the following important characteristic.
■ It is computationally infeasible to determine the decryption key given only
knowledge of the cryptographic algorithm and the encryption key.
In addition, some algorithms, such as RSA, also exhibit the following characteristic.
■ Either of the two related keys can be used for encryption, with the other used
for decryption.
A public-key encryption scheme has six ingredients (Figure 9.1a; compare with Figure 3.1).
■ Plaintext: This is the readable message or data that is fed into the algorithm as input.
■ Encryption algorithm: The encryption algorithm performs various transfor- mations on the plaintext.
1Diffie and Hellman first publicly introduced the concepts of public-key cryptography in 1976. Hellman credits Merkle with independently discovering the concept at that same time, although Merkle did not publish until 1978 [MERK78]. In fact, the first unclassified document describing public-key distribution and public-key cryptography was a 1974 project proposal by Merkle (http://merkle.com/1974). However, this is not the true beginning. Admiral Bobby Inman, while director of the National Security Agency (NSA), claimed that public-key cryptography had been discovered at NSA in the mid-1960s [SIMM93]. The first documented introduction of these concepts came in 1970, from the Communications-Electronics Security Group, Britain’s counterpart to NSA, in a classified report by James Ellis [ELLI70]. Ellis re- ferred to the technique as nonsecret encryption and describes the discovery in [ELLI99].
9.1 / PRINCIPLES OF PUBLIC-KEY CRYPTOSYSTEMS 287
■ Public and private keys: This is a pair of keys that have been selected so that if one is used for encryption, the other is used for decryption. The exact transfor-
mations performed by the algorithm depend on the public or private key that
is provided as input.
■ Ciphertext: This is the encrypted message produced as output. It depends on the plaintext and the key. For a given message, two different keys will produce
two different ciphertexts.
Figure 9.1 Public-Key Cryptography
Plaintext input
Bobs's public-key
ring
Transmitted ciphertext
Plaintext outputEncryption algorithm
(e.g., RSA) Decryption algorithm
Joy
Mike
Mike Bob
Ted
Alice
Alice's public key
Alice's private key
(a) Encryption with public key
Plaintext input
Transmitted ciphertext
Plaintext outputEncryption algorithm
(e.g., RSA) Decryption algorithm
Bob's private key
Bob
Bob's public key
Alice's public key
ring
Joy Ted
(b) Encryption with private key
X
X
PUa
PUb
PRa
PRb
Y = E[PUa, X]
Y = E[PRb, X]
X = D[PRa, Y]
X = D[PUb, Y]
Alice
Bob Alice
288 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA
■ Decryption algorithm: This algorithm accepts the ciphertext and the matching key and produces the original plaintext.
The essential steps are the following.
1. Each user generates a pair of keys to be used for the encryption and decryp- tion of messages.
2. Each user places one of the two keys in a public register or other accessible file. This is the public key. The companion key is kept private. As Figure 9.1a
suggests, each user maintains a collection of public keys obtained from others.
3. If Bob wishes to send a confidential message to Alice, Bob encrypts the mes- sage using Alice’s public key.
4. When Alice receives the message, she decrypts it using her private key. No other recipient can decrypt the message because only Alice knows Alice’s pri-
vate key.
With this approach, all participants have access to public keys, and private
keys are generated locally by each participant and therefore need never be distrib-
uted. As long as a user’s private key remains protected and secret, incoming com-
munication is secure. At any time, a system can change its private key and publish
the companion public key to replace its old public key.
Table 9.2 summarizes some of the important aspects of symmetric and public-
key encryption. To discriminate between the two, we refer to the key used in sym-
metric encryption as a secret key. The two keys used for asymmetric encryption are referred to as the public key and the private key.2 Invariably, the private key is kept secret, but it is referred to as a private key rather than a secret key to avoid confu-
sion with symmetric encryption.
Let us take a closer look at the essential elements of a public-key encryption
scheme, using Figure 9.2 (compare with Figure 3.2). There is some source A that
produces a message in plaintext, X = [X1, X2, c , XM]. The M elements of X are letters in some finite alphabet. The message is intended for destination B. B gener-
ates a related pair of keys: a public key, PUb, and a private key, PRb. PRb is known only to B, whereas PUb is publicly available and therefore accessible by A.
With the message X and the encryption key PUb as input, A forms the cipher- text Y = [Y1, Y2, c , YN]:
Y = E(PUb, X)
The intended receiver, in possession of the matching private key, is able to invert
the transformation:
X = D(PRb,Y)
2The following notation is used consistently throughout. A secret key is represented by Km, where m is some modifier; for example, Ka is a secret key owned by user A. A public key is represented by PUa, for user A, and the corresponding private key is PRa. Encryption of plaintext X can be performed with a secret key, a public key, or a private key, denoted by E(Ka, X), E(PUa, X), and E(PRa, X), respectively. Similarly, decryption of ciphertext Y can be performed with a secret key, a public key, or a private key, denoted by D(Ka, Y), D(PUa, Y), and D(PRa, Y), respectively.
9.1 / PRINCIPLES OF PUBLIC-KEY CRYPTOSYSTEMS 289
An adversary, observing Y and having access to PUb, but not having access to PRb or X, must attempt to recover X and/or PRb. It is assumed that the adversary does have knowledge of the encryption (E) and decryption (D) algorithms. If the ad-
versary is interested only in this particular message, then the focus of effort is to
recover X by generating a plaintext estimate Xn . Often, however, the adversary is interested in being able to read future messages as well, in which case an attempt is
made to recover PRb by generating an estimate PRnb.
Conventional Encryption Public-Key Encryption
Needed to Work:
1. The same algorithm with the same key is
used for encryption and decryption.
2. The sender and receiver must share the
algorithm and the key.
Needed for Security:
1. The key must be kept secret.
2. It must be impossible or at least impractical
to decipher a message if the key is kept
secret.
3. Knowledge of the algorithm plus samples of
ciphertext must be insufficient to determine
the key.
Needed to Work:
1. One algorithm is used for encryption and a related
algorithm for decryption with a pair of keys, one for
encryption and one for decryption.
2. The sender and receiver must each have one of the
matched pair of keys (not the same one).
Needed for Security:
1. One of the two keys must be kept secret.
2. It must be impossible or at least impractical to
decipher a message if one of the keys is kept secret.
3. Knowledge of the algorithm plus one of the keys
plus samples of ciphertext must be insufficient to
determine the other key.
Table 9.2 Conventional and Public-Key Encryption
Figure 9.2 Public-Key Cryptosystem: Confidentiality
Message source
Cryptanalyst
Key pair source
Destination X
P̂Rb
PUb
Encryption algorithm
Decryption algorithm
PRb
X̂
Source A Destination B
Y = E[PUb, X] X = D[PRb, Y]
Hiva-Network.Com
290 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA
Figure 9.3 Public-Key Cryptosystem: Authentication
Message source
Cryptanalyst
Key pair source
Destination X
^
PRa
PRa
PUa
Encryption algorithm
Decryption algorithm
Source A Destination B
Y = E[PRa, X] X = D[PUa, Y]
We mentioned earlier that either of the two related keys can be used for en-
cryption, with the other being used for decryption. This enables a rather differ-
ent cryptographic scheme to be implemented. Whereas the scheme illustrated in
Figure 9.2 provides confidentiality, Figures 9.1b and 9.3 show the use of public-key
encryption to provide authentication:
Y = E(PRa,X) X = D(PUa,Y)
In this case, A prepares a message to B and encrypts it using A’s private key
before transmitting it. B can decrypt the message using A’s public key. Because the
message was encrypted using A’s private key, only A could have prepared the mes-
sage. Therefore, the entire encrypted message serves as a digital signature. In addi- tion, it is impossible to alter the message without access to A’s private key, so the
message is authenticated both in terms of source and in terms of data integrity.
In the preceding scheme, the entire message is encrypted, which, although val-
idating both author and contents, requires a great deal of storage. Each document
must be kept in plaintext to be used for practical purposes. A copy also must be
stored in ciphertext so that the origin and contents can be verified in case of a dis-
pute. A more efficient way of achieving the same results is to encrypt a small block
of bits that is a function of the document. Such a block, called an authenticator,
must have the property that it is infeasible to change the document without chang-
ing the authenticator. If the authenticator is encrypted with the sender’s private
key, it serves as a signature that verifies origin, content, and sequencing. Chapter 13
examines this technique in detail.
9.1 / PRINCIPLES OF PUBLIC-KEY CRYPTOSYSTEMS 291
It is important to emphasize that the encryption process depicted in Figures 9.1b
and 9.3 does not provide confidentiality. That is, the message being sent is safe from
alteration but not from eavesdropping. This is obvious in the case of a signature
based on a portion of the message, because the rest of the message is transmitted in
the clear. Even in the case of complete encryption, as shown in Figure 9.3, there is
no protection of confidentiality because any observer can decrypt the message by
using the sender’s public key.
It is, however, possible to provide both the authentication function and confi-
dentiality by a double use of the public-key scheme (Figure 9.4):
Z = E(PUb, E(PRa,X)) X = D(PUa, D(PRb,Z))
In this case, we begin as before by encrypting a message, using the sender’s private
key. This provides the digital signature. Next, we encrypt again, using the receiver’s
public key. The final ciphertext can be decrypted only by the intended receiver, who
alone has the matching private key. Thus, confidentiality is provided. The disadvan-
tage of this approach is that the public-key algorithm, which is complex, must be
exercised four times rather than two in each communication.
Applications for Public-Key Cryptosystems
Before proceeding, we need to clarify one aspect of public-key cryptosystems that
is otherwise likely to lead to confusion. Public-key systems are characterized by the
use of a cryptographic algorithm with two keys, one held private and one available
publicly. Depending on the application, the sender uses either the sender’s private
key or the receiver’s public key, or both, to perform some type of cryptographic
Figure 9.4 Public-Key Cryptosystem: Authentication and Secrecy
Message source
Message dest.
X Encryption algorithm
Ke y pair source
PUb PRb
Source A Destination B
Key pair source
PRa PUa
Y Encryption algorithm
Z Decryption algorithm
Y Decryption algorithm
X
292 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA
function. In broad terms, we can classify the use of public-key cryptosystems into three categories
■ Encryption/decryption: The sender encrypts a message with the recipient’s public key, and the recipient decrypts the message with the recipient’s private
key.
■ Digital signature: The sender “signs” a message with its private key. Signing is achieved by a cryptographic algorithm applied to the message or to a small
block of data that is a function of the message.
■ Key exchange: Two sides cooperate to exchange a session key, which is a secret key for symmetric encryption generated for use for a particular transaction (or
session) and valid for a short period of time. Several different approaches are
possible, involving the private key(s) of one or both parties; this is discussed in
Chapter 10.
Some algorithms are suitable for all three applications, whereas others can be
used only for one or two of these applications. Table 9.3 indicates the applications
supported by the algorithms discussed in this book.
Requirements for Public-Key Cryptography
The cryptosystem illustrated in Figures 9.2 through 9.4 depends on a cryptographic
algorithm based on two related keys. Diffie and Hellman postulated this system
without demonstrating that such algorithms exist. However, they did lay out the
conditions that such algorithms must fulfill [DIFF76b].
1. It is computationally easy for a party B to generate a key pair (public key PUb, private key PRb).
2. It is computationally easy for a sender A, knowing the public key and the mes- sage to be encrypted, M, to generate the corresponding ciphertext:
C = E(PUb, M)
3. It is computationally easy for the receiver B to decrypt the resulting ciphertext using the private key to recover the original message:
M = D(PRb, C) = D[PRb, E(PUb, M)]
4. It is computationally infeasible for an adversary, knowing the public key, PUb, to determine the private key, PRb.
Algorithm Encryption/Decryption Digital Signature Key Exchange
RSA Yes Yes Yes
Elliptic Curve Yes Yes Yes
Diffie–Hellman No No Yes
DSS No Yes No
Table 9.3 Applications for Public-Key Cryptosystems
9.1 / PRINCIPLES OF PUBLIC-KEY CRYPTOSYSTEMS 293
5. It is computationally infeasible for an adversary, knowing the public key, PUb, and a ciphertext, C, to recover the original message, M.
We can add a sixth requirement that, although useful, is not necessary for all
public-key applications:
6. The two keys can be applied in either order:
M = D[PUb, E(PRb, M)] = D[PRb, E(PUb, M)]
These are formidable requirements, as evidenced by the fact that only a few
algorithms (RSA, elliptic curve cryptography, Diffie–Hellman, DSS) have received
widespread acceptance in the several decades since the concept of public-key cryp-
tography was proposed.
Before elaborating on why the requirements are so formidable, let us first re-
cast them. The requirements boil down to the need for a trap-door one-way func-
tion. A one-way function3 is one that maps a domain into a range such that every function value has a unique inverse, with the condition that the calculation of the
function is easy, whereas the calculation of the inverse is infeasible:
Y = f(X) easy X = f -1(Y) infeasible
Generally, easy is defined to mean a problem that can be solved in polynomial time as a function of input length. Thus, if the length of the input is n bits, then the time to compute the function is proportional to na, where a is a fixed constant. Such algorithms are said to belong to the class P. The term infeasible is a much fuzzier concept. In general, we can say a problem is infeasible if the effort to solve it grows
faster than polynomial time as a function of input size. For example, if the length
of the input is n bits and the time to compute the function is proportional to 2n, the problem is considered infeasible. Unfortunately, it is difficult to determine if a
particular algorithm exhibits this complexity. Furthermore, traditional notions of
computational complexity focus on the worst-case or average-case complexity of
an algorithm. These measures are inadequate for cryptography, which requires that
it be infeasible to invert a function for virtually all inputs, not for the worst case or
even average case. A brief introduction to some of these concepts is provided in
Appendix W.
We now turn to the definition of a trap-door one-way function, which is easy to calculate in one direction and infeasible to calculate in the other direction un-
less certain additional information is known. With the additional information the
inverse can be calculated in polynomial time. We can summarize as follows: A trap-
door one-way function is a family of invertible functions fk, such that
Y = fk(X) easy, if k and X are known X = f k
-1(Y) easy, if k and Y are known
X = f k -1(Y) infeasible, if Y is known but k is not known
3Not to be confused with a one-way hash function, which takes an arbitrarily large data field as its argument and maps it to a fixed output. Such functions are used for authentication (see Chapter 11).
294 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA
Thus, the development of a practical public-key scheme depends on discovery of a
suitable trap-door one-way function.
Public-Key Cryptanalysis
As with symmetric encryption, a public-key encryption scheme is vulnerable to a
brute-force attack. The countermeasure is the same: Use large keys. However, there
is a tradeoff to be considered. Public-key systems depend on the use of some sort of
invertible mathematical function. The complexity of calculating these functions may
not scale linearly with the number of bits in the key but grow more rapidly than that.
Thus, the key size must be large enough to make brute-force attack impractical but
small enough for practical encryption and decryption. In practice, the key sizes that
have been proposed do make brute-force attack impractical but result in encryp-
tion/decryption speeds that are too slow for general-purpose use. Instead, as was
mentioned earlier, public-key encryption is currently confined to key management
and signature applications.
Another form of attack is to find some way to compute the private key given
the public key. To date, it has not been mathematically proven that this form of at-
tack is infeasible for a particular public-key algorithm. Thus, any given algorithm,
including the widely used RSA algorithm, is suspect. The history of cryptanalysis
shows that a problem that seems insoluble from one perspective can be found to
have a solution if looked at in an entirely different way.
Finally, there is a form of attack that is peculiar to public-key systems. This is,
in essence, a probable-message attack. Suppose, for example, that a message were
to be sent that consisted solely of a 56-bit DES key. An adversary could encrypt all
possible 56-bit DES keys using the public key and could discover the encrypted key
by matching the transmitted ciphertext. Thus, no matter how large the key size of the
public-key scheme, the attack is reduced to a brute-force attack on a 56-bit key. This
attack can be thwarted by appending some random bits to such simple messages.
9.2 THE RSA ALGORITHM
The pioneering paper by Diffie and Hellman [DIFF76b] introduced a new approach
to cryptography and, in effect, challenged cryptologists to come up with a crypto-
graphic algorithm that met the requirements for public-key systems. A number of
algorithms have been proposed for public-key cryptography. Some of these, though
initially promising, turned out to be breakable.4
One of the first successful responses to the challenge was developed in 1977
by Ron Rivest, Adi Shamir, and Len Adleman at MIT and first published in 1978
[RIVE78].5 The Rivest-Shamir-Adleman (RSA) scheme has since that time reigned
supreme as the most widely accepted and implemented general-purpose approach
to public-key encryption.
4The most famous of the fallen contenders is the trapdoor knapsack proposed by Ralph Merkle. We describe this in Appendix J. 5Apparently, the first workable public-key system for encryption/decryption was put forward by Clifford Cocks of Britain’s CESG in 1973 [COCK73]; Cocks’ method is virtually identical to RSA.
9.2 / THE RSA ALGORITHM 295
The RSA scheme is a cipher in which the plaintext and ciphertext are integers between 0 and n - 1 for some n. A typical size for n is 1024 bits, or 309 decimal digits. That is, n is less than 21024. We examine RSA in this section in some detail, beginning with an explanation of the algorithm. Then we examine some of the com-
putational and cryptanalytical implications of RSA.
Description of the Algorithm
RSA makes use of an expression with exponentials. Plaintext is encrypted in blocks,
with each block having a binary value less than some number n. That is, the block size must be less than or equal to log 2(n) + 1; in practice, the block size is i bits, where 2i 6 n … 2i + 1. Encryption and decryption are of the following form, for some plaintext block M and ciphertext block C.
C = Me mod n M = C d mod n = (Me)d mod n = Med mod n
Both sender and receiver must know the value of n. The sender knows the value of e, and only the receiver knows the value of d. Thus, this is a public- key encryption algorithm with a public key of PU = {e, n} and a private key of PR = {d, n}. For this algorithm to be satisfactory for public-key encryption, the fol- lowing requirements must be met.
1. It is possible to find values of e, d, and n such that Med mod n = M for all M 6 n. 2. It is relatively easy to calculate Me mod n and C d mod n for all values of M 6 n. 3. It is infeasible to determine d given e and n.
For now, we focus on the first requirement and consider the other questions
later. We need to find a relationship of the form
Med mod n = M
The preceding relationship holds if e and d are multiplicative inverses modulo f(n), where f(n) is the Euler totient function. It is shown in Chapter 2 that for p, q prime, f(pq) = (p - 1)(q - 1). The relationship between e and d can be expressed as
ed mod f(n) = 1 (9.1)
This is equivalent to saying
ed K 1 mod f(n) d K e-1 mod f(n)
That is, e and d are multiplicative inverses mod f(n). Note that, according to the rules of modular arithmetic, this is true only if d (and therefore e) is relatively prime to f(n). Equivalently, gcd(f(n), d) = 1. See Appendix R for a proof that Equation (9.1) satisfies the requirement for RSA.
We are now ready to state the RSA scheme. The ingredients are the following:
p, q, two prime numbers (private, chosen) n = pq (public, calculated) e, with gcd(f(n), e) = 1; 1 6 e 6 f(n) (public, chosen) d K e-1 (mod f(n)) (private, calculated)
296 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA
The private key consists of {d, n} and the public key consists of {e, n}. Suppose that user A has published its public key and that user B wishes to send the message
M to A. Then B calculates C = Me mod n and transmits C. On receipt of this ci- phertext, user A decrypts by calculating M = C d mod n.
Figure 9.5 summarizes the RSA algorithm. It corresponds to Figure 9.1a: Alice
generates a public/private key pair; Bob encrypts using Alice’s public key; and Alice
decrypts using her private key. An example from [SING99] is shown in Figure 9.6.
For this example, the keys were generated as follows.
1. Select two prime numbers, p = 17 and q = 11. 2. Calculate n = pq = 17 * 11 = 187. 3. Calculate f(n) = (p - 1)(q - 1) = 16 * 10 = 160. 4. Select e such that e is relatively prime to f(n) = 160 and less than f(n); we
choose e = 7. 5. Determine d such that de K 1 (mod 160) and d 6 160. The correct value is
d = 23, because 23 * 7 = 161 = (1 * 160) + 1; d can be calculated using the extended Euclid’s algorithm (Chapter 2).
The resulting keys are public key PU = {7, 187} and private key PR = {23, 187}. The example shows the use of these keys for a plaintext input of M = 88. For encryption, we need to calculate C = 887 mod 187. Exploiting the properties of modular arithmetic, we can do this as follows.
887 mod 187 = [(884 mod 187) * (882 mod 187) * (881 mod 187)] mod 187 881 mod 187 = 88
882 mod 187 = 7744 mod 187 = 77
884 mod 187 = 59,969,536 mod 187 = 132
887 mod 187 = (88 * 77 * 132) mod 187 = 894,432 mod 187 = 11
For decryption, we calculate M = 1123 mod 187:
1123 mod 187 = [(111 mod 187) * (112 mod 187) * (114 mod 187) * (118 mod 187) * (118 mod 187)] mod 187 111 mod 187 = 11
112 mod 187 = 121
114 mod 187 = 14,641 mod 187 = 55
118 mod 187 = 214,358,881 mod 187 = 33
1123 mod 187 = (11 * 121 * 55 * 33 * 33) mod 187 = 79,720,245 mod 187 = 88
We now look at an example from [HELL79], which shows the use of RSA to
process multiple blocks of data. In this simple example, the plaintext is an alpha-
numeric string. Each plaintext symbol is assigned a unique code of two decimal
9.2 / THE RSA ALGORITHM 297
digits (e.g., a = 00, A = 26).6 A plaintext block consists of four decimal digits, or two alphanumeric characters. Figure 9.7a illustrates the sequence of events for the
encryption of multiple blocks, and Figure 9.7b gives a specific example. The circled
numbers indicate the order in which operations are performed.
Computational Aspects
We now turn to the issue of the complexity of the computation required to use
RSA. There are actually two issues to consider: encryption/decryption and key
generation. Let us look first at the process of encryption and decryption and then
consider key generation.
6 The complete mapping of alphanumeric characters to decimal digits is at box.com/Crypto7e in the doc- ument RSAexample.pdf.
Figure 9.6 Example of RSA Algorithm
Encryption
Plaintext 88
Plaintext 88
Ciphertext 11
88 mod 187 = 11
PU = 7, 187
Decryption
7 11 mod 187 = 88
PR � 23, 187
23
Figure 9.5 The RSA Algorithm
Key Generation by Alice
Select p, q p and q both prime, p ≠ q
Calculate n = p * q
Calcuate f(n) = (p - 1)(q - 1)
Select integer e gcd (f(n), e) = 1; 1 6 e 6 f(n)
Calculate d d K e-1 (mod f(n))
Public key PU = {e, n}
Private key PR = {d, n}
Encryption by Bob with Alice’s Public Key
Plaintext: M 6 n
Ciphertext: C = Me mod n
Decryption by Alice with Alice’s Public Key
Ciphertext: C
Plaintext: M = C d mod n
298 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA
EXPONENTIATION IN MODULAR ARITHMETIC Both encryption and decryption in RSA involve raising an integer to an integer power, mod n. If the exponentiation is done over the integers and then reduced modulo n, the intermediate values would be gargantuan. Fortunately, as the preceding example shows, we can make use of a
property of modular arithmetic:
[(a mod n) * (b mod n)] mod n = (a * b) mod n
Thus, we can reduce intermediate results modulo n. This makes the calculation practical.
Another consideration is the efficiency of exponentiation, because with RSA,
we are dealing with potentially large exponents. To see how efficiency might be in-
creased, consider that we wish to compute x16. A straightforward approach requires 15 multiplications:
x16 = x * x * x * x * x * x * x * x * x * x * x * x * x * x * x * x
Figure 9.7 RSA Processing of Multiple Blocks
Plaintext P
Decimal string
Sender
Receiver
(a) General approach (b) Example
Blocks of numbers
Transmit
P1, P2,
P1 = C1 d mod n
P2 = C2 d mod n
Ciphertext C
C1 = P1 e mod n
C2 = P2 e mod n
Recovered decimal text
n = pq
Random number generator
e, p, q
Private key d, n
Public key e, n
How_are_you?
33 14 22 62 00 17 04 62 24 14 20 66
Sender
Receiver
Transmit
P1 = 3314 P2 = 2262 P3 = 0017 P4 = 0462 P5 = 2414 P6 = 2066
C1 = 3314 11 mod 11023 = 10260
C2 = 2262 11 mod 11023 = 9489
C3 = 17 11 mod 11023 = 1782
C4 = 462 11 mod 11023 = 727
C5 = 2414 11 mod 11023 = 10032
C6 = 2066 11 mod 11023 = 2253
P1 = 10260 5891 mod 11023 = 3314
P2 = 9489 5891 mod 11023 = 2262
P3 = 1782 5891 mod 11023 = 0017
P4 = 727 5891 mod 11023 = 0462
P5 = 10032 5891 mod 11023 = 2414
P6 = 2253 5891 mod 11023 = 2066
11023 = 73 151
5891 = 11–1 mod 10800 10800 = (73 – 1)(151 – 1) 11023 = 73 51
Random number generator
e = 11 n = 11023
d = 5891 n = 11023
e = 11 p = 73, q = 151
1
2
6
3
4
5
7
1
2
6
3
4
5
7
d = e–1 mod f(n) f(n) = (p – 1)(q – 1)
n = pq
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9.2 / THE RSA ALGORITHM 299
However, we can achieve the same final result with only four multiplications if we
repeatedly take the square of each partial result, successively forming (x2, x4, x8, x16). As another example, suppose we wish to calculate x11 mod n for some integers x and n. Observe that x11 = x1 + 2 + 8 = (x)(x2)(x8). In this case, we compute x mod n, x2 mod n, x4 mod n, and x8 mod n and then calculate [(x mod n) * (x2 mod n) * (x8 mod n)] mod n.
More generally, suppose we wish to find the value ab mod n with a, b, and m positive integers. If we express b as a binary number bkbk - 1 c b0, then we have
b = a bi≠0
2i
Therefore,
ab = a ¢ Σ2i
bi≠0 ≤ = q
bi≠0 a(2
i)
ab mod n = J q bi≠0
a(2 i) R mod n = ¢ q
bi≠0 Ja(2i) mod n R ≤ mod n
We can therefore develop the algorithm7 for computing ab mod n, shown in Figure 9.8. Table 9.4 shows an example of the execution of this algorithm. Note that
the variable c is not needed; it is included for explanatory purposes. The final value of c is the value of the exponent.
EFFICIENT OPERATION USING THE PUBLIC KEY To speed up the operation of the RSA algorithm using the public key, a specific choice of e is usually made. The most common choice is 65537 (216 + 1); two other popular choices are 3 and 17. Each of these choices has only two 1 bits, so the number of multiplications required to per-
form exponentiation is minimized.
7The algorithm has a long history; this particular pseudocode expression is from [CORM09].
Figure 9.8 Algorithm for Computing ab mod n
c 0; f 1
c 2 × cdo
bi = 1
then c c + 1
if
f (f × f) mod n
f (f × a) mod n
for i k downto 0
return f
Note: The integer b is expressed as a binary number bkbk - 1cb0.
300 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA
However, with a very small public key, such as e = 3, RSA becomes vulner- able to a simple attack. Suppose we have three different RSA users who all use
the value e = 3 but have unique values of n, namely (n1, n2, n3). If user A sends the same encrypted message M to all three users, then the three ciphertexts are C1 = M
3 mod n1, C2 = M 3 mod n2, and C3 = M
3 mod n3. It is likely that n1, n2, and n3 are pairwise relatively prime. Therefore, one can use the Chinese remainder theorem (CRT) to compute M3 mod (n1n2n3). By the rules of the RSA algorithm, M is less than each of the ni; therefore M
3 6 n1n2n3. Accordingly, the attacker need only compute the cube root of M3. This attack can be countered by adding a unique pseudorandom bit string as padding to each instance of M to be encrypted. This ap- proach is discussed subsequently.
The reader may have noted that the definition of the RSA algorithm
(Figure 9.5) requires that during key generation the user selects a value of e that is relatively prime to f(n). Thus, if a value of e is selected first and the primes p and q are generated, it may turn out that gcd(f(n), e) ≠ 1. In that case, the user must reject the p, q values and generate a new p, q pair.
EFFICIENT OPERATION USING THE PRIVATE KEY We cannot similarly choose a small constant value of d for efficient operation. A small value of d is vulnerable to a brute-force attack and to other forms of cryptanalysis [WIEN90]. However, there
is a way to speed up computation using the CRT. We wish to compute the value
M = C d mod n. Let us define the following intermediate results:
Vp = C d mod p Vq = C
d mod q
Following the CRT using Equation (8.8), define the quantities
Xp = q * (q-1 mod p) Xq = p * (p-1 mod q)
The CRT then shows, using Equation (8.9), that
M = (VpXp + VqXq) mod n
Furthermore, we can simplify the calculation of Vp and Vq using Fermat’s theorem, which states that ap - 1 K 1 (mod p) if p and a are relatively prime. Some thought should convince you that the following are valid.
Vp = C d mod p = C d mod(p - 1) mod p Vq = C
d mod q = C d mod(q - 1) mod q
i 9 8 7 6 5 4 3 2 1 0
bi 1 0 0 0 1 1 0 0 0 0
c 1 2 4 8 17 35 70 140 280 560
f 7 49 157 526 160 241 298 166 67 1
Table 9.4 Result of the Fast Modular Exponentiation Algorithm for ab mod n, where a = 7, b = 560 = 1000110000, and n = 561
9.2 / THE RSA ALGORITHM 301
The quantities d mod (p - 1) and d mod (q - 1) can be precalculated. The end result is that the calculation is approximately four times as fast as evaluating
M = C d mod n directly [BONE02].
KEY GENERATION Before the application of the public-key cryptosystem, each par- ticipant must generate a pair of keys. This involves the following tasks.
■ Determining two prime numbers, p and q.
■ Selecting either e or d and calculating the other.
First, consider the selection of p and q. Because the value of n = pq will be known to any potential adversary, in order to prevent the discovery of p and q by exhaustive methods, these primes must be chosen from a sufficiently large set
(i.e., p and q must be large numbers). On the other hand, the method used for find- ing large primes must be reasonably efficient.
At present, there are no useful techniques that yield arbitrarily large primes,
so some other means of tackling the problem is needed. The procedure that is gen-
erally used is to pick at random an odd number of the desired order of magnitude
and test whether that number is prime. If not, pick successive random numbers until
one is found that tests prime.
A variety of tests for primality have been developed (e.g., see [KNUT98] for
a description of a number of such tests). Almost invariably, the tests are probabi-
listic. That is, the test will merely determine that a given integer is probably prime. Despite this lack of certainty, these tests can be run in such a way as to make the
probability as close to 1.0 as desired. As an example, one of the more efficient
and popular algorithms, the Miller–Rabin algorithm, is described in Chapter 2.
With this algorithm and most such algorithms, the procedure for testing whether
a given integer n is prime is to perform some calculation that involves n and a randomly chosen integer a. If n “fails” the test, then n is not prime. If n “passes” the test, then n may be prime or nonprime. If n passes many such tests with many different randomly chosen values for a, then we can have high confidence that n is, in fact, prime.
In summary, the procedure for picking a prime number is as follows.
1. Pick an odd integer n at random (e.g., using a pseudorandom number generator).
2. Pick an integer a 6 n at random. 3. Perform the probabilistic primality test, such as Miller–Rabin, with a as a
parameter. If n fails the test, reject the value n and go to step 1.
4. If n has passed a sufficient number of tests, accept n; otherwise, go to step 2.
This is a somewhat tedious procedure. However, remember that this process is per-
formed relatively infrequently: only when a new pair (PU, PR) is needed. It is worth noting how many numbers are likely to be rejected before a
prime number is found. A result from number theory, known as the prime number
theorem, states that the primes near N are spaced on the average one every
302 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA
ln (N) integers. Thus, on average, one would have to test on the order of ln(N) integers before a prime is found. Actually, because all even integers can be im-
mediately rejected, the correct figure is ln(N)/2. For example, if a prime on the order of magnitude of 2200 were sought, then about ln(2200)/2 = 70 trials would be needed to find a prime.
Having determined prime numbers p and q, the process of key generation is completed by selecting a value of e and calculating d or, alternatively, selecting a value of d and calculating e. Assuming the former, then we need to select an e such that gcd(f(n), e) = 1 and then calculate d K e-1 (mod f(n)). Fortunately, there is a single algorithm that will, at the same time, calculate the greatest common divi-
sor of two integers and, if the gcd is 1, determine the inverse of one of the integers
modulo the other. The algorithm, referred to as the extended Euclid’s algorithm,
is explained in Chapter 2. Thus, the procedure is to generate a series of random
numbers, testing each against f(n) until a number relatively prime to f(n) is found. Again, we can ask the question: How many random numbers must we test to find
a usable number, that is, a number relatively prime to f(n)? It can be shown easily that the probability that two random numbers are relatively prime is about 0.6; thus,
very few tests would be needed to find a suitable integer (see Problem 2.18).
The Security of RSA
Five possible approaches to attacking the RSA algorithm are
■ Brute force: This involves trying all possible private keys.
■ Mathematical attacks: There are several approaches, all equivalent in effort to factoring the product of two primes.
■ Timing attacks: These depend on the running time of the decryption algorithm.
■ Hardware fault-based attack: This involves inducing hardware faults in the processor that is generating digital signatures.
■ Chosen ciphertext attacks: This type of attack exploits properties of the RSA algorithm.
The defense against the brute-force approach is the same for RSA as for other
cryptosystems, namely, to use a large key space. Thus, the larger the number of bits
in d, the better. However, because the calculations involved, both in key generation and in encryption/decryption, are complex, the larger the size of the key, the slower
the system will run.
In this subsection, we provide an overview of mathematical and timing attacks.
THE FACTORING PROBLEM We can identify three approaches to attacking RSA mathematically.
1. Factor n into its two prime factors. This enables calculation of f(n) = (p - 1) * (q - 1), which in turn enables determination of d K e-1 (mod f(n)).
2. Determine f(n) directly, without first determining p and q. Again, this enables determination of d K e-1 (mod f(n)).
3. Determine d directly, without first determining f(n).
9.2 / THE RSA ALGORITHM 303
Most discussions of the cryptanalysis of RSA have focused on the task of
factoring n into its two prime factors. Determining f(n) given n is equivalent to factoring n [RIBE96]. With presently known algorithms, determining d given e and n appears to be at least as time-consuming as the factoring problem [KALI95]. Hence, we can use factoring performance as a benchmark against which to evaluate
the security of RSA.
For a large n with large prime factors, factoring is a hard problem, but it is not as hard as it used to be. A striking illustration of this is the following. In 1977, the
three inventors of RSA dared Scientific American readers to decode a cipher they printed in Martin Gardner’s “Mathematical Games” column [GARD77]. They of-
fered a $100 reward for the return of a plaintext sentence, an event they predicted
might not occur for some 40 quadrillion years. In April of 1994, a group working
over the Internet claimed the prize after only eight months of work [LEUT94]. This
challenge used a public key size (length of n) of 129 decimal digits, or around 428 bits. In the meantime, just as they had done for DES, RSA Laboratories had issued
challenges for the RSA cipher with key sizes of 100, 110, 120, and so on, digits. The
latest challenge to be met is the RSA-768 challenge with a key length of 232 decimal
digits, or 768 bits. Table 9.5 shows the results.
A striking fact about the progress reflected in Table 9.5 concerns the method
used. Until the mid-1990s, factoring attacks were made using an approach known
as the quadratic sieve. The attack on RSA-130 used a newer algorithm, the gen-
eralized number field sieve (GNFS), and was able to factor a larger number than
RSA-129 at only 20% of the computing effort.
The threat to larger key sizes is twofold: the continuing increase in computing
power and the continuing refinement of factoring algorithms. We have seen that
the move to a different algorithm resulted in a tremendous speedup. We can expect
further refinements in the GNFS, and the use of an even better algorithm is also
a possibility. In fact, a related algorithm, the special number field sieve (SNFS),
Number of Decimal Digits Number of Bits Date Achieved
100 332 April 1991
110 365 April 1992
120 398 June 1993
129 428 April 1994
130 431 April 1996
140 465 February 1999
155 512 August 1999
160 530 April 2003
174 576 December 2003
200 663 May 2005
193 640 November 2005
232 768 December 2009
Table 9.5 Progress in RSA Factorization
304 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA
can factor numbers with a specialized form considerably faster than the generalized
number field sieve. Figure 9.9 compares the performance of the two algorithms. It is
reasonable to expect a breakthrough that would enable a general factoring perfor-
mance in about the same time as SNFS, or even better [ODLY95]. Thus, we need
to be careful in choosing a key size for RSA. The team that produced the 768-bit
factorization [KLEI10] observed that factoring a 1024-bit RSA modulus would be
about a thousand times harder than factoring a 768-bit modulus, and a 768-bit RSA
modulus is several thousands times harder to factor than a 512-bit one. Based on the
amount of time between the 512-bit and 768-bit factorization successes, the team
felt it to be reasonable to expect that the 1024-bit RSA moduli could be factored
well within the next decade by a similar academic effort. Thus, they recommended
phasing out usage of 1024-bit RSA within the next few years (from 2010).
Figure 9.9 MIPS-years Needed to Factor
1022
M IP
S -y
ea rs
n ee
de d
to f
ac to
r
200018001600140012001000800600 Bits
General number field sieve
Special number field sieve
1020
1018
1016
1014
1012
1010
108
106
104
102
100
9.2 / THE RSA ALGORITHM 305
In addition to specifying the size of n, a number of other constraints have been suggested by researchers. To avoid values of n that may be factored more easily, the algorithm’s inventors suggest the following constraints on p and q.
1. p and q should differ in length by only a few digits. Thus, for a 1024-bit key (309 decimal digits), both p and q should be on the order of magnitude of 1075 to 10100.
2. Both (p - 1) and (q - 1) should contain a large prime factor. 3. gcd(p - 1, q - 1) should be small.
In addition, it has been demonstrated that if e 6 n and d 6 n1/4, then d can be easily determined [WIEN90].
TIMING ATTACKS If one needed yet another lesson about how difficult it is to assess the security of a cryptographic algorithm, the appearance of timing attacks provides
a stunning one. Paul Kocher, a cryptographic consultant, demonstrated that a
snooper can determine a private key by keeping track of how long a computer takes
to decipher messages [KOCH96, KALI96b]. Timing attacks are applicable not just
to RSA, but to other public-key cryptography systems. This attack is alarming for
two reasons: It comes from a completely unexpected direction, and it is a ciphertext-
only attack.
A timing attack is somewhat analogous to a burglar guessing the combi- nation of a safe by observing how long it takes for someone to turn the dial
from number to number. We can explain the attack using the modular expo-
nentiation algorithm of Figure 9.8, but the attack can be adapted to work with
any implementation that does not run in fixed time. In this algorithm, modular
exponentiation is accomplished bit by bit, with one modular multiplication per-
formed at each iteration and an additional modular multiplication performed
for each 1 bit.
As Kocher points out in his paper, the attack is simplest to understand in an
extreme case. Suppose the target system uses a modular multiplication function that
is very fast in almost all cases but in a few cases takes much more time than an entire
average modular exponentiation. The attack proceeds bit-by-bit starting with the
leftmost bit, bk. Suppose that the first j bits are known (to obtain the entire exponent, start with j = 0 and repeat the attack until the entire exponent is known). For a given ciphertext, the attacker can complete the first j iterations of the for loop. The operation of the subsequent step depends on the unknown exponent bit. If the bit
is set, d d (d * a) mod n will be executed. For a few values of a and d, the modu- lar multiplication will be extremely slow, and the attacker knows which these are.
Therefore, if the observed time to execute the decryption algorithm is always slow
when this particular iteration is slow with a 1 bit, then this bit is assumed to be 1.
If a number of observed execution times for the entire algorithm are fast, then this
bit is assumed to be 0.
In practice, modular exponentiation implementations do not have such ex-
treme timing variations, in which the execution time of a single iteration can ex-
ceed the mean execution time of the entire algorithm. Nevertheless, there is enough
variation to make this attack practical. For details, see [KOCH96].
306 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA
Although the timing attack is a serious threat, there are simple countermea-
sures that can be used, including the following.
■ Constant exponentiation time: Ensure that all exponentiations take the same amount of time before returning a result. This is a simple fix but does degrade
performance.
■ Random delay: Better performance could be achieved by adding a random delay to the exponentiation algorithm to confuse the timing attack. Kocher
points out that if defenders don’t add enough noise, attackers could still suc-
ceed by collecting additional measurements to compensate for the random
delays.
■ Blinding: Multiply the ciphertext by a random number before performing ex- ponentiation. This process prevents the attacker from knowing what cipher-
text bits are being processed inside the computer and therefore prevents the
bit-by-bit analysis essential to the timing attack.
RSA Data Security incorporates a blinding feature into some of its products.
The private-key operation M = Cd mod n is implemented as follows.
1. Generate a secret random number r between 0 and n - 1. 2. Compute C′ = C(r e) mod n, where e is the public exponent. 3. Compute M′ = (C′)d mod n with the ordinary RSA implementation. 4. Compute M = M′r -1 mod n. In this equation, r -1 is the multiplicative inverse
of r mod n; see Chapter 2 for a discussion of this concept. It can be demon- strated that this is the correct result by observing that r ed mod n = r mod n.
RSA Data Security reports a 2 to 10% performance penalty for blinding.
FAULT-BASED ATTACK Still another unorthodox approach to attacking RSA is re- ported in [PELL10]. The approach is an attack on a processor that is generating
RSA digital signatures. The attack induces faults in the signature computation by
reducing the power to the processor. The faults cause the software to produce in-
valid signatures, which can then be analyzed by the attacker to recover the private
key. The authors show how such an analysis can be done and then demonstrate it
by extracting a 1024-bit private RSA key in approximately 100 hours, using a com-
mercially available microprocessor.
The attack algorithm involves inducing single-bit errors and observing the re-
sults. The details are provided in [PELL10], which also references other proposed
hardware fault-based attacks against RSA.
This attack, while worthy of consideration, does not appear to be a serious
threat to RSA. It requires that the attacker have physical access to the target ma-
chine and that the attacker is able to directly control the input power to the pro-
cessor. Controlling the input power would for most hardware require more than
simply controlling the AC power, but would also involve the power supply control
hardware on the chip.
9.2 / THE RSA ALGORITHM 307
CHOSEN CIPHERTEXT ATTACK AND OPTIMAL ASYMMETRIC ENCRYPTION PADDING The basic RSA algorithm is vulnerable to a chosen ciphertext attack (CCA). CCA is defined as an attack in which the adversary chooses a number of ciphertexts and
is then given the corresponding plaintexts, decrypted with the target’s private key.
Thus, the adversary could select a plaintext, encrypt it with the target’s public key,
and then be able to get the plaintext back by having it decrypted with the private
key. Clearly, this provides the adversary with no new information. Instead, the ad-
versary exploits properties of RSA and selects blocks of data that, when processed
using the target’s private key, yield information needed for cryptanalysis.
A simple example of a CCA against RSA takes advantage of the following
property of RSA:
E(PU, M1) * E(PU, M2) = E(PU, [M1 * M2]) (9.2)
We can decrypt C = Me mod n using a CCA as follows.
1. Compute X = (C * 2e) mod n. 2. Submit X as a chosen ciphertext and receive back Y = Xd mod n.
But now note that
X = (C mod n) * (2e mod n) = (Me mod n) * (2e mod n) = (2M)e mod n
Therefore, Y = (2M) mod n. From this, we can deduce M. To overcome this simple attack, practical RSA-based cryptosystems randomly pad the plaintext prior
to encryption. This randomizes the ciphertext so that Equation (9.2) no longer
holds. However, more sophisticated CCAs are possible, and a simple padding with a
random value has been shown to be insufficient to provide the desired security. To
counter such attacks, RSA Security Inc., a leading RSA vendor and former holder
of the RSA patent, recommends modifying the plaintext using a procedure known
as optimal asymmetric encryption padding (OAEP). A full discussion of the threats and OAEP are beyond our scope; see [POIN02] for an introduction and [BELL94]
for a thorough analysis. Here, we simply summarize the OAEP procedure.
Figure 9.10 depicts OAEP encryption. As a first step, the message M to be en-
crypted is padded. A set of optional parameters, P, is passed through a hash func- tion, H.8 The output is then padded with zeros to get the desired length in the overall
data block (DB). Next, a random seed is generated and passed through another hash
function, called the mask generating function (MGF). The resulting hash value is bit-
by-bit XORed with DB to produce a maskedDB. The maskedDB is in turn passed
through the MGF to form a hash that is XORed with the seed to produce the masked-
seed. The concatenation of the maskedseed and the maskedDB forms the encoded
message EM. Note that the EM includes the padded message, masked by the seed,
and the seed, masked by the maskedDB. The EM is then encrypted using RSA.
8A hash function maps a variable-length data block or message into a fixed-length value called a hash code. Hash functions are discussed in depth in Chapter 11.
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308 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA
9.3 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
Figure 9.10 Encryption Using Optimal Asymmetric Encryption Padding (OAEP)
Seed
Maskedseed
DB
MaskedDB
M
EM
Padding
H(P)
MGF
MGF
P
P = encoding parameters M = message to be encoded H = hash function
DB = data block MGF = mask generating function EM = encoded message
Key Terms
chosen ciphertext attack
(CCA)
digital signature
key exchange
one-way function
optimal asymmetric encryption
padding (OAEP)
private key
public key
public-key cryptography
public-key cryptosystems
public-key encryption
RSA
timing attack
trap-door one-way function
Review Questions
9.1 What is a public key certificate? 9.2 What are the roles of the public and private key? 9.3 What are three broad categories of applications of public-key cryptosystems?
9.3 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 309
9.4 What requirements must a public-key cryptosystems fulfill to be a secure algorithm? 9.5 How can a probable-message attack be used for public-key cryptanalysis? 9.6 List the different approaches to attack the RSA algorithm. 9.7 Describe the countermeasures to be used against the timing attack.
Problems
9.1 Prior to the discovery of any specific public-key schemes, such as RSA, an existence proof was developed whose purpose was to demonstrate that public-key encryption is possible in theory. Consider the functions f1(x1) = z1; f2(x2, y2) = z2; f3(x3, y3) = z3, where all values are integers with 1 … xi, yi, zi … N. Function f1 can be represented by a vector M1 of length N, in which the kth entry is the value of f1(k). Similarly, f2 and f3 can be represented by N * N matrices M2 and M3. The intent is to represent the encryption/decryption process by table lookups for tables with very large values of N. Such tables would be impractically huge but could be constructed in principle. The scheme works as follows: Construct M1 with a random permutation of all inte- gers between 1 and N; that is, each integer appears exactly once in M1. Construct M2 so that each row contains a random permutation of the first N integers. Finally, fill in M3 to satisfy the following condition:
f 3 (f 2 (f 1 (k), p ), k) = p for all k, p with 1 … k, p , … N
To summarize, 1. M1 takes an input k and produces an output x. 2. M2 takes inputs x and p giving output z. 3. M3 takes inputs z and k and produces p. The three tables, once constructed, are made public. a. It should be clear that it is possible to construct M3 to satisfy the preceding condi-
tion. As an example, fill in M3 for the following simple case:
4 3 5 2 4 1
3 4 2 5 3 1
M1 = 2 M2 = 5 4 3 1 2 M3 =
5 1 3 2 5 4
1 2 1 4 3 5
Convention: The ith element of M1 corresponds to k = i. The ith row of M2 cor- responds to x = i; the jth column of M2 corresponds to p = j. The ith row of M3 corresponds to z = i; the jth column of M3 corresponds to k = j.
b. Describe the use of this set of tables to perform encryption and decryption be- tween two users.
c. Argue that this is a secure scheme. 9.2 Perform encryption and decryption using the RSA algorithm, as in Figure 9.5, for the
following: a. p = 3 ; q = 7 , e = 5 ; M = 1 0 b. p = 5 ; q = 1 3 , e = 5 ; M = 8 c. p = 7 ; q = 1 7 , e = 1 1 ; M = 1 1 d. p = 7 ; q = 1 3 , e = 1 1 ; M = 2 e. p = 1 7 ; q = 2 3 , e = 9 ; M = 7 Hint: Decryption is not as hard as you think; use some finesse.
9.3 In a public-key system using RSA, you intercept the ciphertext C = 2 0 sent to a user whose public key is e = 1 3 , n = 7 7 . What is the plaintext M?
310 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA
9.4 In an RSA system, the public key of a given user is e = 6 5 , n = 2 8 8 1 . What is the private key of this user? Hint: First use trial-and-error to determine p and q; then use the extended Euclidean algorithm to find the multiplicative inverse of 31 modulo f(n).
9.5 In using the RSA algorithm, if a small number of repeated encodings give back the plaintext, what is the likely cause?
9.6 Suppose we have a set of blocks encoded with the RSA algorithm and we don’t have the private key. Assume n = pq, e is the public key. Suppose also someone tells us they know one of the plaintext blocks has a common factor with n. Does this help us in any way?
9.7 In the RSA public-key encryption scheme, each user has a public key, e, and a private key, d. Suppose Bob leaks his private key. Rather than generating a new modulus, he decides to generate a new public and a new private key. Is this safe?
9.8 Suppose Bob uses the RSA cryptosystem with a very large modulus n for which the factorization cannot be found in a reasonable amount of time. Suppose Alice sends a message to Bob by representing each alphabetic character as an integer between 0 and 25 (A S 0, c , Z S 25) and then encrypting each number separately using RSA with large e and large n. Is this method secure? If not, describe the most effi- cient attack against this encryption method.
9.9 Using a spreadsheet (such as Excel) or a calculator, perform the operations described below. Document results of all intermediate modular multiplications. Determine a number of modular multiplications per each major transformation (such as encryp- tion, decryption, primality testing, etc.).
a. Test all odd numbers in the range from 215 to 223 for primality using the Miller– Rabin test with base 2.
b. Encrypt the message block M = 2 using RSA with the following parameters: e = 23 and n = 233 * 241.
c. Compute a private key (d, p, q) corresponding to the given above public key (e, n). d. Perform the decryption of the obtained ciphertext
1. without using the Chinese Remainder Theorem, and 2. using the Chinese Remainder Theorem.
9.10 Assume that you generate an authenticated and encrypted message by first applying the RSA transformation determined by your private key, and then enciphering the mes- sage using recipient’s public key (note that you do NOT use hash function before the first transformation). Will this scheme work correctly [i.e., give the possibility to recon- struct the original message at the recipient’s side, for all possible relations between the sender’s modulus nS and the recipient’s modulus nR (nS 7 nR, nS 6 nR, nS = nR)]? Explain your answer. In case your answer is “no,” how would you correct this scheme?
9.11 “I want to tell you, Holmes,” Dr. Watson’s voice was enthusiastic, “that your recent activities in network security have increased my interest in cryptography. And just yesterday I found a way to make one-time pad encryption practical.”
“Oh, really?” Holmes’ face lost its sleepy look.
“Yes, Holmes. The idea is quite simple. For a given one-way function F, I gener- ate a long pseudorandom sequence of elements by applying F to some standard se- quence of arguments. The cryptanalyst is assumed to know F and the general nature of the sequence, which may be as simple as S, S + 1, S + 2, c , but not secret S. And due to the one-way nature of F, no one is able to extract S given F(S + i) for some i, thus even if he somehow obtains a certain segment of the sequence, he will not be able to determine the rest.”
“I am afraid, Watson, that your proposal isn’t without flaws and at least it needs some additional conditions to be satisfied by F. Let’s consider, for instance, the RSA encryption function, that is F(M) = MK mod N, K is secret. This function is believed to be one-way, but I wouldn’t recommend its use, for example, on the sequence M = 2, 3, 4, 5, 6, . . . ”
9.3 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 311
“But why, Holmes?” Dr. Watson apparently didn’t understand. “Why do you think that the resulting sequence 2K mod N, 3K mod N, 4K mod N, . . . is not appropri- ate for one-time pad encryption if K is kept secret?”
“Because it is—at least partially—predictable, dear Watson, even if K is kept se- cret. You have said that the cryptanalyst is assumed to know F and the general nature of the sequence. Now let’s assume that he will obtain somehow a short segment of the output sequence. In crypto circles, this assumption is generally considered to be a vi- able one. And for this output sequence, knowledge of just the first two elements will allow him to predict quite a lot of the next elements of the sequence, even if not all of them, thus this sequence can’t be considered to be cryptographically strong. And with the knowledge of a longer segment he could predict even more of the next elements of the sequence. Look, knowing the general nature of the sequence and its first two elements 2K mod N and 3K mod N, you can easily compute its following elements.”
Show how this can be done.
9.12 Show how RSA can be represented by matrices M1, M2, and M3 of Problem 9.1. 9.13 Consider the following scheme:
1. Pick an odd number, E. 2. Pick two prime numbers, P and Q, where (P - 1)(Q - 1) - 1 is evenly divisible
by E. 3. Multiply P and Q to get N.
4. Calculate D = (P - 1)(Q - 1)(E - 1) + 1
E
Is this scheme equivalent to RSA? Show why or why not.
9.14 Consider the following scheme by which B encrypts a message for A. 1. A chooses two large primes P and Q that are also relatively prime to (P - 1)
and (Q - 1). 2. A publishes N = PQ as its public key. 3. A calculates P= and Q= such that PP= K 1 (mod Q - 1) and QQ= K 1 (mod P - 1). 4. B encrypts message M as C = MN mod N. 5. A finds M by solving M K C P
= (mod Q) and M K C Q
= (mod P).
a. Explain how this scheme works. b. How does it differ from RSA? c. Is there any particular advantage to RSA compared to this scheme? d. Show how this scheme can be represented by matrices M1, M2, and M3 of
Problem 9.1.
9.15 “This is a very interesting case, Watson,” Holmes said. “The young man loves a girl, and she loves him too. However, her father is a strange fellow who insists that his would-be son-in-law must design a simple and secure protocol for an appropriate public-key cryptosystem he could use in his company’s computer network. The young man came up with the following protocol for communication between two parties. For example, user A wishing to send message M to user B: (messages exchanged are in the format sender’s name, text, receiver’s name)” 1. A sends B the following block: (A, E(PUb, [M, A]), B). 2. B acknowledges receipt by sending to A the following block: (B, E(PUa, [M, B]), A). “You can see that the protocol is really simple. But the girl’s father claims that the young man has not satisfied his call for a simple protocol, because the proposal con- tains a certain redundancy and can be further simplified to the following:” 1. A sends B the block: (A, E(PUb, M), B). 2. B acknowledges receipt by sending to A the block: (B, E(PUa, M), A). “On the basis of that, the girl’s father refuses to allow his daughter to marry the young man, thus making them both unhappy. The young man was just here to ask me for help.”
312 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA
“Hmm, I don’t see how you can help him.” Watson was visibly unhappy with the idea that the sympathetic young man has to lose his love.
“Well, I think I could help. You know, Watson, redundancy is sometimes good to ensure the security of protocol. Thus, the simplification the girl’s father has proposed could make the new protocol vulnerable to an attack the original protocol was able to resist,” mused Holmes. “Yes, it is so, Watson. Look, all an adversary needs is to be one of the users of the network and to be able to intercept messages exchanged between A and B. Being a user of the network, he has his own public encryption key and is able to send his own messages to A or to B and to receive theirs. With the help of the simplified protocol, he could then obtain message M user A has previously sent to B using the following procedure:”
Complete the description.
9.16 Use the fast exponentiation algorithm of Figure 9.8 to determine 6 4 7 2 mod 3415. Show the steps involved in the computation.
9.17 Here is another realization of the fast exponentiation algorithm. Demonstrate that it is equivalent to the one in Figure 9.8. 1. f d 1; T d a; E d b 2. if odd(E) then f d f : T 3. E d [ E/2 ] 4. T d T : T 5. if E + 0 then goto 2 6. output f
9.18 This problem illustrates a simple application of the chosen ciphertext attack. Bob intercepts a ciphertext C intended for Alice and encrypted with Alice’s public key e. Bob wants to obtain the original message M = C d mod n. Bob chooses a random value r less than n and computes
Z = r e mod n X = ZC mod n t = r -1 mod n
Next, Bob gets Alice to authenticate (sign) X with her private key (as in Figure 9.3), thereby decrypting X. Alice returns Y = Xd mod n. Show how Bob can use the infor- mation now available to him to determine M.
9.19 Show the OAEP decoding operation used for decryption that corresponds to the encoding operation of Figure 9.10.
9.20 Improve on algorithm P1 in Appendix W. a. Develop an algorithm that requires 2n multiplications and n + 1 additions. Hint:
xi + 1 = xi * x. b. Develop an algorithm that requires only n + 1 multiplications and n + 1 addi-
tions. Hint: P(x) = a0 + x * q(x), where q(x) is a polynomial of degree (n - 1). Note: The remaining problems concern the knapsack public-key algorithm described in Appendix J.
9.21 What items are in the knapsack in Figure F.1? 9.22 Perform encryption and decryption using the knapsack algorithm for the following:
a. a= = (1, 5, 7, 14); w = 11; m = 30; x = 1011 b. a= = (1, 2, 7, 12, 23, 38, 116, 248); w = 201; m = 457; x = 10101010 c. a= = (2, 4, 7, 15, 29); w = 36; m = 47; x = 10011 d. a= = (15, 92, 108, 279, 563, 1172, 2243, 4468); w = 2033; m = 8764; x = 10110011
9.23 Why is it a requirement that m 7 a n
1 = 1 a= i?
313
Other Public-Key Cryptosystems
10.1 Diffie–Hellman Key Exchange
The Algorithm
Key Exchange Protocols
Man-in-the-Middle Attack
10.2 Elgamal Cryptographic System
10.3 Elliptic Curve Arithmetic
Abelian Groups
Elliptic Curves over Real Numbers
Elliptic Curves over Zp Elliptic Curves over GF(2m)
10.4 Elliptic Curve Cryptography
Analog of Diffie–Hellman Key Exchange
Elliptic Curve Encryption/Decryption
Security of Elliptic Curve Cryptography
10.5 Pseudorandom Number Generation Based on an Asymmetric Cipher
PRNG Based on RSA
PRNG Based on Elliptic Curve Cryptography
10.6 Key Terms, Review Questions, and Problems
CHAPTER
314 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS
This chapter begins with a description of one of the earliest and simplest PKCS:
Diffie–Hellman key exchange. The chapter then looks at another important scheme,
the Elgamal PKCS. Next, we look at the increasingly important PKCS known as elliptic
curve cryptography. Finally, the use of public-key algorithms for pseudorandom num-
ber generation is examined.
10.1 DIFFIE–HELLMAN KEY EXCHANGE
The first published public-key algorithm appeared in the seminal paper by Diffie
and Hellman that defined public-key cryptography [DIFF76b] and is generally re-
ferred to as Diffie–Hellman key exchange.1 A number of commercial products em-
ploy this key exchange technique.
The purpose of the algorithm is to enable two users to securely exchange a
key that can then be used for subsequent symmetric encryption of messages. The
algorithm itself is limited to the exchange of secret values.
The Diffie–Hellman algorithm depends for its effectiveness on the difficulty
of computing discrete logarithms. Briefly, we can define the discrete logarithm in
the following way. Recall from Chapter 2 that a primitive root of a prime number p is one whose powers modulo p generate all the integers from 1 to p - 1. That is, if a is a primitive root of the prime number p, then the numbers
a mod p, a2 mod p, c , ap - 1 mod p
are distinct and consist of the integers from 1 through p - 1 in some permutation. For any integer b and a primitive root a of prime number p, we can find a
unique exponent i such that
b K ai (mod p) where 0 … i … (p - 1)
1Williamson of Britain’s CESG published the identical scheme a few months earlier in a classified docu- ment [WILL76] and claims to have discovered it several years prior to that; see [ELLI99] for a discussion.
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Define Diffie–Hellman key exchange.
◆ Understand the man-in-the-middle attack.
◆ Present an overview of the Elgamal cryptographic system.
◆ Understand elliptic curve arithmetic.
◆ Present an overview of elliptic curve cryptography.
◆ Present two techniques for generating pseudorandom numbers using an asymmetric cipher.
10.1 / DIFFIE–HELLMAN KEY EXCHANGE 315
The exponent i is referred to as the discrete logarithm of b for the base a, mod p. We express this value as dloga,p(b). See Chapter 2 for an extended discussion of discrete logarithms.
The Algorithm
Figure 10.1 summarizes the Diffie–Hellman key exchange algorithm. For this
scheme, there are two publicly known numbers: a prime number q and an inte- ger a that is a primitive root of q. Suppose the users A and B wish to create a shared key.
User A selects a random integer XA 6 q and computes YA = aXA mod q. Similarly, user B independently selects a random integer XB 6 q and computes YB = a
XB mod q. Each side keeps the X value private and makes the Y value avail- able publicly to the other side. Thus, XA is A’s private key and YA is A’s correspond- ing public key, and similarly for B. User A computes the key as K = (YB)
XA mod q and user B computes the key as K = (YA)
XB mod q. These two calculations produce identical results:
Figure 10.1 The Diffie–Hellman Key Exchange
Alice Bob
Alice and Bob share a prime number q and an integer A, such that A < q and A is a primitive root of q
Alice generates a private key XA such that XA < q
Alice calculates a public key YA = AXA mod q
Alice receives Bob’s public key YB in plaintext
Alice calculates shared secret key K = (YB)XA mod q
Bob calculates shared secret key K = (YA)XB mod q
Bob receives Alice’s public key YA in plaintext
Bob calculates a public key YB = AXB mod q
Bob generates a private key XB such that XB < q
Alice and Bob share a prime number q and an integer A, such that A < q and A is a primitive root of q
YA Y B
316 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS
K = (YB) XA mod q
= (aXB mod q)XA mod q = (aXB)XA mod q by the rules of modular arithmetic = aXBXA mod q = (aXA)XB mod q = (aXA mod q)XB mod q = (YA)
XB mod q
The result is that the two sides have exchanged a secret value. Typically, this
secret value is used as shared symmetric secret key. Now consider an adversary who
can observe the key exchange and wishes to determine the secret key K. Because XA and XB are private, an adversary only has the following ingredients to work with: q, a, YA, and YB. Thus, the adversary is forced to take a discrete logarithm to deter- mine the key. For example, to determine the private key of user B, an adversary
must compute
XB = dloga,q(YB)
The adversary can then calculate the key K in the same manner as user B calculates it. That is, the adversary can calculate K as
K = (YA) XB mod q
The security of the Diffie–Hellman key exchange lies in the fact that, while
it is relatively easy to calculate exponentials modulo a prime, it is very difficult
to calculate discrete logarithms. For large primes, the latter task is considered
infeasible.
Here is an example. Key exchange is based on the use of the prime number
q = 353 and a primitive root of 353, in this case a = 3. A and B select private keys XA = 97 and XB = 233, respectively. Each computes its public key:
A computes YA = 3 97 mod 353 = 40.
B computes YB = 3 233 mod 353 = 248.
After they exchange public keys, each can compute the common secret key:
A computes K = (YB) XA mod 353 = 24897 mod 353 = 160.
B computes K = (YA) XB mod 353 = 40233 mod 353 = 160.
We assume an attacker would have available the following information:
q = 353; a = 3; YA = 40; YB = 248
In this simple example, it would be possible by brute force to determine the secret
key 160. In particular, an attacker E can determine the common key by discovering
a solution to the equation 3a mod 353 = 40 or the equation 3b mod 353 = 248. The brute-force approach is to calculate powers of 3 modulo 353, stopping when the re-
sult equals either 40 or 248. The desired answer is reached with the exponent value
of 97, which provides 397 mod 353 = 40. With larger numbers, the problem becomes impractical.
Hiva-Network.Com
10.1 / DIFFIE–HELLMAN KEY EXCHANGE 317
Key Exchange Protocols
Figure 10.1 shows a simple protocol that makes use of the Diffie–Hellman calcula-
tion. Suppose that user A wishes to set up a connection with user B and use a secret
key to encrypt messages on that connection. User A can generate a one-time pri-
vate key XA, calculate YA, and send that to user B. User B responds by generating a private value XB, calculating YB, and sending YB to user A. Both users can now calculate the key. The necessary public values q and a would need to be known ahead of time. Alternatively, user A could pick values for q and a and include those in the first message.
As an example of another use of the Diffie–Hellman algorithm, suppose that a
group of users (e.g., all users on a LAN) each generate a long-lasting private value Xi (for user i) and calculate a public value Yi. These public values, together with global public values for q and a, are stored in some central directory. At any time, user j can access user i’s public value, calculate a secret key, and use that to send an en- crypted message to user A. If the central directory is trusted, then this form of com-
munication provides both confidentiality and a degree of authentication. Because
only i and j can determine the key, no other user can read the message (confidential- ity). Recipient i knows that only user j could have created a message using this key (authentication). However, the technique does not protect against replay attacks.
Man-in-the-Middle Attack
The protocol depicted in Figure 10.1 is insecure against a man-in-the-middle attack.
Suppose Alice and Bob wish to exchange keys, and Darth is the adversary. The at-
tack proceeds as follows (Figure 10.2).
1. Darth prepares for the attack by generating two random private keys XD1 and XD2 and then computing the corresponding public keys YD1 and YD2.
2. Alice transmits YA to Bob.
3. Darth intercepts YA and transmits YD1 to Bob. Darth also calculates K2 = (YA)
XD2 mod q.
4. Bob receives YD1 and calculates K1 = (YD1) XB mod q.
5. Bob transmits YB to Alice.
6. Darth intercepts YB and transmits YD2 to Alice. Darth calculates K1 = (YB)
XD1 mod q.
7. Alice receives YD2 and calculates K2 = (YD2) XA mod q.
At this point, Bob and Alice think that they share a secret key, but instead
Bob and Darth share secret key K1 and Alice and Darth share secret key K2. All future communication between Bob and Alice is compromised in the following way.
1. Alice sends an encrypted message M: E(K2, M).
2. Darth intercepts the encrypted message and decrypts it to recover M.
3. Darth sends Bob E(K1, M) or E(K1, M=), where M= is any message. In the first case, Darth simply wants to eavesdrop on the communication without altering
it. In the second case, Darth wants to modify the message going to Bob.
318 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS
The key exchange protocol is vulnerable to such an attack because it does not
authenticate the participants. This vulnerability can be overcome with the use of digital
signatures and public-key certificates; these topics are explored in Chapters 13 and 14.
10.2 ELGAMAL CRYPTOGRAPHIC SYSTEM
In 1984, T. Elgamal announced a public-key scheme based on discrete logarithms,
closely related to the Diffie–Hellman technique [ELGA84, ELGA85]. The Elgamal2
cryptosystem is used in some form in a number of standards including the digital
signature standard (DSS), which is covered in Chapter 13, and the S/MIME email
standard (Chapter 19).
2For no apparent reason, most of the literature uses the term ElGamal, although Mr. Elgamal’s last name does not have a capital letter G.
Figure 10.2 Man-in-the-Middle Attack
Alice Darth Bob
Private key XA Public key YA = AXA mod q
Private key XB Public key YB = AXB mod q
Private keys XD1, XD2 Public keys YD1 = AXD1 mod q YD2 = AXD2 mod q
YA
Secret key K2 = (YA)XD2 mod q
Secret key K1 = (YB)XD1 mod q
Secret key K1 = (YD1)XB mod q
Secret key K2 = (YD2)XA mod q
Alice and Darth share K2
Bob and Darth share K1
YD2 YD1
YB
10.2 / ELGAMAL CRYPTOGRAPHIC SYSTEM 319
As with Diffie–Hellman, the global elements of Elgamal are a prime number q and a, which is a primitive root of q. User A generates a private/public key pair as follows:
1. Generate a random integer XA, such that 1 6 XA 6 q - 1. 2. Compute YA = a
XA mod q.
3. A’s private key is XA and A’s public key is {q, a, YA}.
Any user B that has access to A’s public key can encrypt a message as follows:
1. Represent the message as an integer M in the range 0 … M … q - 1. Longer messages are sent as a sequence of blocks, with each block being an integer
less than q.
2. Choose a random integer k such that 1 … k … q - 1. 3. Compute a one-time key K = (YA)
k mod q.
4. Encrypt M as the pair of integers (C1, C2) where
C1 = a k mod q; C2 = KM mod q
User A recovers the plaintext as follows:
1. Recover the key by computing K = (C1) XA mod q.
2. Compute M = (C2K -1) mod q.
These steps are summarized in Figure 10.3. It corresponds to Figure 9.1a:
Alice generates a public/private key pair; Bob encrypts using Alice’s public key; and
Alice decrypts using her private key.
Let us demonstrate why the Elgamal scheme works. First, we show how K is recovered by the decryption process:
K = (YA) k mod q K is defined during the encryption process
K = (aXA mod q)k mod q substitute using YA = a XA mod q
K = akXA mod q by the rules of modular arithmetic K = (C1)
XA mod q substitute using C1 = a k mod q
Next, using K, we recover the plaintext as
C2 = KM mod q (C2K
-1) mod q = KMK-1 mod q = M mod q = M
We can restate the Elgamal process as follows, using Figure 10.3.
1. Bob generates a random integer k.
2. Bob generates a one-time key K using Alice’s public-key components YA, q, and k.
3. Bob encrypts k using the public-key component a, yielding C1. C1 provides sufficient information for Alice to recover K.
4. Bob encrypts the plaintext message M using K.
5. Alice recovers K from C1 using her private key.
6. Alice uses K-1 to recover the plaintext message from C2.
320 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS
Thus, K functions as a one-time key, used to encrypt and decrypt the message. For example, let us start with the prime field GF(19); that is, q = 19. It has
primitive roots {2, 3, 10, 13, 14, 15}, as shown in Table 2.7. We choose a = 10. Alice generates a key pair as follows:
1. Alice chooses XA = 5. 2. Then YA = a
XA mod q = a5 mod 19 = 3 (see Table 2.7). 3. Alice’s private key is 5 and Alice’s public key is {q, a, YA} = {19, 10, 3}.
Suppose Bob wants to send the message with the value M = 17. Then:
Figure 10.3 The Elgamal Cryptosystem
Global Public Elements
q prime number
a a 6 q and a a primitive root of q
Key Generation by Alice
Select private XA XA 6 q - 1
Calculate YA YA = a XA mod q
Public key {q, a, YA}
Private key XA
Encryption by Bob with Alice’s Public Key
Plaintext: M 6 q
Select random integer k k 6 q
Calculate K K = (YA) k mod q
Calculate C1 C1 = a k mod q
Calculate C2 C2 = KM mod q
Ciphertext: (C1, C2)
Decryption by Alice with Alice’s Private Key
Ciphertext: (C1, C2)
Calculate K K = (C1) XA mod q
Plaintext: M = (C2K -1) mod q
10.3 / ELLIPTIC CURVE ARITHMETIC 321
1. Bob chooses k = 6. 2. Then K = (YA)
k mod q = 36 mod 19 = 729 mod 19 = 7. 3. So
C1 = a k mod q = a6 mod 19 = 11
C2 = KM mod q = 7 * 17 mod 19 = 119 mod 19 = 5 4. Bob sends the ciphertext (11, 5).
For decryption:
1. Alice calculates K = (C1) XA mod q = 115 mod 19 = 161051 mod 19 = 7.
2. Then K-1 in GF(19) is 7-1 mod 19 = 11. 3. Finally, M = (C2K
-1) mod q = 5 * 11 mod 19 = 55 mod 19 = 17.
If a message must be broken up into blocks and sent as a sequence of encrypted
blocks, a unique value of k should be used for each block. If k is used for more than one block, knowledge of one block M1 of the message enables the user to compute other blocks as follows. Let
C1,1 = a k mod q; C2,1 = KM1 mod q
C1,2 = a k mod q; C2,2 = KM2 mod q
Then,
C2,1 C2,2
= KM1 mod q
KM2 mod q =
M1 mod q
M2 mod q
If M1 is known, then M2 is easily computed as
M2 = (C2,1) -1 C2,2 M1 mod q
The security of Elgamal is based on the difficulty of computing discrete
logarithms. To recover A’s private key, an adversary would have to compute
XA = dloga,q(YA). Alternatively, to recover the one-time key K, an adversary would have to determine the random number k, and this would require computing the discrete logarithm k = dloga,q(C1). [STIN06] points out that these calculations are regarded as infeasible if p is at least 300 decimal digits and q - 1 has at least one “large” prime factor.
10.3 ELLIPTIC CURVE ARITHMETIC
Most of the products and standards that use public-key cryptography for encryp-
tion and digital signatures use RSA. As we have seen, the key length for secure
RSA use has increased over recent years, and this has put a heavier processing
load on applications using RSA. This burden has ramifications, especially for elec-
tronic commerce sites that conduct large numbers of secure transactions. A com-
peting system challenges RSA: elliptic curve cryptography (ECC). ECC is showing
up in standardization efforts, including the IEEE P1363 Standard for Public-Key
Cryptography.
322 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS
The principal attraction of ECC, compared to RSA, is that it appears to offer
equal security for a far smaller key size, thereby reducing processing overhead.
ECC is fundamentally more difficult to explain than either RSA or Diffie–
Hellman, and a full mathematical description is beyond the scope of this book. This
section and the next give some background on elliptic curves and ECC. We begin
with a brief review of the concept of abelian group. Next, we examine the concept
of elliptic curves defined over the real numbers. This is followed by a look at el-
liptic curves defined over finite fields. Finally, we are able to examine elliptic curve
ciphers.
The reader may wish to review the material on finite fields in Chapter 5 before
proceeding.
Abelian Groups
Recall from Chapter 5 that an abelian group G, sometimes denoted by {G, # }, is a set of elements with a binary operation, denoted by # , that associates to each or- dered pair (a, b) of elements in G an element (a # b) in G, such that the following axioms are obeyed:3
(A1) Closure: If a and b belong to G, then a # b is also in G. (A2) Associative: a # (b # c) = (a # b) # c for all a, b, c in G. (A3) Identity element: There is an element e in G such that a # e = e # a = a
for all a in G.
(A4) Inverse element: For each a in G there is an element a′ in G such that a # a′ = a′ # a = e.
(A5) Commutative: a # b = b # a for all a, b in G. A number of public-key ciphers are based on the use of an abelian group.
For example, Diffie–Hellman key exchange involves multiplying pairs of nonzero
integers modulo a prime number q. Keys are generated by exponentiation over the group, with exponentiation defined as repeated multiplication. For example,
ak mod q = (a * a * c * a) mod q. To attack Diffie–Hellman, the attacker must
k times determine k given a and ak; this is the discrete logarithm problem.
For elliptic curve cryptography, an operation over elliptic curves, called addi-
tion, is used. Multiplication is defined by repeated addition. For example,
a * k = (a + a + c + a)
k times where the addition is performed over an elliptic curve. Cryptanalysis involves deter-
mining k given a and (a * k).
3The operator # is generic and can refer to addition, multiplication, or some other mathematical operation.
v
v
10.3 / ELLIPTIC CURVE ARITHMETIC 323
An elliptic curve is defined by an equation in two variables with coefficients. For cryptography, the variables and coefficients are restricted to elements in a finite
field, which results in the definition of a finite abelian group. Before looking at this,
we first look at elliptic curves in which the variables and coefficients are real num-
bers. This case is perhaps easier to visualize.
Elliptic Curves over Real Numbers
Elliptic curves are not ellipses. They are so named because they are described by
cubic equations, similar to those used for calculating the circumference of an ellipse.
In general, cubic equations for elliptic curves take the following form, known as a
Weierstrass equation:
y2 + axy + by = x3 + cx2 + dx + e
where a, b, c, d, e are real numbers and x and y take on values in the real numbers.4 For our purpose, it is sufficient to limit ourselves to equations of the form
y2 = x3 + ax + b (10.1)
Such equations are said to be cubic, or of degree 3, because the highest ex-
ponent they contain is a 3. Also included in the definition of an elliptic curve is a
single element denoted O and called the point at infinity or the zero point, which we discuss subsequently. To plot such a curve, we need to compute
y = 2x3 + ax + b
For given values of a and b, the plot consists of positive and negative values of y for each value of x. Thus, each curve is symmetric about y = 0. Figure 10.4 shows two examples of elliptic curves. As you can see, the formula sometimes produces weird-
looking curves.
Now, consider the set of points E(a, b) consisting of all of the points (x, y) that satisfy Equation (10.1) together with the element O. Using a different value of the pair (a, b) results in a different set E(a, b). Using this terminology, the two curves in Figure 10.4 depict the sets E( - 1, 0) and E(1, 1), respectively.
GEOMETRIC DESCRIPTION OF ADDITION It can be shown that a group can be defined based on the set E(a, b) for specific values of a and b in Equation (10.1), provided the following condition is met:
4a3 + 27b2 ≠ 0 (10.2)
To define the group, we must define an operation, called addition and denoted by
+ , for the set E(a, b), where a and b satisfy Equation (10.2). In geometric terms, the rules for addition can be stated as follows: If three points on an elliptic curve lie on a
straight line, their sum is O. From this definition, we can define the rules of addition over an elliptic curve.
4Note that x and y are true variables, which take on values. This is in contrast to our discussion of polyno- mial rings and fields in Chapter 5, where was treated as an indeterminate.
324 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS
1. O serves as the additive identity. Thus O = - O; for any point P on the elliptic curve, P + O = P. In what follows, we assume P ≠ O and Q ≠ O.
2. The negative of a point P is the point with the same x coordinate but the nega- tive of the y coordinate; that is, if P = (x, y), then - P = (x, - y). Note that these two points can be joined by a vertical line. Note that P + ( - P) = P - P = O.
3. To add two points P and Q with different x coordinates, draw a straight line between them and find the third point of intersection R. It is easily seen that there is a unique point R that is the point of intersection (unless the line is tangent to the curve at either P or Q, in which case we take R = P or R = Q, respectively). To form a group structure, we need to define addition on these
three points: P + Q = - R. That is, we define P + Q to be the mirror image
Figure 10.4 Example of Elliptic Curves
-4
-2
0
2
4
-4
-2
0
2
4
543210-1-2
543210-1-2
(a) y2 = x3 - x
(b) y2 = x3 + x + 1
P
P
Q
Q
-(P + Q)
-(P + Q)
(P + Q)
(P + Q)
10.3 / ELLIPTIC CURVE ARITHMETIC 325
(with respect to the x axis) of the third point of intersection. Figure 10.4 illus- trates this construction.
4. The geometric interpretation of the preceding item also applies to two points, P and - P, with the same x coordinate. The points are joined by a vertical line, which can be viewed as also intersecting the curve at the infinity point. We
therefore have P + ( - P) = O, which is consistent with item (2). 5. To double a point Q, draw the tangent line and find the other point of intersec-
tion S. Then Q + Q = 2Q = - S.
With the preceding list of rules, it can be shown that the set E(a, b) is an abe- lian group.
ALGEBRAIC DESCRIPTION OF ADDITION In this subsection, we present some results that enable calculation of additions over elliptic curves.5 For two distinct points,
P = (xP, yP) and Q = (xQ, yQ), that are not negatives of each other, the slope of the line l that joins them is ∆ = (yQ - yP)/(xQ - xP). There is exactly one other point where l intersects the elliptic curve, and that is the negative of the sum of P and Q. After some algebraic manipulation, we can express the sum R = P + Q as
xR = ∆ 2 - xP - xQ
yR = - yP + ∆(xP - xR) (10.3)
We also need to be able to add a point to itself: P + P = 2P = R. When yP ≠ 0, the expressions are
xR = ¢ 3xP2 + a 2yP
≤2 - 2xP yR = ¢ 3xP2 + a
2yP ≤(xP - xR) - yP (10.4)
Elliptic Curves over Zp Elliptic curve cryptography makes use of elliptic curves in which the variables and coefficients are all restricted to elements of a finite field. Two families of elliptic
curves are used in cryptographic applications: prime curves over Zp and binary curves over GF(2m). For a prime curve over Zp, we use a cubic equation in which the variables and coefficients all take on values in the set of integers from 0 through
p - 1 and in which calculations are performed modulo p. For a binary curve de- fined over GF(2m), the variables and coefficients all take on values in GF(2m) and
in calculations are performed over GF(2m). [FERN99] points out that prime curves
are best for software applications, because the extended bit-fiddling operations
needed by binary curves are not required; and that binary curves are best for hard-
ware applications, where it takes remarkably few logic gates to create a powerful,
fast cryptosystem. We examine these two families in this section and the next.
5For derivations of these results, see [KOBL94] or other mathematical treatments of elliptic curves.
Hiva-Network.Com
326 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS
There is no obvious geometric interpretation of elliptic curve arithmetic over
finite fields. The algebraic interpretation used for elliptic curve arithmetic over real
numbers does readily carry over, and this is the approach we take.
For elliptic curves over Zp, as with real numbers, we limit ourselves to equa-
tions of the form of Equation (10.1), but in this case with coefficients and variables
limited to Zp:
y2 mod p = (x3 + ax + b) mod p (10.5)
For example, Equation (10.5) is satisfied for a = 1, b = 1, x = 9, y = 7, p = 23:
72 mod 23 = (93 + 9 + 1) mod 23 49 mod 23 = 739 mod 23
3 = 3
Now consider the set E p(a, b) consisting of all pairs of integers (x, y) that sat- isfy Equation (10.5), together with a point at infinity O. The coefficients a and b and the variables x and y are all elements of Zp.
For example, let p = 23 and consider the elliptic curve y2 = x3 + x + 1. In this case, a = b = 1. Note that this equation is the same as that of Figure 10.4b. The figure shows a continuous curve with all of the real points that satisfy the equation.
For the set E 23(1, 1), we are only interested in the nonnegative integers in the quad-
rant from (0, 0) through (p - 1, p - 1) that satisfy the equation mod p. Table 10.1 lists the points (other than O) that are part of E 23(1, 1). Figure 10.5 plots the points of E 23(1, 1); note that the points, with one exception, are symmetric about y = 11.5.
It can be shown that a finite abelian group can be defined based on the set
E p(a, b) provided that (x 3 + ax + b) mod p has no repeated factors. This is equiva-
lent to the condition
(4a3 + 27b2) mod p ≠ 0 mod p (10.6)
Note that Equation (10.6) has the same form as Equation (10.2).
The rules for addition over E p(a, b), correspond to the algebraic technique de- scribed for elliptic curves defined over real numbers. For all points P, Q ∈ E p(a, b):
(0, 1) (6, 4) (12, 19)
(0, 22) (6, 19) (13, 7)
(1, 7) (7, 11) (13, 16)
(1, 16) (7, 12) (17, 3)
(3, 10) (9, 7) (17, 20)
(3, 13) (9, 16) (18, 3)
(4, 0) (11, 3) (18, 20)
(5, 4) (11, 20) (19, 5)
(5, 19) (12, 4) (19, 18)
Table 10.1 Points (other than O) on the Elliptic Curve E 23(1, 1)
10.3 / ELLIPTIC CURVE ARITHMETIC 327
1. P + O = P. 2. If P = (xP, yP), then P + (xP, - yP) = O. The point (xP, - yP) is the nega-
tive of P, denoted as - P. For example, in E 23(1, 1), for P = (13, 7), we have - P = (13, - 7). But - 7 mod 23 = 16. Therefore, - P = (13, 16), which is also in E23(1, 1).
3. If P = (xp, yp) and Q = (xQ, yQ) with P ≠ - Q, then R = P + Q = (xR, yR) is determined by the following rules:
xR = (l 2 - xP - xQ) mod p
yR = (l(xP - xR) - yP) mod p
where
l = e a yQ - yP xQ - xP
b mod p if P ≠ Q
a3xP 2 + a
2yP b mod p if P = Q
Figure 10.5 The Elliptic Curve E 23(1, 1)
0 0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20 21 22
1 2 3 4 5 6 7 8 9 10 11 x
y
12 13 14 15 16 17 18 19 20 21 22
328 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS
4. Multiplication is defined as repeated addition; for example, 4P = P + P + P + P.
For example, let P = (3, 10) and Q = (9, 7) in E23(1, 1). Then
l = a7 - 10 9 - 3
b mod 23 = a - 3 6 b mod 23 = a - 1
2 b mod 23 = 11
xR = (11 2 - 3 - 9) mod 23 = 109 mod 23 = 17
yR = (11(3 - 17) - 10) mod 23 = - 164 mod 23 = 20
So P + Q = (17, 20). To find 2P,
l = ¢ 3(32) + 1 2 * 10
≤ mod 23 = a 5 20 b mod 23 = a1
4 b mod 23 = 6
The last step in the preceding equation involves taking the multiplicative in-
verse of 4 in Z23. This can be done using the extended Euclidean algorithm defined
in Section 4.4. To confirm, note that (6 * 4) mod 23 = 24 mod 23 = 1.
xR = (6 2 - 3 - 3) mod 23 = 30 mod 23 = 7
yR = (6(3 - 7) - 10) mod 23 = ( - 34) mod 23 = 12
and 2P = (7, 12). For determining the security of various elliptic curve ciphers, it is of some in-
terest to know the number of points in a finite abelian group defined over an elliptic
curve. In the case of the finite group EP(a, b), the number of points N is bounded by
p + 1 - 22p … N … p + 1 + 22p
Note that the number of points in Ep(a, b) is approximately equal to the number of elements in Zp, namely p elements.
Elliptic Curves over GF(2m)
Recall from Chapter 5 that a finite field GF(2m) consists of 2m elements, together with addition and multiplication operations that can be defined over polynomials.
For elliptic curves over GF(2m), we use a cubic equation in which the variables and
coefficients all take on values in GF(2m) for some number m and in which calcula- tions are performed using the rules of arithmetic in GF(2m).
(0, 1) (g5, g3) (g9, g13)
(1, g6) (g5, g11) (g10, g)
(1, g13) (g6, g8) (g10, g8)
(g3, g8) (g6, g14) (g12, 0)
(g3, g13) (g9, g10) (g12, g12)
Table 10.2 Points (other than O) on the Elliptic Curve E 24(g
4, 1)
10.3 / ELLIPTIC CURVE ARITHMETIC 329
It turns out that the form of cubic equation appropriate for cryptographic
applications for elliptic curves is somewhat different for GF(2m) than for Zp. The form is
y2 + xy = x3 + ax2 + b (10.7)
where it is understood that the variables x and y and the coefficients a and b are ele- ments of GF(2m) and that calculations are performed in GF(2m).
Now consider the set E2m(a, b) consisting of all pairs of integers (x, y) that sat- isfy Equation (10.7), together with a point at infinity O.
For example, let us use the finite field GF(24) with the irreducible polynomial
f(x) = x4 + x + 1. This yields a generator g that satisfies f(g) = 0 with a value of g4 = g + 1, or in binary, g = 0010. We can develop the powers of g as follows.
g0 = 0001 g4 = 0011 g8 = 0101 g12 = 1111
g1 = 0010 g5 = 0110 g9 = 1010 g13 = 1101
g2 = 0100 g6 = 1100 g10 = 0111 g14 = 1001
g3 = 1000 g7 = 1011 g11 = 1110 g15 = 0001
For example, g5 = (g4)(g) = (g + 1)(g) = g2 + g = 0110. Now consider the elliptic curve y2 + xy = x3 + g4x2 + 1. In this case, a = g4
and b = g0 = 1. One point that satisfies this equation is (g5, g3):
(g3)2 + (g5)(g3) = (g5)3 + (g4)(g5)2 + 1 g6 + g8 = g15 + g14 + 1 1100 + 0101 = 0001 + 1001 + 0001 1001 = 1001
Table 10.2 lists the points (other than O) that are part of E 24(g 4, 1). Figure 10.6 plots
the points of E 24(g 4, 1).
It can be shown that a finite abelian group can be defined based on the set
E 2m(a, b), provided that b ≠ 0. The rules for addition can be stated as follows. For all points P, Q ∈ E2m(a, b):
1. P + O = P. 2. If P = (xP, yP), then P + (xP, xP + yP) = O. The point (xP, xP + yP) is the
negative of P, which is denoted as - P. 3. If P = (xP, yP) and Q = (xQ, yQ) with P ≠ - Q and P ≠ Q, then
R = P + Q = (xR, yR) is determined by the following rules:
xR = l 2 + l + xP + xQ + a
yR = l(xP + xR) + xR + yP
where
l = yQ + yP xQ + xP
330 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS
4. If P = (xP, yP) then R = 2P = (xR, yR) is determined by the following rules:
xR = l 2 + l + a
yR = xP 2 + (l + 1)xR
where
l = xP + yP xP
10.4 ELLIPTIC CURVE CRYPTOGRAPHY
The addition operation in ECC is the counterpart of modular multiplication in
RSA, and multiple addition is the counterpart of modular exponentiation. To form
a cryptographic system using elliptic curves, we need to find a “hard problem” cor-
responding to factoring the product of two primes or taking the discrete logarithm.
Consider the equation Q = kP where Q, P ∈ EP(a, b) and k 6 p. It is rela- tively easy to calculate Q given k and P, but it is hard to determine k given Q and P. This is called the discrete logarithm problem for elliptic curves.
We give an example taken from the Certicom Web site (www.certicom.
com). Consider the group E 23(9,17). This is the group defined by the equation
y2 mod 23 = (x3 + 9x + 17) mod 23. What is the discrete logarithm k of Q = (4, 5) to the base P = (16, 5)? The brute-force method is to compute multiples of P until Q is found. Thus,
P = (16,5); 2P = (20, 20); 3P = (14, 14); 4P = (19, 20); 5P = (13, 10); 6P = (7, 3); 7P = (8, 7); 8P = (12, 17); 9P = (4, 5)
Figure 10.6 The Elliptic Curve E 24(g 4, 1)
1 1 g
g2 g3 g4 g5 g6 g7 g8 g9
g10 g11 g12 g13 g14
0
g g2 g3 g4 g5 g6 g7 g8 g9 g10 g11
x
y
g12 g13 g14 0
10.4 / ELLIPTIC CURVE CRYPTOGRAPHY 331
Because 9P = (4, 5) = Q, the discrete logarithm Q = (4, 5) to the base P = (16, 5) is k = 9. In a real application, k would be so large as to make the brute- force approach infeasible.
In the remainder of this section, we show two approaches to ECC that give the
flavor of this technique.
Analog of Diffie–Hellman Key Exchange
Key exchange using elliptic curves can be done in the following manner. First pick
a large integer q, which is either a prime number p or an integer of the form 2m, and elliptic curve parameters a and b for Equation (10.5) or Equation (10.7). This defines the elliptic group of points Eq(a, b). Next, pick a base point G = (x1, y1) in Ep(a, b) whose order is a very large value n. The order n of a point G on an elliptic curve is the smallest positive integer n such that nG = 0 and G are parameters of the cryptosystem known to all participants.
A key exchange between users A and B can be accomplished as follows
(Figure 10.7).
1. A selects an integer nA less than n. This is A’s private key. A then generates a public key PA = nA * G; the public key is a point in Eq(a, b).
2. B similarly selects a private key nB and computes a public key PB.
3. A generates the secret key k = nA * PB. B generates the secret key k = nB * PA.
The two calculations in step 3 produce the same result because
nA * PB = nA * (nB * G) = nB * (nA * G) = nB * PA
To break this scheme, an attacker would need to be able to compute k given G and kG, which is assumed to be hard.
As an example,6 take p = 211; E p(0, - 4), which is equivalent to the curve y2 = x3 - 4; and G = (2, 2). One can calculate that 240G = O. A’s private key is nA = 121, so A’s public key is PA = 121(2, 2) = (115, 48). B’s private key is nB = 203, so B’s public key is 203(2, 3) = (130, 203). The shared secret key is 121(130, 203) = 203(115, 48) = (161, 69).
Note that the secret key is a pair of numbers. If this key is to be used as a ses-
sion key for conventional encryption, then a single number must be generated. We
could simply use the x coordinates or some simple function of the x coordinate.
Elliptic Curve Encryption/Decryption
Several approaches to encryption/decryption using elliptic curves have been ana-
lyzed in the literature. In this subsection, we look at perhaps the simplest. The
first task in this system is to encode the plaintext message m to be sent as an (x, y) point Pm.
6Provided by Ed Schaefer of Santa Clara University.
332 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS
It is the point Pm that will be encrypted as a ciphertext and subsequently decrypted. Note that we cannot simply encode the message as the x or y coordinate of a point, because not all such coordinates are in Eq(a, b); for example, see Table 10.1. Again, there are several approaches to this encoding, which we will not address here, but
suffice it to say that there are relatively straightforward techniques that can be
used.
As with the key exchange system, an encryption/decryption system requires a
point G and an elliptic group Eq(a, b) as parameters. Each user A selects a private key nA and generates a public key PA = nA * G.
To encrypt and send a message Pm to B, A chooses a random positive integer k and produces the ciphertext Cm consisting of the pair of points:
Cm = {kG, Pm + kPB}
Note that A has used B’s public key PB. To decrypt the ciphertext, B multiplies the first point in the pair by B’s private key and subtracts the result from the second
point:
Pm + kPB - nB(kG) = Pm + k(nBG) - nB(kG) = Pm
Figure 10.7 ECC Diffie–Hellman Key Exchange
Global Public Elements
E q(a, b) elliptic curve with parameters a, b, and q, where q is a prime or an integer of the form 2m
G point on elliptic curve whose order is large value n
User A Key Generation
Select private nA nA 6 n
Calculate public PA PA = nA * G
User B Key Generation
Select private nB nB 6 n
Calculate public PB PB = nB * G
Calculation of Secret Key by User A
K = nA * PB
Calculation of Secret Key by User B
K = nB * PA
10.4 / ELLIPTIC CURVE CRYPTOGRAPHY 333
A has masked the message Pm by adding kPB to it. Nobody but A knows the value of k, so even though Pb is a public key, nobody can remove the mask kPB. However, A also includes a “clue,” which is enough to remove the mask if one knows the private key nB. For an attacker to recover the message, the attacker would have to compute k given G and kG, which is assumed to be hard.
Let us consider a simple example. The global public elements are q = 257; E q(a, b) = E 257(0, - 4), which is equivalent to the curve y2 = x3 - 4; and G = (2, 2). Bob’s private key is nB = 101, and his public key is PB = nBG = 101(2, 2) = (197, 167). Alice wishes to send a message to Bob that is encoded in the elliptic
point Pm = (112, 26). Alice chooses random integer k = 41 and computes kG = 41(2, 2) = (136, 128), kPB = 41(197, 167) = (68, 84) and Pm + kPB = (112, 26) + (68, 84) = (246, 174). Alice sends the ciphertext Cm = (C1, C2) = {(136, 128), (246, 174)} to Bob. Bob receives the ciphertext and computes C2 - nBC1 = (246, 174) - 101(136, 128) = (246, 174) - (68, 84) = (112, 26).
Security of Elliptic Curve Cryptography
The security of ECC depends on how difficult it is to determine k given kP and P. This is referred to as the elliptic curve logarithm problem. The fastest known tech-
nique for taking the elliptic curve logarithm is known as the Pollard rho method.
Table 10.3, from NIST SP 800-57 (Recommendation for Key Management—Part 1: General, September 2015), compares various algorithms by showing comparable key sizes in terms of computational effort for cryptanalysis. As can be seen, a con-
siderably smaller key size can be used for ECC compared to RSA.
Based on this analysis, SP 800-57 recommends that at least through 2030, ac-
ceptable key lengths are from 3072 to 14,360 bits for RSA and 256 to 512 bits for
ECC. Similarly, the European Union Agency for Network and Information Security
(ENISA) recommends in their 2014 report (Algorithms, Key Size and Parameters report—2014, November 2014) minimum key lengths for future system of 3072 bits and 256 bits for RSA and ECC, respectively.
Symmetric Key Algorithms
Diffie–Hellman, Digital Signature Algorithm
RSA (size of n in bits)
ECC (modulus size in bits)
80 L = 1024 N = 160
1024 160–223
112 L = 2048 N = 224
2048 224–255
128 L = 3072 N = 256
3072 256–383
192 L = 7680 N = 384
7680 384–511
256 L = 15,360 N = 512
15,360 512 +
Note: L = size of public key, N = size of private key.
Table 10.3 Comparable Key Sizes in Terms of Computational Effort for Cryptanalysis (NIST SP-800-57)
334 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS
Analysis indicates that for equal key lengths, the computational effort re-
quired for ECC and RSA is comparable [JURI97]. Thus, there is a computational
advantage to using ECC with a shorter key length than a comparably secure RSA.
10.5 PSEUDORANDOM NUMBER GENERATION BASED ON AN ASYMMETRIC CIPHER
We noted in Chapter 8 that because a symmetric block cipher produces an appar-
ently random output, it can serve as the basis of a pseudorandom number generator
(PRNG). Similarly, an asymmetric encryption algorithm produces apparently ran-
dom output and can be used to build a PRNG. Because asymmetric algorithms are
typically much slower than symmetric algorithms, asymmetric algorithms are not
used to generate open-ended PRNG bit streams. Rather, the asymmetric approach
is useful for creating a pseudorandom function (PRF) for generating a short pseu-
dorandom bit sequence.
In this section, we examine two PRNG designs based on pseudorandom
functions.
PRNG Based on RSA
For a sufficient key length, the RSA algorithm is considered secure and is a good
candidate to form the basis of a PRNG. Such a PRNG, known as the Micali–Schnorr
PRNG [MICA91], is recommended in the ANSI standard X9.82 (Random Number Generation) and in the ISO standard 18031 (Random Bit Generation).
The PRNG is illustrated in Figure 10.8. As can be seen, this PRNG has much
the same structure as the output feedback (OFB) mode used as a PRNG (see Figure
8.4b and the portion of Figure 7.6a enclosed with a dashed box). In this case, the
encryption algorithm is RSA rather than a symmetric block cipher. Also, a portion
of the output is fed back to the next iteration of the encryption algorithm and the
remainder of the output is used as pseudorandom bits. The motivation for this sepa-
ration of the output into two distinct parts is so that the pseudorandom bits from
one stage do not provide input to the next stage. This separation should contribute
to forward unpredictability.
Figure 10.8 Micali–Schnorr Pseudorandom Bit Generator
Seed = x0
x1 = r most significant bits
z1 = k least significant bits
y1 = x0 mod n e
n, e, r, k n, e, r, k n, e, r, k
x2 = r most significant bits
z2 = k least significant bits
x3 = r most significant bits
z3 = k least significant bits
y2 = x1 mod n e y3 = x2 mod n
e Encrypt Encrypt Encrypt
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10.5 / PSEUDORANDOM NUMBER GENERATION BASED ON AN ASYMMETRIC CIPHER 335
We can define the PRNG as follows.
Setup Select p, q primes; n = pq; f(n) = (p - 1)(q - 1). Select e such that gcd(e, f(n)) = 1. These are the standard RSA setup selections (see Figure 9.5). In addition, let N = [log2n] + 1 (the bitlength of n). Select r, k such that r + k = N.
Seed Select a random seed x0 of bitlength r.
Generate Generate a pseudorandom sequence of length k * m using the loop for i from 1 to m do
yi = xi - 1 e mod n
xi = r most significant bits of yi zi = k least significant bits of yi
Output The output sequence is z1 }z2 } c }zm.
The parameters n, r, e, and k are selected to satisfy the following six requirements.
1. n = pq n is chosen as the product of two primes to have the cryptographic strength required of
RSA.
2. 1 6 e 6 f(n); gcd (e, f(n)) = 1 Ensures that the mapping s S se mod n is 1 to 1.
3. re Ú 2N Ensures that the exponentiation requires a full modular reduction.
4. r Ú 2 * strength Protects against a cryptographic attacks. 5. k, r are multiples of 8 An implementation convenience.
6. k Ú 8; r + k = N All bits are used.
The variable strength in requirement 4 is defined in NIST SP 800-90 as fol- lows: A number associated with the amount of work (that is, the number of opera-
tions) required to break a cryptographic algorithm or system; a security strength
is specified in bits and is a specific value from the set (112, 128, 192, 256) for this
Recommendation. The amount of work needed is 2strength.
There is clearly a tradeoff between r and k. Because RSA is computation- ally intensive compared to a block cipher, we would like to generate as many
pseudorandom bits per iteration as possible and therefore would like a large
value of k. However, for cryptographic strength, we would like r to be as large as possible.
For example, if e = 3 and N = 1024, then we have the inequality 3r 7 1024, yielding a minimum required size for r of 683 bits. For r set to that size, k = 341 bits are generated for each exponentiation (each RSA encryption). In this case,
each exponentiation requires only one modular squaring of a 683-bit number and
one modular multiplication. That is, we need only calculate (xi * (xi2 mod n)) mod n.
336 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS
PRNG Based on Elliptic Curve Cryptography
In this subsection, we briefly summarize a technique developed by the U.S. National
Security Agency (NSA) known as dual elliptic curve PRNG (DEC PRNG). This
technique is recommended in NIST SP 800-90, the ANSI standard X9.82, and the
ISO standard 18031. There has been some controversy regarding both the security
and efficiency of this algorithm compared to other alternatives (e.g., see [SCHO06],
[BROW07]).
[SCHO06] summarizes the algorithm as follows: Let P and Q be two known
points on a given elliptic curve. The seed of the DEC PRNG is a random integer
s0 ∈ {0, 1, c , #E(GF(p)) - 1}, where # E(GF(p)) denotes the number of points on the curve. Let x denote a function that gives the x-coordinate of a point of the curve. Let lsbi(s) denote the i least significant bits of an integer s. The DEC PRNG transforms the seed into the pseudorandom sequence of length 240k, k 7 0, as follows.
for i = 1 to k do Set si d x(Si-1 P) Set ri d lsb240 (x(si Q))
end for Return r1,...,rk
Given the security concerns expressed for this PRNG, the only motivation for
its use would be that it is used in a system that already implements ECC but does
not implement any other symmetric, asymmetric, or hash cryptographic algorithm
that could be used to build a PRNG.
10.6 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
Key Terms
abelian group
binary curve
cubic equation
Diffie–Hellman key exchange
discrete logarithm
elliptic curve
elliptic curve arithmetic
elliptic curve cryptography
finite field
man-in-the-middle attack
Micali–Schnorr
prime curve
primitive root
zero point
Review Questions
10.1 Briefly explain Diffie–Hellman key exchange. 10.2 What is an elliptic curve? 10.3 What is the zero point of an elliptic curve? 10.4 What is the sum of three points on an elliptic curve that lie on a straight line?
10.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 337
Problems
10.1 Alice and Bob use the Diffie–Hellman key exchange technique with a common prime q = 1 5 7 and a primitive root a = 5. a. If Alice has a private key XA = 15, find her public key YA. b. If Bob has a private key XB = 27, find his public key YB. c. What is the shared secret key between Alice and Bob?
10.2 Alice and Bob use the Diffie-Hellman key exchange technique with a common prime q = 2 3 and a primitive root a = 5 . a. If Bob has a public key YB = 1 0 , what is Bob’s private key YB? b. If Alice has a public key YA = 8 , what is the shared key K with Bob? c. Show that 5 is a primitive root of 23.
10.3 In the Diffie–Hellman protocol, each participant selects a secret number x and sends the other participant ax mod q for some public number a. What would happen if the participants sent each other xa for some public number a instead? Give at least one method Alice and Bob could use to agree on a key. Can Eve break your system with- out finding the secret numbers? Can Eve find the secret numbers?
10.4 This problem illustrates the point that the Diffie–Hellman protocol is not secure without the step where you take the modulus; i.e. the “Indiscrete Log Problem” is not a hard problem! You are Eve and have captured Alice and Bob and imprisoned them. You overhear the following dialog.
Bob: Oh, let’s not bother with the prime in the Diffie–Hellman protocol, it will make things easier.
Alice: Okay, but we still need a base a to raise things to. How about a = 3?
Bob: All right, then my result is 27.
Alice: And mine is 243.
What is Bob’s private key XB and Alice’s private key XA? What is their secret com- bined key? (Don’t forget to show your work.)
10.5 Section 10.1 describes a man-in-the-middle attack on the Diffie–Hellman key exchange protocol in which the adversary generates two public–private key pairs for the attack. Could the same attack be accomplished with one pair? Explain.
10.6 Suppose Alice and Bob use an Elgamal scheme with a common prime q = 1 5 7 and a primitive root a = 5 . a. If Bob has public key YB = 1 0 and Alice chose the random integer k = 3 , what
is the ciphertext of M = 9 ? b. If Alice now chooses a different value of k so that the encoding of M = 9 is
C = (2 5 , C2 ), what is the integer C2? 10.7 Rule (5) for doing arithmetic in elliptic curves over real numbers states that to double
a point Q2, draw the tangent line and find the other point of intersection S. Then Q + Q = 2Q = - S. If the tangent line is not vertical, there will be exactly one point of intersection. However, suppose the tangent line is vertical? In that case, what is the value 2Q? What is the value 3Q?
10.8 Demonstrate that the two elliptic curves of Figure 10.4 each satisfy the conditions for a group over the real numbers.
10.9 Is (5, 12) a point on the elliptic curve y2 = x 3 + 4 x - 1 over real numbers?
10.10 On the elliptic curve over the real numbers y2 = x3 - 17
12 x + 1, Let P = (0,1) and
Q = (1.5,1.5). Find P + Q and 2P. 10.11 Does the elliptic curve equation y2 = x 3 + x + 2 define a group over Z7?
338 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS
10.12 Consider the elliptic curve E7(2,1); that is, the curve is defined by y 2 = x 3 + 2 x + 1
with a modulus of p = 7 . Determine all of the points in E7(2, 1). Hint: Start by calcu- lating the right-hand side of the equation for all values of x.
10.13 What are the negatives of the following elliptic curve points over Z7? P = (3, 5); Q = (2, 5); R = (5, 0).
10.14 For E11(1, 7), consider the point G = (3, 2). Compute the multiple of G from 2G through 13G.
10.15 This problem performs elliptic curve encryption/decryption using the scheme out- lined in Section 10.4. The cryptosystem parameters are E11(1, 7) and G = (3, 2). B’s private key is nB = 7. a. Find B’s public key PB. b. A wishes to encrypt the message Pm = (10, 7) and chooses the random value
k = 5. Determine the ciphertext Cm. c. Show the calculation by which B recovers Pm from Cm.
10.16 The following is a first attempt at an elliptic curve signature scheme. We have a global elliptic curve, prime p, and “generator” G. Alice picks a private signing key XA and forms the public verifying key YA = XAG. To sign a message M:
■ Alice picks a value k. ■ Alice sends Bob M, k, and the signature S = M - kXAG. ■ Bob verifies that M = S + kYA.
a. Show that this scheme works. That is, show that the verification process produces an equality if the signature is valid.
b. Show that the scheme is unacceptable by describing a simple technique for forging a user’s signature on an arbitrary message.
10.17 Here is an improved version of the scheme given in the previous problem. As before, we have a global elliptic curve, prime p, and “generator” G. Alice picks a private signing key XA and forms the public verifying key YA = XAG. To sign a message M:
■ Bob picks a value k. ■ Bob sends Alice C1 = kG. ■ Alice sends Bob M and the signature S = M - XAC1. ■ Bob verifies that M = S + kYA.
a. Show that this scheme works. That is, show that the verification process produces an equality if the signature is valid.
b. Show that forging a message in this scheme is as hard as breaking (Elgamal) elliptic curve cryptography. (Or find an easier way to forge a message?)
c. This scheme has an extra “pass” compared to other cryptosystems and signature schemes we have looked at. What are some drawbacks to this?
PART FOUR: CRYPTOGRAPHIC DATA INTEGRITY ALGORITHMS
CHAPTER
Cryptographic Hash Functions 11.1 Applications of Cryptographic Hash Functions
Message Authentication
Digital Signatures
Other Applications
11.2 Two Simple Hash Functions
11.3 Requirements and Security
Security Requirements for Cryptographic Hash Functions
Brute-Force Attacks
Cryptanalysis
11.4 Hash Functions Based on Cipher Block Chaining
11.5 Secure Hash Algorithm (SHA)
SHA-512 Logic
SHA-512 Round Function
Example
11.6 SHA-3
The Sponge Construction
The SHA-3 Iteration Function f
11.7 Key Terms, Review Questions, and Problems
339
340 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
A hash function H accepts a variable-length block of data M as input and produces a fixed-size hash value h = H(M). A “good” hash function has the property that the results of applying the function to a large set of inputs will produce outputs that are
evenly distributed and apparently random. In general terms, the principal object of
a hash function is data integrity. A change to any bit or bits in M results, with high probability, in a change to the hash value.
The kind of hash function needed for security applications is referred to as a
cryptographic hash function. A cryptographic hash function is an algorithm for which it is computationally infeasible (because no attack is significantly more efficient than
brute force) to find either (a) a data object that maps to a pre-specified hash result
(the one-way property) or (b) two data objects that map to the same hash result (the
collision-free property). Because of these characteristics, hash functions are often used
to determine whether or not data has changed.
Figure 11.1 depicts the general operation of a cryptographic hash function.
Typically, the input is padded out to an integer multiple of some fixed length
(e.g., 1024 bits), and the padding includes the value of the length of the original mes-
sage in bits. The length field is a security measure to increase the difficulty for an
attacker to produce an alternative message with the same hash value, as explained
subsequently.
This chapter begins with a discussion of the wide variety of applications for
cryptographic hash functions. Next, we look at the security requirements for such
functions. Then we look at the use of cipher block chaining to implement a crypto-
graphic hash function. The remainder of the chapter is devoted to the most important
and widely used family of cryptographic hash functions, the Secure Hash Algorithm
(SHA) family.
Appendix N describes MD5, a well-known cryptographic hash function with
similarities to SHA-1.
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Summarize the applications of cryptographic hash functions.
◆ Explain why a hash function used for message authentication needs to be secured.
◆ Understand the differences among preimage resistant, second preimage resistant, and collision resistant properties.
◆ Present an overview of the basic structure of cryptographic hash functions.
◆ Describe how cipher block chaining can be used to construct a hash function.
◆ Understand the operation of SHA-512.
◆ Understand the birthday paradox and present an overview of the birthday attack.
11.1 / APPLICATIONS OF CRYPTOGRAPHIC HASH FUNCTIONS 341
11.1 APPLICATIONS OF CRYPTOGRAPHIC HASH FUNCTIONS
Perhaps the most versatile cryptographic algorithm is the cryptographic hash func-
tion. It is used in a wide variety of security applications and Internet protocols.
To better understand some of the requirements and security implications for cryp-
tographic hash functions, it is useful to look at the range of applications in which it
is employed.
Message Authentication
Message authentication is a mechanism or service used to verify the integrity of
a message. Message authentication assures that data received are exactly as sent
(i.e., there is no modification, insertion, deletion, or replay). In many cases, there is
a requirement that the authentication mechanism assures that purported identity of
the sender is valid. When a hash function is used to provide message authentication,
the hash function value is often referred to as a message digest.1
The essence of the use of a hash function for message integrity is as follows.
The sender computes a hash value as a function of the bits in the message and trans-
mits both the hash value and the message. The receiver performs the same hash cal-
culation on the message bits and compares this value with the incoming hash value.
Figure 11.1 Cryptographic Hash Function; h = H(M)
Message or data block M (variable length) P, L
P, L = padding plus length field
L bits
Hash value h (fixed length)
H
1The topic of this section is invariably referred to as message authentication. However, the concepts and techniques apply equally to data at rest. For example, authentication techniques can be applied to a file in storage to assure that the file is not tampered with.
342 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
If there is a mismatch, the receiver knows that the message (or possibly the hash
value) has been altered (Figure 11.2a).
The hash value must be transmitted in a secure fashion. That is, the hash value
must be protected so that if an adversary alters or replaces the message, it is not
feasible for adversary to also alter the hash value to fool the receiver. This type
of attack is shown in Figure 11.2b. In this example, Alice transmits a data block
and attaches a hash value. Darth intercepts the message, alters or replaces the data
block, and calculates and attaches a new hash value. Bob receives the altered data
with the new hash value and does not detect the change. To prevent this attack, the
hash value generated by Alice must be protected.
Figure 11.2 Attack Against Hash Function
(b) Man-in-the-middle attack
Alice
Darth
Bob
BobAlice
COMPARE
data
data
data
H
data
data
data
H
H
(a) Use of hash function to check data integrity
COMPARE
data
data
data
H H
11.1 / APPLICATIONS OF CRYPTOGRAPHIC HASH FUNCTIONS 343
Figure 11.3 illustrates a variety of ways in which a hash code can be used to
provide message authentication, as follows.
a. The message plus concatenated hash code is encrypted using symmetric encryption. Because only A and B share the secret key, the message must have
come from A and has not been altered. The hash code provides the structure or
redundancy required to achieve authentication. Because encryption is applied
to the entire message plus hash code, confidentiality is also provided.
b. Only the hash code is encrypted, using symmetric encryption. This reduces the processing burden for those applications that do not require confidentiality.
Figure 11.3 Simplified Examples of the Use of a Hash Function for Message Authentication
E
K
M
H
| | D
K
M
H(M )
H
Compare
(a)
M
H
| |
K
(b)
M
D
H
CompareK
E
E(K, [M || H(M )])
E(K, H(M ))
Destination BSource A
| |S M
H
| |
S (c)
| |
M
H(M || S)
H(M || S)
H
Compare
M
H
| |
S (d)
| |
E
K
| |S H
Compare
MD
K E(K, [M || H(M || S)])
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344 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
c. It is possible to use a hash function but no encryption for message authentica- tion. The technique assumes that the two communicating parties share a common
secret value S. A computes the hash value over the concatenation of M and S and appends the resulting hash value to M. Because B possesses S, it can recompute the hash value to verify. Because the secret value itself is not sent, an opponent
cannot modify an intercepted message and cannot generate a false message.
d. Confidentiality can be added to the approach of method (c) by encrypting the entire message plus the hash code.
When confidentiality is not required, method (b) has an advantage over
methods (a) and (d), which encrypts the entire message, in that less computa-
tion is required. Nevertheless, there has been growing interest in techniques that
avoid encryption (Figure 11.3c). Several reasons for this interest are pointed out
in [TSUD92].
■ Encryption software is relatively slow. Even though the amount of data to be
encrypted per message is small, there may be a steady stream of messages into
and out of a system.
■ Encryption hardware costs are not negligible. Low-cost chip implementations
of DES are available, but the cost adds up if all nodes in a network must have
this capability.
■ Encryption hardware is optimized toward large data sizes. For small blocks of
data, a high proportion of the time is spent in initialization/invocation overhead.
■ Encryption algorithms may be covered by patents, and there is a cost associ-
ated with licensing their use.
More commonly, message authentication is achieved using a message authentication code (MAC), also known as a keyed hash function. Typically, MACs are used between two parties that share a secret key to authenticate information
exchanged between those parties. A MAC function takes as input a secret key and
a data block and produces a hash value, referred to as the MAC, which is associ-
ated with the protected message. If the integrity of the message needs to be checked,
the MAC function can be applied to the message and the result compared with the
associated MAC value. An attacker who alters the message will be unable to alter the
associated MAC value without knowledge of the secret key. Note that the verifying
party also knows who the sending party is because no one else knows the secret key.
Note that the combination of hashing and encryption results in an overall
function that is, in fact, a MAC (Figure 11.3b). That is, E(K, H(M)) is a function of a variable-length message M and a secret key K, and it produces a fixed-size output that is secure against an opponent who does not know the secret key. In practice,
specific MAC algorithms are designed that are generally more efficient than an
encryption algorithm.
We discuss MACs in Chapter 12.
Digital Signatures
Another important application, which is similar to the message authentication
application, is the digital signature. The operation of the digital signature is similar to that of the MAC. In the case of the digital signature, the hash value of a message
11.1 / APPLICATIONS OF CRYPTOGRAPHIC HASH FUNCTIONS 345
is encrypted with a user’s private key. Anyone who knows the user’s public key can
verify the integrity of the message that is associated with the digital signature. In
this case, an attacker who wishes to alter the message would need to know the user’s
private key. As we shall see in Chapter 14, the implications of digital signatures go
beyond just message authentication.
Figure 11.4 illustrates, in a simplified fashion, how a hash code is used to
provide a digital signature.
a. The hash code is encrypted, using public-key encryption with the sender’s private key. As with Figure 11.3b, this provides authentication. It also provides
a digital signature, because only the sender could have produced the encrypted
hash code. In fact, this is the essence of the digital signature technique.
b. If confidentiality as well as a digital signature is desired, then the message plus the private-key-encrypted hash code can be encrypted using a symmetric
secret key. This is a common technique.
Other Applications
Hash functions are commonly used to create a one-way password file. Chapter 21 explains a scheme in which a hash of a password is stored by an operating system
rather than the password itself. Thus, the actual password is not retrievable by a
hacker who gains access to the password file. In simple terms, when a user enters a
password, the hash of that password is compared to the stored hash value for veri-
fication. This approach to password protection is used by most operating systems.
Hash functions can be used for intrusion detection and virus detection. Store H(F) for each file on a system and secure the hash values (e.g., on a CD-R that is
Figure 11.4 Simplified Examples of Digital Signatures
M
H
| | E
E
K
D
K
M
D
H
Compare
(b) E(PRa, H(M ))
E(K, [M || E(PRa, H(M ))])
Destination BSource A
PRa
PRa
PUa
PUa
M
H
| |
(a)
M
E D
H
Compare
E(PRa, H(M ))
346 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
kept secure). One can later determine if a file has been modified by recomputing
H(F). An intruder would need to change F without changing H(F).
A cryptographic hash function can be used to construct a pseudorandom function (PRF) or a pseudorandom number generator (PRNG). A common application for a hash-based PRF is for the generation of symmetric keys. We discuss
this application in Chapter 12.
11.2 TWO SIMPLE HASH FUNCTIONS
To get some feel for the security considerations involved in cryptographic hash
functions, we present two simple, insecure hash functions in this section. All hash
functions operate using the following general principles. The input (message, file,
etc.) is viewed as a sequence of n -bit blocks. The input is processed one block at a time in an iterative fashion to produce an n-bit hash function.
One of the simplest hash functions is the bit-by-bit exclusive-OR (XOR) of
every block. This can be expressed as
Ci = bi1 ⊕ bi2 ⊕ g ⊕ bim
where
Ci = ith bit of the hash code, 1 … i … n m = number of n@bit blocks in the input bij = ith bit in jth block ⊕ = XOR operation
This operation produces a simple parity bit for each bit position and is known
as a longitudinal redundancy check. It is reasonably effective for random data as a
data integrity check. Each n-bit hash value is equally likely. Thus, the probability that a data error will result in an unchanged hash value is 2-n. With more predict-
ably formatted data, the function is less effective. For example, in most normal text
files, the high-order bit of each octet is always zero. So if a 128-bit hash value is
used, instead of an effectiveness of 2-128, the hash function on this type of data has
an effectiveness of 2-112.
A simple way to improve matters is to perform a one-bit circular shift, or
rotation, on the hash value after each block is processed. The procedure can be
summarized as follows.
1. Initially set the n-bit hash value to zero.
2. Process each successive n-bit block of data as follows:
a. Rotate the current hash value to the left by one bit. b. XOR the block into the hash value.
This has the effect of “randomizing” the input more completely and overcoming
any regularities that appear in the input. Figure 11.5 illustrates these two types of
hash functions for 16-bit hash values.
11.2 / TWO SIMPLE HASH FUNCTIONS 347
Although the second procedure provides a good measure of data integrity, it is
virtually useless for data security when an encrypted hash code is used with a plain-
text message, as in Figures 11.3b and 11.4a. Given a message, it is an easy matter
to produce a new message that yields that hash code: Simply prepare the desired
alternate message and then append an n-bit block that forces the new message plus block to yield the desired hash code.
Although a simple XOR or rotated XOR (RXOR) is insufficient if only the
hash code is encrypted, you may still feel that such a simple function could be
useful when the message together with the hash code is encrypted (Figure 11.3a).
But you must be careful. A technique originally proposed by the National
Bureau of Standards used the simple XOR applied to 64-bit blocks of the mes-
sage and then an encryption of the entire message that used the cipher block
chaining (CBC) mode. We can define the scheme as follows: Given a message M consisting of a sequence of 64-bit blocks X1, X2, c , XN, define the hash code
Figure 11.5 Two Simple Hash Functions
XOR of every 16-bit blockXOR with 1-bit r otation to the right
16 bits
348 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
h = H(M) as the block-by-block XOR of all blocks and append the hash code as the final block:
h = XN + 1 = X1 ⊕ X2 ⊕ c ⊕ XN
Next, encrypt the entire message plus hash code using CBC mode to produce the
encrypted message Y1, Y2, c , YN + 1. [JUEN85] points out several ways in which the ciphertext of this message can be manipulated in such a way that it is not detect-
able by the hash code. For example, by the definition of CBC (Figure 6.4), we have
X1 = IV ⊕ D(K,Y1) Xi = Yi - 1 ⊕ D(K, Yi)
XN + 1 = YN ⊕ D(K, YN + 1)
But XN + 1 is the hash code:
XN + 1 = X1 ⊕ X2 ⊕ c ⊕ XN = [IV ⊕ D(K, Y1)] ⊕ [Y1 ⊕ D(K, Y2)] ⊕ c ⊕ [YN - 1 ⊕ D(K, YN)]
Because the terms in the preceding equation can be XORed in any order, it follows
that the hash code would not change if the ciphertext blocks were permuted.
11.3 REQUIREMENTS AND SECURITY
Before proceeding, we need to define two terms. For a hash value h = H(x), we say that x is the preimage of h. That is, x is a data block whose hash value, using the function H, is h. Because H is a many-to-one mapping, for any given hash value h, there will in general be multiple preimages. A collision occurs if we have x ≠ y and H(x) = H(y). Because we are using hash functions for data integrity, collisions are clearly undesirable.
Let us consider how many preimages are there for a given hash value, which is
a measure of the number of potential collisions for a given hash value. Suppose the
length of the hash code is n bits, and the function H takes as input messages or data blocks of length b bits with b 7 n. Then, the total number of possible messages is 2b and the total number of possible hash values is 2n. On average, each hash value
corresponds to 2b - n preimages. If H tends to uniformly distribute hash values then,
in fact, each hash value will have close to 2b - n preimages. If we now allow inputs
of arbitrary length, not just a fixed length of some number of bits, then the number
of preimages per hash value is arbitrarily large. However, the security risks in the
use of a hash function are not as severe as they might appear from this analysis.
To understand better the security implications of cryptographic hash functions, we
need precisely define their security requirements.
Security Requirements for Cryptographic Hash Functions
Table 11.1 lists the generally accepted requirements for a cryptographic hash func-
tion. The first three properties are requirements for the practical application of a
hash function.
11.3 / REQUIREMENTS AND SECURITY 349
The fourth property, preimage resistant, is the one-way property: it is easy to generate a code given a message, but virtually impossible to generate a message
given a code. This property is important if the authentication technique involves the
use of a secret value (Figure 11.3c). The secret value itself is not sent. However, if
the hash function is not one way, an attacker can easily discover the secret value:
If the attacker can observe or intercept a transmission, the attacker obtains the
message M, and the hash code h = H(S }M). The attacker then inverts the hash function to obtain S }M = H-1(MDM). Because the attacker now has both M and SAB }M, it is a trivial matter to recover SAB.
The fifth property, second preimage resistant, guarantees that it is infeasible to find an alternative message with the same hash value as a given message. This pre-
vents forgery when an encrypted hash code is used (Figures 11.3b and 11.4a). If this
property were not true, an attacker would be capable of the following sequence:
First, observe or intercept a message plus its encrypted hash code; second, generate
an unencrypted hash code from the message; third, generate an alternate message
with the same hash code.
A hash function that satisfies the first five properties in Table 11.1 is referred
to as a weak hash function. If the sixth property, collision resistant, is also satis- fied, then it is referred to as a strong hash function. A strong hash function protects
against an attack in which one party generates a message for another party to sign.
For example, suppose Bob writes an IOU message, sends it to Alice, and she signs
it. Bob finds two messages with the same hash, one of which requires Alice to pay a
small amount and one that requires a large payment. Alice signs the first message,
and Bob is then able to claim that the second message is authentic.
Figure 11.6 shows the relationships among the three resistant properties.
A function that is collision resistant is also second preimage resistant, but the
reverse is not necessarily true. A function can be collision resistant but not preim-
age resistant and vice versa. A function can be preimage resistant but not second
preimage resistant and vice versa. See [MENE97] for a discussion.
Requirement Description
Variable input size H can be applied to a block of data of any size.
Fixed output size H produces a fixed-length output.
Efficiency H(x) is relatively easy to compute for any given x, making both hardware and software implementations practical.
Preimage resistant (one-way property) For any given hash value h, it is computationally infeasible to find y such that H(y) = h.
Second preimage resistant (weak collision
resistant)
For any given block x, it is computationally infeasible to find y ≠ x with H(y) = H(x).
Collision resistant (strong collision resistant) It is computationally infeasible to find any pair
(x, y) with x ≠ y, such that H(x) = H(y).
Pseudorandomness Output of H meets standard tests for
pseudorandomness.
Table 11.1 Requirements for a Cryptographic Hash Function H
350 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
Table 11.2 shows the resistant properties required for various hash function
applications.
The final requirement in Table 11.1, pseudorandomness, has not tradition- ally been listed as a requirement of cryptographic hash functions but is more or
less implied. [JOHN05] points out that cryptographic hash functions are commonly
used for key derivation and pseudorandom number generation, and that in message
integrity applications, the three resistant properties depend on the output of the
hash function appearing to be random. Thus, it makes sense to verify that in fact a
given hash function produces pseudorandom output.
Brute-Force Attacks
As with encryption algorithms, there are two categories of attacks on hash func-
tions: brute-force attacks and cryptanalysis. A brute-force attack does not depend
on the specific algorithm but depends only on bit length. In the case of a hash func-
tion, a brute-force attack depends only on the bit length of the hash value. A crypt-
analysis, in contrast, is an attack based on weaknesses in a particular cryptographic
algorithm. We look first at brute-force attacks.
Figure 11.6 Relationship Among Hash Function Properties
Second preimage resistant
Preimage resistant
Collision resistant
Preimage Resistant Second Preimage
Resistant Collision Resistant
Hash + digital signature yes yes yes*
Intrusion detection and virus
detection
yes
Hash + symmetric encryption
One-way password file yes
MAC yes yes yes*
Table 11.2 Hash Function Resistance Properties Required for Various Data Integrity Applications
*Resistance required if attacker is able to mount a chosen message attack
11.3 / REQUIREMENTS AND SECURITY 351
PREIMAGE AND SECOND PREIMAGE ATTACKS For a preimage or second preimage attack, an adversary wishes to find a value y such that H(y) is equal to a given hash value h. The brute-force method is to pick values of y at random and try each value until a collision occurs. For an m-bit hash value, the level of effort is proportional to 2m. Specifically, the adversary would have to try, on average, 2m - 1 values of y to
find one that generates a given hash value h. This result is derived in Appendix U [Equation (U.1)].
COLLISION RESISTANT ATTACKS For a collision resistant attack, an adversary wishes to find two messages or data blocks, x and y, that yield the same hash function: H(x) = H(y). This turns out to require considerably less effort than a preimage or second preimage attack. The effort required is explained by a mathematical result
referred to as the birthday paradox. In essence, if we choose random variables from a uniform distribution in the range 0 through N - 1, then the probability that a repeated element is encountered exceeds 0.5 after 2N choices have been made. Thus, for an m-bit hash value, if we pick data blocks at random, we can expect to find two data blocks with the same hash value within 22m = 2m/2 attempts. The mathematical derivation of this result is found in Appendix U.
Yuval proposed the following strategy to exploit the birthday paradox in a
collision resistant attack [YUVA79].
1. The source, A, is prepared to sign a legitimate message x by appending the appropriate m-bit hash code and encrypting that hash code with A’s private key (Figure 11.4a).
2. The opponent generates 2m/2 variations x′ of x, all of which convey essentially the same meaning, and stores the messages and their hash values.
3. The opponent prepares a fraudulent message y for which A’s signature is desired.
4. The opponent generates minor variations y′ of y, all of which convey essen- tially the same meaning. For each y′, the opponent computes H(y′), checks for matches with any of the H(x′) values, and continues until a match is found. That is, the process continues until a y′ is generated with a hash value equal to the hash value of one of the x′ values.
5. The opponent offers the valid variation to A for signature. This signature can then be attached to the fraudulent variation for transmission to the intended
recipient. Because the two variations have the same hash code, they will pro-
duce the same signature; the opponent is assured of success even though the
encryption key is not known.
Thus, if a 64-bit hash code is used, the level of effort required is only on the
order of 232 [see Appendix U, Equation (U.7)].
The generation of many variations that convey the same meaning is not diffi-
cult. For example, the opponent could insert a number of “space-space- backspace”
character pairs between words throughout the document. Variations could then
be generated by substituting “space-backspace-space” in selected instances.
352 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
Alternatively, the opponent could simply reword the message but retain the
meaning. Figure 11.7 provides an example.
To summarize, for a hash code of length m, the level of effort required, as we have seen, is proportional to the following.
Preimage resistant 2m
Second preimage resistant 2m
Collision resistant 2m/2
Figure 11.7 A Letter in 238 Variations
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11.3 / REQUIREMENTS AND SECURITY 353
If collision resistance is required (and this is desirable for a general-purpose
secure hash code), then the value 2m/2 determines the strength of the hash code
against brute-force attacks. Van Oorschot and Wiener [VANO94] presented
a design for a $10 million collision search machine for MD5, which has a 128-bit hash
length, that could find a collision in 24 days. Thus, a 128-bit code may be viewed as
inadequate. The next step up, if a hash code is treated as a sequence of 32 bits,
is a 160-bit hash length. With a hash length of 160 bits, the same search machine
would require over four thousand years to find a collision. With today’s technology,
the time would be much shorter, so that 160 bits now appears suspect.
Cryptanalysis
As with encryption algorithms, cryptanalytic attacks on hash functions seek to
exploit some property of the algorithm to perform some attack other than an
exhaustive search. The way to measure the resistance of a hash algorithm to crypt-
analysis is to compare its strength to the effort required for a brute-force attack.
That is, an ideal hash algorithm will require a cryptanalytic effort greater than or
equal to the brute-force effort.
In recent years, there has been considerable effort, and some successes,
in developing cryptanalytic attacks on hash functions. To understand these, we
need to look at the overall structure of a typical secure hash function, indicated
in Figure 11.8. This structure, referred to as an iterated hash function, was pro-
posed by Merkle [MERK79, MERK89] and is the structure of most hash func-
tions in use today, including SHA, which is discussed later in this chapter. The
hash function takes an input message and partitions it into L fixed-sized blocks of b bits each. If necessary, the final block is padded to b bits. The final block also includes the value of the total length of the input to the hash function. The
inclusion of the length makes the job of the opponent more difficult. Either the
opponent must find two messages of equal length that hash to the same value or
two messages of differing lengths that, together with their length values, hash to
the same value.
Figure 11.8 General Structure of Secure Hash Code
f fn n n
IV = CV0 CV1
b
n
CVL–1
CVLn
b
Y0 Y1 YL–1
IV = Initial value CVi = Chaining variable Yi = ith input block f = Compression algorithm
L = Number of input blocks n = Length of hash code b = Length of input block
b
f
354 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
The hash algorithm involves repeated use of a compression function, f, that takes two inputs (an n-bit input from the previous step, called the chaining variable, and a b-bit block) and produces an n-bit output. At the start of hashing, the chaining variable has an initial value that is specified as part of the algorithm. The final value
of the chaining variable is the hash value. Often, b 7 n; hence the term compression. The hash function can be summarized as
CV0 = IV = initial n@bit value CVi = f(CVi - 1, Yi - 1) 1 … i … L
H(M) = CVL
where the input to the hash function is a message M consisting of the blocks Y0, Y1, c , YL - 1.
The motivation for this iterative structure stems from the observation by
Merkle [MERK89] and Damgard [DAMG89] that if the length field is included in
the input, and if the compression function is collision resistant, then so is the resul-
tant iterated hash function.2 Therefore, the structure can be used to produce
a secure hash function to operate on a message of any length. The problem of
designing a secure hash function reduces to that of designing a collision-resistant
compression function that operates on inputs of some fixed size.
Cryptanalysis of hash functions focuses on the internal structure of f and is
based on attempts to find efficient techniques for producing collisions for a single
execution of f. Once that is done, the attack must take into account the fixed value
of IV. The attack on f depends on exploiting its internal structure. Typically, as with
symmetric block ciphers, f consists of a series of rounds of processing, so that the
attack involves analysis of the pattern of bit changes from round to round.
Keep in mind that for any hash function there must exist collisions, because
we are mapping a message of length at least equal to twice the block size b (because we must append a length field) into a hash code of length n, where b Ú n. What is required is that it is computationally infeasible to find collisions.
The attacks that have been mounted on hash functions are rather complex and
beyond our scope here. For the interested reader, [DOBB96] and [BELL97] are
recommended.
11.4 HASH FUNCTIONS BASED ON CIPHER BLOCK CHAINING
A number of proposals have been made for hash functions based on using a cipher
block chaining technique, but without using the secret key. One of the first such
proposals was that of Rabin [RABI78]. Divide a message M into fixed-size blocks M1, M2, c , MN and use a symmetric encryption system such as DES to compute the hash code G as
H0 = initial value Hi = E(Mi, Hi - 1) G = HN
2The converse is not necessarily true.
11.5 / SECURE HASH ALGORITHM (SHA) 355
This is similar to the CBC technique, but in this case, there is no secret key. As with
any hash code, this scheme is subject to the birthday attack, and if the encryp-
tion algorithm is DES and only a 64-bit hash code is produced, then the system
is vulnerable.
Furthermore, another version of the birthday attack can be used even if the
opponent has access to only one message and its valid signature and cannot obtain
multiple signings. Here is the scenario: We assume that the opponent intercepts
a message with a signature in the form of an encrypted hash code and that the
unencrypted hash code is m bits long.
1. Use the algorithm defined at the beginning of this subsection to calculate the unencrypted hash code G.
2. Construct any desired message in the form Q1, Q2, c , QN - 2. 3. Compute Hi = E(Qi, Hi - 1) for 1 … i … (N - 2). 4. Generate 2m/2 random blocks; for each block X, compute E(X, HN - 2).
Generate an additional 2m/2 random blocks; for each block Y, compute D(Y, G), where D is the decryption function corresponding to E.
5. Based on the birthday paradox, with high probability there will be an X and Y such that E(X, HN - 2) = D(Y, G).
6. Form the message Q1, Q2, c , QN - 2, X, Y. This message has the hash code G and therefore can be used with the intercepted encrypted signature.
This form of attack is known as a meet-in-the-middle-attack. A number of researchers have proposed refinements intended to strengthen the basic block
chaining approach. For example, Davies and Price [DAVI89] describe the variation:
Hi = E(Mi, Hi - 1) ⊕ Hi - 1
Another variation, proposed in [MEYE88], is
Hi = E(Hi - 1, Mi) ⊕ Mi
However, both of these schemes have been shown to be vulnerable to a variety
of attacks [MIYA90]. More generally, it can be shown that some form of birthday
attack will succeed against any hash scheme involving the use of cipher block chain-
ing without a secret key, provided that either the resulting hash code is small enough
(e.g., 64 bits or less) or that a larger hash code can be decomposed into independent
subcodes [JUEN87].
Thus, attention has been directed at finding other approaches to hashing.
Many of these have also been shown to have weaknesses [MITC92].
11.5 SECURE HASH ALGORITHM (SHA)
In recent years, the most widely used hash function has been the Secure Hash
Algorithm (SHA). Indeed, because virtually every other widely used hash function
had been found to have substantial cryptanalytic weaknesses, SHA was more or
less the last remaining standardized hash algorithm by 2005. SHA was developed
356 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
by the National Institute of Standards and Technology (NIST) and published as a
federal information processing standard (FIPS 180) in 1993. When weaknesses were
discovered in SHA, now known as SHA-0, a revised version was issued as FIPS 180-1 in 1995 and is referred to as SHA-1. The actual standards document is entitled “Secure Hash Standard.” SHA is based on the hash function MD4, and its design
closely models MD4.
SHA-1 produces a hash value of 160 bits. In 2002, NIST produced a revised
version of the standard, FIPS 180-2, that defined three new versions of SHA, with
hash value lengths of 256, 384, and 512 bits, known as SHA-256, SHA-384, and
SHA-512, respectively. Collectively, these hash algorithms are known as SHA-2. These new versions have the same underlying structure and use the same types of
modular arithmetic and logical binary operations as SHA-1. A revised document
was issued as FIP PUB 180-3 in 2008, which added a 224-bit version (Table 11.3).
In 2015, NIST issued FIPS 180-4, which added two additional algorithms:
SHA-512/224 and SHA-512/256. SHA-1 and SHA-2 are also specified in RFC
6234, which essentially duplicates the material in FIPS 180-3 but adds a C code
implementation.
In 2005, NIST announced the intention to phase out approval of SHA-1 and
move to a reliance on SHA-2 by 2010. Shortly thereafter, a research team described
an attack in which two separate messages could be found that deliver the same
SHA-1 hash using 269 operations, far fewer than the 280 operations previously
thought needed to find a collision with an SHA-1 hash [WANG05]. This result
should hasten the transition to SHA-2.
In this section, we provide a description of SHA-512. The other versions are
quite similar.
SHA-512 Logic
The algorithm takes as input a message with a maximum length of less than 2128 bits
and produces as output a 512-bit message digest. The input is processed in 1024-bit
blocks. Figure 11.9 depicts the overall processing of a message to produce a digest.
Algorithm Message Size Block Size Word Size Message
Digest Size
SHA-1 6 264 512 32 160
SHA-224 6 264 512 32 224
SHA-256 6 264 512 32 256
SHA-384 6 2128 1024 64 384
SHA-512 6 2128 1024 64 512
SHA-512/224 6 2128 1024 64 224
SHA-512/256 6 2128 1024 64 256
Note: All sizes are measured in bits.
Table 11.3 Comparison of SHA Parameters
11.5 / SECURE HASH ALGORITHM (SHA) 357
This follows the general structure depicted in Figure 11.8. The processing consists
of the following steps.
Step 1 Append padding bits. The message is padded so that its length is congruent to 896 modulo 1024 [length K 896(mod 1024)]. Padding is always added, even if the message is already of the desired length. Thus, the number of
padding bits is in the range of 1 to 1024. The padding consists of a single 1 bit
followed by the necessary number of 0 bits.
Step 2 Append length. A block of 128 bits is appended to the message. This block is treated as an unsigned 128-bit integer (most significant byte first) and
contains the length of the original message in bits (before the padding).
The outcome of the first two steps yields a message that is an integer
multiple of 1024 bits in length. In Figure 11.9, the expanded message is rep-
resented as the sequence of 1024-bit blocks M1, M2, c , MN, so that the total length of the expanded message is N * 1024 bits.
Step 3 Initialize hash buffer. A 512-bit buffer is used to hold intermediate and final results of the hash function. The buffer can be represented as eight 64-bit
registers (a, b, c, d, e, f, g, h). These registers are initialized to the following
64-bit integers (hexadecimal values):
a = 6A09E667F3BCC908 e = 510E527FADE682D1
b = BB67AE8584CAA73B f = 9B05688C2B3E6C1F
c = 3C6EF372FE94F82B g = 1F83D9ABFB41BD6B
d = A54FF53A5F1D36F1 h = 5BE0CD19137E2179
Figure 11.9 Message Digest Generation Using SHA-512
N 1024 bits
M1
H1
M2 MN
F
IV = H0
Message
hash code
1024 bits
512 bits 512 bits 512 bits
1024 bits 1024 bits
L bits
L
128 bits
1000000, . . . ,0
+
H2
F
+
HN
F
+
+ = word-by-word addition mod 264
358 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
These values are stored in big-endian format, which is the most significant byte of a word in the low-address (leftmost) byte position. These words
were obtained by taking the first sixty-four bits of the fractional parts of the
square roots of the first eight prime numbers.
Step 4 Process message in 1024-bit (128-byte) blocks. The heart of the algorithm is a module that consists of 80 rounds; this module is labeled F in Figure 11.9.
The logic is illustrated in Figure 11.10.
Each round takes as input the 512-bit buffer value, abcdefgh, and
updates the contents of the buffer. At input to the first round, the buffer
has the value of the intermediate hash value, Hi - 1. Each round t makes use of a 64-bit value Wt, derived from the current 1024-bit block being pro- cessed (Mi). These values are derived using a message schedule described subsequently. Each round also makes use of an additive constant Kt, where 0 … t … 79 indicates one of the 80 rounds. These words represent the first 64 bits of the fractional parts of the cube roots of the first 80 prime numbers.
The constants provide a “randomized” set of 64-bit patterns, which should
eliminate any regularities in the input data. Table 11.4 shows these constants
in hexadecimal format (from left to right).
Figure 11.10 SHA-512 Processing of a Single 1024-Bit Block
64
Mi
Wt
Hi
Hi–1
W0
W79
Kt
K0
K79
a b c
Round 0
d e f g h
a b c
Round t
d e f g h
Message schedule
a b c
Round 79
d e f g h
+ + + + + + + +
11.5 / SECURE HASH ALGORITHM (SHA) 359
The output of the eightieth round is added to the input to the first
round (Hi - 1) to produce Hi. The addition is done independently for each of the eight words in the buffer with each of the corresponding words in Hi - 1, using addition modulo 264.
Step 5 Output. After all N 1024-bit blocks have been processed, the output from the Nth stage is the 512-bit message digest.
We can summarize the behavior of SHA-512 as follows:
H0 = IV Hi = SUM64(Hi - 1, abcdefghi)
MD = HN
where
IV = initial value of the abcdefgh buffer, defined in step 3 abcdefghi = the output of the last round of processing of the ith message block N = the number of blocks in the message (including padding and
length fields)
SUM64 = addition modulo 2 64 performed separately on each word of the
pair of inputs
MD = final message digest value
428a2f98d728ae22 7137449123ef65cd b5c0fbcfec4d3b2f e9b5dba58189dbbc
3956c25bf348b538 59f111f1b605d019 923f82a4af194f9b ab1c5ed5da6d8118
d807aa98a3030242 12835b0145706fbe 243185be4ee4b28c 550c7dc3d5ffb4e2
72be5d74f27b896f 80deb1fe3b1696b1 9bdc06a725c71235 c19bf174cf692694
e49b69c19ef14ad2 efbe4786384f25e3 0fc19dc68b8cd5b5 240ca1cc77ac9c65
2de92c6f592b0275 4a7484aa6ea6e483 5cb0a9dcbd41fbd4 76f988da831153b5
983e5152ee66dfab a831c66d2db43210 b00327c898fb213f bf597fc7beef0ee4
c6e00bf33da88fc2 d5a79147930aa725 06ca6351e003826f 142929670a0e6e70
27b70a8546d22ffc 2e1b21385c26c926 4d2c6dfc5ac42aed 53380d139d95b3df
650a73548baf63de 766a0abb3c77b2a8 81c2c92e47edaee6 92722c851482353b
a2bfe8a14cf10364 a81a664bbc423001 c24b8b70d0f89791 c76c51a30654be30
d192e819d6ef5218 d69906245565a910 f40e35855771202a 106aa07032bbd1b8
19a4c116b8d2d0c8 1e376c085141ab53 2748774cdf8eeb99 34b0bcb5e19b48a8
391c0cb3c5c95a63 4ed8aa4ae3418acb 5b9cca4f7763e373 682e6ff3d6b2b8a3
748f82ee5defb2fc 78a5636f43172f60 84c87814a1f0ab72 8cc702081a6439ec
90befffa23631e28 a4506cebde82bde9 bef9a3f7b2c67915 c67178f2e372532b
ca273eceea26619c d186b8c721c0c207 eada7dd6cde0eb1e f57d4f7fee6ed178
06f067aa72176fba 0a637dc5a2c898a6 113f9804bef90dae 1b710b35131c471b
28db77f523047d84 32caab7b40c72493 3c9ebe0a15c9bebc 431d67c49c100d4c
4cc5d4becb3e42b6 597f299cfc657e2a 5fcb6fab3ad6faec 6c44198c4a475817
Table 11.4 SHA-512 Constants
360 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
SHA-512 Round Function
Let us look in more detail at the logic in each of the 80 steps of the processing
of one 512-bit block (Figure 11.11). Each round is defined by the following set of
equations:
T1 = h + Ch(e, f, g) + ( a 512 1 e) + Wt + Kt
T2 = ( a 512 0 a) + Maj(a, b, c)
h = g g = f f = e e = d + T1 d = c c = b b = a a = T1 + T2
where
t = step number; 0 … t … 79 Ch(e, f, g) = (e AND f) ⊕ (NOT e AND g)
the conditional function: If e then f else g
Maj(a, b, c) = (a AND b) ⊕ (a AND c) ⊕ (b AND c) the function is true only of the majority (two or three) of the
arguments are true
(Σ5120 a) = ROTR 28(a) ⊕ ROTR34(a) ⊕ ROTR39(a)
(Σ5121 e) = ROTR 14(e) ⊕ ROTR18(e) ⊕ ROTR41(e)
ROTRn(x) = circular right shift (rotation) of the 64-bit argument x by n bits
Figure 11.11 Elementary SHA-512 Operation (single round)
a b c d e f g h
a b c d 512 bits
e f g h
Ch
Kt
Wt
Maj
+
+ +
+
+
+
+
11.5 / SECURE HASH ALGORITHM (SHA) 361
Wt = a 64-bit word derived from the current 1024-bit input block Kt = a 64-bit additive constant + = addition modulo 264
Two observations can be made about the round function.
1. Six of the eight words of the output of the round function involve simply per- mutation (b, c, d, f, g, h) by means of rotation. This is indicated by shading in Figure 11.11.
2. Only two of the output words (a, e) are generated by substitution. Word e is a function of input variables (d, e, f, g, h), as well as the round word Wt and the constant Kt. Word a is a function of all of the input variables except d, as well as the round word Wt and the constant Kt.
It remains to indicate how the 64-bit word values Wt are derived from the 1024-bit message. Figure 11.12 illustrates the mapping. The first 16 values of Wt are taken directly from the 16 words of the current block. The remaining values are
defined as
Wt = s1 512(Wt - 2) + Wt - 7 + s0512(Wt - 15) + Wt - 16
where
s0 512(x) = ROTR1(x) ⊕ ROTR8(x) ⊕ SHR7(x)
s1 512(x) = ROTR19(x) ⊕ ROTR61(x) ⊕ SHR6(x)
ROTRn(x) = circular right shift (rotation) of the 64-bit argument x by n bits SHRn(x) = right shift of the 64-bit argument x by n bits with padding by zeros on
the left
+ = addition modulo 264
Thus, in the first 16 steps of processing, the value of Wt is equal to the cor- responding word in the message block. For the remaining 64 steps, the value of
Wt consists of the circular left shift by one bit of the XOR of four of the preced- ing values of Wt, with two of those values subjected to shift and rotate operations.
Figure 11.12 Creation of 80-word Input Sequence for SHA-512 Processing of Single Block
1024 bits
64 bits
Wt–16W0 W1 W9 W14 W63 W64 W72 W77Wt–15 Wt–7 Wt–2
W0 W1 W15 W16 Wt
Mi
W79
+
s0 s1 s0 s1 s0 s1
+ +
Hiva-Network.Com
362 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
This introduces a great deal of redundancy and interdependence into the message
blocks that are compressed, which complicates the task of finding a different
message block that maps to the same compression function output. Figure 11.13
summarizes the SHA-512 logic.
The SHA-512 algorithm has the property that every bit of the hash code is a
function of every bit of the input. The complex repetition of the basic function F
produces results that are well mixed; that is, it is unlikely that two messages cho-
sen at random, even if they exhibit similar regularities, will have the same hash
code. Unless there is some hidden weakness in SHA-512, which has not so far been
published, the difficulty of coming up with two messages having the same message
digest is on the order of 2256 operations, while the difficulty of finding a message
with a given digest is on the order of 2512 operations.
Example
We include here an example based on one in FIPS 180. We wish to hash a one-block
message consisting of three ASCII characters: “abc,” which is equivalent to the
following 24-bit binary string:
01100001 01100010 01100011
Recall from step 1 of the SHA algorithm, that the message is padded to a
length congruent to 896 modulo 1024. In this case of a single block, the padding
consists of 896 - 24 = 872 bits, consisting of a “1” bit followed by 871 “0” bits. Then a 128-bit length value is appended to the message, which contains the length
of the original message in bits (before the padding). The original length is 24 bits,
or a hexadecimal value of 18. Putting this all together, the 1024-bit message block,
in hexadecimal, is
6162638000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000018
This block is assigned to the words W0, . . . , W15 of the message schedule,
which appears as follows.
W0 = 6162638000000000 W8 = 0000000000000000 W1 = 0000000000000000 W9 = 0000000000000000 W2 = 0000000000000000 W10 = 0000000000000000 W3 = 0000000000000000 W11 = 0000000000000000 W4 = 0000000000000000 W12 = 0000000000000000 W5 = 0000000000000000 W13 = 0000000000000000 W6 = 0000000000000000 W14 = 0000000000000000 W7 = 0000000000000000 W15 = 0000000000000018
11.5 / SECURE HASH ALGORITHM (SHA) 363
The padded message consists blocks M1, M2, c , MN. Each message block Mi consists of 16 64-bit words Mi,0, Mi,1, c , Mi,15. All addition is performed modulo 264.
H0,0 = 6A09E667F3BCC908 H0,4 = 510E527FADE682D1 H0,1 = BB67AE8584CAA73B H0,5 = 9B05688C2B3E6C1F H0,2 = 3C6EF372FE94F82B H0,6 = 1F83D9ABFB41BD6B H0,3 = A54FF53A5F1D36F1 H0,7 = 5BE0CD19137E2179
for i = 1 to N 1. Prepare the message schedule W
for t = 0 to 15 Wt = Mi,t
for t = 16 to 79 Wt = s1
512(Wt - 2) + Wt - 7 + s0512(Wt - 15) + Wt - 16 2. Initialize the working variables
a = Hi - 1, 0 e = Hi - 1, 4 b = Hi - 1, 1 f = Hi - 1, 5 c = Hi - 1, 2 g = Hi - 1, 6 d = Hi - 1, 3 h = Hi - 1, 7
3. Perform the main hash computation for t = 0 to 79
T1 = h + Ch(e, f, g) + ¢Σ5121 e≤ + Wt + Kt T2 = ¢Σ5120 a≤ + Maj(a, b, c) h = g g = f f = e e = d + T1 d = c c = b b = a a = T1 + T2
4. Compute the intermediate hash value
Hi, 0 = a + Hi - 1, 0 Hi, 4 = e + Hi - 1,4 Hi, 1 = b + Hi - 1, 1 Hi, 5 = f + Hi - 1, 5 Hi, 2 = c + Hi - 1, 2 Hi, 6 = g + Hi - 1, 6 Hi, 3 = d + Hi - 1, 3 Hi, 7 = h + Hi - 1, 7
return {HN, 0 }HN, 1 }HN, 2 }HN, 3 }HN, 4 }HN, 5 }HN, 6 }HN, 7}
Figure 11.13 SHA-512 Logic
364 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
As indicated in Figure 11.13, the eight 64-bit variables, a through h, are initialized to values H0,0 through H0,7. The following table shows the initial values of these variables and their values after each of the first two rounds.
a 6a09e667f3bcc908 f6afceb8bcfcddf5 1320f8c9fb872cc0
b bb67ae8584caa73b 6a09e667f3bcc908 f6afceb8bcfcddf5
c 3c6ef372fe94f82b bb67ae8584caa73b 6a09e667f3bcc908
d a54ff53a5f1d36f1 3c6ef372fe94f82b bb67ae8584caa73b
e 510e527fade682d1 58cb02347ab51f91 c3d4ebfd48650ffa
f 9b05688c2b3e6c1f 510e527fade682d1 58cb02347ab51f91
g 1f83d9abfb41bd6b 9b05688c2b3e6c1f 510e527fade682d1
h 5be0cd19137e2179 1f83d9abfb41bd6b 9b05688c2b3e6c1f
Note that in each of the rounds, six of the variables are copied directly from
variables from the preceding round.
The process continues through 80 rounds. The output of the final round is
73a54f399fa4b1b2 10d9c4c4295599f6 d67806db8b148677 654ef9abec389ca9 d08446aa79693ed7 9bb4d39778c07f9e 25c96a7768fb2aa3 ceb9fc3691ce8326
The hash value is then calculated as
H1,0 = 6a09e667f3bcc908 + 73a54f399fa4b1b2 = ddaf35a193617aba H1,1 = bb67ae8584caa73b + 10d9c4c4295599f6 = cc417349ae204131 H1,2 = 3c6ef372fe94f82b + d67806db8b148677 = 12e6fa4e89a97ea2 H1,3 = a54ff53a5f1d36f1 + 654ef9abec389ca9 = 0a9eeee64b55d39a H1,4 = 510e527fade682d1 + d08446aa79693ed7 = 2192992a274fc1a8 H1,5 = 9b05688c2b3e6c1f + 9bb4d39778c07f9e = 36ba3c23a3feebbd H1,6 = 1f83d9abfb41bd6b + 25c96a7768fb2aa3 = 454d4423643ce80e H1,7 = 5be0cd19137e2179 + ceb9fc3691ce8326 = 2a9ac94fa54ca49f
The resulting 512-bit message digest is
ddaf35a193617aba cc417349ae204131 12e6fa4e89a97ea2 0a9eeee64b55d39a 2192992a274fc1a8 36ba3c23a3feebbd 454d4423643ce80e 2a9ac94fa54ca49f
Suppose now that we change the input message by one bit, from “abc” to
“cbc.” Then, the 1024-bit message block is
6362638000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000018
And the resulting 512-bit message digest is
531668966ee79b70 0b8e593261101354 4273f7ef7b31f279 2a7ef68d53f93264 319c165ad96d9187 55e6a204c2607e27 6e05cdf993a64c85 ef9e1e125c0f925f
The number of bit positions that differ between the two hash values is 253,
almost exactly half the bit positions, indicating that SHA-512 has a good avalanche
effect.
11.6 / SHA-3 365
11.6 SHA-3
As of this writing, the Secure Hash Algorithm (SHA-1) has not yet been “broken.”
That is, no one has demonstrated a technique for producing collisions in a practical
amount of time. However, because SHA-1 is very similar, in structure and in the
basic mathematical operations used, to MD5 and SHA-0, both of which have been
broken, SHA-1 is considered insecure and has been phased out for SHA-2.
SHA-2, particularly the 512-bit version, would appear to provide unassailable
security. However, SHA-2 shares the same structure and mathematical operations
as its predecessors, and this is a cause for concern. Because it will take years to find
a suitable replacement for SHA-2, should it become vulnerable, NIST decided to
begin the process of developing a new hash standard.
Accordingly, NIST announced in 2007 a competition to produce the next gen-
eration NIST hash function, to be called SHA-3. The winning design for SHA-3
was announced by NIST in October 2012 and published as FIP 102 in August 2015.
SHA-3 is a cryptographic hash function that is intended to complement SHA-2 as
the approved standard for a wide range of applications.
Appendix V looks at the evaluation criteria used by NIST to select from
among the candidates for AES, plus the rationale for picking Keccak, which was
the winning candidate. This material is useful in understanding not just the SHA-3
design but also the criteria by which to judge any cryptographic hash algorithm.
The Sponge Construction
The underlying structure of SHA-3 is a scheme referred to by its designers as a
sponge construction [BERT07, BERT11]. The sponge construction has the same general structure as other iterated hash functions (Figure 11.8). The sponge func-
tion takes an input message and partitions it into fixed-size blocks. Each block is
processed in turn with the output of each iteration fed into the next iteration, finally
producing an output block.
The sponge function is defined by three parameters:
f = the internal function used to process each input block3
r = the size in bits of the input blocks, called the bitrate pad = the padding algorithm
A sponge function allows both variable length input and output, making it a
flexible structure that can be used for a hash function (fixed-length output), a pseu-
dorandom number generator (fixed-length input), and other cryptographic func-
tions. Figure 11.14 illustrates this point. An input message of n bits is partitioned into k fixed-size blocks of r bits each. The message is padded to achieve a length that is an integer multiple of r bits. The resulting partition is the sequence of blocks P0, P1, c , Pk - 1, with length k * r. For uniformity, padding is always added, so
3The Keccak documentation refers to f as a permutation. As we shall see, it involves both permutations and substitutions. We refer to f as the iteration function, because it is the function that is executed once for each iteration, that is, once for each block of the message that is processed.
366 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
that if n mod r = 0, a padding block of r bits is added. The actual padding algorithm is a parameter of the function. The sponge specification [BERT11] proposes two
padding schemes:
■ Simple padding: Denoted by pad10*, appends a single bit 1 followed by the minimum number of bits 0 such that the length of the result is a multiple of the
block length.
■ Multirate padding: Denoted by pad10*1, appends a single bit 1 followed by the minimum number of bits 0 followed by a single bit 1 such that the length
of the result is a multiple of the block length. This is the simplest padding
scheme that allows secure use of the same f with different rates r. FIPS 202 uses multirate padding.
After processing all of the blocks, the sponge function generates a sequence
of output blocks Z0, Z1, c , Zj - 1. The number of output blocks generated is determined by the number of output bits desired. If the desired output is / bits, then j blocks are produced, such that (j - 1) * r 6 / … j * r.
Figure 11.15 shows the iterated structure of the sponge function. The sponge
construction operates on a state variable s of b = r + c bits, which is initialized to all zeros and modified at each iteration. The value r is called the bitrate. This value is the block size used to partition the input message. The term bitrate re- flects the fact that r is the number of bits processed at each iteration: the larger the value of r, the greater the rate at which message bits are processed by the sponge
Figure 11.14 Sponge Function Input and Output
k r bits
(a) Input
(b) Output
P0 P1
Z0 Z1
Zj–1
Pk–1
message pad
r bits r bits r bits
r bits r bits r bits
l bits
n bits
11.6 / SHA-3 367
construction. The value c is referred to as the capacity. A discussion of the secu- rity implications of the capacity is beyond our scope. In essence, the capacity is a
measure of the achievable complexity of the sponge construction and therefore the
achievable level of security. A given implementation can trade claimed security for
speed by increasing the capacity c and decreasing the bitrate r accordingly, or vice versa. The default values for Keccak are c = 1024 bits, r = 576 bits, and therefore b = 1600 bits.
The sponge construction consists of two phases. The absorbing phase proceeds as follows: For each iteration, the input block to be processed is padded with zeroes
to extend its length from r bits to b bits. Then, the bitwise XOR of the extended
Figure 11.15 Sponge Construction
(a) Absorbing phase
(b) Squeezing phase
f
r c
0c
0c
0c
0r 0c
P0
P1
P2
f
s
f
s
f
s
0cPk–1
b r c
b
r c
Z0
r
Z1
368 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
message block and s is formed to create a b-bit input to the iteration function f. The output of f is the value of s for the next iteration.
If the desired output length / satisfies / … b, then at the completion of the absorbing phase, the first / bits of s are returned and the sponge construction termi- nates. Otherwise, the sponge construction enters the squeezing phase. To begin, the first / bits of s are retained as block Z0. Then, the value of s is updated with repeated executions of f, and at each iteration, the first / bits of s are retained as block Zi and concatenated with previously generated blocks. The process continues through
(j - 1) iterations until we have (j - 1) * r 6 / … j * r. At this point the first / bits of the concatenated block Z are returned.
Note that the absorbing phase has the structure of a typical hash function.
A common case will be one in which the desired hash length is less than or equal
to the input block length; that is, / … r. In that case, the sponge construction termi- nates after the absorbing phase. If a longer output than b bits is required, then the squeezing phase is employed. Thus the sponge construction is quite flexible. For
example, a short message with a length r could be used as a seed and the sponge construction would function as a pseudorandom number generator.
To summarize, the sponge construction is a simple iterated construction for
building a function F with variable-length input and arbitrary output length based on a fixed-length transformation or permutation f operating on a fixed number b of bits. The sponge construction is defined formally in [BERT11] as follows:
Algorithm The sponge construction SPONGE[f, pad, r] Require: r < b
Interface: Z = sponge(M,/) with M ∈ Z2*, integer / > 0 and Z ∈ Z2 /
P = M }pad[r](|M|) s = 0b
for i = 0 to |P|r − 1 do s = s ⊕ (Pi}0b − r) s = f(s) end for Z =:s;r while |Z|r r < / do s = f (s) Z = Z} :s;r end while return :Z;ℓ
In the algorithm definition, the following notation is used: �M� is the length in bits of a bit string M. A bit string M can be considered as a sequence of blocks of some fixed length x, where the last block may be shorter. The number of blocks of M is denoted by �M� x. The blocks of M are denoted by Mi and the index ranges from 0 to �M� x - 1. The expression :M;/ denotes the truncation of M to its first / bits.
11.6 / SHA-3 369
Message Digest Size 224 256 384 512
Message Size no maximum no maximum no maximum no maximum
Block Size (bitrate r) 1152 1088 832 576
Word Size 64 64 64 64
Number of Rounds 24 24 24 24
Capacity c 448 512 768 1024
Collision Resistance 2112 2128 2192 2256
Second Preimage Resistance 2224 2256 2384 2512
Note: All sizes and security levels—are measured in bits.
Table 11.5 SHA-3 Parameters
SHA-3 makes use of the iteration function f, labeled Keccak-f, which is described in the next section. The overall SHA-3 function is a sponge function
expressed as Keccak[r, c] to reflect that SHA-3 has two operational parameters, r, the message block size, and c, the capacity, with the default of r + c = 1600 bits. Table 11.5 shows the supported values of r and c. As Table 11.5 shows, the hash function security associated with the sponge construction is a function of the
capacity c. In terms of the sponge algorithm defined above, Keccak[r, c] is defined as
Keccak [r, c]∆ SPONGE [Keccak@f [r + c], pad 10*1, r]
We now turn to a discussion of the iteration function Keccak-f.
The SHA-3 Iteration Function f
We now examine the iteration function Keccak-f used to process each successive block of the input message. Recall that f takes as input a 1600-bit variable s consist- ing of r bits, corresponding to the message block size followed by c bits, referred to as the capacity. For internal processing within f, the input state variable s is orga- nized as a 5 * 5 * 64 array a. The 64-bit units are referred to as lanes. For our purposes, we generally use the notation a[x, y, z] to refer to an individual bit with the state array. When we are more concerned with operations that affect entire
lanes, we designate the 5 * 5 matrix as L[x, y], where each entry in L is a 64-bit lane. The use of indices within this matrix is shown in Figure 11.16.4 Thus, the col-
umns are labeled x = 0 through x = 4, the rows are labeled y = 0 through y = 4, and the individual bits within a lane are labeled z = 0 through z = 63. The mapping between the bits of s and those of a is
s[64(5y + x) + z] = a[x, y, z]
4Note that the first index (x) designates a column and the second index (y) designates a row. This is in conflict with the convention used in most mathematics sources, where the first index designates a row and the second index designates a column (e.g., Knuth, D. The Art of Computing Programming, Volume 1, Fundamental Algorithms; and Korn, G., and Korn, T. Mathematical Handbook for Scientists and Engineers).
370 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
We can visualize this with respect to the matrix in Figure 11.16. When treat-
ing the state as a matrix of lanes, the first lane in the lower left corner, L[0, 0], cor- responds to the first 64 bits of s. The lane in the second column, lowest row, L[1, 0], corresponds to the next 64 bits of s. Thus, the array a is filled with the bits of s starting with row y = 0 and proceeding row by row.
STRUCTURE OF f The function f is executed once for each input block of the message to be hashed. The function takes as input the 1600-bit state variable and converts
it into a 5 * 5 matrix of 64-bit lanes. This matrix then passes through 24 rounds of processing. Each round consists of five steps, and each step updates the state matrix
by permutation or substitution operations. As shown in Figure 11.17, the rounds are
identical with the exception of the final step in each round, which is modified by a
round constant that differs for each round.
The application of the five steps can be expressed as the composition5 of
functions:
R = i o x o p o r o u
Table 11.6 summarizes the operation of the five steps. The steps have a sim-
ple description leading to a specification that is compact and in which no trapdoor
can be hidden. The operations on lanes in the specification are limited to bitwise
Boolean operations (XOR, AND, NOT) and rotations. There is no need for table
lookups, arithmetic operations, or data-dependent rotations. Thus, SHA-3 is easily
and efficiently implemented in either hardware or software.
We examine each of the step functions in turn.
Figure 11.16 SHA-3 State Matrix
L[0, 4]
x = 0 x = 1 x = 2 x = 3 x = 4
L[0, 3]
L[0, 2]
L[0, 1]
L[0, 0]
a[x, y, 0] a[x, y, 1] a[x, y, 2]
y = 1
y = 0
y = 2
y = 3
y = 4 L[1, 4]
L[1, 3]
L[1, 2]
L[1, 1]
L[1, 0]
L[2, 4]
L[2, 3]
L[2, 2]
L[2, 1]
L[2, 0]
(a) State variable as 5 5 matrix A of 64-bit words
(b) Bit labeling of 64-bit words
L[3, 4]
L[3, 3]
L[3, 2]
L[4, 1]
L[3, 0]
L[4, 4]
L[4, 3]
L[4, 2]
L[4, 1]
L[4, 0]
a[x, y, 63]a[x, y, 62]a[x, y, z]
5If f and g are two functions, then the function F with the equation y = F(x) = g[f(x)] is called the composition of f and g and is denoted as F = g o f.
Hiva-Network.Com
11.6 / SHA-3 371
Figure 11.17 SHA-3 Iteration Function f
theta (u) step
s
s
rho (r) step
pi (p) step
chi (x) step
R ou
nd 0
iota (i) step RC[0]
rot(x, y)
theta (u) step
rho (r) step
pi (p) step
chi (x) step
R ou
nd 2
3
iota (i) step RC[23]
rot(x, y)
Function Type Description
u Substitution New value of each bit in each word depends on its current
value and on one bit in each word of preceding column
and one bit of each word in succeeding column.
r Permutation The bits of each word are permuted using a circular bit
shift. W[0, 0] is not affected.
p Permutation Words are permuted in the 5 * 5 matrix. W[0, 0] is not affected.
x Substitution New value of each bit in each word depends on its current
value and on one bit in next word in the same row and one
bit in the second next word in the same row.
i Substitution W[0, 0] is updated by XOR with a round constant.
Table 11.6 Step Functions in SHA-3
THETA STEP FUNCTION The Keccak reference defines the u function as follows. For bit z in column x, row y,
u: a[x, y, z] d a[x, y, z] ⊕ a 4
y== 0 a[(x - 1), y=, z] ⊕ a
4
y== 0 a[(x + 1), y=, (z - 1)] (11.1)
372 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
where the summations are XOR operations. We can see more clearly what this
operation accomplishes with reference to Figure 11.18a. First, define the bitwise
XOR of the lanes in column x as
C[x] = L[x, 0] ⊕ L[x, 1] ⊕ L[x, 2] ⊕ L[x, 3] ⊕ L[x, 4]
Consider lane L[x, y] in column x, row y. The first summation in Equation 11.1 performs a bitwise XOR of the lanes in column (x - 1) mod 4 to form the 64-bit lane C[x - 1]. The second summation performs a bitwise XOR of the lanes in column (x + 1) mod 4, and then rotates the bits within the 64-bit lane so that the bit in position z is mapped into position z + 1 mod 64. This forms the lane ROT (C[x + 1], 1). These two lanes and L[x, y] are combined by bitwise XOR to form the updated value of L[x, y]. This can be expressed as
L[x, y] d L[x, y] ⊕ C[x - 1] ⊕ ROT(C[x + 1], 1)
Figure 11.18.a illustrates the operation on L[3, 2]. The same operation is performed on all of the other lanes in the matrix.
Figure 11.18 Theta and Chi Step Functions
(a) u step function
L[2, 3]L[2, 3] ROT(C[3], 1)C[1]
L[0, 4]
x = 0 x = 1 x = 2 x = 3 x = 4
L[0, 3]
L[0, 2]
L[0, 1]
L[0, 0]
y = 1
y = 0
y = 2
y = 3
y = 4 L[1, 4]
L[1, 3]
L[1, 2]
L[1, 1]
L[1, 0]
L[2, 4]
L[2, 3]
L[2, 2]
L[2, 1]
L[2, 0]
L[3, 4]
L[3, 3]
L[3, 2]
L[4, 1]
L[3, 0]
L[4, 4]
L[4, 3]
L[4, 2]
L[4, 1]
L[4, 0]
(b) x step function
L[2, 3]L[2, 3] L[3, 3] AND L[4, 3]
L[0, 4]
x = 0 x = 1 x = 2 x = 3 x = 4
L[0, 3]
L[0, 2]
L[0, 1]
L[0, 0]
y = 1
y = 0
y = 2
y = 3
y = 4 L[1, 4]
L[1, 3]
L[1, 2]
L[1, 1]
L[1, 0]
L[2, 4]
L[2, 3]
L[2, 2]
L[2, 1]
L[2, 0]
L[3, 4]
L[3, 3]
L[3, 2]
L[4, 1]
L[3, 0]
L[4, 4]
L[4, 3]
L[4, 2]
L[4, 1]
L[4, 0]
11.6 / SHA-3 373
Several observations are in order. Each bit in a lane is updated using the bit itself
and one bit in the same bit position from each lane in the preceding column and one
bit in the adjacent bit position from each lane in the succeeding column. Thus the up-
dated value of each bit depends on 11 bits. This provides good mixing. Also, the theta
step provides good diffusion, as that term was defined in Chapter 4. The designers of
Keccak state that the theta step provides a high level of diffusion on average and that
without theta, the round function would not provide diffusion of any significance.
RHO STEP FUNCTION The r function is defined as follows:
r: a[x, y, z] d a[x, y, z] if x = y = 0
otherwise,
r: a[x, y, z] d aJx, y, az - (t + 1)(t + 2) 2
b R (11.2) with t satisfying 0 … t 6 24 and ¢0 1
2 3 ≤t¢1
0 ≤ = ¢x
y ≤ in GF(5)2 * 2
It is not immediately obvious what this step performs, so let us look at the
process in detail.
1. The lane in position (x, y) = (0, 0), that is L[0, 0], is unaffected. For all other words, a circular bit shift within the lane is performed.
2. The variable t, with 0 … t 6 24, is used to determine both the amount of the circular bit shift and which lane is assigned which shift value.
3. The 24 individual bit shifts that are performed have the respective values
(t + 1)(t + 2) 2
mod 64.
4. The shift determined by the value of t is performed on the lane in position (x, y) in the 5 * 5 matrix of lanes. Specifically, for each value of t, the corre-
sponding matrix position is defined by ¢x y ≤ = ¢0 1
2 3 ≤t¢1
0 ≤. For example, for
t = 3, we have
¢x y ≤ = ¢0 1
2 3 ≤3 ¢1
0 ≤ mod 5
= ¢0 1 2 3
≤ ¢0 1 2 3
≤ ¢0 1 2 3
≤ ¢1 0 ≤ mod 5
= ¢0 1 2 3
≤ ¢0 1 2 3
≤ ¢0 2 ≤ mod 5
= ¢0 1 2 3
≤ ¢2 6 ≤ mod 5 = ¢0 1
2 3 ≤ ¢2
1 ≤ mod 5
= ¢1 7 ≤ mod 5 = ¢1
2 ≤
374 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
Table 11.7 shows the calculations that are performed to determine the amount
of the bit shift and the location of each bit shift value. Note that all of the rotation
amounts are different.
The r function thus consists of a simple permutation (circular shift) within
each lane. The intent is to provide diffusion within each lane. Without this function,
diffusion between lanes would be very slow.
PI STEP FUNCTION The p function is defined as follows:
p: a[x, y] d a[x=, y=], with¢x y ≤ = ¢0 1
2 3 ≤ ¢x=
y= ≤ (11.3)
This can be rewritten as (x, y) * (y, (2x + 3y)). Thus, the lanes within the 5 * 5 matrix are moved so that the new x position equals the old y position and the
Table 11.7 Rotation Values Used in SHA-3
t g(t) g (t) mod 64 x, y
0 1 1 1, 0
1 3 3 0, 2
2 6 6 2, 1
3 10 10 1, 2
4 15 15 2, 3
5 21 21 3, 3
6 28 28 3, 0
7 36 36 0, 1
8 45 45 1, 3
9 55 55 3, 1
10 66 2 1, 4
11 78 14 4, 4
(b) Rotation values by word position in matrix
x = 0 x = 1 x = 2 x = 3 x = 4
y = 4 18 2 61 56 14
y = 3 41 45 15 21 8
y = 2 3 10 43 25 39
y = 1 36 44 6 55 20
y = 0 0 1 62 28 27
t g(t) g (t) mod 64 x, y
12 91 27 4, 0
13 105 41 0, 3
14 120 56 3, 4
15 136 8 4, 3
16 153 25 3, 2
17 171 43 2, 2
18 190 62 2, 0
19 210 18 0, 4
20 231 39 4, 2
21 253 61 2, 4
22 276 20 4, 1
23 300 44 1, 1
(a) Calculation of values and positions
Note: g(t) = (t + 1)(t + 2)/2
¢x y ≤ = ¢0 1
2 3 ≤t¢1
0 ≤ mod 5
11.6 / SHA-3 375
Figure 11.19 Pi Step Function
Z[0, 4]
x = 0 x = 1 x = 2
(a) Lane position at start of step
(b) Lane position after permutation
x = 3 x = 4
Z[0, 3]
Z[0, 2]
Z[0, 1]
Z[0, 0]
y = 1
y = 0
y = 2
y = 3
y = 4 Z[1, 4]
Z[1, 3]
Z[1, 2]
Z[1, 1]
Z[1, 0]
Z[2, 4]
Z[2, 3]
Z[2, 2]
Z[2, 1]
Z[2, 0]
Z[3, 4]
Z[3, 3]
Z[3, 2]
Z[3, 1]
Z[3, 0]
Z[4, 4]
row 0row
3 row
1 row
4 row
2
row 2
row 4
row 1
row 3
Z[4, 3]
Z[4, 2]
Z[4, 1]
Z[4, 0]
Z[2, 0]
x = 0 x = 1 x = 2 x = 3 x = 4
Z[4, 0]
Z[1, 0]
Z[3, 0]
Z[0, 0]
y = 1
y = 0
y = 2
y = 3
y = 4 Z[3, 1]
Z[0, 1]
Z[2, 1]
Z[4, 1]
Z[1, 1]
Z[4, 2]
Z[1, 2]
Z[3, 2]
Z[0, 2]
Z[2, 2]
Z[0, 3]
Z[2, 3]
Z[4, 3]
Z[1, 3]
Z[3, 3]
Z[1, 4]
Z[3, 4]
Z[0, 4]
Z[2, 4]
Z[4, 4]
new y position is determined by (2x + 3y) mod 5. Figure 11.19 helps in visualizing this permutation. Lanes that are along the same diagonal (increasing in y value, going from left to right) prior to p are arranged on the same row in the matrix after
p is executed. Note that the position of L[0, 0] is unchanged. Thus the p step is a permutation of lanes: The lanes move position within the
5 * 5 matrix. The r step is a permutation of bits: Bits within a lane are rotated. Note that the p step matrix positions are calculated in the same way that, for the r
step, the one-dimensional sequence of rotation constants is mapped to the lanes of
the matrix.
CHI STEP FUNCTION The x function is defined as follows:
x: a[x] d a[x] ⊕ ((a[x + 1] ⊕ 1) AND a[x + 2]) (11.4)
This function operates to update each bit based on its current value and the
value of the corresponding bit position in the next two lanes in the same row. The
376 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
Round Constant
(hexadecimal) Number of 1 bits
0 0000000000000001 1
1 0000000000008082 3
2 800000000000808A 5
3 8000000080008000 3
4 000000000000808B 5
5 0000000080000001 2
6 8000000080008081 5
7 8000000000008009 4
8 000000000000008A 3
9 0000000000000088 2
10 0000000080008009 4
11 000000008000000A 3
Table 11.8 Round Constants in SHA-3
Round Constant
(hexadecimal) Number of 1 bits
12 000000008000808B 6
13 800000000000008B 5
14 8000000000008089 5
15 8000000000008003 4
16 8000000000008002 3
17 8000000000000080 2
18 000000000000800A 3
19 800000008000000A 4
20 8000000080008081 5
21 8000000000008080 3
22 0000000080000001 2
23 8000000080008008 4
operation is more clearly seen if we consider a single bit a[x, y, z] and write out the Boolean expression:
a[x, y, z] d a[x, y, z] ⊕ (NOT(a[x + 1, y, z])) AND (a[x + 2, y, z])
Figure 11.18b illustrates the operation of the x function on the bits of the
lane L[3, 2]. This is the only one of the step functions that is a nonlinear mapping. Without it, the SHA-3 round function would be linear.
IOTA STEP FUNCTION The i function is defined as follows:
i: a d a ⊕ RC[ir] (11.5)
This function combines an array element with a round constant that differs for
each round. It breaks up any symmetry induced by the other four step functions. In
fact, Equation 11.5 is somewhat misleading. The round constant is applied only to
the first lane of the internal state array. We express this is as follows:
L[0, 0] d L[0, 0] ⊕ RC[ir] 0 … ir … 24
Table 11.8 lists the 24 64-bit round constants. Note that the Hamming weight,
or number of 1 bits, in the round constants ranges from 1 to 6. Most of the bit posi-
tions are zero and thus do not change the corresponding bits in L[0, 0]. If we take the cumulative OR of all 24 round constants, we get
RC[0] OR RC[1] OR c OR RC[23] = 800000008000808B
Thus, only 7 bit positions are active and can affect the value of L[0, 0]. Of course, from round to round, the permutations and substitutions propagate the
effects of the i function to all of the lanes and all of the bit positions in the matrix.
It is easily seen that the disruption diffuses through u and x to all lanes of the state
after a single round.
11.7 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 377
11.7 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
absorbing phase
big endian
birthday attack
birthday paradox
bitrate
capacity
Chi step function collision
resistant
compression function
cryptographic hash function
hash code
hash function
hash value
Iota step function
Keccak
keyed hash function
lane
little endian
MD4
MD5
message authentication code
(MAC)
message digest
one-way hash function
Pi step function
preimage resistant
Rho step function
second preimage resistant
SHA-1
SHA-224
SHA-256
SHA-3
SHA-384
SHA-512
sponge construction
squeezing phase
strong collision resistance
Theta step function
weak collision resistance
Key Terms
Review Questions 11.1 What characteristics are needed in a secure hash function? 11.2 What is the difference between weak and strong collision resistance? 11.3 What is the role of a compression function in a hash function? 11.4 What is the difference between little-endian and big-endian format? 11.5 What basic arithmetical and logical functions are used in SHA? 11.6 Describe the set of criteria used by NIST to evaluate SHA-3 candidates. 11.7 Define the term sponge construction. 11.8 Briefly describe the internal structure of the iteration function f. 11.9 List and briefly describe the step functions that comprise the iteration function f.
Problems 11.1 The high-speed transport protocol XTP (Xpress Transfer Protocol) uses a 32-bit
checksum function defined as the concatenation of two 16-bit functions: XOR and RXOR, defined in Section 11.4 as “two simple hash functions” and illustrated in Figure 11.5. a. Will this checksum detect all errors caused by an odd number of error bits?
Explain. b. Will this checksum detect all errors caused by an even number of error bits? If not,
characterize the error patterns that will cause the checksum to fail. c. Comment on the effectiveness of this function for use as a hash function for
authentication.
11.2 a. Consider the Davies and Price hash code scheme described in Section 11.4 and assume that DES is used as the encryption algorithm:
Hi = Hi - 1 ⊕ E(Mi, Hi - 1)
378 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
Recall the complementarity property of DES (Problem 3.14): If Y = E(K, X), then Y′ = E(K′, X′). Use this property to show how a message consisting of blocks M1, M2, c , MN can be altered without altering its hash code.
b. Show that a similar attack will succeed against the scheme proposed in [MEYE88]:
Hi = Mi ⊕ E(Hi - 1, Mi)
11.3 a. Consider the following hash function. Messages are in the form of a sequence of
numbers in Zn, M = (a1, a2, c at). The hash value h is calculated as ¢ at i = 1
ai≤ for some predefined value n. Does this hash function satisfy any of the requirements for a hash function listed in Table 11.1? Explain your answer.
b. Repeat part (a) for the hash function h = ¢ at i = 1
(ai) 2≤ mod n.
c. Calculate the hash function of part (b) for M = (189, 632, 900, 722, 349) and n = 989.
11.4 It is possible to use a hash function to construct a block cipher with a structure similar to DES. Because a hash function is one way and a block cipher must be reversible (to decrypt), how is it possible?
11.5 Now consider the opposite problem: using an encryption algorithm to construct a one-way hash function. Consider using RSA with a known key. Then process a message consisting of a sequence of blocks as follows: Encrypt the first block, XOR the result with the second block and encrypt again, etc. Show that this scheme is not secure by solving the following problem. Given a two-block message B1, B2, and its hash
RSAH(B1,B2) = RSA(RSA(B1) ⊕ B2)
Given an arbitrary block C1, choose C2 so that RSAH(C1, C2) = RSAH(B1, B2). Thus, the hash function does not satisfy weak collision resistance.
11.6 Suppose H(m) is a collision-resistant hash function that maps a message of arbitrary bit length into an n-bit hash value. Is it true that, for all messages x, x′ with x ≠ x′, we have H(x) ≠ H(x′) Explain your answer.
11.7 In Figure 11.12, it is assumed that an array of 80 64-bit words is available to store the values of Wt, so that they can be precomputed at the beginning of the processing of a block. Now assume that space is at a premium. As an alternative, consider the use of a 16-word circular buffer that is initially loaded with W0 through W15. Design an algorithm that, for each step t, computes the required input value Wt.
11.8 For SHA-512, show the equations for the values of W16, W18, W23, and W31. 11.9 State the value of the padding field in SHA-512 if the length of the message is
a. 2942 bits b. 2943 bits c. 2944 bits
11.10 State the value of the length field in SHA-512 if the length of the message is a. 2942 bits b. 2943 bits c. 2944 bits
11.11 Suppose a1a2a3a4 are the 4 bytes in a 32-bit word. Each ai can be viewed as an integer in the range 0 to 255, represented in binary. In a big-endian architecture, this word represents the integer
a12 24 + a2216 + a328 + a4
11.7 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 379
In a little-endian architecture, this word represents the integer
a42 24 + a3216 + a228 + a1
a. Some hash functions, such as MD5, assume a little-endian architecture. It is impor- tant that the message digest be independent of the underlying architecture. There- fore, to perform the modulo 2 addition operation of MD5 or RIPEMD-160 on a big-endian architecture, an adjustment must be made. Suppose X = x1 x2 x3 x4 and Y = y1 y2 y3 y4. Show how the MD5 addition operation (X + Y) would be carried out on a big-endian machine.
b. SHA assumes a big-endian architecture. Show how the operation (X + Y) for SHA would be carried out on a little-endian machine.
11.12 This problem introduces a hash function similar in spirit to SHA that operates on letters instead of binary data. It is called the toy tetragraph hash (tth).6 Given a mes- sage consisting of a sequence of letters, tth produces a hash value consisting of four letters. First, tth divides the message into blocks of 16 letters, ignoring spaces, punc- tuation, and capitalization. If the message length is not divisible by 16, it is padded out with nulls. A four-number running total is maintained that starts out with the value (0, 0, 0, 0); this is input to the compression function for processing the first block. The compression function consists of two rounds.
Round 1 Get the next block of text and arrange it as a row-wise 4 * 4 block of text and convert it to numbers (A = 0, B = 1, etc.). For example, for the block ABCDEFGHIJKLMNOP, we have
A B C D
E F G H
I J K L
M N O P
0 1 2 3
4 5 6 7
8 9 10 11
12 13 14 15
Then, add each column mod 26 and add the result to the running total, mod 26. In this example, the running total is (24, 2, 6, 10).
Round 2 Using the matrix from round 1, rotate the first row left by 1, second row left by 2, third row left by 3, and reverse the order of the fourth row. In our example:
B C D A
G H E F
L I J K
P O N M
1 2 3 0
6 7 4 5
11 8 9 10
15 14 13 12
Now, add each column mod 26 and add the result to the running total. The new run- ning total is (5, 7, 9, 11). This running total is now the input into the first round of the compression function for the next block of text. After the final block is processed, convert the final running total to letters. For example, if the message is ABCDEF- GHIJKLMNOP, then the hash is FHJL.
6I thank William K. Mason, of the magazine staff of The Cryptogram, for providing this example.
Hiva-Network.Com
380 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS
a. Draw figures comparable to Figures 11.9 and 11.10 to depict the overall tth logic and the compression function logic.
b. Calculate the hash function for the 22-letter message “Practice makes us perfect.” c. To demonstrate the weakness of tth, find a message of length 32-letter to produces
the same hash.
11.13 For each of the possible capacity values of SHA-3 (Table 11.5), which lanes in the internal 55 state matrix start out as lanes of all zeros?
11.14 Consider the SHA-3 option with a block size of 1024 bits and assume that each of the lanes in the first message block (P0) has at least one nonzero bit. To start, all of the lanes in the internal state matrix that correspond to the capacity portion of the initial state are all zeros. Show how long it will take before all of these lanes have at least one nonzero bit. Note: Ignore the permutation. That is, keep track of the original zero lanes even after they have changed position in the matrix.
11.15 Consider the state matrix as illustrated in Figure 11.16a. Now rearrange the rows and columns of the matrix so that L[0, 0] is in the center. Specifically, arrange the columns in the left-to-right order (x = 3, x = 4, x = 0, x = 1, x = 2) and arrange the rows in the top-to-bottom order (y = 2, y = 1, y = 0, y = 4, y = 6). This should give you some insight into the permutation algorithm used for the function and for permut- ing the rotation constants in the function. Using this rearranged matrix, describe the permutation algorithm.
11.16 The function only affects L[0, 0]. Section 11.6 states that the changes to L[0, 0] diffuse through u and to all lanes of the state after a single round. a. Show that this is so. b. How long before all of the bit positions in the matrix are affected by the changes
to L[0, 0]?
381
Message Authentication Codes
12.1 Message Authentication Requirements
12.2 Message Authentication Functions
Message Encryption
Message Authentication Code
12.3 Requirements for Message Authentication Codes
12.4 Security of MACs
Brute-Force Attacks
Cryptanalysis
12.5 MACs Based on Hash Functions: HMAC
HMAC Design Objectives
HMAC Algorithm
Security of HMAC
12.6 MACs Based on Block Ciphers: DAA and CMAC
Data Authentication Algorithm
Cipher-Based Message Authentication Code (CMAC)
12.7 Authenticated Encryption: CCM and GCM
Counter with Cipher Block Chaining-Message Authentication Code
Galois/Counter Mode
12.8 Key Wrapping
Background
The Key Wrapping Algorithm
Key Unwrapping
12.9 Pseudorandom Number Generation Using Hash Functions and MACs
PRNG Based on Hash Function
PRNG Based on MAC Function
12.10 Key Terms, Review Questions, and Problems
CHAPTER
382 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
One of the most fascinating and complex areas of cryptography is that of message
authentication and the related area of digital signatures. It would be impossible, in
anything less than book length, to exhaust all the cryptographic functions and proto-
cols that have been proposed or implemented for message authentication and digital
signatures. Instead, the purpose of this chapter and the next is to provide a broad
overview of the subject and to develop a systematic means of describing the various
approaches.
This chapter begins with an introduction to the requirements for authen-
tication and digital signature and the types of attacks to be countered. Then the
basic approaches are surveyed. The remainder of the chapter deals with the funda-
mental approach to message authentication known as the message authentication
code (MAC). Following an overview of this topic, the chapter looks at security
considerations for MACs. This is followed by a discussion of specific MACs in
two categories: those built from cryptographic hash functions and those built using
a block cipher mode of operation. Next, we look at a relatively recent approach
known as authenticated encryption. Finally, we look at the use of cryptographic
hash functions and MACs for pseudorandom number generation.
12.1 MESSAGE AUTHENTICATION REQUIREMENTS
In the context of communications across a network, the following attacks can be
identified.
1. Disclosure: Release of message contents to any person or process not possess- ing the appropriate cryptographic key.
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ List and explain the possible attacks that are relevant to message authentication.
◆ Define the term message authentication code.
◆ List and explain the requirements for a message authentication code.
◆ Present an overview of HMAC.
◆ Present an overview of CMAC.
◆ Explain the concept of authenticated encryption.
◆ Present an overview of CCM.
◆ Present an overview of GCM.
◆ Discuss the concept of key wrapping and explain its use.
◆ Understand how a hash function or a message authentication code can be used for pseudorandom number generation.
12.2 / MESSAGE AUTHENTICATION FUNCTIONS 383
2. Traffic analysis: Discovery of the pattern of traffic between parties. In a connection-oriented application, the frequency and duration of connec-
tions could be determined. In either a connection-oriented or connectionless
environment, the number and length of messages between parties could be
determined.
3. Masquerade: Insertion of messages into the network from a fraudulent source. This includes the creation of messages by an opponent that are purported to
come from an authorized entity. Also included are fraudulent acknowledg-
ments of message receipt or nonreceipt by someone other than the message
recipient.
4. Content modification: Changes to the contents of a message, including inser- tion, deletion, transposition, and modification.
5. Sequence modification: Any modification to a sequence of messages between parties, including insertion, deletion, and reordering.
6. Timing modification: Delay or replay of messages. In a connection-oriented application, an entire session or sequence of messages could be a replay of
some previous valid session, or individual messages in the sequence could be
delayed or replayed. In a connectionless application, an individual message
(e.g., datagram) could be delayed or replayed.
7. Source repudiation: Denial of transmission of message by source.
8. Destination repudiation: Denial of receipt of message by destination.
Measures to deal with the first two attacks are in the realm of message
confidentiality and are dealt with in Part One. Measures to deal with items
(3) through (6) in the foregoing list are generally regarded as message authentica-
tion. Mechanisms for dealing specifically with item (7) come under the heading of
digital signatures. Generally, a digital signature technique will also counter some
or all of the attacks listed under items (3) through (6). Dealing with item (8) may
require a combination of the use of digital signatures and a protocol designed to
counter this attack.
In summary, message authentication is a procedure to verify that received
messages come from the alleged source and have not been altered. Message au-
thentication may also verify sequencing and timeliness. A digital signature is an
authentication technique that also includes measures to counter repudiation by the
source.
12.2 MESSAGE AUTHENTICATION FUNCTIONS
Any message authentication or digital signature mechanism has two levels of func-
tionality. At the lower level, there must be some sort of function that produces an
authenticator: a value to be used to authenticate a message. This lower-level func-
tion is then used as a primitive in a higher-level authentication protocol that enables
a receiver to verify the authenticity of a message.
This section is concerned with the types of functions that may be used to pro-
duce an authenticator. These may be grouped into three classes.
384 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
■ Hash function: A function that maps a message of any length into a fixed-length hash value, which serves as the authenticator
■ Message encryption: The ciphertext of the entire message serves as its authenticator
■ Message authentication code (MAC): A function of the message and a secret key that produces a fixed-length value that serves as the authenticator
Hash functions, and how they may serve for message authentication, are dis-
cussed in Chapter 11. The remainder of this section briefly examines the remaining
two topics. The remainder of the chapter elaborates on the topic of MACs.
Message Encryption
Message encryption by itself can provide a measure of authentication. The analysis
differs for symmetric and public-key encryption schemes.
SYMMETRIC ENCRYPTION Consider the straightforward use of symmetric encryption (Figure 12.1a). A message M transmitted from source A to destination B is encrypted using a secret key K shared by A and B. If no other party knows the key, then confi- dentiality is provided: No other party can recover the plaintext of the message.
Figure 12.1 Basic Uses of Message Encryption
Destination BSource A
M
K K
E
(a) Symmetric encryption: confidentiality and authentication
D M
PUb (b) Public-key encryption: confidentiality
E(K, M)
M E D M
E(PUb, M)
E(PRa, M) E(PRa, M)E(PUb, E(PRa, M))
M E D M
(c) Public-key encryption: authentication and signature
(d) Public-key encryption: confidentiality, authentication, and signature
E D
PRb
PRa
M E D M
E(PRa, M)
PRa
PUa
PUaPUb PRb
12.2 / MESSAGE AUTHENTICATION FUNCTIONS 385
In addition, B is assured that the message was generated by A. Why? The
message must have come from A, because A is the only other party that possesses
K and therefore the only other party with the information necessary to construct ciphertext that can be decrypted with K. Furthermore, if M is recovered, B knows that none of the bits of M have been altered, because an opponent that does not know K would not know how to alter bits in the ciphertext to produce the desired changes in the plaintext.
So we may say that symmetric encryption provides authentication as well as
confidentiality. However, this flat statement needs to be qualified. Consider exactly
what is happening at B. Given a decryption function D and a secret key K, the destination will accept any input X and produce output Y = D(K, X). If X is the ciphertext of a legitimate message M produced by the corresponding encryption function, then Y is some plaintext message M. Otherwise, Y will likely be a mean- ingless sequence of bits. There may need to be some automated means of determin-
ing at B whether Y is legitimate plaintext and therefore must have come from A. The implications of the line of reasoning in the preceding paragraph are pro-
found from the point of view of authentication. Suppose the message M can be any arbitrary bit pattern. In that case, there is no way to determine automatically, at the
destination, whether an incoming message is the ciphertext of a legitimate message.
This conclusion is incontrovertible: If M can be any bit pattern, then regardless of the value of X, the value Y = D(K, X) is some bit pattern and therefore must be accepted as authentic plaintext.
Thus, in general, we require that only a small subset of all possible bit patterns
be considered legitimate plaintext. In that case, any spurious ciphertext is unlikely
to produce legitimate plaintext. For example, suppose that only one bit pattern in
106 is legitimate plaintext. Then the probability that any randomly chosen bit pat-
tern, treated as ciphertext, will produce a legitimate plaintext message is only 10-6.
For a number of applications and encryption schemes, the desired conditions
prevail as a matter of course. For example, suppose that we are transmitting English-
language messages using a Caesar cipher with a shift of one (K = 1). A sends the following legitimate ciphertext:
nbsftfbupbutboeepftfbupbutboemjuumfmbnctfbujwz
B decrypts to produce the following plaintext:
mareseatoatsanddoeseatoatsandlittlelambseativy
A simple frequency analysis confirms that this message has the profile of ordinary
English. On the other hand, if an opponent generates the following random se-
quence of letters:
zuvrsoevgqxlzwigamdvnmhpmccxiuureosfbcebtqxsxq
this decrypts to
ytuqrndufpwkyvhfzlcumlgolbbwhttqdnreabdaspwrwp
which does not fit the profile of ordinary English.
386 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
It may be difficult to determine automatically if incoming ciphertext de- crypts to intelligible plaintext. If the plaintext is, say, a binary object file or digi-
tized X-rays, determination of properly formed and therefore authentic plaintext
may be difficult. Thus, an opponent could achieve a certain level of disruption
simply by issuing messages with random content purporting to come from a
legitimate user.
One solution to this problem is to force the plaintext to have some structure
that is easily recognized but that cannot be replicated without recourse to the en-
cryption function. We could, for example, append an error-detecting code, also
known as a frame check sequence (FCS) or checksum, to each message before en-
cryption, as illustrated in Figure 12.2a. A prepares a plaintext message M and then provides this as input to a function F that produces an FCS. The FCS is appended to
M and the entire block is then encrypted. At the destination, B decrypts the incom- ing block and treats the results as a message with an appended FCS. B applies the
same function F to attempt to reproduce the FCS. If the calculated FCS is equal to
the incoming FCS, then the message is considered authentic. It is unlikely that any
random sequence of bits would exhibit the desired relationship.
Note that the order in which the FCS and encryption functions are performed
is critical. The sequence illustrated in Figure 12.2a is referred to in [DIFF79] as
internal error control, which the authors contrast with external error control (Figure 12.2b). With internal error control, authentication is provided because an
opponent would have difficulty generating ciphertext that, when decrypted, would
have valid error control bits. If instead the FCS is the outer code, an opponent can
construct messages with valid error-control codes. Although the opponent cannot
know what the decrypted plaintext will be, he or she can still hope to create confu-
sion and disrupt operations.
Figure 12.2 Internal and External Error Control
(b) External error control
Destination BSource A
K K
M | |
F
(a) Internal error control
MD F
Compare
EM
F(M) F(M) E(K, [M || F(M)])
M | |E D
K
F
Compare
K
F
E(K, M)
F(E(K, M))
E(K, M)
M
12.2 / MESSAGE AUTHENTICATION FUNCTIONS 387
An error-control code is just one example; in fact, any sort of structuring
added to the transmitted message serves to strengthen the authentication capability.
Such structure is provided by the use of a communications architecture consisting
of layered protocols. As an example, consider the structure of messages transmit-
ted using the TCP/IP protocol architecture. Figure 12.3 shows the format of a TCP
segment, illustrating the TCP header. Now suppose that each pair of hosts shared
a unique secret key, so that all exchanges between a pair of hosts used the same
key, regardless of application. Then we could simply encrypt all of the datagram ex-
cept the IP header. Again, if an opponent substituted some arbitrary bit pattern for
the encrypted TCP segment, the resulting plaintext would not include a meaning-
ful header. In this case, the header includes not only a checksum (which covers the
header) but also other useful information, such as the sequence number. Because
successive TCP segments on a given connection are numbered sequentially, encryp-
tion assures that an opponent does not delay, misorder, or delete any segments.
PUBLIC-KEY ENCRYPTION The straightforward use of public-key encryption (Figure 12.1b) provides confidentiality but not authentication. The source (A) uses
the public key PUb of the destination (B) to encrypt M. Because only B has the cor- responding private key PRb, only B can decrypt the message. This scheme provides no authentication, because any opponent could also use B’s public key to encrypt a
message and claim to be A.
To provide authentication, A uses its private key to encrypt the message, and
B uses A’s public key to decrypt (Figure 12.1c). This provides authentication using
the same type of reasoning as in the symmetric encryption case: The message must
have come from A because A is the only party that possesses PRa and therefore the only party with the information necessary to construct ciphertext that can be
decrypted with PUa. Again, the same reasoning as before applies: There must be some internal structure to the plaintext so that the receiver can distinguish between
well-formed plaintext and random bits.
Figure 12.3 TCP Segment
Source port Destination port
Checksum Urgent pointer
Sequence number
Acknowledgment number
Options + padding
Application data
Reserved Flags WindowDataoffset
0Bit: 4 10 16 31
20 o
ct et
s
388 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
Assuming there is such structure, then the scheme of Figure 12.1c does pro-
vide authentication. It also provides what is known as digital signature.1 Only A
could have constructed the ciphertext because only A possesses PRa. Not even B, the recipient, could have constructed the ciphertext. Therefore, if B is in possession
of the ciphertext, B has the means to prove that the message must have come from
A. In effect, A has “signed” the message by using its private key to encrypt. Note
that this scheme does not provide confidentiality. Anyone in possession of A’s pub-
lic key can decrypt the ciphertext.
To provide both confidentiality and authentication, A can encrypt M first using its private key, which provides the digital signature, and then using B’s pub-
lic key, which provides confidentiality (Figure 12.1d). The disadvantage of this ap-
proach is that the public-key algorithm, which is complex, must be exercised four
times rather than two in each communication.
Message Authentication Code
An alternative authentication technique involves the use of a secret key to generate
a small fixed-size block of data, known as a cryptographic checksum or MAC, that is appended to the message. This technique assumes that two communicating parties,
say A and B, share a common secret key K. When A has a message to send to B, it calculates the MAC as a function of the message and the key:
MAC = C(K, M)
where
M = input message C = MAC function K = shared secret key MAC = message authentication code
The message plus MAC are transmitted to the intended recipient. The recipient
performs the same calculation on the received message, using the same secret key,
to generate a new MAC. The received MAC is compared to the calculated MAC
(Figure 12.4a). If we assume that only the receiver and the sender know the identity
of the secret key, and if the received MAC matches the calculated MAC, then
1. The receiver is assured that the message has not been altered. If an attacker al- ters the message but does not alter the MAC, then the receiver’s calculation of
the MAC will differ from the received MAC. Because the attacker is assumed
not to know the secret key, the attacker cannot alter the MAC to correspond
to the alterations in the message.
2. The receiver is assured that the message is from the alleged sender. Because no one else knows the secret key, no one else could prepare a message with a
proper MAC.
1This is not the way in which digital signatures are constructed, as we shall see, but the principle is the same.
Hiva-Network.Com
12.2 / MESSAGE AUTHENTICATION FUNCTIONS 389
3. If the message includes a sequence number (such as is used with HDLC, X.25, and TCP), then the receiver can be assured of the proper sequence because an
attacker cannot successfully alter the sequence number.
A MAC function is similar to encryption. One difference is that the MAC
algorithm need not be reversible, as it must be for decryption. In general, the MAC
function is a many-to-one function. The domain of the function consists of messages
of some arbitrary length, whereas the range consists of all possible MACs and all
possible keys. If an n-bit MAC is used, then there are 2n possible MACs, whereas there are N possible messages with N W 2n. Furthermore, with a k-bit key, there are 2k possible keys.
For example, suppose that we are using 100-bit messages and a 10-bit MAC.
Then, there are a total of 2100 different messages but only 210 different MACs. So,
on average, each MAC value is generated by a total of 2100/210 = 290 different mes- sages. If a 5-bit key is used, then there are 25 = 32 different mappings from the set of messages to the set of MAC values.
It turns out that, because of the mathematical properties of the authentication
function, it is less vulnerable to being broken than encryption.
The process depicted in Figure 12.4a provides authentication but not confiden-
tiality, because the message as a whole is transmitted in the clear. Confidentiality
can be provided by performing message encryption either after (Figure 12.4b) or
before (Figure 12.4c) the MAC algorithm. In both these cases, two separate keys are
Figure 12.4 Basic Uses of Message Authentication code (MAC)
Destination BSource A
M | |
K
C
(a) Message authentication
M E
| |
(c) Message authentication and confidentiality; authentication tied to ciphertext
M
C(K, M)
E(K2, [M || C(K1, M)])
C(K1, E(K2, M))
E(K2, M)
C
CompareK
EM | |
K1
K1
K2
K2
K2 K1
K1
K2
C
(b) Message authentication and confidentiality; authentication tied to plaintext
MD C
Compare
C
C
Compare
D M
C(K1, M)
390 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
needed, each of which is shared by the sender and the receiver. In the first case, the
MAC is calculated with the message as input and is then concatenated to the mes-
sage. The entire block is then encrypted. In the second case, the message is encrypted
first. Then the MAC is calculated using the resulting ciphertext and is concatenated
to the ciphertext to form the transmitted block. Typically, it is preferable to tie the
authentication directly to the plaintext, so the method of Figure 12.4b is used.
Because symmetric encryption will provide authentication and because it is
widely used with readily available products, why not simply use this instead of a
separate message authentication code? [DAVI89] suggests three situations in which
a message authentication code is used.
1. There are a number of applications in which the same message is broadcast to a number of destinations. Examples are notification to users that the network
is now unavailable or an alarm signal in a military control center. It is cheaper
and more reliable to have only one destination responsible for monitoring au-
thenticity. Thus, the message must be broadcast in plaintext with an associated
message authentication code. The responsible system has the secret key and
performs authentication. If a violation occurs, the other destination systems
are alerted by a general alarm.
2. Another possible scenario is an exchange in which one side has a heavy load and cannot afford the time to decrypt all incoming messages. Authentication is
carried out on a selective basis, messages being chosen at random for checking.
3. Authentication of a computer program in plaintext is an attractive service. The computer program can be executed without having to decrypt it every time,
which would be wasteful of processor resources. However, if a message au-
thentication code were attached to the program, it could be checked whenever
assurance was required of the integrity of the program.
Three other rationales may be added.
4. For some applications, it may not be of concern to keep messages secret, but it is important to authenticate messages. An example is the Simple Network
Management Protocol Version 3 (SNMPv3), which separates the functions of
confidentiality and authentication. For this application, it is usually important
for a managed system to authenticate incoming SNMP messages, particularly
if the message contains a command to change parameters at the managed sys-
tem. On the other hand, it may not be necessary to conceal the SNMP traffic.
5. Separation of authentication and confidentiality functions affords architec- tural flexibility. For example, it may be desired to perform authentication at
the application level but to provide confidentiality at a lower level, such as the
transport layer.
6. A user may wish to prolong the period of protection beyond the time of recep- tion and yet allow processing of message contents. With message encryption, the
protection is lost when the message is decrypted, so the message is protected
against fraudulent modifications only in transit but not within the target system.
Finally, note that the MAC does not provide a digital signature, because both
sender and receiver share the same key.
12.3 / REQUIREMENTS FOR MESSAGE AUTHENTICATION CODES 391
12.3 REQUIREMENTS FOR MESSAGE AUTHENTICATION CODES
A MAC, also known as a cryptographic checksum, is generated by a function C of
the form
T = MAC(K, M)
where M is a variable-length message, K is a secret key shared only by sender and re- ceiver, and MAC(K, M) is the fixed-length authenticator, sometimes called a tag. The tag is appended to the message at the source at a time when the message is assumed or
known to be correct. The receiver authenticates that message by recomputing the tag.
When an entire message is encrypted for confidentiality, using either symmet-
ric or asymmetric encryption, the security of the scheme generally depends on the
bit length of the key. Barring some weakness in the algorithm, the opponent must
resort to a brute-force attack using all possible keys. On average, such an attack will
require 2(k - 1) attempts for a k-bit key. In particular, for a ciphertext-only attack, the opponent, given ciphertext C, performs Pi = D(Ki, C) for all possible key values Ki until a Pi is produced that matches the form of acceptable plaintext.
In the case of a MAC, the considerations are entirely different. In general,
the MAC function is a many-to-one function, due to the many-to-one nature of
the function. Using brute-force methods, how would an opponent attempt to dis-
cover a key? If confidentiality is not employed, the opponent has access to plain-
text messages and their associated MACs. Suppose k 7 n; that is, suppose that the key size is greater than the MAC size. Then, given a known M1 and T1, with T1 = MAC(K, M1), the cryptanalyst can perform Ti = MAC(Ki, M1) for all pos- sible key values ki. At least one key is guaranteed to produce a match of Ti = T1. Note that a total of 2k tags will be produced, but there are only 2n 6 2k different tag values. Thus, a number of keys will produce the correct tag and the opponent has no
way of knowing which is the correct key. On average, a total of 2k/2n = 2(k - n) keys will produce a match. Thus, the opponent must iterate the attack.
■ Round 1
Given: M1, T1 = MAC(K, M1) Compute Ti = MAC(Ki, M1) for all 2
k keys
Number of matches L 2(k - n)
■ Round 2
Given: M2, T2 = MAC(K, M2) Compute Ti = MAC(Ki, M2) for the 2
(k - n) keys resulting from Round 1
Number of matches L 2(k - 2 * n)
And so on. On average, a rounds will be needed k = a * n. For example, if an 80-bit key is used and the tag is 32 bits, then the first round will produce about 248
possible keys. The second round will narrow the possible keys to about 216 possibili-
ties. The third round should produce only a single key, which must be the one used
by the sender.
392 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
If the key length is less than or equal to the tag length, then it is likely that a
first round will produce a single match. It is possible that more than one key will
produce such a match, in which case the opponent would need to perform the same
test on a new (message, tag) pair.
Thus, a brute-force attempt to discover the authentication key is no less ef-
fort and may be more effort than that required to discover a decryption key of the
same length. However, other attacks that do not require the discovery of the key
are possible.
Consider the following MAC algorithm. Let M = (X1 }X2 } c }Xm) be a message that is treated as a concatenation of 64-bit blocks Xi. Then define
∆(M) = X1 ⊕ X2 ⊕ c ⊕ Xm MAC(K, M) = E(K, ∆(M))
where ⊕ is the exclusive-OR (XOR) operation and the encryption algorithm is DES in electronic codebook mode. Thus, the key length is 56 bits, and the tag
length is 64 bits. If an opponent observes {M }MAC(K, M)}, a brute-force attempt to determine K will require at least 256 encryptions. But the opponent can attack the system by replacing X1 through Xm - 1 with any desired values Y1 through Ym - 1 and replacing Xm with Ym, where Ym is calculated as
Ym = Y1 ⊕ Y2 ⊕ g ⊕ Ym - 1 ⊕ ∆(M)
The opponent can now concatenate the new message, which consists of Y1 through Ym, using the original tag to form a message that will be accepted as authen- tic by the receiver. With this tactic, any message of length 64 * (m - 1) bits can be fraudulently inserted.
Thus, in assessing the security of a MAC function, we need to consider the
types of attacks that may be mounted against it. With that in mind, let us state the
requirements for the function. Assume that an opponent knows the MAC func-
tion but does not know K. Then the MAC function should satisfy the following requirements.
1. If an opponent observes M and MAC(K, M), it should be computationally infeasible for the opponent to construct a message M′ such that
MAC(K, M′) = MAC(K, M)
2. MAC(K, M) should be uniformly distributed in the sense that for randomly chosen messages, M and M′, the probability that MAC(K, M) = MAC(K, M′) is 2-n, where n is the number of bits in the tag.
3. Let M′ be equal to some known transformation on M. That is, M′ = f(M). For example, f may involve inverting one or more specific bits. In that case,
Pr [MAC(K, M) = MAC(K, M′)] = 2-n
The first requirement speaks to the earlier example, in which an opponent is
able to construct a new message to match a given tag, even though the opponent
does not know and does not learn the key. The second requirement deals with the
need to thwart a brute-force attack based on chosen plaintext. That is, if we assume
12.4 / SECURITY OF MACs 393
that the opponent does not know K but does have access to the MAC function and can present messages for MAC generation, then the opponent could try various
messages until finding one that matches a given tag. If the MAC function exhibits
uniform distribution, then a brute-force method would require, on average, 2(n - 1)
attempts before finding a message that fits a given tag.
The final requirement dictates that the authentication algorithm should not be
weaker with respect to certain parts or bits of the message than others. If this were
not the case, then an opponent who had M and MAC(K, M) could attempt varia- tions on M at the known “weak spots” with a likelihood of early success at produc- ing a new message that matched the old tags.
12.4 SECURITY OF MACs
Just as with encryption algorithms and hash functions, we can group attacks on
MACs into two categories: brute-force attacks and cryptanalysis.
Brute-Force Attacks
A brute-force attack on a MAC is a more difficult undertaking than a brute-force
attack on a hash function because it requires known message-tag pairs. Let us see
why this is so. To attack a hash code, we can proceed in the following way. Given
a fixed message x with n-bit hash code h = H(x), a brute-force method of finding a collision is to pick a random bit string y and check if H(y) = H(x). The attacker can do this repeatedly off line. Whether an off-line attack can be used on a MAC
algorithm depends on the relative size of the key and the tag.
To proceed, we need to state the desired security property of a MAC algo-
rithm, which can be expressed as follows.
■ Computation resistance: Given one or more text-MAC pairs [xi, MAC(K, xi)], it is computationally infeasible to compute any text-MAC pair [x, MAC(K, x)] for any new input x ≠ xi.
In other words, the attacker would like to come up with the valid MAC code for a
given message x. There are two lines of attack possible: attack the key space and at- tack the MAC value. We examine each of these in turn.
If an attacker can determine the MAC key, then it is possible to generate a
valid MAC value for any input x. Suppose the key size is k bits and that the attacker has one known text-tag pair. Then the attacker can compute the n-bit tag on the known text for all possible keys. At least one key is guaranteed to produce the cor-
rect tag, namely, the valid key that was initially used to produce the known text-tag
pair. This phase of the attack takes a level of effort proportional to 2k (that is, one
operation for each of the 2k possible key values). However, as was described earlier,
because the MAC is a many-to-one mapping, there may be other keys that produce
the correct value. Thus, if more than one key is found to produce the correct value,
additional text-tag pairs must be tested. It can be shown that the level of effort
drops off rapidly with each additional text-MAC pair and that the overall level of
effort is roughly 2k [MENE97].
394 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
An attacker can also work on the tag without attempting to recover the key.
Here, the objective is to generate a valid tag for a given message or to find a message
that matches a given tag. In either case, the level of effort is comparable to that for
attacking the one-way or weak collision-resistant property of a hash code, or 2n.
In the case of the MAC, the attack cannot be conducted off line without further
input; the attacker will require chosen text-tag pairs or knowledge of the key.
To summarize, the level of effort for brute-force attack on a MAC algorithm
can be expressed as min(2k, 2n). The assessment of strength is similar to that for
symmetric encryption algorithms. It would appear reasonable to require that the
key length and tag length satisfy a relationship such as min(k, n) Ú N, where N is perhaps in the range of 128 bits.
Cryptanalysis
As with encryption algorithms and hash functions, cryptanalytic attacks on MAC
algorithms seek to exploit some property of the algorithm to perform some attack
other than an exhaustive search. The way to measure the resistance of a MAC algo-
rithm to cryptanalysis is to compare its strength to the effort required for a brute-
force attack. That is, an ideal MAC algorithm will require a cryptanalytic effort
greater than or equal to the brute-force effort.
There is much more variety in the structure of MACs than in hash functions,
so it is difficult to generalize about the cryptanalysis of MACs. Furthermore, far less
work has been done on developing such attacks. A useful survey of some methods
for specific MACs is [PREN96].
12.5 MACs BASED ON HASH FUNCTIONS: HMAC
Later in this chapter, we look at examples of a MAC based on the use of a symmetric
block cipher. This has traditionally been the most common approach to constructing
a MAC. In recent years, there has been increased interest in developing a MAC de-
rived from a cryptographic hash function. The motivations for this interest are
1. Cryptographic hash functions such as MD5 and SHA generally execute faster in software than symmetric block ciphers such as DES.
2. Library code for cryptographic hash functions is widely available.
With the development of AES and the more widespread availability of code
for encryption algorithms, these considerations are less significant, but hash-based
MACs continue to be widely used.
A hash function such as SHA was not designed for use as a MAC and can-
not be used directly for that purpose, because it does not rely on a secret key.
There have been a number of proposals for the incorporation of a secret key into
an existing hash algorithm. The approach that has received the most support is
HMAC [BELL96a, BELL96b]. HMAC has been issued as RFC 2104, has been
chosen as the mandatory-to-implement MAC for IP security, and is used in other
Internet protocols, such as SSL. HMAC has also been issued as a NIST standard
(FIPS 198).
12.5 / MACs BASED ON HASH FUNCTIONS: HMAC 395
HMAC Design Objectives
RFC 2104 lists the following design objectives for HMAC.
■ To use, without modifications, available hash functions. In particular, to use
hash functions that perform well in software and for which code is freely and
widely available.
■ To allow for easy replaceability of the embedded hash function in case faster
or more secure hash functions are found or required.
■ To preserve the original performance of the hash function without incurring a
significant degradation.
■ To use and handle keys in a simple way.
■ To have a well understood cryptographic analysis of the strength of the au-
thentication mechanism based on reasonable assumptions about the embed-
ded hash function.
The first two objectives are important to the acceptability of HMAC. HMAC
treats the hash function as a “black box.” This has two benefits. First, an existing im-
plementation of a hash function can be used as a module in implementing HMAC.
In this way, the bulk of the HMAC code is prepackaged and ready to use without
modification. Second, if it is ever desired to replace a given hash function in an
HMAC implementation, all that is required is to remove the existing hash function
module and drop in the new module. This could be done if a faster hash function
were desired. More important, if the security of the embedded hash function were
compromised, the security of HMAC could be retained simply by replacing the em-
bedded hash function with a more secure one (e.g., replacing SHA-2 with SHA-3).
The last design objective in the preceding list is, in fact, the main advantage
of HMAC over other proposed hash-based schemes. HMAC can be proven secure
provided that the embedded hash function has some reasonable cryptographic
strengths. We return to this point later in this section, but first we examine the struc-
ture of HMAC.
HMAC Algorithm
Figure 12.5 illustrates the overall operation of HMAC. Define the following terms.
H = embedded hash function (e.g., MD5, SHA-1, RIPEMD-160) IV = initial value input to hash function M = message input to HMAC (including the padding specified in the embedded
hash function)
Yi = i th block of M, 0 … i … (L - 1) L = number of blocks in M b = number of bits in a block n = length of hash code produced by embedded hash function K = secret key; recommended length is Ú n; if key length is greater than b, the
key is input to the hash function to produce an n-bit key K+ = K padded with zeros on the left so that the result is b bits in length
396 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
ipad = 00110110 (36 in hexadecimal) repeated b/8 times opad = 01011100 (5C in hexadecimal) repeated b/8 times
Then HMAC can be expressed as
HMAC(K, M) = H[(K+ ⊕ opad) }H[(K+ ⊕ ipad) }M]]
We can describe the algorithm as follows.
1. Append zeros to the left end of K to create a b-bit string K+ (e.g., if K is of length 160 bits and b = 512, then K will be appended with 44 zeroes).
2. XOR (bitwise exclusive-OR) K+ with ipad to produce the b-bit block Si.
3. Append M to Si.
4. Apply H to the stream generated in step 3.
5. XOR K+ with opad to produce the b-bit block So.
6. Append the hash result from step 4 to So.
7. Apply H to the stream generated in step 6 and output the result.
Note that the XOR with ipad results in flipping one-half of the bits of K. Similarly, the XOR with opad results in flipping one-half of the bits of K, using a
Figure 12.5 HMAC Structure
K+
Si
So
Y0 Y1 YL–1
b bits
b bits
b bits b bits
ipad
K+ opad
HashIV n bits
n bits
Pad to b bits
HashIV n bits
n bits
HMAC(K, M)
H(Si || M)
12.5 / MACs BASED ON HASH FUNCTIONS: HMAC 397
different set of bits. In effect, by passing Si and So through the compression function of the hash algorithm, we have pseudorandomly generated two keys from K.
HMAC should execute in approximately the same time as the embedded hash
function for long messages. HMAC adds three executions of the hash compression
function (for Si, So, and the block produced from the inner hash). A more efficient implementation is possible, as shown in Figure 12.6. Two
quantities are precomputed:
f(IV, (K+ ⊕ ipad)) f(IV, (K+ ⊕ opad))
where f(cv, block) is the compression function for the hash function, which takes as
arguments a chaining variable of n bits and a block of b bits and produces a chain- ing variable of n bits. These quantities only need to be computed initially and every time the key changes. In effect, the precomputed quantities substitute for the initial
value (IV) in the hash function. With this implementation, only one additional in- stance of the compression function is added to the processing normally produced
Figure 12.6 Efficient Implementation of HMAC
b bits b bits b bits
Precomputed Computed per message
HashIV n bits
b bits
n bits
Pad to b bits
n bits
n bits
HMAC(K, M)
f
IV
b bits
f f
K+
Si
So
Y0 Y1
ipad
K+ opad
YL–1
H(Si || M)
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398 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
by the hash function. This more efficient implementation is especially worthwhile if
most of the messages for which a MAC is computed are short.
Security of HMAC
The security of any MAC function based on an embedded hash function depends
in some way on the cryptographic strength of the underlying hash function. The
appeal of HMAC is that its designers have been able to prove an exact relation-
ship between the strength of the embedded hash function and the strength of
HMAC.
The security of a MAC function is generally expressed in terms of the prob-
ability of successful forgery with a given amount of time spent by the forger and
a given number of message-tag pairs created with the same key. In essence, it is
proved in [BELL96a] that for a given level of effort (time, message–tag pairs) on
messages generated by a legitimate user and seen by the attacker, the probability
of successful attack on HMAC is equivalent to one of the following attacks on the
embedded hash function.
1. The attacker is able to compute an output of the compression function even with an IV that is random, secret, and unknown to the attacker.
2. The attacker finds collisions in the hash function even when the IV is random and secret.
In the first attack, we can view the compression function as equivalent to the
hash function applied to a message consisting of a single b-bit block. For this attack, the IV of the hash function is replaced by a secret, random value of n bits. An attack on this hash function requires either a brute-force attack on the key, which is a level
of effort on the order of 2n, or a birthday attack, which is a special case of the second
attack, discussed next.
In the second attack, the attacker is looking for two messages M and M′ that produce the same hash: H(M) = H(M′). This is the birthday attack discussed in Chapter 11. We have shown that this requires a level of effort of 2n/2 for a hash
length of n. On this basis, the security of MD5 is called into question, because a level of effort of 264 looks feasible with today’s technology. Does this mean that
a 128-bit hash function such as MD5 is unsuitable for HMAC? The answer is no,
because of the following argument. To attack MD5, the attacker can choose any
set of messages and work on these off line on a dedicated computing facility to
find a collision. Because the attacker knows the hash algorithm and the default IV, the attacker can generate the hash code for each of the messages that the attacker
generates. However, when attacking HMAC, the attacker cannot generate mes-
sage/code pairs off line because the attacker does not know K. Therefore, the at- tacker must observe a sequence of messages generated by HMAC under the same
key and perform the attack on these known messages. For a hash code length of
128 bits, this requires 264 observed blocks (272 bits) generated using the same key.
On a 1-Gbps link, one would need to observe a continuous stream of messages
with no change in key for about 150,000 years in order to succeed. Thus, if speed
is a concern, it is fully acceptable to use MD5 rather than SHA-1 as the embedded
hash function for HMAC.
12.6 / MACs BASED ON BLOCK CIPHERS: DAA AND CMAC 399
12.6 MACs BASED ON BLOCK CIPHERS: DAA AND CMAC
In this section, we look at two MACs that are based on the use of a block cipher
mode of operation. We begin with an older algorithm, the Data Authentication
Algorithm (DAA), which is now obsolete. Then we examine CMAC, which is de-
signed to overcome the deficiencies of DAA.
Data Authentication Algorithm
The Data Authentication Algorithm (DAA), based on DES, has been one of the most widely used MACs for a number of years. The algorithm is both a FIPS pub-
lication (FIPS PUB 113) and an ANSI standard (X9.17). However, as we discuss
subsequently, security weaknesses in this algorithm have been discovered, and it is
being replaced by newer and stronger algorithms.
The algorithm can be defined as using the cipher block chaining (CBC) mode
of operation of DES (Figure 6.4) with an initialization vector of zero. The data (e.g.,
message, record, file, or program) to be authenticated are grouped into contiguous
64-bit blocks: D1, D2, c , DN. If necessary, the final block is padded on the right with zeroes to form a full 64-bit block. Using the DES encryption algorithm E and a
secret key K, a data authentication code (DAC) is calculated as follows (Figure 12.7).
O1 = E(K, D) O2 = E(K, [D2 ⊕ O1]) O3 = E(K, [D3 ⊕ O2])# # # ON = E(K, [DN ⊕ ON - 1])
Figure 12.7 Data Authentication Algorithm (FIPS PUB 113)
Time = 1
DES encrypt
K (56 bits)
Time = 2
K
+ + +
K K
Time = NTime = N – 1
O1 (64 bits)
O2
D1 (64 bits) D2 DN–1
ON
DN
ON–1
DAC (16 to 64 bits)
DES encrypt
DES encrypt
DES encrypt
400 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
The DAC consists of either the entire block ON or the leftmost M bits of the block, with 16 … M … 64.
Cipher-Based Message Authentication Code (CMAC)
As was mentioned, DAA has been widely adopted in government and industry.
[BELL00] demonstrated that this MAC is secure under a reasonable set of security
criteria, with the following restriction. Only messages of one fixed length of mn bits are processed, where n is the cipher block size and m is a fixed positive integer. As a simple example, notice that given the CBC MAC of a one-block message X, say T = MAC(K, X), the adversary immediately knows the CBC MAC for the two- block message X }(X ⊕ T) since this is once again T.
Black and Rogaway [BLAC00] demonstrated that this limitation could be
overcome using three keys: one key K of length k to be used at each step of the cipher block chaining and two keys of length b, where b is the cipher block length. This proposed construction was refined by Iwata and Kurosawa so that the two
n-bit keys could be derived from the encryption key, rather than being provided separately [IWAT03]. This refinement, adopted by NIST, is the Cipher-based Message Authentication Code (CMAC) mode of operation for use with AES and triple DES. It is specified in NIST Special Publication 800-38B.
First, let us define the operation of CMAC when the message is an integer
multiple n of the cipher block length b. For AES, b = 128, and for triple DES, b = 64. The message is divided into n blocks (M1, M2, c , Mn). The algorithm makes use of a k-bit encryption key K and a b-bit constant, K1. For AES, the key size k is 128, 192, or 256 bits; for triple DES, the key size is 112 or 168 bits. CMAC is calculated as follows (Figure 12.8).
C1 = E(K, M1) C2 = E(K, [M2 ⊕ C1]) C3 = E(K, [M3 ⊕ C2]) # # #
Cn = E(K, [Mn ⊕ Cn - 1 ⊕ K1]) T = MSBTlen(Cn)
where
T = message authentication code, also referred to as the tag Tlen = bit length of T MSBs(X) = the s leftmost bits of the bit string X
If the message is not an integer multiple of the cipher block length, then the
final block is padded to the right (least significant bits) with a 1 and as many 0s as
necessary so that the final block is also of length b. The CMAC operation then pro- ceeds as before, except that a different b-bit key K2 is used instead of K1.
12.6 / MACs BASED ON HASH FUNCTIONS: HMAC 401
The two b-bit keys are derived from the k-bit encryption key as follows.
L = E(K, 0b) K1 = L # x K2 = L # x2 = (L # x) # x
where multiplication ( # ) is done in the finite field GF(2b) and x and x2 are first- and second-order polynomials that are elements of GF(2b). Thus, the binary represen-
tation of x consists of b - 2 zeros followed by 10; the binary representation of x2 consists of b - 3 zeros followed by 100. The finite field is defined with respect to an irreducible polynomial that is lexicographically first among all such polynomials
with the minimum possible number of nonzero terms. For the two approved block
sizes, the polynomials are x64 + x4 + x3 + x + 1 and x128 + x7 + x2 + x + 1. To generate K1 and K2, the block cipher is applied to the block that consists
entirely of 0 bits. The first subkey is derived from the resulting ciphertext by a
left shift of one bit and, conditionally, by XORing a constant that depends on the
block size. The second subkey is derived in the same manner from the first subkey.
This property of finite fields of the form GF(2b) was explained in the discussion of
MixColumns in Chapter 6.
Figure 12.8 Cipher-Based Message Authentication Code (CMAC)
EncryptK K K
T
Encrypt Encrypt
MSB(Tlen)
M1
K1
K2
M2 Mn
(a) Message length is integer multiple of block size
EncryptK K K
T
Encrypt Encrypt
MSB(Tlen)
10...0
(b) Message length is not integer multiple of block size
b
k
MnM1 M2
402 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
12.7 AUTHENTICATED ENCRYPTION: CCM AND GCM
Authenticated encryption (AE) is a term used to describe encryption systems that
simultaneously protect confidentiality and authenticity (integrity) of communica-
tions. Many applications and protocols require both forms of security, but until re-
cently the two services have been designed separately.
There are four common approaches to providing both confidentiality and en-
cryption for a message M.
■ Hashing followed by encryption (H S E): First compute the cryptographic hash function over M as h = H(M). Then encrypt the message plus hash func- tion: E(K, (M }h)).
■ Authentication followed by encryption (A S E): Use two keys. First authen- ticate the plaintext by computing the MAC value as T = MAC(K1, M). Then encrypt the message plus tag: E(K2, [M }T ]). This approach is taken by the SSL/TLS protocols (Chapter 17).
■ Encryption followed by authentication (E S A): Use two keys. First encrypt the message to yield the ciphertext C = E(K2, M). Then authenticate the ciphertext with T = MAC(K1, C) to yield the pair (C, T). This approach is used in the IPSec protocol (Chapter 20).
■ Independently encrypt and authenticate (E + A). Use two keys. Encrypt the message to yield the ciphertext C = E(K2, M). Authenticate the plain- text with T = MAC(K1, M) to yield the pair (C, T). These operations can be performed in either order. This approach is used by the SSH protocol
(Chapter 17).
Both decryption and verification are straightforward for each approach. For
H S E, A S E, and E + A, decrypt first, then verify. For E S A, verify first, then decrypt. There are security vulnerabilities with all of these approaches. The H S E approach is used in the Wired Equivalent Privacy (WEP) protocol to protect WiFi
networks. This approach had fundamental weaknesses and led to the replacement of
the WEP protocol. [BLAC05] and [BELL00] point out that there are security con-
cerns in each of the three encryption/MAC approaches listed above. Nevertheless,
with proper design, any of these approaches can provide a high level of security.
This is the goal of the two approaches discussed in this section, both of which have
been standardized by NIST.
Counter with Cipher Block Chaining-Message Authentication Code
The CCM mode of operation was standardized by NIST specifically to sup-
port the security requirements of IEEE 802.11 WiFi wireless local area networks
(Chapter 18), but can be used in any networking application requiring authenti-
cated encryption. CCM is a variation of the encrypt-and-MAC approach to authen-
ticated encryption. It is defined in NIST SP 800-38C.
The key algorithmic ingredients of CCM are the AES encryption algorithm
(Chapter 6), the CTR mode of operation (Chapter 7), and the CMAC authentication
12.7 / AUTHENTICATED ENCRYPTION: CCM AND GCM 403
algorithm (Section 12.6). A single key K is used for both encryption and MAC algo- rithms. The input to the CCM encryption process consists of three elements.
1. Data that will be both authenticated and encrypted. This is the plaintext mes- sage P of data block.
2. Associated data A that will be authenticated but not encrypted. An example is a protocol header that must be transmitted in the clear for proper protocol
operation but which needs to be authenticated.
3. A nonce N that is assigned to the payload and the associated data. This is a unique value that is different for every instance during the lifetime of a pro-
tocol association and is intended to prevent replay attacks and certain other
types of attacks.
Figure 12.9 illustrates the operation of CCM. For authentication, the input
includes the nonce, the associated data, and the plaintext. This input is formatted
as a sequence of blocks B0 through Br. The first block contains the nonce plus some formatting bits that indicate the lengths of the N, A, and P elements. This is fol- lowed by zero or more blocks that contain A, followed by zero of more blocks that contain P. The resulting sequence of blocks serves as input to the CMAC algorithm, which produces a MAC value with length Tlen, which is less than or equal to the block length (Figure 12.9a).
For encryption, a sequence of counters is generated that must be independent
of the nonce. The authentication tag is encrypted in CTR mode using the single
counter Ctr0. The Tlen most significant bits of the output are XORed with the tag to produce an encrypted tag. The remaining counters are used for the CTR mode en-
cryption of the plaintext (Figure 7.7). The encrypted plaintext is concatenated with
the encrypted tag to form the ciphertext output (Figure 12.9b).
SP 800-38C defines the authentication/encryption process as follows.
1. Apply the formatting function to (N, A, P) to produce the blocks B0, B1, c , Br. 2. Set Y0 = E(K, B0). 3. For i = 1 to r, do Yi = E(K, (Bi ⊕ Yi - 1)). 4. Set T = MSBTlen(Yr). 5. Apply the counter generation function to generate the counter blocks
Ctr0, Ctr1, c , Ctrm, where m = <Plen/128=. 6. For j = 0 to m, do Sj = E(K, Ctrj). 7. Set S = S1 }S2 } g }Sm. 8. Return C = (P ⊕ MSBPlen(S)) }(T ⊕ MSBTlen(S0)).
For decryption and verification, the recipient requires the following input: the
ciphertext C, the nonce N, the associated data A, the key K, and the initial counter Ctr0. The steps are as follows.
1. If Clen … Tlen, then return INVALID. 2. Apply the counter generation function to generate the counter blocks
Ctr0, Ctr1, c , Ctrm, where m = <Clen/128=. 3. For j = 0 to m, do Sj = E(K, Ctrj).
404 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
4. Set S = S1 }S2 } g }Sm. 5. Set P = MSBClen - Tlen(C) ⊕ MSBClen - Tlen(S). 6. Set T = LSBTlen(C) ⊕ MSBTlen(S0). 7. Apply the formatting function to N, A, P) to produce the blocks B0, B1, c , Br. 8. Set Y0 = E(K, B0). 9. For i = 1 to r do Yi = E(K, (Bi ⊕ Yi - 1)).
10. If T ≠ MSBTlen(Yr), then return INVALID, else return P.
Figure 12.9 Counter with Cipher Block Chaining-Message Authentication Code (CCM)
(a) Authentication
(b) Encryption
B0
Ctr0
B1 B2 Br
Tag
Tag
Nonce Plaintext
Plaintext
Ciphertext
Ass. Data
K CMAC
MSB(Tlen) K
CTRCtr1, Ctr2, ..., Ctrm
EncryptK
12.7 / AUTHENTICATED ENCRYPTION: CCM AND GCM 405
CCM is a relatively complex algorithm. Note that it requires two complete
passes through the plaintext, once to generate the MAC value, and once for encryp-
tion. Further, the details of the specification require a tradeoff between the length
of the nonce and the length of the tag, which is an unnecessary restriction. Also note
that the encryption key is used twice with the CTR encryption mode: once to gener-
ate the tag and once to encrypt the plaintext plus tag. Whether these complexities
add to the security of the algorithm is not clear. In any case, two analyses of the
algorithm ([JONS02] and [ROGA03]) conclude that CCM provides a high level of
security.
Galois/Counter Mode
The GCM mode of operation, standardized by NIST in NIST SP 800-38D, is de-
signed to be parallelizable so that it can provide high throughput with low cost and
low latency. In essence, the message is encrypted in variant of CTR mode. The re-
sulting ciphertext is multiplied with key material and message length information
over GF(2128) to generate the authenticator tag. The standard also specifies a mode
of operation that supplies the MAC only, known as GMAC.
The GCM mode makes use of two functions: GHASH, which is a keyed hash
function, and GCTR, which is essentially the CTR mode with the counters deter-
mined by a simple increment by one operation.
GHASHH(X) takes a input the hash key H and a bit string X such that len(X) = 128m bits for some positive integer m and produces a 128-bit MAC value. The function may be specified as follows (Figure 12.10a).
1. Let X1, X2, c , Xm - 1, Xm denote the unique sequence of blocks such that X = X1 }X2 } g }Xm - 1 }Xm.
2. Let Y0 be a block of 128 zeros, designated as 0 128.
3. For i = 1, c , m, let Yi = (Yi - 1 ⊕ Xi) # H, where # designates multiplication in GF(2128).
4. Return Ym.
The GHASHH(X) function can be expressed as
(X1 # Hm) ⊕ (X2 # Hm - 1) ⊕ g ⊕ (Xm - 1 # H2) ⊕ (Xm # H) This formulation has desirable performance implications. If the same hash key
is to be used to authenticate multiple messages, then the values H2, H3, c can be precalculated one time for use with each message to be authenticated. Then, the
blocks of the data to be authenticated (X1, X2, c , Xm) can be processed in paral- lel, because the computations are independent of one another.
GCTRK(ICB, X) takes a input a secret key K and a bit string X arbitrary length and returns a ciphertext Y of bit length (X). The function may be specified as follows (Figure 12.10b).
1. If X is the empty string, then return the empty string as Y.
2. Let n = <(len(X)/128)=. That is, n is the smallest integer greater than or equal to (X)/128.
406 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
3. Let X1, X2, c , Xn - 1, Xn * denote the unique sequence of bit strings such that
X = X1 }X2 } g }Xn - 1 }Xn *;
X1, X2, c , Xn - 1 are complete 128@bit blocks.
4. Let CB1 = ICB. 5. For, i = 2 to n let CBi = inc32(CBi - 1), where the inc32(S) function increments
the rightmost 32 bits of S by 1 mod 232, and the remaining bits are unchanged.
6. For i = 1 to n - 1, do Yi = Xi ⊕ E(K, CBi). 7. Let Y n
* = Xn * ⊕ MSBlen(Xn*)(E(K, CBn)).
8. Let Y = Y1 }Y2 } c }Yn - 1 }Y n *
9. Return Y.
Note that the counter values can be quickly generated and that the encryption
operations can be performed in parallel.
Figure 12.10 GCM Authentication and Encryption Functions
(a) GHASHH(X1 || X2 || . . . || Xm) = Ym
X1
X1
X2
ICB
Xm
Y1
Y1
Y2 Ym
H
E
inc
H H
K
X2
CB2
Y2
EK
Xn–1
CBn–1
Yn–1
E
inc
K
Xn
CBn
Yn
E
MSB
K
*
(b) GCTRK (ICB, X1 || X2 || . . . || Xn) = Y1 || Y2 || . . . ||Yn**
*
Hiva-Network.Com
12.7 / AUTHENTICATED ENCRYPTION: CCM AND GCM 407
We can now define the overall authenticated encryption function
(Figure 12.11). The input consists of a secret key K, an initialization vector IV, a plaintext P, and additional authenticated data A. The notation [x]s means the s-bit binary representation of the nonnegative integer x. The steps are as follows.
1. Let H = E(K, 0128). 2. Define a block, J0, as
If len(IV) = 96, then let J0 = IV }0 31 }1.
If len (IV) ≠ 96, then let s = 128<len(IV)/128= - len(IV), and let J0 = GHASHH(IV }0
s + 64 }[len(IV)]64). 3. Let C = GCTRK(inc32(J0), P). 4. Let u = 128<len(C)/128= - len(C) and let v = 128<len(A)/128= - len(A). 5. Define a block, S, as
S = GHASHH(A }0 v }C }0u }[len(A)]64 }[len(C)]64)
6. Let T = MSBt(GCTRK(J0, S)), where t is the supported tag length. 7. Return (C, T).
Figure 12.11 Galois Counter—Message Authentication Code (GCM)
IV
J0
J0
Plaintext
K
K
GCTR
encode
incr
GCTR
Tag
GHASH
A = Ass. Data C = Ciphertext [len(A)]64 [len(C )]640v 0u
MSBt
EK
H
0
408 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
In step 1, the hash key is generated by encrypting a block of all zeros with
the secret key K. In step 2, the pre-counter block (J0) is generated from the IV. In particular, when the length of the IV is 96 bits, then the padding string 031 }1 is appended to the IV to form the pre-counter block. Otherwise, the IV is padded with the minimum number of 0 bits, possibly none, so that the length of the result-
ing string is a multiple of 128 bits (the block size); this string in turn is appended
with 64 additional 0 bits, followed by the 64-bit representation of the length of
the IV, and the GHASH function is applied to the resulting string to form the pre-counter block.
Thus, GCM is based on the CTR mode of operation and adds a MAC that au-
thenticates both the message and additional data that requires only authentication.
The function that computes the hash uses only multiplication in a Galois field. This
choice was made because the operation of multiplication is easy to perform within a
Galois field and is easily implemented in hardware [MCGR05].
[MCGR04] examines the available block cipher modes of operation and
shows that a CTR-based authenticated encryption approach is the most efficient
mode of operation for high-speed packet networks. The paper further demonstrates
that GCM meets a high level of security requirements.
12.8 KEY WRAPPING
Background
The most recent block cipher mode of operation defined by NIST is the Key Wrap
(KW) mode of operation (SP 800-38F), which uses AES or triple DEA as the un-
derlying encryption algorithm. The AES version is also documented in RFC 3394.
The purpose of key wrapping is to securely exchange a symmetric key to be
shared by two parties, using a symmetric key already shared by those parties. The
latter key is called a key encryption key (KEK). Two questions need to be addressed at this point. First, why do we need to
use a symmetric key already known to two parties to encrypt a new symmetric key?
Such a requirement is found in a number of protocols described in this book, such
as the key management portion of IEEE 802.11 and IPsec. Quite often, a protocol
calls for a hierarchy of keys, with keys lower on the hierarchy used more frequently,
and changed more frequently to thwart attacks. A higher-level key, which is used in-
frequently and therefore more resistant to cryptanalysis, is used to encrypt a newly
created lower-level key so that it can be exchanged between parties that share the
higher-level key.
The second question is, why do we need a new mode? The intent of the new
mode is to operate on keys whose length is greater than the block size of the encryp-
tion algorithm. For example, AES uses a block size of 128 bits but can use a key
size of 128, 192, or 256 bits. In the latter two cases, encryption of the key involves
multiple blocks. We consider the value of key data to be greater than the value of
other data, because the key will be used multiple times, and compromise of the
key compromises all of the data encrypted with the key. Therefore, NIST desired
12.8 / KEY WRAPPING 409
a robust encryption mode. KW is robust in the sense that each bit of output can be
expected to depend in a nontrivial fashion on each bit of input. This is not the case
for any of the other modes of operation that we have described. For example, in
all of the modes so far described, the last block of plaintext only influences the last
block of ciphertext. Similarly, the first block of ciphertext is derived only from the
first block of plaintext.
To achieve this robust operation, KW achieves a considerably lower through-
put than the other modes, but the tradeoff may be appropriate for some key
management applications. Also, KW is only used for small amounts of plaintext
compared to, say, the encryption of a message or a file.
The Key Wrapping Algorithm
The key wrapping algorithm operates on blocks of 64 bits. The input to the algo-
rithm consists of a 64-bit constant, discussed subsequently, and a plaintext key that
is divided into blocks of 64 bits. We use the following notation:
MSB64(W) most significant 64 bits of W
LSB64(W) least significant 64 bits of W
W temporary value; output of encryption function
bitwise exclusive-OR
} concatenation
K key encryption key
n number of 64-bit key data blocks
s number of stages in the wrapping process; s = 6n Pi ith plaintext key data block; 1 … i … n Ci ith ciphertext data block; 0 … i … n A(t) 64-bit integrity check register after encryption stage t; 1 … t … s A(0) initial integrity check value (ICV); in hexadecimal:
A6A6A6A6A6A6A6A6
R(t, i) 64-bit register i after encryption stage t; 1 … t … s; 1 … i … n
We now describe the key wrapping algorithm:
Inputs: Plaintext, n 64-bit values (P1, P2, c , Pn) Key encryption key, K
Outputs: Ciphertext, (n + 1) 64-bit values (C0, C1, c , Cn)
1. Initialize variables.
A(0) = A6A6A6A6A6A6A6A6
for i = 1 to n
R(0, i) = Pi
⊕
410 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
2. Calculate intermediate values.
for t = 1 to s
W = E(K, [A(t−1) } R(t−1, 1)]) A(t) = t ⊕ MSB64(W) R(t, n) = LSB64(W)
for i = 1 to n−1
R(t, i) = R(t−1, i+1)
3. Output results.
C0 = A(s)
for i = 1 to n
Ci = R(s, i)
Note that the ciphertext is one block longer than the plaintext key, to ac-
commodate the ICV. Upon unwrapping (decryption), both the 64-bit ICV and the
plaintext key are recovered. If the recovered ICV differs from the input value of
hexadecimal A6A6A6A6A6A6A6A6, then an error or alteration has been detected
and the plaintext key is rejected. Thus, the key wrap algorithm provides not only
confidentiality but also data integrity.
Figure 12.12 illustrated the key wrapping algorithm for encrypting a 256-bit
key. Each box represents one encryption stage (one value of t). Note that the A output is fed as input to the next stage (t + 1), whereas the R output skips forward n stages (t + n), which in this example is n = 4. This arrangement further increases the avalanche effect and the mixing of bits. To achieve this skipping of stages, a slid-
ing buffer is used, so that the R output from stage t is shifted in the buffer one posi- tion for each stage, until it becomes the input for stage t + n. This might be clearer if we expand the inner for loop for a 256-bit key (n = 4). Then the assignments are as follows:
R(t, 1) = R(t - 1, 2) R(t, 2) = R(t - 1, 3) R(t, 3) = R(t - 1, 4)
For example, consider that at stage 5, the R output has a value of R(5, 4) = x. At stage 6, we execute R(6, 3) = R(5, 4) = x. At stage 7, we execute R(7, 2) = R (6, 3) = x. At stage 8, we execute R(8, 1) = R(7, 2) = x. So, at stage 9, the input value of R(t - 1, 1) is R(8, 1) = x.
Figure 12.13 depicts the operation of stage t for a 256-bit key. The dashed feedback lines indicate the assignment of new values to the stage variables.
Key Unwrapping
The key unwrapping algorithm can be defined as follows:
Inputs: Ciphertext, (n + 1) 64-bit values (C0, C1, c , Cn) Key encryption key, K
Outputs: Plaintext, n 64-bit values (P1, P2, c , Pn), ICV
12.8 / KEY WRAPPING 411
1. Initialize variables.
A(s) = C0 for i = 1 to n
R(s, i) = Ci
2. Calculate intermediate values.
for t = s to 1
W = D(K, [(A(t) ⊕ t)} R(t, n)])
Figure 12.12 Key Wrapping Operation for 256-Bit Key
t = 1 t = 2 t = 3 t = 4
t = 5 t = 6 t = 7 t = 8
t = 9 t = 10 t = 11 t = 12
t = 13 t = 14 t = 15 t = 16
t = 17 t = 18 t = 19 t = 20
t = 21
C0 = A(24)
A(0) A(1)
C1 = R(24, 1) = R(21, 4)
P1 = R(0, 1)
P2 = R(0, 2)
P3 = R(0, 3)
P4 = R(0, 4) A(2) A(3)
C2 = R(24, 2) = R(22, 4)
C3 = R(24, 3) = R(23, 4)
C4 = R(24, 4)
t = 22 t = 23 t = 24
412 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
A(t–1) = MSB64(W)
R(t–1, 1) = LSB64(W)
for i = 2 to n
R(t–1, i) = R(t, i–1)
3. Output results.
if A(0) = A6A6A6A6A6A6A6A6
then
for i = 1 to n
P(i) = R(0, i)
else
return error
Note that the decryption function is used in the unwrapping algorithm.
We now demonstrate that the unwrap function is the inverse of the wrap func-
tion, that is, that the unwrap function recovers the plaintext key and the ICV. First,
note that because the index variable t is counted down from s to 1 for unwrapping, stage t of the unwrap algorithm corresponds to stage t of the wrap algorithm. The input variables to stage t of the wrap algorithm are indexed at t - 1 and the output variables of stage t of the unwrap algorithm are indexed at t - 1. Thus, to demon- strate that the two algorithms are inverses of each other, we need only demonstrate
that the output variables of stage t of the unwrap algorithm are equal to the input variables to stage t of the wrap algorithm.
This demonstration is in two parts. First we demonstrate that the calculation
of A and R variables prior to the for loop are inverses. To do this, let us simplify the notation a bit. Define the 128-bit value T to be the 64-bit value t followed by 64 zeros. Then, the first three lines of step 2 of the wrap algorithm can be written as the
following single line:
A(t) }R(t, n) = T ⊕ E(K, [A(t - 1) }R(t - 1, 1)]) (12.1)
The first three lines of step 2 of the unwrap algorithm can be written as:
A(t - 1) }R(t - 1, 1) = D(K, ([A(t) }R(t, n)] ⊕ T)) (12.2)
Figure 12.13 Key Wrapping Operation for 256-Bit Key: Stage t
A(t – 1)
Encrypt
MSB
K
t LSB
R(t – 1, 1) R(t – 1, 2)
R(t – 1, 3)
R(t – 1, 4)
12.9 / PSEUDORANDOM NUMBER GENERATION USING HASH FUNCTIONS 413
Expanding the right-hand side by substituting from Equation 12.1,
D(K, ([A(t) }R(t, n)] ⊕ T)) = D(K, ([T ⊕ E(K, [A(t - 1) }R(t - 1, 1)])] ⊕ T))
Now we recognize that T ⊕ T = 0 and that for any x, x ⊕ 0 = x. So,
D(K, ([A(t) }R(t, n)] ⊕ T)) = D(K, ([E(K, [A(t - 1) }R(t - 1, 1)])) = A(t - 1) }R(t - 1, 1)
The second part of the demonstration is to show that the for loops in step 2 of the wrap and unwrap algorithms are inverses. For stage k of the wrap algorithm, the variables R(t - 1, 1) through R(t - 1, n) are input. R(t - 1, 1) is used in the encryption calculation. R(t - 1, 2) through R(t - 1, n) are mapped, respectively into R(t, 1) through R(t, n - 1), and R(t, n) is output from the encryption function. For stage k of the unwrap algorithm, the variables R(t, 1) through R(t, n) are input. R(t, n) is input to the decryption function to produce R(t - 1, 1). The remaining variables R(t - 1, 2) through R(t - 1, n) are generated by the for loop, such that they are mapped, respectively, from R(t, 1) through R(t, n - 1).
Thus, we have shown that the output variables of stage k of the unwrap algo- rithm equal the input variables of stage k of the wrap algorithm.
12.9 PSEUDORANDOM NUMBER GENERATION USING HASH FUNCTIONS AND MACs
The essential elements of any pseudorandom number generator (PRNG) are a seed
value and a deterministic algorithm for generating a stream of pseudorandom bits.
If the algorithm is used as a pseudorandom function (PRF) to produce a required
value, such as a session key, then the seed should only be known to the user of the
PRF. If the algorithm is used to produce a stream encryption function, then the seed
has the role of a secret key that must be known to the sender and the receiver.
We noted in Chapters 8 and 10 that, because an encryption algorithm pro-
duces an apparently random output, it can serve as the basis of a (PRNG). Similarly,
a hash function or MAC produces apparently random output and can be used to
build a PRNG. Both ISO standard 18031 (Random Bit Generation) and NIST SP 800-90 (Recommendation for Random Number Generation Using Deterministic Random Bit Generators) define an approach for random number generation using a cryptographic hash function. SP 800-90 also defines a random number generator
based on HMAC. We look at these two approaches in turn.
PRNG Based on Hash Function
Figure 12.14a shows the basic strategy for a hash-based PRNG specified in SP 800-
90 and ISO 18031. The algorithm takes as input:
V = seed seedlen = bit length of V Ú K + 64, where k is a desired security level expressed in bits
n = desired number of output bits
414 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
The algorithm uses the cryptographic hash function H with an hash value out-
put of outlen bits. The basic operation of the algorithm is
m = <n/outlen= data = V
W = the null string
For i = 1 to m
wi = H (data)
W = } wi data = (data + 1) mod 2seedlen
Return leftmost n bits of W
Thus, the pseudorandom bit stream is w1 }w2 } c }wm with the final block truncated if required.
The SP 800-90 specification also provides for periodically updating V to en- hance security. The specification also indicates that there are no known or suspected
weaknesses in the hash-based approach for a strong cryptographic hash algorithm,
such as SHA-2.
Figure 12.14 Basic Structure of Hash-Based PRNGs (SP 800-90)
(a) PRNG using cryptographic hash function
(b) PRNG using HMAC
V
K
Cryptographic hash function
Pseudorandom output
+1
V
HMAC
Pseudorandom output
12.9 / PSEUDORANDOM NUMBER GENERATION USING HASH FUNCTIONS 415
PRNG Based on MAC Function
Although there are no known or suspected weaknesses in the use of a cryptographic
hash function for a PRNG in the manner of Figure 12.14a, a higher degree of con-
fidence can be achieved by using a MAC. Almost invariably, HMAC is used for
constructing a MAC-based PRNG. This is because HMAC is a widely used stan-
dardized MAC function and is implemented in many protocols and applications. As
SP 800-90 points out, the disadvantage of this approach compared to the hash-based
approach is that the execution time is twice as long, because HMAC involves two
executions of the underlying hash function for each output block. The advantage of
the HMAC approach is that it provides a greater degree of confidence in its secu-
rity, compared to a pure hash-based approach.
For the MAC-based approach, there are two inputs: a key K and a seed V. In effect, the combination of K and V form the overall seed for the PRNG specified in SP 800-90. Figure 12.14b shows the basic structure of the PRNG mechanism, and
the leftmost column of Figure 12.15 shows the logic. Note that the key remains the
same for each block of output, and the data input for each block is equal to the tag
output of the previous block. The SP 800-90 specification also provides for periodi-
cally updating K and V to enhance security. It is instructive to compare the SP 800-90 recommendation with the use of
HMAC for a PRNG in some applications, and this is shown in Figure 12.15. For the
IEEE 802.11i wireless LAN security standard (Chapter 18), the data input consists
of the seed concatenated with a counter. The counter is incremented for each block
wi of output. This approach would seem to offer enhanced security compared to the SP 800-90 approach. Consider that for SP 800-90, the data input for output block
wi is just the output wi - 1 of the previous execution of HMAC. Thus, an opponent who is able to observe the pseudorandom output knows both the input and output
of HMAC. Even so, with the assumption that HMAC is secure, knowledge of the
input and output should not be sufficient to recover K and hence not sufficient to
predict future pseudorandom bits.
The approach taken by the Transport Layer Security protocol (Chapter 17)
and the Wireless Transport Layer Security Protocol (Chapter 18) involves invoking
HMAC twice for each block of output wi. As with IEEE 802.11, this is done in such a way that the output does not yield direct information about the input. The double
use of HMAC doubles the execution burden and would seem to be security overkill.
Figure 12.15 Three PRNGs Based on HMAC
m = <n/outlen= w0 = V W = the null string For i = 1 to m
wi = MAC(K, wi - 1) W = W } wi
Return leftmost n bits of W
m = <n/outlen= W = the null string For i = 1 to m
wi = MAC(K, (V } i)) W = W } wi
Return leftmost n bits of W
m = <n/outlen= A(0) = V W = the null string For i = 1 to m A(i) = MAC(K, A(i - 1)) wi = MAC(K, (A(i) } V) W = W } wi Return leftmost n bits of W
NIST SP 800-90 IEEE 802.11i TLS/WTLS
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416 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
12.10 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
Key Terms
authenticator
Cipher-Based Message
Authentication Code
(CMAC)
CMAC
Counter with Cipher Block
Chaining-Message
Authentication Code
(CCM)
cryptographic checksum
cryptographic hash
function
Data Authentication
Algorithm (DAA)
Galois/Counter Mode
(GCM)
HMAC
key encryption key
Key Wrap mode
key wrapping
message authentication
message authentication code
(MAC)
Review Questions
12.1 What types of attacks are addressed by message authentication? 12.2 What two levels of functionality comprise a message authentication or digital signa-
ture mechanism?
12.3 What are some approaches to producing message authentication? 12.4 When a combination of symmetric encryption and an error control code is used for
message authentication, in what order must the two functions be performed?
12.5 What is a message authentication code? 12.6 What is the difference between a message authentication code and a one-way hash
function?
12.7 In what ways can a hash value be secured so as to provide message authentication? 12.8 Is it necessary to recover the secret key in order to attack a MAC algorithm? 12.9 What changes in HMAC are required in order to replace one underlying hash func-
tion with another?
Problems
12.1 If F is an error-detection function, either internal or external use (Figure 12.2) will provide error-detection capability. If any bit of the transmitted message is altered, this will be reflected in a mismatch of the received FCS and the calculated FCS, whether the FCS function is performed inside or outside the encryption function. Some codes also provide an error-correction capability. Depending on the nature of the function, if one or a small number of bits is altered in transit, the error-correction code contains sufficient redundant information to determine the errored bit or bits and correct them. Clearly, an error-correction code will provide error correction ca- pability when used external to the encryption function. Will it also provide this capa- bility if used internal to the encryption function?
12.2 The data authentication algorithm, described in Section 12.6, can be defined as using the cipher block chaining (CBC) mode of operation of DES with an initialization vec- tor of zero (Figure 12.7). Show that the same result can be produced using the cipher feedback mode.
12.10 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 417
12.3 At the beginning of Section 12.6, it was noted that given the CBC MAC of a one- block message X, say T = MAC(K, X), the adversary immediately knows the CBC MAC for the two-block message X} (X ⊕ T) since this is once again T. Justify this statement.
12.4 In this problem, we demonstrate that for CMAC, a variant that XORs the second key after applying the final encryption doesn’t work. Let us consider this for the case of the message being an integer multiple of the block size. Then, the variant can be expressed as VMAC(K, M) = CBC(K, M) ⊕ K1. Now suppose an adver- sary is able to ask for the MACs of three messages: the message 0 = 0n, where n is the cipher block size; the message 1 = 1n; and the message 1} 0. As a result of these three queries, the adversary gets T0 = CBC(K, 0) ⊕ K1; T1 = CBC(K, 1) ⊕ K1 and T2 = CBC(K, [CBC(K, 1)]) ⊕ K1. Show that the adversary can compute the correct MAC for the (unqueried) message 0} (T0 ⊕ T1).
12.5 In the discussion of subkey generation in CMAC, it states that the block cipher is ap- plied to the block that consists entirely of 0 bits. The first subkey is derived from the resulting string by a left shift of one bit and, conditionally, by XORing a constant that depends on the block size. The second subkey is derived in the same manner from the first subkey. a. What constants are needed for block sizes of 192-bits and 256 bits? b. Explain how the left shift and XOR accomplishes the desired result.
12.6 Section 12.7 listed four general approaches to provide confidentiality and message encryption: H S E, A S E, E S A, and E + A. a. Which of the above performs decryption before verification? b. Which of the above performs verification before decryption?
12.7 Show that the GHASH function calculates
(X1 # Hm) ⊕ (X2 # Hm - 1) ⊕ g ⊕ (Xm - 1 # H2) ⊕ (Xm # H)
12.8 Draw a figure similar to Figure 12.11 that shows authenticated decryption. 12.9 Alice want to send a single bit of information (a yes or a no) to Bob by means of a
word of length 2. Alice and Bob have four possible keys available to perform mes- sage authentication. The following matrix shows the 2-bit word sent for each message under each key:
Message
Key 0 1
1 00 11
2 01 10
3 10 01
4 11 00
a. The preceding matrix is in a useful form for Alice. Construct a matrix with the same information that would be more useful for Bob.
b. What is the probability that someone else can successfully impersonate Alice? c. What is the probability that someone can replace an intercepted message with
another message successfully?
12.10 Draw figures similar to Figures 12.12 and 12.13 for the unwrap algorithm.
418 CHAPTER 12 / MESSAGE AUTHENTICATION CODES
12.11 Consider the following key wrapping algorithm:
1. Initialize variables. A = A6A6A6A6A6A6A6A6 for i = 1 to n R(i) = Pi
2. Calculate intermediate values. for j = 0 to 5 for i = 1 to n B = E(K, [A } R(i)]) t = (n × j) + i A = t ⊕ MSB64(B) R(i) = LSB64(B)
3. Output results. C0 = A for i = 1 to n Ci = R(i)
a. Compare this algorithm, functionally, with the algorithm specified in SP 800-38F and described in Section 12.8.
b. Write the corresponding unwrap algorithm.
419
13.1 Digital Signatures
Properties
Attacks and Forgeries
Digital Signature Requirements
Direct Digital Signature
13.2 Elgamal Digital Signature Scheme
13.3 Schnorr Digital Signature Scheme
13.4 NIST Digital Signature Algorithm
The DSA Approach
The Digital Signature Algorithm
13.5 Elliptic Curve Digital Signature Algorithm
Global Domain Parameters
Key Generation
Digital Signature Generation and Authentication
13.6 RSA-PSS Digital Signature Algorithm
Mask Generation Function
The Signing Operation
Signature Verification
13.7 Key Terms, Review Questions, and Problems
CHAPTER
Digital Signatures
420 CHAPTER 13 / DIGITAL SIGNATURES
The most important development from the work on public-key cryptography is the
digital signature. The digital signature provides a set of security capabilities that would
be difficult to implement in any other way.
Figure 13.1 is a generic model of the process of constructing and using digital
signatures. All of the digital signature schemes discussed in this chapter have this
structure. Suppose that Bob wants to send a message to Alice. Although it is not
important that the message be kept secret, he wants Alice to be certain that the
message is indeed from him. For this purpose, Bob uses a secure hash function, such
as SHA-512, to generate a hash value for the message. That hash value, together
with Bob’s private key serves as input to a digital signature generation algorithm,
which produces a short block that functions as a digital signature. Bob sends the message with the signature attached. When Alice receives the message plus signa-
ture, she (1) calculates a hash value for the message; (2) provides the hash value and
Bob’s public key as inputs to a digital signature verification algorithm. If the algo-
rithm returns the result that the signature is valid, Alice is assured that the message
must have been signed by Bob. No one else has Bob’s private key and therefore no
one else could have created a signature that could be verified for this message with
Bob’s public key. In addition, it is impossible to alter the message without access to
Bob’s private key, so the message is authenticated both in terms of source and in
terms of data integrity.
We begin this chapter with an overview of digital signatures. We then present the
Elgamal and Schnorr digital signature schemes, understanding of which makes it easier
to understand the NIST Digital Signature Algorithm (DSA). The chapter then cov-
ers the two other important standardized digital signature schemes: the Elliptic Curve
Digital Signature Algorithm (ECDSA) and the RSA Probabilistic Signature Scheme
(RSA-PSS).
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Present an overview of the digital signature process.
◆ Understand the Elgamal digital signature scheme.
◆ Understand the Schnorr digital signature scheme.
◆ Understand the NIST digital signature scheme.
◆ Compare and contrast the NIST digital signature scheme with the Elgamal and Schnorr digital signature schemes.
◆ Understand the elliptic curve digital signature scheme.
◆ Understand the RSA-PSS digital signature scheme.
13.1 / DIGITAL SIGNATURES 421
13.1 DIGITAL SIGNATURES
Properties
Message authentication protects two parties who exchange messages from any third
party. However, it does not protect the two parties against each other. Several forms
of dispute between the two parties are possible.
Figure 13.1 Simplified Depiction of Essential Elements of Digital Signature Process
Bob Alice
Bob’s signature
for M
Message M
Cryptographic hash
function
Digital signature
generation algorithm
Digital signature
verification algorithm
h
Message M
Cryptographic hash
function
h
S
Message M S Return signature
valid or not valid
Bob’s private
key
(a) Bob signs a message (b) Alice verifies the signature
Bob’s public key
422 CHAPTER 13 / DIGITAL SIGNATURES
For example, suppose that John sends an authenticated message to Mary,
using one of the schemes of Figure 12.1. Consider the following disputes that could
arise.
1. Mary may forge a different message and claim that it came from John. Mary would simply have to create a message and append an authentication code
using the key that John and Mary share.
2. John can deny sending the message. Because it is possible for Mary to forge a message, there is no way to prove that John did in fact send the message.
Both scenarios are of legitimate concern. Here is an example of the first
scenario: An electronic funds transfer takes place, and the receiver increases the
amount of funds transferred and claims that the larger amount had arrived from
the sender. An example of the second scenario is that an electronic mail message
contains instructions to a stockbroker for a transaction that subsequently turns out
badly. The sender pretends that the message was never sent.
In situations where there is not complete trust between sender and receiver,
something more than authentication is needed. The most attractive solution to
this problem is the digital signature. The digital signature must have the following
properties:
■ It must verify the author and the date and time of the signature.
■ It must authenticate the contents at the time of the signature.
■ It must be verifiable by third parties, to resolve disputes.
Thus, the digital signature function includes the authentication function.
Attacks and Forgeries
[GOLD88] lists the following types of attacks, in order of increasing severity. Here
A denotes the user whose signature method is being attacked, and C denotes the
attacker.
■ Key-only attack: C only knows A’s public key.
■ Known message attack: C is given access to a set of messages and their signatures.
■ Generic chosen message attack: C chooses a list of messages before attempt- ing to breaks A’s signature scheme, independent of A’s public key. C then
obtains from A valid signatures for the chosen messages. The attack is generic,
because it does not depend on A’s public key; the same attack is used against
everyone.
■ Directed chosen message attack: Similar to the generic attack, except that the list of messages to be signed is chosen after C knows A’s public key but before
any signatures are seen.
■ Adaptive chosen message attack: C is allowed to use A as an “oracle.” This means that C may request from A signatures of messages that depend on
previously obtained message-signature pairs.
13.1 / DIGITAL SIGNATURES 423
[GOLD88] then defines success at breaking a signature scheme as an outcome
in which C can do any of the following with a non-negligible probability:
■ Total break: C determines A’s private key.
■ Universal forgery: C finds an efficient signing algorithm that provides an equivalent way of constructing signatures on arbitrary messages.
■ Selective forgery: C forges a signature for a particular message chosen by C.
■ Existential forgery: C forges a signature for at least one message. C has no control over the message. Consequently, this forgery may only be a minor
nuisance to A.
Digital Signature Requirements
On the basis of the properties and attacks just discussed, we can formulate the
following requirements for a digital signature.
■ The signature must be a bit pattern that depends on the message being signed.
■ The signature must use some information only known to the sender to prevent
both forgery and denial.
■ It must be relatively easy to produce the digital signature.
■ It must be relatively easy to recognize and verify the digital signature.
■ It must be computationally infeasible to forge a digital signature, either by
constructing a new message for an existing digital signature or by constructing
a fraudulent digital signature for a given message.
■ It must be practical to retain a copy of the digital signature in storage.
A secure hash function, embedded in a scheme such as that of Figure 13.1, provides
a basis for satisfying these requirements. However, care must be taken in the design
of the details of the scheme.
Direct Digital Signature
The term direct digital signature refers to a digital signature scheme that involves only the communicating parties (source, destination). It is assumed that the destina-
tion knows the public key of the source.
Confidentiality can be provided by encrypting the entire message plus
signature with a shared secret key (symmetric encryption). Note that it is important
to perform the signature function first and then an outer confidentiality function.
In case of dispute, some third party must view the message and its signature. If the
signature is calculated on an encrypted message, then the third party also needs
a ccess to the decryption key to read the original message. However, if the signature
is the inner operation, then the recipient can store the plaintext message and its
signature for later use in dispute resolution.
The validity of the scheme just described depends on the security of the send-
er’s private key. If a sender later wishes to deny sending a particular message, the
sender can claim that the private key was lost or stolen and that someone else forged
his or her signature. Administrative controls relating to the security of private keys
424 CHAPTER 13 / DIGITAL SIGNATURES
can be employed to thwart or at least weaken this ploy, but the threat is still there,
at least to some degree. One example is to require every signed message to include
a timestamp (date and time) and to require prompt reporting of compromised keys to a central authority.
Another threat is that a private key might actually be stolen from X at time T.
The opponent can then send a message signed with X’s signature and stamped with
a time before or equal to T.
The universally accepted technique for dealing with these threats is the use
of a digital certificate and certificate authorities. We defer a discussion of this topic
until Chapter 14, and focus in this chapter on digital signature algorithms.
13.2 ELGAMAL DIGITAL SIGNATURE SCHEME
Before examining the NIST Digital Signature Algorithm, it will be helpful to under-
stand the Elgamal and Schnorr signature schemes. Recall from Chapter 10, that the
Elgamal encryption scheme is designed to enable encryption by a user’s public key
with decryption by the user’s private key. The Elgamal signature scheme involves
the use of the private key for digital signature generation and the public key for
digital signature verification [ELGA84, ELGA85].
Before proceeding, we need a result from number theory. Recall from Chapter 2
that for a prime number q, if a is a primitive root of q, then
a, a2, c , aq - 1
are distinct (mod q). It can be shown that, if a is a primitive root of q, then
1. For any integer m, am K 1 (mod q) if and only if m K 0 (mod q - 1). 2. For any integers, i, j, ai K aj (mod q) if and only if i K j (mod q - 1).
As with Elgamal encryption, the global elements of Elgamal digital signature are a prime number q and a, which is a primitive root of q. User A generates a private/public key pair as follows.
1. Generate a random integer XA, such that 1 6 XA 6 q - 1. 2. Compute YA = a
XA mod q.
3. A’s private key is XA; A’s pubic key is {q, a, YA}.
To sign a message M, user A first computes the hash m = H(M), such that m is an integer in the range 0 … m … q - 1. A then forms a digital signature as follows.
1. Choose a random integer K such that 1 … K … q - 1 and gcd(K, q - 1) = 1. That is, K is relatively prime to q - 1.
2. Compute S1 = a K mod q. Note that this is the same as the computation of C1
for Elgamal encryption.
3. Compute K-1 mod (q - 1). That is, compute the inverse of K modulo q - 1. 4. Compute S2 = K
-1(m - XAS1) mod (q - 1). 5. The signature consists of the pair (S1, S2).
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13.3 / SCHNORR DIGITAL SIGNATURE SCHEME 425
Any user B can verify the signature as follows.
1. Compute V1 = a m mod q.
2. Compute V2 = (YA) S1(S1)
S2 mod q.
The signature is valid if V1 = V2. Let us demonstrate that this is so. Assume that the equality is true. Then we have
am mod q = (YA) S1(S1)
S2 mod q assume V1 = V2 am mod q = aXAS1aKS2 mod q substituting for YA and S1 am - XAS1 mod q = aKS2 mod q rearranging terms m - XAS1 K KS2 mod (q - 1) property of primitive roots m - XAS1 K KK-1 (m - XAS1) mod (q - 1) substituting for S2
For example, let us start with the prime field GF(19); that is, q = 19. It has primitive roots {2, 3, 10, 13, 14, 15}, as shown in Table 2.7. We choose a = 10.
Alice generates a key pair as follows:
1. Alice chooses XA = 16. 2. Then YA = a
XA mod q = a16 mod 19 = 4. 3. Alice’s private key is 16; Alice’s pubic key is {q, a, YA} = {19, 10, 4}.
Suppose Alice wants to sign a message with hash value m = 14.
1. Alice chooses K = 5, which is relatively prime to q - 1 = 18. 2. S1 = a
K mod q = 105 mod 19 = 3 (see Table 2.7). 3. K-1 mod (q - 1) = 5-1 mod 18 = 11. 4. S2 = K
-1 (m - XAS1) mod (q - 1) = 11 (14 - (16)(3)) mod 18 = - 374 mod 18 = 4.
Bob can verify the signature as follows.
1. V1 = a m mod q = 1014 mod 19 = 16.
2. V2 = (YA) S1(S1)
S2 mod q = (43)(34) mod 19 = 5184 mod 19 = 16.
Thus, the signature is valid because V1 = V2.
13.3 SCHNORR DIGITAL SIGNATURE SCHEME
As with the Elgamal digital signature scheme, the Schnorr signature scheme is
based on discrete logarithms [SCHN89, SCHN91]. The Schnorr scheme minimizes
the message-dependent amount of computation required to generate a signature.
The main work for signature generation does not depend on the message and can
be done during the idle time of the processor. The message-dependent part of the
signature generation requires multiplying a 2n-bit integer with an n-bit integer. The scheme is based on using a prime modulus p, with p - 1 having a prime
factor q of appropriate size; that is, p - 1 K 0 (mod q). Typically, we use p ≈ 21024 and q ≈ 2160. Thus, p is a 1024-bit number, and q is a 160-bit number, which is also the length of the SHA-1 hash value.
426 CHAPTER 13 / DIGITAL SIGNATURES
The first part of this scheme is the generation of a private/public key pair,
which consists of the following steps.
1. Choose primes p and q, such that q is a prime factor of p - 1. 2. Choose an integer a, such that aq = 1 mod p. The values a, p, and q comprise a
global public key that can be common to a group of users.
3. Choose a random integer s with 0 6 s 6 q. This is the user’s private key. 4. Calculate v = a-s mod p. This is the user’s public key.
A user with private key s and public key v generates a signature as follows.
1. Choose a random integer r with 0 6 r 6 q and compute x = ar mod p. This computation is a preprocessing stage independent of the message M to be signed.
2. Concatenate the message with x and hash the result to compute the value e:
e = H(M }x)
3. Compute y = (r + se) mod q. The signature consists of the pair (e, y).
Any other user can verify the signature as follows.
1. Compute x′ = ayve mod p. 2. Verify that e = H (M }x′).
To see that the verification works, observe that
x′ K ayve K aya-se K ay - se K ar K x (mod p)
Hence, H (M }x′) = H (M }x).
13.4 NIST DIGITAL SIGNATURE ALGORITHM
The National Institute of Standards and Technology (NIST) has published
Federal Information Processing Standard FIPS 186, known as the Digital
Signature Algorithm (DSA). The DSA makes use of the Secure Hash Algorithm
(SHA) described in Chapter 12. The DSA was originally proposed in 1991 and
revised in 1993 in response to public feedback concerning the security of the
scheme. There was a further minor revision in 1996. In 2000, an expanded version
of the standard was issued as FIPS 186-2, subsequently updated to FIPS 186-3 in
2009, and FIPS 186-4 in 2013. This latest version also incorporates digital signa-
ture algorithms based on RSA and on elliptic curve cryptography. In this section,
we discuss DSA.
The DSA Approach
The DSA uses an algorithm that is designed to provide only the digital signa-
ture function. Unlike RSA, it cannot be used for encryption or key exchange.
Nevertheless, it is a public-key technique.
13.4 / NIST DIGITAL SIGNATURE ALGORITHM 427
Figure 13.2 contrasts the DSA approach for generating digital signatures to
that used with RSA. In the RSA approach, the message to be signed is input to a
hash function that produces a secure hash code of fixed length. This hash code is
then encrypted using the sender’s private key to form the signature. Both the mes-
sage and the signature are then transmitted. The recipient takes the message and
produces a hash code. The recipient also decrypts the signature using the sender’s
public key. If the calculated hash code matches the decrypted signature, the signa-
ture is accepted as valid. Because only the sender knows the private key, only the
sender could have produced a valid signature.
The DSA approach also makes use of a hash function. The hash code is pro-
vided as input to a signature function along with a random number k generated for this particular signature. The signature function also depends on the sender’s private
key (PRa) and a set of parameters known to a group of communicating principals. We can consider this set to constitute a global public key (PUG).
1 The result is a
signature consisting of two components, labeled s and r. At the receiving end, the hash code of the incoming message is generated. The
hash code and the signature are inputs to a verification function. The verification
function also depends on the global public key as well as the sender’s public key
(PUa), which is paired with the sender’s private key. The output of the verification function is a value that is equal to the signature component r if the signature is valid. The signature function is such that only the sender, with knowledge of the private
key, could have produced the valid signature.
We turn now to the details of the algorithm.
1It is also possible to allow these additional parameters to vary with each user so that they are a part of a user’s public key. In practice, it is more likely that a global public key will be used that is separate from each user’s public key.
Figure 13.2 Two Approaches to Digital Signatures
(a) RSA approach
M
H
| | M
Sig Ver
H
Compare
k
s r
(b) DSA approach
M
H
| | M
E D
H
ComparePRa
PRaPUG PUaPUG
PUa
E(PRa, H(M)]
428 CHAPTER 13 / DIGITAL SIGNATURES
The Digital Signature Algorithm
DSA is based on the difficulty of computing discrete logarithms (see Chapter 2)
and is based on schemes originally presented by Elgamal [ELGA85] and Schnorr
[SCHN91].
Figure 13.3 summarizes the algorithm. There are three parameters that are
public and can be common to a group of users. An N-bit prime number q is chosen. Next, a prime number p is selected with a length between 512 and 1024 bits such that q divides (p - 1). Finally, g is chosen to be of the form h(p - 1)/q mod p, where h is an integer between 1 and (p - 1) with the restriction that g must be greater than 1.2 Thus, the global public-key components of DSA are the same as in the
Schnorr signature scheme.
With these parameters in hand, each user selects a private key and generates
a public key. The private key x must be a number from 1 to (q - 1) and should be chosen randomly or pseudorandomly. The public key is calculated from the
private key as y = gx mod p. The calculation of y given x is relatively straight- forward. However, given the public key y, it is believed to be computationally infeasible to determine x, which is the discrete logarithm of y to the base g, mod p (see Chapter 2).
2In number-theoretic terms, g is of order q mod p; see Chapter 2.
Global Public-Key Components
p prime number where 2L - 1 6 p 6 2L for 512 … L … 1024 and L a multiple of 64; i.e., bit length L between 512 and 1024 bits in increments of 64 bits
q prime divisor of (p - 1), where 2N - 1 6 q 6 2N i.e., bit length of N bits
g = h(p - 1)/q is an exponent mod p, where h is any integer with 1 6 h 6 (p - 1) such that h(p - 1)/q mod p 7 1
User’s Private Key
x random or pseudorandom integer with 0 6 x 6 q
User’s Public Key
y = gx mod p
User’s Per-Message Secret Number
k random or pseudorandom integer with 0 6 k 6 q
Signing
r = (gk mod p) mod q
s = [k -1 (H(M) + xr)] mod q
Signature = (r, s)
Verifying
w = (s′)-1 mod q
u1 = [H(M′)w] mod q
u2 = (r′)w mod q
v = [(gu1yu2) mod p] mod q
TEST: v = r′
M = message to be signed
H(M) = hash of M using SHA-1
M′, r′, s′ = received versions of M, r, s
Figure 13.3 The Digital Signature Algorithm (DSA)
13.4 / NIST DIGITAL SIGNATURE ALGORITHM 429
The signature of a message M consists of the pair of numbers r and s, which are functions of the public key components (p, q, g), the user’s private key (x), the hash code of the message H(M), and an additional integer k that should be generated randomly or pseudorandomly and be unique for each signing.
Let M, r′, and s′ be the received versions of M, r, and s, respectively. Verification is performed using the formulas shown in Figure 13.3. The receiver
generates a quantity v that is a function of the public key components, the sender’s public key, the hash code of the incoming message, and the received versions of r and s. If this quantity matches the r component of the signature, then the signature is validated.
Figure 13.4 depicts the functions of signing and verifying.
Figure 13.4 DSA Signing and Verifying
(a) Signing
(b) Verifying
M
s r
Mœ
rœ
H
r = (gk mod p) mod q
q
s = [k–1 (H(M) + xr)] mod q
w = (sœ)–1 mod q
k k
k
q
x x
M
H(M)
H H(Mœ)
p g
q
q
y
v
g
u1 = [H(M œ)w)] mod q
u2 = (r œ)w mod q
v = [(gu1yu2) mod p] mod q
signature verification
rœ = v?
rœ
rœ
w
sœ
430 CHAPTER 13 / DIGITAL SIGNATURES
The structure of the algorithm, as revealed in Figure 13.4, is quite interesting.
Note that the test at the end is on the value r, which does not depend on the mes- sage at all. Instead, r is a function of k and the three global public-key components. The multiplicative inverse of k (mod q) is passed to a function that also has as inputs the message hash code and the user’s private key. The structure of this function is
such that the receiver can recover r using the incoming message and signature, the public key of the user, and the global public key. It is certainly not obvious from
Figure 13.3 or Figure 13.4 that such a scheme would work. A proof is provided in
Appendix K.
Given the difficulty of taking discrete logarithms, it is infeasible for an
opponent to recover k from r or to recover x from s. Another point worth noting is that the only computationally demanding
task in signature generation is the exponential calculation gk mod p. Because this value does not depend on the message to be signed, it can be computed ahead of
time. Indeed, a user could precalculate a number of values of r to be used to sign documents as needed. The only other somewhat demanding task is the determi-
nation of a multiplicative inverse, k -1. Again, a number of these values can be precalculated.
13.5 ELLIPTIC CURVE DIGITAL SIGNATURE ALGORITHM
As was mentioned, the 2009 version of FIPS 186 includes a new digital signature
technique based on elliptic curve cryptography, known as the Elliptic Curve Digital Signature Algorithm (ECDSA). ECDSA is enjoying increasing acceptance due to the efficiency advantage of elliptic curve cryptography, which yields security
comparable to that of other schemes with a smaller key bit length.
First we give a brief overview of the process involved in ECDSA. In essence,
four elements are involved.
1. All those participating in the digital signature scheme use the same global domain parameters, which define an elliptic curve and a point of origin on the curve.
2. A signer must first generate a public, private key pair. For the private key, the signer selects a random or pseudorandom number. Using that random number
and the point of origin, the signer computes another point on the elliptic curve.
This is the signer’s public key.
3. A hash value is generated for the message to be signed. Using the private key, the domain parameters, and the hash value, a signature is generated. The
signature consists of two integers, r and s.
4. To verify the signature, the verifier uses as input the signer’s public key, the domain parameters, and the integer s. The output is a value v that is compared to r. The signature is verified if v = r.
Let us examine each of these four elements in turn.
13.5 / ELLIPTIC CURVE DIGITAL SIGNATURE ALGORITHM 431
Global Domain Parameters
Recall from Chapter 10 that two families of elliptic curves are used in cryptographic
applications: prime curves over Zp and binary curves over GF(2 m). For ECDSA,
prime curves are used. The global domain parameters for ECDSA are the following:
q a prime number
a, b integers that specify the elliptic curve equation defined over Zq with the equation y2 = x3 + ax + b
G a base point represented by G = (xg, yg) on the elliptic curve equation n order of point G; that is, n is the smallest positive integer such that
nG = O. This is also the number of points on the curve.
Key Generation
Each signer must generate a pair of keys, one private and one public. The signer,
let us call him Bob, generates the two keys using the following steps:
1. Select a random integer d, d ∈ [1, n - 1] 2. Compute Q = dG. This is a point in E q(a, b) 3. Bob’s public key is Q and private key is d.
Digital Signature Generation and Authentication
With the public domain parameters and a private key in hand, Bob generates
a digital signature of 320 bytes for message m using the following steps:
1. Select a random or pseudorandom integer k, k ∈ [1, n - 1] 2. Compute point P = (x, y) = kG and r = x mod n. If r = 0 then goto step 1 3. Compute t = k -1 mod n 4. Compute e = H(m), where H is one of the SHA-2 or SHA-3 hash functions. 5. Compute s = k -1(e + dr) mod n. If s = O then goto step 1 6. The signature of message m is the pair (r, s).
Alice knows the public domain parameters and Bob’s public key. Alice is
presented with Bob’s message and digital signature and verifies the signature using
the following steps:
1. Verify that r and s are integers in the range 1 through n - 1 2. Using SHA, compute the 160-bit hash value e = H(m) 3. Compute w = s-1 mod n 4. Compute u1 = ew and u2 = rw 5. Compute the point X = (x1, y1) = u1G + u2Q 6. If X = O, reject the signature else compute v = x1 mod n 7. Accept Bob’s signature if and only if v = r
432 CHAPTER 13 / DIGITAL SIGNATURES
Figure 13.5 illustrates the signature authentication process. We can verify that
this process is valid as follows. If the message received by Alice is in fact signed by
Bob, then
s = k -1(e + dr) mod n
Then
k = s-1(e + dr) mod n k = (s-1e + s-1dr) mod n k = (we + wdr) mod n k = (u1 + u2d) mod n
Now consider that
u1G + u2Q = u1G + u2dG = (u1 + u2d)G = kG
Figure 13.5 ECDSA Signing and Verifying
Q
Yes
No
No
No
No
No
Yes
Yes
Yes
Yes
r, s
Accept signature
Reject signature
Bob Alice
Generate k (x, y) = kG r = x mod n
Generate private key d. Public key Q = dG
q, a, b, G, n are shared global variables
Signature of m is r, s
v = x1 mod n
e = H(m) s = k–1 (e + dr) mod n
e = H(m) w = s–1 mod n u1 = ew, u2 = rw X = (x1, x2) = u1G + u2Q
r = 0?
s = 0?
X = O?
v = r?
r, s integers in range [1, n–1]?
13.6 / RSA-PSS DIGITAL SIGNATURE ALGORITHM 433
In step 6 of the verification process, we have v = x1 mod n, where point X = (x1, y1) = u1G + u2Q. Thus we see that v = r since r = x mod n and x is the x coordinate of the point kG and we have already seen that u1G + u2Q = kG.
13.6 RSA-PSS DIGITAL SIGNATURE ALGORITHM
In addition to the NIST Digital Signature Algorithm and ECDSA, the 2009 version
of FIPS 186 also includes several techniques based on RSA, all of which were devel-
oped by RSA Laboratories and are in wide use. A worked-out example, using RSA,
is available at this book’s Web site.
In this section, we discuss the RSA Probabilistic Signature Scheme (RSA-PSS),
which is the latest of the RSA schemes and the one that RSA Laboratories recom-
mends as the most secure of the RSA schemes.
Because the RSA-based schemes are widely deployed in many applications,
including financial applications, there has been great interest in demonstrating that
such schemes are secure. The three main RSA signature schemes differ mainly in
the padding format the signature generation operation employs to embed the hash
value into a message representative, and in how the signature verification opera-
tion determines that the hash value and the message representative are consistent.
For all of the schemes developed prior to PSS, it has not been possible to develop
a mathematical proof that the signature scheme is as secure as the underlying RSA
encryption/decryption primitive [KALI01]. The PSS approach was first proposed by
Bellare and Rogaway [BELL96c, BELL98]. This approach, unlike the other RSA-
based schemes, introduces a randomization process that enables the security of the
method to be shown to be closely related to the security of the RSA algorithm itself.
This makes RSA-PSS more desirable as the choice for RSA-based digital signature
applications.
Mask Generation Function
Before explaining the RSA-PSS operation, we need to describe the mask gener-
ation function (MGF) used as a building block. MGF(X, maskLen) is a pseudo- random function that has as input parameters a bit string X of any length and the desired length L in octets of the output. MGFs are typically based on a secure cryptographic hash function such as SHA-1. An MGF based on a hash function is
intended to be a cryptographically secure way of generating a message digest, or
hash, of variable length based on an underlying cryptographic hash function that
produces a fixed-length output.
The MGF function used in the current specification for RSA-PSS is MGF1,
with the following parameters:
Options Hash hash function with output hLen octets
Input X octet string to be masked
maskLen length in octets of the mask
Output mask an octet string of length maskLen
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434 CHAPTER 13 / DIGITAL SIGNATURES
MGF1 is defined as follows:
1. Initialize variables.
T = empty string
k = <maskLen/hLen= - 1 2. Calculate intermediate values.
for counter = 0 to k
Represent counter as a 32-bit string C
T = T } Hash(X } C) 3. Output results.
mask = the leading maskLen octets of T
In essence, MGF1 does the following. If the length of the desired output is
equal to the length of the hash value (maskLen = hLen), then the output is the hash of the input value X concatenated with a 32-bit counter value of 0. If maskLen is greater than hLen, the MGF1 keeps iterating by hashing X concatenated with the counter and appending that to the current string T. So that the output is
Hash (X }0) }Hash(X }1) } c }Hash(X }k)
This is repeated until the length of T is greater than or equal to maskLen, at which point the output is the first maskLen octets of T.
The Signing Operation
MESSAGE ENCODING The first stage in generating an RSA-PSS signature of a message M is to generate from M a fixed-length message digest, called an encoded message (EM). Figure 13.6 illustrates this process. We define the following parameters and functions:
Options Hash hash function with output hLen octets. The current preferred alternative is SHA-1, which produces a 20-octet
hash value.
MGF mask generation function. The current specification calls
for MGF1.
sLen length in octets of the salt. Typically sLen = hLen, which for the current version is 20 octets.
Input M message to be encoded for signing.
emBits This value is one less than the length in bits of the RSA modulus n.
Output EM encoded message. This is the message digest that will be encrypted to form the digital signature.
Parameters emLen length of EM in octets = <emBits/8=. padding1 hexadecimal string 00 00 00 00 00 00 00 00; that is, a string
of 64 zero bits.
13.6 / RSA-PSS DIGITAL SIGNATURE ALGORITHM 435
padding2 hexadecimal string of 00 octets with a length
(emLen - sLen - hLen - 2) octets, followed by the hexadecimal octet with value 01.
salt a pseudorandom number.
bc the hexadecimal value BC.
The encoding process consists of the following steps.
1. Generate the hash value of M: mHash = Hash(M) 2. Generate a pseudorandom octet string salt and form block M′ = padding1 }
mHash }salt 3. Generate the hash value of M′: H = Hash(M′) 4. Form data block DB = padding2 }salt 5. Calculate the MGF value of H: dbMask = MGF(H, emLen - hLen - 1) 6. Calculate maskedDB = DB ⊕ dbMsk 7. Set the leftmost 8emLen - emBits bits of the leftmost octet in maskedDB to 0 8. EM = maskedDB }H}0xbc
We make several comments about the complex nature of this message
digest algorithm. All of the RSA-based standardized digital signature schemes
involve appending one or more constants (e.g., padding 1 and padding 2) in the
process of forming the message digest. The objective is to make it more difficult
for an adversary to find another message that maps to the same message digest
Figure 13.6 RSA-PSS Encoding
Hash
Hash
MGF
M
mHash saltpadding1
bcmaskedDB
saltpadding2
Mœ =
DB =
EM = H
436 CHAPTER 13 / DIGITAL SIGNATURES
as a given message or to find two messages that map to the same message digest.
RSA-PSS also incorporates a pseudorandom number, namely the salt. Because the
salt changes with every use, signing the same message twice using the same private
key will yield two different signatures. This is an added measure of security.
FORMING THE SIGNATURE We now show how the signature is formed by a signer with private key {d, n} and public key {e, n} (see Figure 9.5). Treat the octet string EM as an unsigned, nonnegative binary integer m. The signature s is formed by encrypting m as follows:
s = md mod n
Let k be the length in octets of the RSA modulus n. For example if the key size for RSA is 2048 bits, then k = 2048/8 = 256. Then convert the signature value s into the octet string S of length k octets.
Signature Verification
DECRYPTION For signature verification, treat the signature S as an unsigned, nonnegative binary integer s. The message digest m is recovered by decrypting s as follows:
m = se mod n
Then, convert the message representative m to an encoded message EM of length emLen = <(modBits - 1)/8= octets, where modBits is the length in bits of the RSA modulus n.
EM VERIFICATION EM verification can be described as follows:
Options Hash hash function with output hLen octets.
MGF mask generation function.
sLen length in octets of the salt.
Input M message to be verified.
EM the octet string representing the decrypted signature, with length emLen = <emBits/8=.
emBits This value is one less than the length in bits of the RSA modulus n.
Parameters padding1 hexadecimal string 00 00 00 00 00 00 00 00; that is, a string of 64 zero bits.
padding2 hexadecimal string of 00 octets with a length
(emLen - sLen - hLen - 2) octets, followed by the hexadecimal octet with value 01.
1. Generate the hash value of M: mHash = Hash(M) 2. If emLen 6 hLen + sLen + 2, output “inconsistent” and stop 3. If the rightmost octet of EM does not have hexadecimal value BC, output
“ inconsistent” and stop
13.6 / RSA-PSS DIGITAL SIGNATURE ALGORITHM 437
4. Let maskedDB be the leftmost emLen - hLen - 1 octets of EM, and let H be the next hLen octets
5. If the leftmost 8emLen - emBits bits of the leftmost octet in maskedDB are not all equal to zero, output “inconsistent” and stop
6. Calculate dbMask = MGF (H, emLen - hLen - 1) 7. Calculate DB = maskedDB ⊕ dbMsk 8. Set the leftmost 8emLen - emBits bits of the leftmost octet in DB to zero 9. If the leftmost (emLen - hLen - sLen - 1) octets of DB are not equal to
padding2, output “inconsistent” and stop
10. Let salt be the last sLen octets of DB
11. Form block M′ = padding1 }mHash }salt 12. Generate the hash value of M′: H′ = Hash(M′) 13. If H = H′, output “consistent.” Otherwise, output “inconsistent”
Figure 13.7 illustrates the process. The shaded boxes labeled H and H′ cor- respond, respectively, to the value contained in the decrypted signature and the
value generated from the message M associated with the signature. The remaining three shaded areas contain values generated from the decrypted signature and com-
pared to known constants. We can now see more clearly the different roles played
by the constants and the pseudorandom value salt, all of which are embedded in the
Figure 13.7 RSA-PSS EM Verification
Hash
Hash
MGF
M
mHash saltpadding1
maskedDB
dbMask
salt
= Mœ
DB =
EM = H
Hœ
438 CHAPTER 13 / DIGITAL SIGNATURES
EM generated by the signer. The constants are known to the verifier, so that the computed constants can be compared to the known constants as an additional check
that the signature is valid (in addition to comparing H and H′). The salt results in a different signature every time a given message is signed with the same private key.
The verifier does not know the value of the salt and does not attempt a comparison.
Thus, the salt plays a similar role to the pseudorandom variable k in the NIST DSA and in ECDSA. In both of those schemes, k is a pseudorandom number generated by the signer, resulting in different signatures from multiple signings of the same mes-
sage with the same private key. A verifier does not and need not know the value of k.
13.7 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
Key Terms
digital signature
Digital Signature Algorithm
(DSA)
direct digital signature
Elgamal digital signature
Elliptic Curve Digital
Signature Algorithm
(ECDSA)
Schnorr digital signature
timestamp
Review Questions 13.1 List two disputes that can arise in the context of message authentication. 13.2 What are the properties a digital signature should have? 13.3 What requirements should a digital signature scheme satisfy? 13.4 What is the difference between direct and arbitrated digital signature? 13.5 In what order should the signature function and the confidentiality function be
applied to a message, and why?
13.6 What are some threats associated with a direct digital signature scheme?
Problems 13.1 Dr. Watson patiently waited until Sherlock Holmes finished. “Some interesting prob-
lem to solve, Holmes?” he asked when Holmes finally logged out.
“Oh, not exactly. I merely checked my email and then made a couple of network experiments instead of my usual chemical ones. I have only one client now and I have already solved his problem. If I remember correctly, you once mentioned cryptology among your other hobbies, so it may interest you.”
“Well, I am only an amateur cryptologist, Holmes. But of course I am interested in the problem. What is it about?”
“My client is Mr. Hosgrave, director of a small but progressive bank. The bank is fully computerized and of course uses network communications extensively. The bank already uses RSA to protect its data and to digitally sign documents that are communicated. Now the bank wants to introduce some changes in its procedures; in particular, it needs to digitally sign some documents by two signatories.
1. The first signatory prepares the document, forms its signature, and passes the document to the second signatory.
13.7 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 439
2. The second signatory as a first step must verify that the document was really signed by the first signatory. She then incorporates her signature into the document’s sig- nature so that the recipient, as well as any member of the public, may verify that the document was indeed signed by both signatories. In addition, only the second signa- tory has to be able to verify the document’s signature after the first step; that is, the recipient (or any member of the public) should be able to verify only the complete document with signatures of both signatories, but not the document in its intermedi- ate form where only one signatory has signed it. Moreover, the bank would like to make use of its existing modules that support RSA-style digital signatures.”
“Hm, I understand how RSA can be used to digitally sign documents by one signatory, Holmes. I guess you have solved the problem of Mr. Hosgrave by appropriate gener- alization of RSA digital signatures.”
“Exactly, Watson,” nodded Sherlock Holmes. “Originally, the RSA digital sig- nature was formed by encrypting the document by the signatory’s private decryption key ‘d’, and the signature could be verified by anyone through its decryption using publicly known encryption key ‘e’. One can verify that the signature S was formed by the person who knows d, which is supposed to be the only signatory. Now the problem of Mr. Hosgrave can be solved in the same way by slight generalization of the process, that is …”
Finish the explanation.
13.2 DSA specifies that if the signature generation process results in a value of s = 0, a new value of k should be generated and the signature should be recalculated. Why?
13.3 What happens if a k value used in creating a DSA signature is compromised? 13.4 The DSA document includes a recommended algorithm for testing a number for
primality. 1. [Choose w] Let w be a random odd integer. Then (w - 1) is even and can be
expressed in the form 2am with m odd. That is, 2a is the largest power of 2 that divides (w - 1).
2. [Generate b] Let b be a random integer in the range 1 6 b 6 w. 3. [Exponentiate] Set j = 0 and z = bm mod w. 4. [Done?] If j = 0 and z = 1, or if z = w - 1, then w passes the test and may be
prime; go to step 8. 5. [Terminate?] If j 7 0 and z = 1, then w is not prime; terminate algorithm for this w. 6. [Increase j] Set j = j + 1. If j 6 a, set z = z2 mod w and go to step 4. 7. [Terminate] w is not prime; terminate algorithm for this w. 8. [Test again?] If enough random values of b have been tested, then accept w as
prime and terminate algorithm; otherwise, go to step 2. a. Explain how the algorithm works. b. Show that it is equivalent to the Miller–Rabin test described in Chapter 2.
13.5 With DSA, because the value of k is generated for each signature, even if the same message is signed twice on different occasions, the signatures will differ. This is not true of RSA signatures. What is the practical implication of this difference?
13.6 Consider the problem of creating domain parameters for DSA. Suppose we have already found primes p and q such that q�(p - 1). Now we need to find g ∈ Zp with g of order q mod p. Consider the following two algorithms:
Algorithm 1 Algorithm 2
repeat repeat select g ∈ Zp select h ∈ Zp h d gq mod p g d h(p - 1)/q mod p until (h = 1 and g ≠ 1) until (g ≠ 1) return g return g
440 CHAPTER 13 / DIGITAL SIGNATURES
a. What happens in Algorithm 1 if ord(q) = q is chosen? b. hat happens in Algorithm 2 if ord(q) = q is chosen? c. Suppose p = 64891 and q = 421. How many loop iterations do you expect
Algorithm 1 to make before it finds a generator? d. If p is 512 bits and q is 128 bits, would you recommend using Algorithm 1 to find g?
Explain. e. Suppose p = 64891 and q = 421. What is the probability that Algorithm 2
computes a generator in its very first loop iteration? (If it is helpful, you may use
the fact that a (d�n)c(d) = n when answering this question.)
13.7 It is tempting to try to develop a variation on Diffie–Hellman that could be used as a digital signature. Here is one that is simpler than DSA and that does not require a secret random number in addition to the private key.
Public elements: q prime number
a a 6 q and a is primitive root of q Private key: X X 6 q Public key: Y = aX mod q mod q
To sign a message M, compute h = H(M), which is the hash code of the message. We require that gcd(h, q - 1) = 1. If not, append the hash to the message and calcu- late a new hash. Continue this process until a hash code is produced that is relatively prime to (q - 1). Then calculate Z to satisfy Z K X * h(mod q - 1). The signa- ture of the message is s = aZ. To verify the signature, a user compute t such that t * h = 1 mod (q - 1) and verifies Y = s t mod q. a. Show that the scheme is unacceptable by describing a simple technique for forging
a user’s signature on an arbitrary message. b. Show that the scheme is unacceptable by describing a simple technique for forging
a user’s signature on an arbitrary message.
13.8 Assume a technique for a digital signature scheme using a cryptographic one-way hash function (H) as follows. To sign an n-bit message, the sender randomly gener- ates in advance 2n 64-bit cryptographic keys:
k1, k2, c , kn k1′, k2′, c , kn′
which are kept private. The sender generates the following two sets of validation parameters which are made public.
v1, v2, c , vn and v1′, v2′, c , vn′
where
vi = H(ki ‘ 0), vi′ = H(ki′‘ 1)
The user sends the appropriate ki or Ki œ according to whether Mi is 0 or 1 respectively.
For example, if the first 3-bits of the message are 011, then the first three keys of the signature are k1, k
œ 2, k
œ 3.
a. How does the receiver validate the message? b. Is the technique secure? c. How many times can the same set of secret keys be safely used for different mes-
sages? d. What, if any, practical problems does this scheme present?
441
14.1 Symmetric Key Distribution Using Symmetric Encryption
A Key Distribution Scenario
Hierarchical Key Control
Session Key Lifetime
A Transparent Key Control Scheme
Decentralized Key Control
Controlling Key Usage
14.2 Symmetric Key Distribution Using Asymmetric Encryption
Simple Secret Key Distribution
Secret Key Distribution with Confidentiality and Authentication
A Hybrid Scheme
14.3 Distribution of Public Keys
Public Announcement of Public Keys
Publicly Available Directory
Public-Key Authority
Public-Key Certificates
14.4 X.509 Certificates
Certificates
X.509 Version 3
14.5 Public-Key Infrastructure
PKIX Management Functions
PKIX Management Protocols
14.6 Key Terms, Review Questions, and Problems
PART FIVE: MUTUAL TRUST
CHAPTER
Key Management and Distribution
442 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION
The topics of cryptographic key management and cryptographic key distribution are
complex, involving cryptographic, protocol, and management considerations. The pur-
pose of this chapter is to give the reader a feel for the issues involved and a broad sur-
vey of the various aspects of key management and distribution. For more information,
the place to start is the three-volume NIST SP 800-57, followed by the recommended
readings listed at the end of this chapter.
14.1 SYMMETRIC KEY DISTRIBUTION USING SYMMETRIC ENCRYPTION
For symmetric encryption to work, the two parties to an exchange must share the
same key, and that key must be protected from access by others. Furthermore, fre-
quent key changes are usually desirable to limit the amount of data compromised
if an attacker learns the key. Therefore, the strength of any cryptographic system
rests with the key distribution technique, a term that refers to the means of delivering a key to two parties who wish to exchange data without allowing others to see the
key. For two parties A and B, key distribution can be achieved in a number of ways,
as follows:
1. A can select a key and physically deliver it to B.
2. A third party can select the key and physically deliver it to A and B.
3. If A and B have previously and recently used a key, one party can transmit the new key to the other, encrypted using the old key.
4. If A and B each has an encrypted connection to a third party C, C can deliver a key on the encrypted links to A and B.
Options 1 and 2 call for manual delivery of a key. For link encryption, this
is a reasonable requirement, because each link encryption device is going to be
exchanging data only with its partner on the other end of the link. However, for
end-to-end encryption over a network, manual delivery is awkward. In a distributed system, any given host or terminal may need to engage in exchanges with many other
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Discuss the concept of a key hierarchy.
◆ Understand the issues involved in using asymmetric encryption to distribute symmetric keys.
◆ Present an overview of approaches to public-key distribution and analyze the risks involved in various approaches.
◆ List and explain the elements in an X.509 certificate.
◆ Present an overview of public-key infrastructure concepts.
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14.1 / SYMMETRIC KEY DISTRIBUTION USING SYMMETRIC ENCRYPTION 443
hosts and terminals over time. Thus, each device needs a number of keys supplied
dynamically. The problem is especially difficult in a wide-area distributed system.
The scale of the problem depends on the number of communicating pairs that
must be supported. If end-to-end encryption is done at a network or IP level, then a
key is needed for each pair of hosts on the network that wish to communicate. Thus,
if there are N hosts, the number of required keys is [N(N - 1)]/2. If encryption is done at the application level, then a key is needed for every pair of users or pro-
cesses that require communication. Thus, a network may have hundreds of hosts
but thousands of users and processes. Figure 14.1 illustrates the magnitude of the
key distribution task for end-to-end encryption.1 A network using node-level
encryption with 1000 nodes would conceivably need to distribute as many as half a
million keys. If that same network supported 10,000 applications, then as many as
50 million keys may be required for application-level encryption.
Returning to our list, option 3 is a possibility for either link encryption or
end-to-end encryption, but if an attacker ever succeeds in gaining access to one key,
then all subsequent keys will be revealed. Furthermore, the initial distribution of
potentially millions of keys still must be made.
1Note that this figure uses a log-log scale, so that a linear graph indicates exponential growth. A basic review of log scales is in the math refresher document at the Computer Science Student Resource Site at WilliamStallings.com/StudentSupport.html.
Figure 14.1 Number of Keys Required to Support Arbitrary Connections between Endpoints
109
108
107
106
103 104 105
N um
be r
of k
ey s
5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9
Number of endpoints
444 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION
For end-to-end encryption, some variation on option 4 has been widely
adopted. In this scheme, a key distribution center is responsible for distributing
keys to pairs of users (hosts, processes, applications) as needed. Each user must
share a unique key with the key distribution center for purposes of key distribution.
The use of a key distribution center is based on the use of a hierarchy of keys.
At a minimum, two levels of keys are used (Figure 14.2). Communication between
end systems is encrypted using a temporary key, often referred to as a session key. Typically, the session key is used for the duration of a logical connection, such as a
frame relay connection or transport connection, and then discarded. Each session
key is obtained from the key distribution center over the same networking facili-
ties used for end-user communication. Accordingly, session keys are transmitted in
encrypted form, using a master key that is shared by the key distribution center and an end system or user.
For each end system or user, there is a unique master key that it shares with
the key distribution center. Of course, these master keys must be securely distrib-
uted in some fashion. However, the scale of the problem is vastly reduced. If there
are N entities that wish to communicate in pairs, then, as was mentioned, as many as [N(N - 1)]/2 session keys are needed at any one time. However, only N master keys are required, one for each entity. Thus, master keys can be distributed in some
non-cryptographic way, such as physical delivery.
A Key Distribution Scenario
The key distribution concept can be deployed in a number of ways. A typical
scenario is illustrated in Figure 14.3, which is based on a figure in [POPE79]. The sce-
nario assumes that each user shares a unique master key with the key distribution
center (KDC).
Let us assume that user A wishes to establish a logical connection with B and
requires a one-time session key to protect the data transmitted over the connection.
Figure 14.2 The Use of a Key Hierarchy
Data Cryptographic protection
Session keys Cryptographic protection
Master keys Non-cryptographic protection
14.1 / SYMMETRIC KEY DISTRIBUTION USING SYMMETRIC ENCRYPTION 445
A has a master key, Ka, known only to itself and the KDC; similarly, B shares the master key Kb with the KDC. The following steps occur.
1. A issues a request to the KDC for a session key to protect a logical connection to B. The message includes the identity of A and B and a unique identifier, N1, for this transaction, which we refer to as a nonce. The nonce may be a timestamp, a counter, or a random number; the minimum requirement is that it differs with
each request. Also, to prevent masquerade, it should be difficult for an opponent
to guess the nonce. Thus, a random number is a good choice for a nonce.
2. The KDC responds with a message encrypted using Ka. Thus, A is the only one who can successfully read the message, and A knows that it originated at the
KDC. The message includes two items intended for A:
■ The one-time session key, Ks, to be used for the session
■ The original request message, including the nonce, to enable A to match
this response with the appropriate request
Thus, A can verify that its original request was not altered before reception by
the KDC and, because of the nonce, that this is not a replay of some previous
request.
In addition, the message includes two items intended for B:
■ The one-time session key, Ks, to be used for the session
■ An identifier of A (e.g., its network address), IDA These last two items are encrypted with Kb (the master key that the KDC shares with B). They are to be sent to B to establish the connection and prove
A’s identity.
Figure 14.3 Key Distribution Scenario
Key Distribution Center (KDC)
Key distribution
steps
Authentication steps
Initiator A Responder B
(1) IDA || IDB || N1
(2) E(Ka, [Ks || IDA || IDB || N1]) || E(Kb, [Ks || IDA])
(3) E(Kb, [Ks || IDA])
(4) E(Ks, N2)
(5) E(Ks, f(N2))
446 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION
3. A stores the session key for use in the upcoming session and forwards to B the information that originated at the KDC for B, namely, E(Kb,[Ks }IDA]). Because this information is encrypted with Kb, it is protected from eavesdrop- ping. B now knows the session key (Ks), knows that the other party is A (from IDA), and knows that the information originated at the KDC (because it is encrypted using Kb).
At this point, a session key has been securely delivered to A and B, and they
may begin their protected exchange. However, two additional steps are desirable:
4. Using the newly minted session key for encryption, B sends a nonce, N2, to A.
5. Also, using Ks, A responds with f(N2), where f is a function that performs some transformation on N2 (e.g., adding one).
These steps assure B that the original message it received (step 3) was not a replay.
Note that the actual key distribution involves only steps 1 through 3, but that
steps 4 and 5, as well as step 3, perform an authentication function.
Hierarchical Key Control
It is not necessary to limit the key distribution function to a single KDC. Indeed, for
very large networks, it may not be practical to do so. As an alternative, a hierarchy
of KDCs can be established. For example, there can be local KDCs, each respon-
sible for a small domain of the overall internetwork, such as a single LAN or a single
building. For communication among entities within the same local domain, the local
KDC is responsible for key distribution. If two entities in different domains desire a
shared key, then the corresponding local KDCs can communicate through a global
KDC. In this case, any one of the three KDCs involved can actually select the key.
The hierarchical concept can be extended to three or even more layers, depending
on the size of the user population and the geographic scope of the internetwork.
A hierarchical scheme minimizes the effort involved in master key distri-
bution, because most master keys are those shared by a local KDC with its local
entities. Furthermore, such a scheme limits the damage of a faulty or subverted
KDC to its local area only.
Session Key Lifetime
The more frequently session keys are exchanged, the more secure they are, because
the opponent has less ciphertext to work with for any given session key. On the
other hand, the distribution of session keys delays the start of any exchange and
places a burden on network capacity. A security manager must try to balance these
competing considerations in determining the lifetime of a particular session key.
For connection-oriented protocols, one obvious choice is to use the same ses-
sion key for the length of time that the connection is open, using a new session key
for each new session. If a logical connection has a very long lifetime, then it would
be prudent to change the session key periodically, perhaps every time the PDU
(protocol data unit) sequence number cycles.
For a connectionless protocol, such as a transaction-oriented protocol, there
is no explicit connection initiation or termination. Thus, it is not obvious how often
one needs to change the session key. The most secure approach is to use a new
14.1 / SYMMETRIC KEY DISTRIBUTION USING SYMMETRIC ENCRYPTION 447
session key for each exchange. However, this negates one of the principal benefits
of connectionless protocols, which is minimum overhead and delay for each transac-
tion. A better strategy is to use a given session key for a certain fixed period only or
for a certain number of transactions.
A Transparent Key Control Scheme
The approach suggested in Figure 14.3 has many variations, one of which is
described in this subsection. The scheme (Figure 14.4) is useful for providing
end-to-end encryption at a network or transport level in a way that is transpar-
ent to the end users. The approach assumes that communication makes use of a
connection- oriented end-to-end protocol, such as TCP. The noteworthy element of
this approach is a session security module (SSM), which may consist of functionality
Figure 14.4 Automatic Key Distribution for Connection-Oriented Protocol
Key distribution
center
Network
1. Host sends packet requesting connection. 2. Security service buffers packet; asks KDC for session key. 3. KDC distributes session key to both hosts. 4. Buffered packet transmitted.
HOST
Application
Security service
HOST
Application
Security service
2
3
4
1
448 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION
at one protocol layer, that performs end-to-end encryption and obtains session keys
on behalf of its host or terminal.
The steps involved in establishing a connection are shown in Figure 14.4. When
one host wishes to set up a connection to another host, it transmits a connection-
request packet (step 1). The SSM saves that packet and applies to the KDC for
permission to establish the connection (step 2). The communication between the
SSM and the KDC is encrypted using a master key shared only by this SSM and
the KDC. If the KDC approves the connection request, it generates the session
key and delivers it to the two appropriate SSMs, using a unique permanent key for
each SSM (step 3). The requesting SSM can now release the connection request
packet, and a connection is set up between the two end systems (step 4). All user
data exchanged between the two end systems are encrypted by their respective SSMs
using the one-time session key.
The automated key distribution approach provides the flexibility and dynamic
characteristics needed to allow a number of terminal users to access a number of
hosts and for the hosts to exchange data with each other.
Decentralized Key Control
The use of a key distribution center imposes the requirement that the KDC be
trusted and be protected from subversion. This requirement can be avoided if key
distribution is fully decentralized. Although full decentralization is not practical for
larger networks using symmetric encryption only, it may be useful within a local
context.
A decentralized approach requires that each end system be able to commu-
nicate in a secure manner with all potential partner end systems for purposes of
session key distribution. Thus, there may need to be as many as [n(n - 1)]/2 master keys for a configuration with n end systems.
A session key may be established with the following sequence of steps
(Figure 14.5).
1. A issues a request to B for a session key and includes a nonce, N1.
2. B responds with a message that is encrypted using the shared master key. The response includes the session key selected by B, an identifier of B, the value
f(N1), and another nonce, N2.
3. Using the new session key, A returns f(N2) to B.
Figure 14.5 Decentralized Key Distribution
(1) IDA || N1
(2) E(Km, [Ks || IDA || IDB || f(N1) || N2 ])
Initiator A
Responder B
(3) E(Ks, f(N2))
14.1 / SYMMETRIC KEY DISTRIBUTION USING SYMMETRIC ENCRYPTION 449
Thus, although each node must maintain at most (n - 1) master keys, as many session keys as required may be generated and used. Because the messages trans-
ferred using the master key are short, cryptanalysis is difficult. As before, session
keys are used for only a limited time to protect them.
Controlling Key Usage
The concept of a key hierarchy and the use of automated key distribution techniques
greatly reduce the number of keys that must be manually managed and distributed.
It also may be desirable to impose some control on the way in which automatically
distributed keys are used. For example, in addition to separating master keys from
session keys, we may wish to define different types of session keys on the basis of
use, such as
■ Data-encrypting key, for general communication across a network
■ PIN-encrypting key, for personal identification numbers (PINs) used in
electronic funds transfer and point-of-sale applications
■ File-encrypting key, for encrypting files stored in publicly accessible locations
To illustrate the value of separating keys by type, consider the risk that a master
key is imported as a data-encrypting key into a device. Normally, the master key is
physically secured within the cryptographic hardware of the key distribution center
and of the end systems. Session keys encrypted with this master key are available to
application programs, as are the data encrypted with such session keys. However,
if a master key is treated as a session key, it may be possible for an unauthorized application to obtain plaintext of session keys encrypted with that master key.
Thus, it may be desirable to institute controls in systems that limit the ways
in which keys are used, based on characteristics associated with those keys. One
simple plan is to associate a tag with each key ([JONE82]; see also [DAVI89]).
The proposed technique is for use with DES and makes use of the extra 8 bits in
each 64-bit DES key. That is, the eight non-key bits ordinarily reserved for parity
checking form the key tag. The bits have the following interpretation:
■ One bit indicates whether the key is a session key or a master key
■ One bit indicates whether the key can be used for encryption
■ One bit indicates whether the key can be used for decryption
■ The remaining bits are spares for future use.
Because the tag is embedded in the key, it is encrypted along with the key when that
key is distributed, thus providing protection. The drawbacks of this scheme are
1. The tag length is limited to 8 bits, limiting its flexibility and functionality.
2. Because the tag is not transmitted in clear form, it can be used only at the point of decryption, limiting the ways in which key use can be controlled.
A more flexible scheme, referred to as the control vector, is described in
[MATY91a and b]. In this scheme, each session key has an associated control vector
consisting of a number of fields that specify the uses and restrictions for that session
key. The length of the control vector may vary.
450 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION
The control vector is cryptographically coupled with the key at the time of
key generation at the KDC. The coupling and decoupling processes are illustrated
in Figure 14.6. As a first step, the control vector is passed through a hash func-
tion that produces a value whose length is equal to the encryption key length. Hash
functions are discussed in detail in Chapter 11. In essence, a hash function maps
values from a larger range into a smaller range with a reasonably uniform spread.
Thus, for example, if numbers in the range 1 to 100 are hashed into numbers in the
range 1 to 10, approximately 10% of the source values should map into each of the
target values.
The hash value is then XORed with the master key to produce an output that
is used as the key input for encrypting the session key. Thus,
Hash value = H = h(CV) Key input = Km ⊕ H
Ciphertext = E([Km ⊕ H], Ks)
where Km is the master key and Ks is the session key. The session key is recovered in plaintext by the reverse operation:
D([Km ⊕ H], E([Km ⊕ H], Ks))
When a session key is delivered to a user from the KDC, it is accompanied
by the control vector in clear form. The session key can be recovered only by using
both the master key that the user shares with the KDC and the control vector. Thus,
the linkage between the session key and its control vector is maintained.
Figure 14.6 Control Vector Encryption and Decryption
Control vector
Master key
Session key
Hashing function
Key input
Key input
Ciphertext input
Plaintext input
Encryption function
Encrypted session key
(a) Control vector encryption
Control vector
Master key
Encrypted session key
Hashing function
Decryption function
Session key
(b) Control vector decryption
14.2 / SYMMETRIC KEY DISTRIBUTION USING ASYMMETRIC ENCRYPTION 451
Use of the control vector has two advantages over use of an 8-bit tag. First,
there is no restriction on length of the control vector, which enables arbitrarily com-
plex controls to be imposed on key use. Second, the control vector is available in
clear form at all stages of operation. Thus, control of key use can be exercised in
multiple locations.
14.2 SYMMETRIC KEY DISTRIBUTION USING ASYMMETRIC ENCRYPTION
Because of the inefficiency of public-key cryptosystems, they are almost never used
for the direct encryption of sizable blocks of data, but are limited to relatively small
blocks. One of the most important uses of a public-key cryptosystem is to encrypt
secret keys for distribution. We see many specific examples of this in Part Five.
Here, we discuss general principles and typical approaches.
Simple Secret Key Distribution
An extremely simple scheme was put forward by Merkle [MERK79], as illustrated
in Figure 14.7. If A wishes to communicate with B, the following procedure is
employed:
1. A generates a public/private key pair {PUa, PRa} and transmits a message to B consisting of PUa and an identifier of A, IDA.
2. B generates a secret key, Ks, and transmits it to A, which is encrypted with A’s public key.
3. A computes D(PRa, E(PUa, Ks)) to recover the secret key. Because only A can decrypt the message, only A and B will know the identity of Ks.
4. A discards PUa and PRa and B discards PUa.
A and B can now securely communicate using conventional encryption and
the session key Ks. At the completion of the exchange, both A and B discard Ks. Despite its simplicity, this is an attractive protocol. No keys exist before the start of
the communication and none exist after the completion of communication. Thus,
the risk of compromise of the keys is minimal. At the same time, the communication
is secure from eavesdropping.
The protocol depicted in Figure 14.7 is insecure against an adversary who can
intercept messages and then either relay the intercepted message or substitute another
message (see Figure 1.3c). Such an attack is known as a man-in-the-middle attack [RIVE84]. We saw this type of attack in Chapter 10 (Figure 10.2). In the present
Figure 14.7 Simple Use of Public-Key Encryption to Establish a Session Key
BA
(1) PUa || IDA
(2) E(PUa, Ks)
Hiva-Network.Com
452 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION
case, if an adversary, D, has control of the intervening communication channel,
then D can compromise the communication in the following fashion without being
detected (Figure 14.8).
1. A generates a public/private key pair {PUa, PRa} and transmits a message intended for B consisting of PUa and an identifier of A, IDA.
2. D intercepts the message, creates its own public/private key pair {PUd, PRd} and transmits PUd }IDA to B.
3. B generates a secret key, Ks, and transmits E(PUd, Ks).
4. D intercepts the message and learns Ks by computing D(PRd, E(PUd, Ks)).
5. D transmits E(PUa, Ks) to A.
Figure 14.8 Another Man-in-the-Middle Attack
Alice Darth Bob
Private key PRa Public key PUa
Private key PRb Public key PUb Secret key Ks
Ks = D(PRd, E(PUd, Ks))
Private key PRd Public key PUd
PUa, IDA
PUd, IDA
E(PUd, Ks)
Alice, Bob, and Darth share K1
E(PUa, Ks)
14.2 / SYMMETRIC KEY DISTRIBUTION USING ASYMMETRIC ENCRYPTION 453
The result is that both A and B know Ks and are unaware that Ks has also been revealed to D. A and B can now exchange messages using Ks. D no longer actively interferes with the communications channel but simply eavesdrops. Knowing Ks, D can decrypt all messages, and both A and B are unaware of the problem. Thus,
this simple protocol is only useful in an environment where the only threat is
eavesdropping.
Secret Key Distribution with Confidentiality and Authentication
Figure 14.9, based on an approach suggested in [NEED78], provides protection
against both active and passive attacks. We begin at a point when it is assumed that
A and B have exchanged public keys by one of the schemes described subsequently
in this chapter. Then the following steps occur.
1. A uses B’s public key to encrypt a message to B containing an identifier of A(IDA) and a nonce (N1), which is used to identify this transaction uniquely.
2. B sends a message to A encrypted with PUa and containing A’s nonce (N1) as well as a new nonce generated by B (N2). Because only B could have decrypted message (1), the presence of N1 in message (2) assures A that the correspondent is B.
3. A returns N2, encrypted using B’s public key, to assure B that its correspon- dent is A.
4. A selects a secret key Ks and sends M = E(PUb, E(PRa, Ks)) to B. Encryption of this message with B’s public key ensures that only B can read it; encryption
with A’s private key ensures that only A could have sent it.
5. B computes D(PUa, D(PRb, M)) to recover the secret key.
The result is that this scheme ensures both confidentiality and authentication
in the exchange of a secret key.
Figure 14.9 Public-Key Distribution of Secret Keys
Initiator A
Responder B
(1) E(PUb, [N1 || IDA])
(4) E(PUb, E(PRa, Ks))
(3) E(PUb, N2)
(2) E(PUa, [N1 || N2])
454 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION
A Hybrid Scheme
Yet another way to use public-key encryption to distribute secret keys is a hybrid
approach in use on IBM mainframes [LE93]. This scheme retains the use of a key
distribution center (KDC) that shares a secret master key with each user and dis-
tributes secret session keys encrypted with the master key. A public-key scheme is
used to distribute the master keys. The following rationale is provided for using this
three-level approach:
■ Performance: There are many applications, especially transaction-oriented applications, in which the session keys change frequently. Distribution of ses-
sion keys by public-key encryption could degrade overall system performance
because of the relatively high computational load of public-key encryption
and decryption. With a three-level hierarchy, public-key encryption is used
only occasionally to update the master key between a user and the KDC.
■ Backward compatibility: The hybrid scheme is easily overlaid on an existing KDC scheme with minimal disruption or software changes.
The addition of a public-key layer provides a secure, efficient means of dis-
tributing master keys. This is an advantage in a configuration in which a single KDC
serves a widely distributed set of users.
14.3 DISTRIBUTION OF PUBLIC KEYS
Several techniques have been proposed for the distribution of public keys. Virtually
all these proposals can be grouped into the following general schemes:
■ Public announcement
■ Publicly available directory
■ Public-key authority
■ Public-key certificates
Public Announcement of Public Keys
On the face of it, the point of public-key encryption is that the public key is public.
Thus, if there is some broadly accepted public-key algorithm, such as RSA, any
participant can send his or her public key to any other participant or broadcast the
key to the community at large (Figure 14.10). For example, because of the growing
popularity of PGP (pretty good privacy, discussed in Chapter 19), which makes use
of RSA, many PGP users have adopted the practice of appending their public key
to messages that they send to public forums, such as USENET newsgroups and
Internet mailing lists.
Although this approach is convenient, it has a major weakness. Anyone can
forge such a public announcement. That is, some user could pretend to be user A
and send a public key to another participant or broadcast such a public key. Until
such time as user A discovers the forgery and alerts other participants, the forger is
able to read all encrypted messages intended for A and can use the forged keys for
authentication (see Figure 9.3).
14.3 / DISTRIBUTION OF PUBLIC KEYS 455
Publicly Available Directory
A greater degree of security can be achieved by maintaining a publicly available
dynamic directory of public keys. Maintenance and distribution of the public
directory would have to be the responsibility of some trusted entity or organization
(Figure 14.11). Such a scheme would include the following elements:
1. The authority maintains a directory with a {name, public key} entry for each participant.
2. Each participant registers a public key with the directory authority. Registration would have to be in person or by some form of secure authenti-
cated communication.
3. A participant may replace the existing key with a new one at any time, either because of the desire to replace a public key that has already been used for
a large amount of data, or because the corresponding private key has been
compromised in some way.
4. Participants could also access the directory electronically. For this purpose, secure, authenticated communication from the authority to the participant is
mandatory.
Figure 14.10 Uncontrolled Public-Key Distribution
PUa
PUa
PUa
PUa
PUb
PUb
PUb
PUb
BA
Figure 14.11 Public-Key Publication
Public-key directory
PUa PUb
A B
456 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION
This scheme is clearly more secure than individual public announcements
but still has vulnerabilities. If an adversary succeeds in obtaining or computing the
private key of the directory authority, the adversary could authoritatively pass out
counterfeit public keys and subsequently impersonate any participant and eaves-
drop on messages sent to any participant. Another way to achieve the same end is
for the adversary to tamper with the records kept by the authority.
Public-Key Authority
Stronger security for public-key distribution can be achieved by providing tighter
control over the distribution of public keys from the directory. A typical scenario is
illustrated in Figure 14.12, which is based on a figure in [POPE79]. As before, the
scenario assumes that a central authority maintains a dynamic directory of public
keys of all participants. In addition, each participant reliably knows a public key for
the authority, with only the authority knowing the corresponding private key. The
following steps (matched by number to Figure 14.12) occur.
1. A sends a timestamped message to the public-key authority containing a request for the current public key of B.
2. The authority responds with a message that is encrypted using the authority’s private key, PRauth. Thus, A is able to decrypt the message using the author- ity’s public key. Therefore, A is assured that the message originated with the
authority. The message includes the following:
■ B’s public key, PUb, which A can use to encrypt messages destined for B
■ The original request used to enable A to match this response with the cor-
responding earlier request and to verify that the original request was not
altered before reception by the authority
■ The original timestamp given so A can determine that this is not an old mes-
sage from the authority containing a key other than B’s current public key
3. A stores B’s public key and also uses it to encrypt a message to B containing an identifier of A (IDA) and a nonce (N1), which is used to identify this trans- action uniquely.
4, 5. B retrieves A’s public key from the authority in the same manner as A retrieved B’s public key.
At this point, public keys have been securely delivered to A and B, and they
may begin their protected exchange. However, two additional steps are desirable:
6. B sends a message to A encrypted with PUa and containing A’s nonce (N1) as well as a new nonce generated by B (N2). Because only B could have decrypted message (3), the presence of N1 in message (6) assures A that the correspondent is B.
7. A returns N2, which is encrypted using B’s public key, to assure B that its correspondent is A.
Thus, a total of seven messages are required. However, the initial five
messages need be used only infrequently because both A and B can save the other’s
public key for future use—a technique known as caching. Periodically, a user should
request fresh copies of the public keys of its correspondents to ensure currency.
14.3 / DISTRIBUTION OF PUBLIC KEYS 457
Public-Key Certificates
The scenario of Figure 14.12 is attractive, yet it has some drawbacks. The public-key
authority could be somewhat of a bottleneck in the system, for a user must appeal
to the authority for a public key for every other user that it wishes to contact.
As before, the directory of names and public keys maintained by the authority is
vulnerable to tampering.
An alternative approach, first suggested by Kohnfelder [KOHN78], is to use
certificates that can be used by participants to exchange keys without contacting a public-key authority, in a way that is as reliable as if the keys were obtained directly
from a public-key authority. In essence, a certificate consists of a public key, an
identifier of the key owner, and the whole block signed by a trusted third party.
Typically, the third party is a certificate authority, such as a government agency or
a financial institution, that is trusted by the user community. A user can present
his or her public key to the authority in a secure manner and obtain a certificate.
The user can then publish the certificate. Anyone needing this user’s public key can
obtain the certificate and verify that it is valid by way of the attached trusted signa-
ture. A participant can also convey its key information to another by transmitting
its certificate. Other participants can verify that the certificate was created by the
authority. We can place the following requirements on this scheme:
1. Any participant can read a certificate to determine the name and public key of the certificate’s owner.
2. Any participant can verify that the certificate originated from the certificate authority and is not counterfeit.
3. Only the certificate authority can create and update certificates.
Figure 14.12 Public-Key Distribution Scenario
Public-key authorityInitiator A Responder B
(1) Request || T1
(2) E(PRauth, [PUb || Request || T1])
(3) E(PUb, [ IDA || N1])
(4) Request || T2
(5) E(PRauth, [PUa || Request || T2])
(6) E(PUa, [ N1 || N2])
(7) E(PUb, N2)
458 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION
These requirements are satisfied by the original proposal in [KOHN78]. Denning
[DENN83] added the following additional requirement:
4. Any participant can verify the time validity of the certificate.
A certificate scheme is illustrated in Figure 14.13. Each participant applies
to the certificate authority, supplying a public key and requesting a certificate.
Application must be in person or by some form of secure authenticated communi-
cation. For participant A, the authority provides a certificate of the form
CA = E(PRauth, [T}IDA }PUa])
where PRauth is the private key used by the authority and T is a timestamp. A may then pass this certificate on to any other participant, who reads and verifies the
certificate as follows:
D(PUauth, CA) = D(PUauth, E(PRauth, [T}IDA }PUa])) = (T}IDA }PUa)
The recipient uses the authority’s public key, PUauth, to decrypt the certifi- cate. Because the certificate is readable only using the authority’s public key, this
verifies that the certificate came from the certificate authority. The elements IDA and PUa provide the recipient with the name and public key of the certificate’s holder. The timestamp T validates the currency of the certificate. The timestamp
Figure 14.13 Exchange of Public-Key Certificates
(a) Obtaining certificates from CA
(b) Exchanging certificates
PUa PUb
A B
Certificate Authority
CA = E(PRauth, [T1 || IDA || PUa])
CB = E(PRauth, [T2 || IDB || PUb])
(1) CA
(2) CB
A B
14.4 / X.509 CERTIFICATES 459
counters the following scenario. A’s private key is learned by an adversary.
A generates a new private/public key pair and applies to the certificate authority
for a new certificate. Meanwhile, the adversary replays the old certificate to B. If B
then encrypts messages using the compromised old public key, the adversary can
read those messages.
In this context, the compromise of a private key is comparable to the loss of a
credit card. The owner cancels the credit card number but is at risk until all possible
communicants are aware that the old credit card is obsolete. Thus, the timestamp
serves as something like an expiration date. If a certificate is sufficiently old, it is
assumed to be expired.
One scheme has become universally accepted for formatting public-key
certificates: the X.509 standard. X.509 certificates are used in most network security
applications, including IP security, transport layer security (TLS), and S/MIME, all
of which are discussed in Part Five. X.509 is examined in detail in the next section.
14.4 X.509 CERTIFICATES
ITU-T recommendation X.509 is part of the X.500 series of recommendations that
define a directory service. The directory is, in effect, a server or distributed set
of servers that maintains a database of information about users. The information
includes a mapping from user name to network address, as well as other attributes
and information about the users.
X.509 defines a framework for the provision of authentication services by the
X.500 directory to its users. The directory may serve as a repository of public-key
certificates of the type discussed in Section 14.3. Each certificate contains the public
key of a user and is signed with the private key of a trusted certification authority.
In addition, X.509 defines alternative authentication protocols based on the use of
public-key certificates.
X.509 is an important standard because the certificate structure and authenti-
cation protocols defined in X.509 are used in a variety of contexts. For example, the
X.509 certificate format is used in S/MIME (Chapter 19), IP Security (Chapter 20),
and SSL/TLS (Chapter 17).
X.509 was initially issued in 1988. The standard was subsequently revised
in 1993 to address some of the security concerns documented in [IANS90] and
[MITC90]. The standard is currently at version 7, issued in 2012.
X.509 is based on the use of public-key cryptography and digital signatures.
The standard does not dictate the use of a specific digital signature algorithm nor a
specific hash function. Figure 14.14 illustrates the overall X.509 scheme for genera-
tion of a public-key certificate. The certificate for Bob’s public key includes unique
identifying information for Bob, Bob’s public key, and identifying information
about the CA, plus other information as explained subsequently. This information
is then signed by computing a hash value of the information and generating a digital
signature using the hash value and the CA’s private key. X.509 indicates that the
signature is formed by encrypting the hash value. This suggests the use of one of the
RSA schemes discussed in Section 13.6. However, the current version of X.509 does
460 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION
not dictate a specific digital signature algorithm. If the NIST DSA (Section 13.4) or
the ECDSA (Section 13.5) scheme is used, then the hash value is not encrypted but
serves as input to a digital signature generation algorithm.
Certificates
The heart of the X.509 scheme is the public-key certificate associated with each
user. These user certificates are assumed to be created by some trusted certification
authority (CA) and placed in the directory by the CA or by the user. The directory
server itself is not responsible for the creation of public keys or for the certifica-
tion function; it merely provides an easily accessible location for users to obtain
certificates.
Figure 14.15a shows the general format of a certificate, which includes the
following elements.
■ Version: Differentiates among successive versions of the certificate format; the default is version 1. If the issuer unique identifier or subject unique identifier are present, the value must be version 2. If one or more extensions are present,
the version must be version 3. Although the X.509 specification is currently at
version 7, no changes have been made to the fields that make up the certificate
since version 3.
■ Serial number: An integer value unique within the issuing CA that is unam- biguously associated with this certificate.
■ Signature algorithm identifier: The algorithm used to sign the certificate together with any associated parameters. Because this information is repeated
in the signature field at the end of the certificate, this field has little, if any, utility.
Figure 14.14 X.509 Public-Key Certificate Use
Unsigned certificate: contains user ID, user's public key
Signed certificate
Recipient can verify signature by comparing hash code values
Generate hash code of unsigned certificate
Encrypt hash code with CA's private key to form signature
H
H
Bob's ID information
CA information
Bob's public key
E D
Decrypt signature with CA's public key to recover hash code
Use certificate to verify Bob's public key
Create signed digital certificate
Hiva-Network.Com
14.4 / X.509 CERTIFICATES 461
■ Issuer name: X.500 name of the CA that created and signed this certificate.
■ Period of validity: Consists of two dates: the first and last on which the certifi- cate is valid.
■ Subject name: The name of the user to whom this certificate refers. That is, this certificate certifies the public key of the subject who holds the corresponding
private key.
■ Subject’s public-key information: The public key of the subject, plus an identi- fier of the algorithm for which this key is to be used, together with any associ-
ated parameters.
■ Issuer unique identifier: An optional-bit string field used to identify uniquely the issuing CA in the event the X.500 name has been reused for different
entities.
■ Subject unique identifier: An optional-bit string field used to identify uniquely the subject in the event the X.500 name has been reused for different entities.
■ Extensions: A set of one or more extension fields. Extensions were added in version 3 and are discussed later in this section.
■ Signature: Covers all of the other fields of the certificate. One component of this field is the digital signature applied to the other fields of the certificate.
This field includes the signature algorithm identifier.
The unique identifier fields were added in version 2 to handle the possible
reuse of subject and/or issuer names over time. These fields are rarely used.
Figure 14.15 X.509 Formats
Certificate serial number
Version
Issuer name
Signature algorithm identifier
Subject name
Extensions
Issuer unique identifier
Subject unique identifier
Algorithm Parameters
Not before
Algorithms Parameters
Key
Algorithms Parameters
Signature of certificate
(a) X.509 certificate
Not after
Subject's public key
info
Signature
Period of validity
V er
si on
1
V er
si on
2
V er
si on
3
A ll
ve rs
io ns
Issuer name
This update date
Next update date
• • •
Signature algorithm identifier
Algorithm Parameters
User certificate serial #
(b) Certificate revocation list
Revocation date
Algorithms Parameters
Signature of certificate Signature
Revoked certificate
User certificate serial # Revocation date
Revoked certificate
462 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION
The standard uses the following notation to define a certificate:
CA V A W = CA {V, SN, AI, CA, UCA, A, UA, Ap, TA}
where
Y V X W = the certificate of user X issued by certification authority Y Y {I} = the signing of I by Y. It consists of I with an encrypted hash
code appended
V = version of the certificate SN = serial number of the certificate AI = identifier of the algorithm used to sign the certificate
CA = name of certificate authority UCA = optional unique identifier of the CA
A = name of user A UA = optional unique identifier of the user A Ap = public key of user A TA = period of validity of the certificate
The CA signs the certificate with its private key. If the corresponding public
key is known to a user, then that user can verify that a certificate signed by the CA is
valid. This is the typical digital signature approach illustrated in Figure 13.2.
OBTAINING A USER’S CERTIFICATE User certificates generated by a CA have the following characteristics:
■ Any user with access to the public key of the CA can verify the user public key
that was certified.
■ No party other than the certification authority can modify the certificate
without this being detected.
Because certificates are unforgeable, they can be placed in a directory without the
need for the directory to make special efforts to protect them.
If all users subscribe to the same CA, then there is a common trust of that CA.
All user certificates can be placed in the directory for access by all users. In addi-
tion, a user can transmit his or her certificate directly to other users. In either case,
once B is in possession of A’s certificate, B has confidence that messages it encrypts
with A’s public key will be secure from eavesdropping and that messages signed
with A’s private key are unforgeable.
If there is a large community of users, it may not be practical for all users to
subscribe to the same CA. Because it is the CA that signs certificates, each partici-
pating user must have a copy of the CA’s own public key to verify signatures. This
public key must be provided to each user in an absolutely secure (with respect
to integrity and authenticity) way so that the user has confidence in the associ-
ated certificates. Thus, with many users, it may be more practical for there to be
a number of CAs, each of which securely provides its public key to some fraction
of the users.
14.4 / X.509 CERTIFICATES 463
Now suppose that A has obtained a certificate from certification authority
X1 and B has obtained a certificate from CA X2. If A does not securely know the
public key of X2, then B’s certificate, issued by X2, is useless to A. A can read B’s
certificate, but A cannot verify the signature. However, if the two CAs have securely
exchanged their own public keys, the following procedure will enable A to obtain
B’s public key.
Step 1 A obtains from the directory the certificate of X2 signed by X1. Because A securely knows X1>s public key, A can obtain X2>s public key from its certificate and verify it by means of X1>s signature on the certificate.
Step 2 A then goes back to the directory and obtains the certificate of B signed by X2. Because A now has a trusted copy of X2>s public key, A can verify the signature and securely obtain B’s public key.
A has used a chain of certificates to obtain B’s public key. In the notation of
X.509, this chain is expressed as
X1 V X2 W X2 V B W
In the same fashion, B can obtain A’s public key with the reverse chain:
X2 V X1 W X1 V A W
This scheme need not be limited to a chain of two certificates. An arbitrarily
long path of CAs can be followed to produce a chain. A chain with N elements would be expressed as
X1 V X2 W X2 V X3 W c XN V B W
In this case, each pair of CAs in the chain (Xi, Xi + 1) must have created certifi-
cates for each other.
All these certificates of CAs by CAs need to appear in the directory, and the
user needs to know how they are linked to follow a path to another user’s public-key
certificate. X.509 suggests that CAs be arranged in a hierarchy so that navigation is
straightforward.
Figure 14.16, taken from X.509, is an example of such a hierarchy. The con-
nected circles indicate the hierarchical relationship among the CAs; the associated
boxes indicate certificates maintained in the directory for each CA entry. The direc-
tory entry for each CA includes two types of certificates:
■ Forward certificates: Certificates of X generated by other CAs
■ Reverse certificates: Certificates generated by X that are the certificates of other CAs
In this example, user A can acquire the following certificates from the direc-
tory to establish a certification path to B:
X V W W W V V W V V Y W Y V Z W Z V B W
When A has obtained these certificates, it can unwrap the certification path in
sequence to recover a trusted copy of B’s public key. Using this public key, A can
send encrypted messages to B. If A wishes to receive encrypted messages back
464 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION
from B, or to sign messages sent to B, then B will require A’s public key, which can
be obtained from the following certification path:
Z V Y W Y V V W V V W W W V X W X V A W
B can obtain this set of certificates from the directory, or A can provide them
as part of its initial message to B.
REVOCATION OF CERTIFICATES Recall from Figure 14.15 that each certificate includes a period of validity, much like a credit card. Typically, a new certificate is issued just
before the expiration of the old one. In addition, it may be desirable on occasion to
revoke a certificate before it expires, for one of the following reasons.
1. The user’s private key is assumed to be compromised.
2. The user is no longer certified by this CA. Reasons for this include that the subject’s name has changed, the certificate is superseded, or the certificate was
not issued in conformance with the CA’s policies.
3. The CA’s certificate is assumed to be compromised.
Each CA must maintain a list consisting of all revoked but not expired
certificates issued by that CA, including both those issued to users and to other
CAs. These lists should also be posted on the directory.
Figure 14.16 X.509 Hierarchy: A Hypothetical Example
U
V
W Y
Z
B
X
C A
U<<V>> V<<U>>
V<<W>> W<<V>>
V<<Y>> Y<<V>>
W<<X>> X<<W>> X<<Z>>
Y<<Z>> Z<<Y>> Z<<X>>
X<<C>> X<<A>> Z<<B>>
14.4 / X.509 CERTIFICATES 465
Each certificate revocation list (CRL) posted to the directory is signed by the
issuer and includes (Figure 14.15b) the issuer’s name, the date the list was created,
the date the next CRL is scheduled to be issued, and an entry for each revoked
certificate. Each entry consists of the serial number of a certificate and revocation
date for that certificate. Because serial numbers are unique within a CA, the serial
number is sufficient to identify the certificate.
When a user receives a certificate in a message, the user must determine
whether the certificate has been revoked. The user could check the directory each
time a certificate is received. To avoid the delays (and possible costs) associated
with directory searches, it is likely that the user would maintain a local cache of
certificates and lists of revoked certificates.
X.509 Version 3
The X.509 version 2 format does not convey all of the information that recent design
and implementation experience has shown to be needed. [FORD95] lists the follow-
ing requirements not satisfied by version 2.
1. The subject field is inadequate to convey the identity of a key owner to a public-key user. X.509 names may be relatively short and lacking in obvious
identification details that may be needed by the user.
2. The subject field is also inadequate for many applications, which typically recognize entities by an Internet email address, a URL, or some other Internet-
related identification.
3. There is a need to indicate security policy information. This enables a security application or function, such as IPSec, to relate an X.509 certificate to a given
policy.
4. There is a need to limit the damage that can result from a faulty or malicious CA by setting constraints on the applicability of a particular certificate.
5. It is important to be able to identify different keys used by the same owner at different times. This feature supports key lifecycle management: in particular,
the ability to update key pairs for users and CAs on a regular basis or under
exceptional circumstances.
Rather than continue to add fields to a fixed format, standards developers
felt that a more flexible approach was needed. Thus, version 3 includes a number
of optional extensions that may be added to the version 2 format. Each extension
consists of an extension identifier, a criticality indicator, and an extension value.
The criticality indicator indicates whether an extension can be safely ignored. If the
indicator has a value of TRUE and an implementation does not recognize the
extension, it must treat the certificate as invalid.
The certificate extensions fall into three main categories: key and policy
information, subject and issuer attributes, and certification path constraints.
KEY AND POLICY INFORMATION These extensions convey additional information about the subject and issuer keys, plus indicators of certificate policy. A certif-
icate policy is a named set of rules that indicates the applicability of a certifi-
cate to a particular community and/or class of application with common security
requirements. For example, a policy might be applicable to the authentication of
466 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION
electronic data interchange (EDI) transactions for the trading of goods within a
given price range.
This area includes:
■ Authority key identifier: Identifies the public key to be used to verify the signature on this certificate or CRL. Enables distinct keys of the same CA to
be differentiated. One use of this field is to handle CA key pair updating.
■ Subject key identifier: Identifies the public key being certified. Useful for sub- ject key pair updating. Also, a subject may have multiple key pairs and, cor-
respondingly, different certificates for different purposes (e.g., digital signature
and encryption key agreement).
■ Key usage: Indicates a restriction imposed as to the purposes for which, and the policies under which, the certified public key may be used. May indicate
one or more of the following: digital signature, nonrepudiation, key encryp-
tion, data encryption, key agreement, CA signature verification on certificates,
CA signature verification on CRLs.
■ Private-key usage period: Indicates the period of use of the private key cor- responding to the public key. Typically, the private key is used over a different
period from the validity of the public key. For example, with digital signature
keys, the usage period for the signing private key is typically shorter than that
for the verifying public key.
■ Certificate policies: Certificates may be used in environments where multiple policies apply. This extension lists policies that the certificate is recognized as
supporting, together with optional qualifier information.
■ Policy mappings: Used only in certificates for CAs issued by other CAs. Policy mappings allow an issuing CA to indicate that one or more of that issuer’s
policies can be considered equivalent to another policy used in the subject
CA’s domain.
CERTIFICATE SUBJECT AND ISSUER ATTRIBUTES These extensions support alterna- tive names, in alternative formats, for a certificate subject or certificate issuer and
can convey additional information about the certificate subject to increase a cer-
tificate user’s confidence that the certificate subject is a particular person or entity.
For example, information such as postal address, position within a corporation, or
picture image may be required.
The extension fields in this area include:
■ Subject alternative name: Contains one or more alternative names, using any of a variety of forms. This field is important for supporting certain applications,
such as electronic mail, EDI, and IPSec, which may employ their own name
forms.
■ Issuer alternative name: Contains one or more alternative names, using any of a variety of forms.
■ Subject directory attributes: Conveys any desired X.500 directory attribute values for the subject of this certificate.
14.5 / PUBLIC-KEY INFRASTRUCTURE 467
CERTIFICATION PATH CONSTRAINTS These extensions allow constraint specifications to be included in certificates issued for CAs by other CAs. The constraints may
restrict the types of certificates that can be issued by the subject CA or that may
occur subsequently in a certification chain.
The extension fields in this area include:
■ Basic constraints: Indicates if the subject may act as a CA. If so, a certification path length constraint may be specified.
■ Name constraints: Indicates a name space within which all subject names in subsequent certificates in a certification path must be located.
■ Policy constraints: Specifies constraints that may require explicit certifi- cate policy identification or inhibit policy mapping for the remainder of the
certification path.
14.5 PUBLIC-KEY INFRASTRUCTURE
RFC 4949 (Internet Security Glossary) defines public-key infrastructure (PKI) as the set of hardware, software, people, policies, and procedures needed to create,
manage, store, distribute, and revoke digital certificates based on asymmetric
cryptography. The principal objective for developing a PKI is to enable secure,
convenient, and efficient acquisition of public keys. The Internet Engineering Task
Force (IETF) Public Key Infrastructure X.509 (PKIX) working group has been the
driving force behind setting up a formal (and generic) model based on X.509 that is
suitable for deploying a certificate-based architecture on the Internet. This section
describes the PKIX model.
Figure 14.17 shows the interrelationship among the key elements of the PKIX
model. These elements are
■ End entity: A generic term used to denote end users, devices (e.g., servers, routers), or any other entity that can be identified in the subject field of a
public-key certificate. End entities typically consume and/or support PKI-
related services.
■ Certification authority (CA): The issuer of certificates and (usually) certifi- cate revocation lists (CRLs). It may also support a variety of administrative
functions, although these are often delegated to one or more Registration
Authorities.
■ Registration authority (RA): An optional component that can assume a num- ber of administrative functions from the CA. The RA is often associated with
the end entity registration process but can assist in a number of other areas
as well.
■ CRL issuer: An optional component that a CA can delegate to publish CRLs.
■ Repository: A generic term used to denote any method for storing certificates and CRLs so that they can be retrieved by end entities.
468 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION
PKIX Management Functions
PKIX identifies a number of management functions that potentially need to be
supported by management protocols. These are indicated in Figure 14.17 and
include the following:
■ Registration: This is the process whereby a user first makes itself known to a CA (directly or through an RA), prior to that CA issuing a certificate or
certificates for that user. Registration begins the process of enrolling in a PKI.
Registration usually involves some offline or online procedure for mutual
authentication. Typically, the end entity is issued one or more shared secret
keys used for subsequent authentication.
■ Initialization: Before a client system can operate securely, it is necessary to install key materials that have the appropriate relationship with keys stored
elsewhere in the infrastructure. For example, the client needs to be securely
initialized with the public key and other assured information of the trusted
CA(s), to be used in validating certificate paths.
■ Certification: This is the process in which a CA issues a certificate for a user’s public key, returns that certificate to the user’s client system, and/or posts that
certificate in a repository.
■ Key pair recovery: Key pairs can be used to support digital signature creation and verification, encryption and decryption, or both. When a key pair is used for
Figure 14.17 PKIX Architectural Model
End entity Certificate/CRL retrieval
Certificate publication
Certificate/CRL publication
CRL publication
Cross certification
C er
ti fic
at e/
C R
L R
ep os
it or
y
Certificate authority
Registration authority
Certificate authority
Registration, initialization, certification, key pair recovery, key pair update revocation request
PKI users
PKI management
entities
CRL issuer
14.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 469
encryption/decryption, it is important to provide a mechanism to recover the
necessary decryption keys when normal access to the keying material is no longer
possible, otherwise it will not be possible to recover the encrypted data. Loss of
access to the decryption key can result from forgotten passwords/PINs, corrupted
disk drives, damage to hardware tokens, and so on. Key pair recovery allows end
entities to restore their encryption/decryption key pair from an authorized key
backup facility (typically, the CA that issued the end entity’s certificate).
■ Key pair update: All key pairs need to be updated regularly (i.e., replaced with a new key pair) and new certificates issued. Update is required when the
certificate lifetime expires and as a result of certificate revocation.
■ Revocation request: An authorized person advises a CA of an abnormal situ- ation requiring certificate revocation. Reasons for revocation include private-
key compromise, change in affiliation, and name change.
■ Cross certification: Two CAs exchange information used in establishing a cross-certificate. A cross-certificate is a certificate issued by one CA to another
CA that contains a CA signature key used for issuing certificates.
PKIX Management Protocols
The PKIX working group has defines two alternative management protocols
between PKIX entities that support the management functions listed in the pre-
ceding subsection. RFC 2510 defines the certificate management protocols (CMP).
Within CMP, each of the management functions is explicitly identified by specific
protocol exchanges. CMP is designed to be a flexible protocol able to accommodate
a variety of technical, operational, and business models.
RFC 2797 defines certificate management messages over CMS (CMC), where
CMS refers to RFC 2630, cryptographic message syntax. CMC is built on earlier work
and is intended to leverage existing implementations. Although all of the PKIX func-
tions are supported, the functions do not all map into specific protocol exchanges.
14.6 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
Key Terms
Review Questions 14.1 Explain why man-in-the-middle attacks are ineffective on the secret key distribution
protocol discussed in Figure 14.3.
14.2 What is the major issue in end to end key distribution? How does the key hierarchy concept address that issue?
14.3 What is a nonce? 14.4 What is a key distribution center? 14.5 What are two different uses of public-key cryptography related to key distribution?
end-to-end encryption
key distribution
key distribution center (KDC)
key management
man-in-the-middle attack
master key
nonce
public-key certificate
public-key directory
X.509 certificate
Hiva-Network.Com
470 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION
14.6 List four general categories of schemes for the distribution of public keys. 14.7 Discuss the potential security issues that arise due to public key directory based
system.
14.8 What is a public-key certificate? 14.9 What are the requirements for the use of a public-key certificate scheme? 14.10 What is the purpose of the X.509 standard? 14.11 What is a chain of certificates? 14.12 How is an X.509 certificate revoked?
Problems 14.1 One local area network vendor provides a key distribution facility, as illustrated in
Figure 14.18. a. Describe the scheme. b. Compare this scheme to that of Figure 14.3. What are the pros and cons?
14.2 “We are under great pressure, Holmes.” Detective Lestrade looked nervous. “We have learned that copies of sensitive government documents are stored in computers of one foreign embassy here in London. Normally these documents exist in electronic form only on a selected few government computers that satisfy the most stringent security requirements. However, sometimes they must be sent through the network connecting all government computers. But all messages in this network are encrypted using a top-secret encryption algorithm certified by our best crypto experts. Even the NSA and the KGB are unable to break it. And now these documents have appeared in hands of diplomats of a small, otherwise insignificant, country. And we have no idea how it could happen.”
“But you do have some suspicion who did it, do you?” asked Holmes.
“Yes, we did some routine investigation. There is a man who has legal access to one of the government computers and has frequent contacts with diplomats from the embassy. But the computer he has access to is not one of the trusted ones where these documents are normally stored. He is the suspect, but we have no idea how he could obtain copies of the documents. Even if he could obtain a copy of an encrypted document, he couldn’t decrypt it.”
Figure 14.18 Figure for Problem 14.1
Key Distribution
Center (KDC)
B A
(1) IDA, E(Ka, Na)
(2) IDA, E(Ka, Na), IDB, E(Kb, Nb)
(4) E(Ka, [Ks, IDB, Na])
(3) E(Kb, [Ks, IDA, Nb]), E(Ka, [Ks, IDB, Na])
14.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 471
“Hmm, please describe the communication protocol used on the network.” Holmes opened his eyes, thus proving that he had followed Lestrade’s talk with an attention that contrasted with his sleepy look.
“Well, the protocol is as follows. Each node N of the network has been assigned a unique secret key Kn. This key is used to secure communication between the node and a trusted server. That is, all the keys are stored also on the server. User A, wishing to send a secret message M to user B, initiates the following protocol:
1. A generates a random number R and sends to the server his name A, destination B, and E(Ka, R).
2. Server responds by sending E(Kb, R) to A. 3. A sends E(R, M) together with E(Kb, R) to B. 4. B knows Kb, thus decrypts E(Kb, R), to get R and will subsequently use R to
decrypt E(R, M) to get M.
You see that a random key is generated every time a message has to be sent. I admit the man could intercept messages sent between the top-secret trusted nodes, but I see no way he could decrypt them.”
“Well, I think you have your man, Lestrade. The protocol isn’t secure because the server doesn’t authenticate users who send him a request. Apparently designers of the protocol have believed that sending E(Kx, R) implicitly authenticates user X as the sender, as only X (and the server) knows Kx. But you know that E(Kx, R) can be intercepted and later replayed. Once you understand where the hole is, you will be able to obtain enough evidence by monitoring the man’s use of the computer he has access to. Most likely he works as follows. After intercepting E(Ka, R) and E(R, M) (see steps 1 and 3 of the protocol), the man, let’s denote him as Z, will continue by pretending to be A and . . .
Finish the sentence for Holmes.
14.3 The 1988 version of X.509 lists properties that RSA keys must satisfy to be secure given current knowledge about the difficulty of factoring large numbers. The discus- sion concludes with a constraint on the public exponent and the modulus n:
It must be ensured that e 7 log2(n) to prevent attack by taking the eth root mod n to disclose the plaintext.
Although the constraint is correct, the reason given for requiring it is incorrect. What is wrong with the reason given and what is the correct reason?
14.4 Find at least one intermediate certification authority’s certificate and one trusted root certification authority’s certificate on your computer (e.g., in the browser). Print screenshots of both the general and details tab for each certificate.
14.5 NIST defines the term cryptoperiod as the time span during which a specific key is authorized for use or in which the keys for a given system or application may remain in effect. One document on key management uses the following time diagram for a shared secret key.
Originator usage period
Recipient usage period
Cryptoperiod
472 CHAPTER 14 / KEY MANAGEMENT AND DISTRIBUTION
Explain the overlap by giving an example application in which the originator’s usage period for the shared secret key begins before the recipient’s usage period and also ends before the recipients usage period.
14.6 Consider the following protocol, designed to let A and B decide on a fresh, shared session key KAB
= . We assume that they already share a long-term key KAB. 1. A S B: A, NA. 2. B S A: E(KAB, [NA, KAB= ]) 3. A S B: E(KAB= , NA) a. We first try to understand the protocol designer’s reasoning: —Why would A and B believe after the protocol ran that they share KAB
= with the other party?
—Why would they believe that this shared key is fresh? In both cases, you should explain both the reasons of both A and B, so your answer should complete the sentences A believes that she shares KAB
= with B since . . . B believes that he shares KAB
= with A since . . . A believes that KAB
= is fresh since . . . B believes that KAB
= is fresh since . . . b. Assume now that A starts a run of this protocol with B. However, the connection
is intercepted by the adversary C. Show how C can start a new run of the protocol using reflection, causing A to believe that she has agreed on a fresh key with B (in spite of the fact that she has only been communicating with C). Thus, in particular, the belief in (a) is false.
c. Propose a modification of the protocol that prevents this attack. 14.7 What are the management functions of a PKI? What is a cross certificate? 14.8 State the significance of key pair recovery. When is the key pair updated?
Note: The remaining problems deal with the a cryptographic product developed by IBM, which is briefly described in a document at box.com/Crypto7e (IBMCrypto.pdf). Try these problems after reviewing the document.
14.9 What is the effect of adding the instruction EMKi
EMKi: X S E(KMHi, X) i = 0, 1
14.10 Suppose N different systems use the IBM Cryptographic Subsystem with host master keys KMH[i](i = 1, 2, c N). Devise a method for communicating between sys- tems without requiring the system to either share a common host master key or to divulge their individual host master keys. Hint: Each system needs three variants of its host master key.
14.11 The principal objective of the IBM Cryptographic Subsystem is to protect transmis- sions between a terminal and the processing system. Devise a procedure, perhaps adding instructions, which will allow the processor to generate a session key KS and distribute it to Terminal i and Terminal j without having to store a key-equivalent variable in the host.
473
CHAPTER
User Authentication 15.1 Remote User-Authentication Principles
The NIST Model for Electronic User Authentication
Means of Authentication
Mutual Authentication
One-Way Authentication
15.2 Remote User-Authentication Using Symmetric Encryption
Mutual Authentication
One-Way Authentication
15.3 Kerberos
Motivation
Kerberos Version 4
Kerberos Version 5
15.4 Remote User-Authentication Using Asymmetric Encryption
Mutual Authentication
One-Way Authentication
15.5 Federated Identity Management
Identity Management
Identity Federation
15.6 Personal Identity Verification
PIV System Model
PIV Documentation
PIV Credentials and Keys
Authentication
15.7 Key Terms, Review Questions, and Problems
474 CHAPTER 15 / USER AUTHENTICATION
This chapter examines some of the authentication functions that have been developed
to support network-based user authentication. The chapter begins with an introduc-
tion to some of the concepts and key considerations for user authentication over a
network or the Internet. The next section examines user-authentication protocols that
rely on symmetric encryption. This is followed by a section on one of the earliest and
also one of the most widely used authentication services: Kerberos. Next, the chapter
looks at user-authentication protocols that rely on asymmetric encryption. This is fol-
lowed by a discussion of the X.509 user-authentication protocol. Finally, the concept of
federated identity is introduced.
15.1 REMOTE USER-AUTHENTICATION PRINCIPLES
In most computer security contexts, user authentication is the fundamental build-
ing block and the primary line of defense. User authentication is the basis for most
types of access control and for user accountability. RFC 4949 (Internet Security Glossary) defines user authentication as the process of verifying an identity claimed by or for a system entity. This process consists of two steps:
■ Identification step: Presenting an identifier to the security system. (Identifiers should be assigned carefully, because authenticated identities are the basis for
other security services, such as access control service.)
■ Verification step: Presenting or generating authentication information that corroborates the binding between the entity and the identifier.
For example, user Alice Toklas could have the user identifier ABTOKLAS.
This information needs to be stored on any server or computer system that Alice
wishes to use and could be known to system administrators and other users.
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Understand the distinction between identification and verification.
◆ Present an overview of techniques for remote user authentication using symmetric encryption.
◆ Give a presentation on Kerberos.
◆ Explain the differences between versions 4 and 5 of Kerberos.
◆ Describe the use of Kerberos in multiple realms.
◆ Present an overview of techniques for remote user authentication using asymmetric encryption.
◆ Understand the need for a federated identity management system.
◆ Explain the use of PIV mechanisms as part of a user authentication system.
15.1 / REMOTE USER-AUTHENTICATION PRINCIPLES 475
A typical item of authentication information associated with this user ID is a pass-
word, which is kept secret (known only to Alice and to the system). If no one is
able to obtain or guess Alice’s password, then the combination of Alice’s user ID
and password enables administrators to set up Alice’s access permissions and audit
her activity. Because Alice’s ID is not secret, system users can send her email, but
because her password is secret, no one can pretend to be Alice.
In essence, identification is the means by which a user provides a claimed
identity to the system; user authentication is the means of establishing the validity
of the claim. Note that user authentication is distinct from message authentication.
As defined in Chapter 12, message authentication is a procedure that allows com-
municating parties to verify that the contents of a received message have not been
altered and that the source is authentic. This chapter is concerned solely with user
authentication.
The NIST Model for Electronic User Authentication
NIST SP 800-63-2 (Electronic Authentication Guideline, August 2013) defines elec- tronic user authentication as the process of establishing confidence in user identi-
ties that are presented electronically to an information system. Systems can use the
authenticated identity to determine if the authenticated individual is authorized to
perform particular functions, such as database transactions or access to system re-
sources. In many cases, the authentication and transaction or other authorized function
takes place across an open network such as the Internet. Equally authentication and
subsequent authorization can take place locally, such as across a local area network.
SP 800-63-2 defines a general model for user authentication that involves a num-
ber of entities and procedures. We discuss this model with reference to Figure 15.1.
The initial requirement for performing user authentication is that the user
must be registered with the system. The following is a typical sequence for registra-
tion. An applicant applies to a registration authority (RA) to become a subscriber
Figure 15.1 The NIST SP 800-63-2 E-Authentication Architectural Model
Registration authority (RA)
Registration, credential issuance, and maintenance
E-Authentication using token and credential
Identity proofing User registration
To ke
n, cre
de nti
al
Re gis
tra tio
n/i ssu
an ce
Authenticated session
Authenticated protocol
Exchange
Authenticated assertion
Registration Confirmation
Token/credential Validation
Relying party (RP)
Verifier
Subscriber/ claimant
Credential service
provider (RA)
476 CHAPTER 15 / USER AUTHENTICATION
of a credential service provider (CSP). In this model, the RA is a trusted entity that establishes and vouches for the identity of an applicant to a CSP. The CSP then
engages in an exchange with the subscriber. Depending on the details of the over-
all authentication system, the CSP issues some sort of electronic credential to the
subscriber. The credential is a data structure that authoritatively binds an identity and additional attributes to a token possessed by a subscriber, and can be verified
when presented to the verifier in an authentication transaction. The token could
be an encryption key or an encrypted password that identifies the subscriber. The
token may be issued by the CSP, generated directly by the subscriber, or provided
by a third party. The token and credential may be used in subsequent authentica-
tion events.
Once a user is registered as a subscriber, the actual authentication process can
take place between the subscriber and one or more systems that perform authen-
tication and, subsequently, authorization. The party to be authenticated is called a
claimant and the party verifying that identity is called a verifier. When a claimant successfully demonstrates possession and control of a token to a verifier through an
authentication protocol, the verifier can verify that the claimant is the subscriber
named in the corresponding credential. The verifier passes on an assertion about the
identity of the subscriber to the relying party (RP). That assertion includes identity information about a subscriber, such as the subscriber name, an identifier assigned
at registration, or other subscriber attributes that were verified in the registration
process. The RP can use the authenticated information provided by the verifier to
make access control or authorization decisions.
An implemented system for authentication will differ from or be more com-
plex than this simplified model, but the model illustrates the key roles and functions
needed for a secure authentication system.
Means of Authentication
There are four general means of authenticating a user’s identity, which can be used
alone or in combination:
■ Something the individual knows: Examples include a password, a personal identification number (PIN), or answers to a prearranged set of questions.
■ Something the individual possesses: Examples include cryptographic keys, electronic keycards, smart cards, and physical keys. This type of authenticator
is referred to as a token.
■ Something the individual is (static biometrics): Examples include recognition by fingerprint, retina, and face.
■ Something the individual does (dynamic biometrics): Examples include recog- nition by voice pattern, handwriting characteristics, and typing rhythm.
All of these methods, properly implemented and used, can provide secure
user authentication. However, each method has problems. An adversary may be
able to guess or steal a password. Similarly, an adversary may be able to forge or
steal a token. A user may forget a password or lose a token. Furthermore, there is a
significant administrative overhead for managing password and token information
on systems and securing such information on systems. With respect to biometric
15.1 / REMOTE USER-AUTHENTICATION PRINCIPLES 477
authenticators, there are a variety of problems, including dealing with false positives
and false negatives, user acceptance, cost, and convenience. For network-based user
authentication, the most important methods involve cryptographic keys and some-
thing the individual knows, such as a password.
Mutual Authentication
An important application area is that of mutual authentication protocols. Such pro-
tocols enable communicating parties to satisfy themselves mutually about each oth-
er’s identity and to exchange session keys. This topic was examined in Chapter 14.
There, the focus was key distribution. We return to this topic here to consider the
wider implications of authentication.
Central to the problem of authenticated key exchange are two issues: confi-
dentiality and timeliness. To prevent masquerade and to prevent compromise of
session keys, essential identification and session-key information must be commu-
nicated in encrypted form. This requires the prior existence of secret or public keys
that can be used for this purpose. The second issue, timeliness, is important because
of the threat of message replays. Such replays, at worst, could allow an opponent to
compromise a session key or successfully impersonate another party. At minimum,
a successful replay can disrupt operations by presenting parties with messages that
appear genuine but are not.
[GONG93] lists the following examples of replay attacks:
1. The simplest replay attack is one in which the opponent simply copies a mes- sage and replays it later.
2. An opponent can replay a timestamped message within the valid time window. If both the original and the replay arrive within then time window, this inci-
dent can be logged.
3. As with example (2), an opponent can replay a timestamped message within the valid time window, but in addition, the opponent suppresses the original
message. Thus, the repetition cannot be detected.
4. Another attack involves a backward replay without modification. This is a re- play back to the message sender. This attack is possible if symmetric encryp-
tion is used and the sender cannot easily recognize the difference between
messages sent and messages received on the basis of content.
One approach to coping with replay attacks is to attach a sequence number to
each message used in an authentication exchange. A new message is accepted only
if its sequence number is in the proper order. The difficulty with this approach is
that it requires each party to keep track of the last sequence number for each claim-
ant it has dealt with. Because of this overhead, sequence numbers are generally not
used for authentication and key exchange. Instead, one of the following two general
approaches is used:
■ Timestamps: Party A accepts a message as fresh only if the message contains a timestamp that, in A’s judgment, is close enough to A’s knowledge of cur- rent time. This approach requires that clocks among the various participants
be synchronized.
478 CHAPTER 15 / USER AUTHENTICATION
■ Challenge/response: Party A, expecting a fresh message from B, first sends B a nonce (challenge) and requires that the subsequent message (response) re- ceived from B contain the correct nonce value.
It can be argued (e.g., [LAM92a]) that the timestamp approach should not be
used for connection-oriented applications because of the inherent difficulties with
this technique. First, some sort of protocol is needed to maintain synchronization
among the various processor clocks. This protocol must be both fault tolerant, to
cope with network errors, and secure, to cope with hostile attacks. Second, the oppor-
tunity for a successful attack will arise if there is a temporary loss of synchronization
resulting from a fault in the clock mechanism of one of the parties. Finally, because
of the variable and unpredictable nature of network delays, distributed clocks cannot
be expected to maintain precise synchronization. Therefore, any timestamp-based
procedure must allow for a window of time sufficiently large to accommodate net-
work delays yet sufficiently small to minimize the opportunity for attack.
On the other hand, the challenge-response approach is unsuitable for a con-
nectionless type of application, because it requires the overhead of a handshake be-
fore any connectionless transmission, effectively negating the chief characteristic of
a connectionless transaction. For such applications, reliance on some sort of secure
time server and a consistent attempt by each party to keep its clocks in synchroniza-
tion may be the best approach (e.g., [LAM92b]).
One-Way Authentication
One application for which encryption is growing in popularity is electronic mail
(email). The very nature of electronic mail, and its chief benefit, is that it is not nec-
essary for the sender and receiver to be online at the same time. Instead, the email
message is forwarded to the receiver’s electronic mailbox, where it is buffered until
the receiver is available to read it.
The “envelope” or header of the email message must be in the clear, so that
the message can be handled by the store-and-forward email protocol, such as the
Simple Mail Transfer Protocol (SMTP) or X.400. However, it is often desirable that
the mail-handling protocol not require access to the plaintext form of the message,
because that would require trusting the mail-handling mechanism. Accordingly, the
email message should be encrypted such that the mail-handling system is not in
possession of the decryption key.
A second requirement is that of authentication. Typically, the recipient wants some assurance that the message is from the alleged sender.
15.2 REMOTE USER-AUTHENTICATION USING SYMMETRIC ENCRYPTION
Mutual Authentication
As was discussed in Chapter 14, a two-level hierarchy of symmetric encryption keys
can be used to provide confidentiality for communication in a distributed environ-
ment. In general, this strategy involves the use of a trusted key distribution center
Hiva-Network.Com
15.2 / REMOTE USER-AUTHENTICATION USING SYMMETRIC ENCRYPTION 479
(KDC). Each party in the network shares a secret key, known as a master key, with
the KDC. The KDC is responsible for generating keys to be used for a short time
over a connection between two parties, known as session keys, and for distribut-
ing those keys using the master keys to protect the distribution. This approach is
quite common. As an example, we look at the Kerberos system in Section 15.3.
The discussion in this subsection is relevant to an understanding of the Kerberos
mechanisms.
Figure 14.3 illustrates a proposal initially put forth by Needham and Schroeder
[NEED78] for secret key distribution using a KDC that, as was mentioned in
Chapter 14, includes authentication features. The protocol can be summarized as
follows.1
1. A S KDC: IDA }IDB }N1 2. KDC S A: E(Ka, [Ks }IDB }N1 }E(Kb, [Ks }IDA])]) 3. A S B: E(Kb, [Ks }IDA]) 4. B S A: E(Ks, N2) 5. A S B: E(Ks, f(N2)) where f() is a generic function that modifies the
value of the nonce.
Secret keys Ka and Kb are shared between A and the KDC and B and the KDC, respectively. The purpose of the protocol is to distribute securely a session
key Ks to A and B. Entity A securely acquires a new session key in step 2. The mes- sage in step 3 can be decrypted, and hence understood, only by B. Step 4 reflects B’s
knowledge of Ks, and step 5 assures B of A’s knowledge of Ks and assures B that this is a fresh message because of the use of the nonce N2. Recall from our discussion in Chapter 14 that the purpose of steps 4 and 5 is to prevent a certain type of replay at-
tack. In particular, if an opponent is able to capture the message in step 3 and replay
it, this might in some fashion disrupt operations at B.
Despite the handshake of steps 4 and 5, the protocol is still vulnerable to a
form of replay attack. Suppose that an opponent, X, has been able to compromise
an old session key. Admittedly, this is a much more unlikely occurrence than that
an opponent has simply observed and recorded step 3. Nevertheless, it is a potential
security risk. X can impersonate A and trick B into using the old key by simply re-
playing step 3. Unless B remembers indefinitely all previous session keys used with
A, B will be unable to determine that this is a replay. If X can intercept the hand-
shake message in step 4, then it can impersonate A’s response in step 5. From this
point on, X can send bogus messages to B that appear to B to come from A using an
authenticated session key.
Denning [DENN81, DENN82] proposes to overcome this weakness by a
modification to the Needham/Schroeder protocol that includes the addition of a
timestamp to steps 2 and 3. Her proposal assumes that the master keys, Ka and Kb, are secure, and it consists of the following steps.
1The portion to the left of the colon indicates the sender and the receiver; the portion to the right indi- cates the contents of the message; the symbol } indicates concatenation.
480 CHAPTER 15 / USER AUTHENTICATION
1. A S KDC: IDA }IDB 2. KDC S A: E(Ka, [Ks }IDB }T}E(Kb, [Ks }IDA }T])]) 3. A S B: E(Kb, [Ks }IDA }T]) 4. B S A: E(Ks, N1) 5. A S B: E(Ks, f(N1))
T is a timestamp that assures A and B that the session key has only just been
generated. Thus, both A and B know that the key distribution is a fresh exchange.
A and B can verify timeliness by checking that
�Clock - T� 6 ∆t1 + ∆t2 where ∆t1 is the estimated normal discrepancy between the KDC’s clock and the local clock (at A or B) and ∆t2 is the expected network delay time. Each node can set its clock against some standard reference source. Because the timestamp T is encrypted using the secure master keys, an opponent, even with knowledge of an
old session key, cannot succeed because a replay of step 3 will be detected by B as
untimely.
A final point: Steps 4 and 5 were not included in the original presentation
[DENN81] but were added later [DENN82]. These steps confirm the receipt of the
session key at B.
The Denning protocol seems to provide an increased degree of security com-
pared to the Needham/Schroeder protocol. However, a new concern is raised:
namely, that this new scheme requires reliance on clocks that are synchronized
throughout the network. [GONG92] points out a risk involved. The risk is based
on the fact that the distributed clocks can become unsynchronized as a result of
sabotage on or faults in the clocks or the synchronization mechanism.2 The problem
occurs when a sender’s clock is ahead of the intended recipient’s clock. In this case,
an opponent can intercept a message from the sender and replay it later when the
timestamp in the message becomes current at the recipient’s site. This replay could
cause unexpected results. Gong refers to such attacks as suppress-replay attacks. One way to counter suppress-replay attacks is to enforce the requirement that
parties regularly check their clocks against the KDC’s clock. The other alternative,
which avoids the need for clock synchronization, is to rely on handshaking protocols
using nonces. This latter alternative is not vulnerable to a suppress-replay attack,
because the nonces the recipient will choose in the future are unpredictable to the
sender. The Needham/Schroeder protocol relies on nonces only but, as we have
seen, has other vulnerabilities.
In [KEHN92], an attempt is made to respond to the concerns about suppress-
replay attacks and at the same time fix the problems in the Needham/Schroeder
protocol. Subsequently, an inconsistency in this latter protocol was noted and an
improved strategy was presented in [NEUM93a].3 The protocol is
2Such things can and do happen. In recent years, flawed chips were used in a number of computers and other electronic systems to track the time and date. The chips had a tendency to skip forward one day. [NEUM90] 3It really is hard to get these things right.
15.2 / REMOTE USER-AUTHENTICATION USING SYMMETRIC ENCRYPTION 481
1. A S B: IDA }Na 2. B S KDC: IDB }Nb }E(Kb, [IDA }Na }Tb]) 3. KDC S A: E(Ka, [IDB }Na }Ks }Tb]) }E(Kb, [IDA }Ks }Tb]) }Nb 4. A S B: E(Kb, [IDA }Ks }Tb]) }E(Ks, Nb)
Let us follow this exchange step by step.
1. A initiates the authentication exchange by generating a nonce, Na, and sending that plus its identifier to B in plaintext. This nonce will be returned to A in an
encrypted message that includes the session key, assuring A of its timeliness.
2. B alerts the KDC that a session key is needed. Its message to the KDC in- cludes its identifier and a nonce, Nb. This nonce will be returned to B in an encrypted message that includes the session key, assuring B of its timeliness.
B’s message to the KDC also includes a block encrypted with the secret key
shared by B and the KDC. This block is used to instruct the KDC to issue
credentials to A; the block specifies the intended recipient of the credentials, a
suggested expiration time for the credentials, and the nonce received from A.
3. The KDC passes on to A B’s nonce and a block encrypted with the secret key that B shares with the KDC. The block serves as a “ticket” that can be used
by A for subsequent authentications, as will be seen. The KDC also sends to
A a block encrypted with the secret key shared by A and the KDC. This block
verifies that B has received A’s initial message (IDB) and that this is a timely message and not a replay (Na), and it provides A with a session key (Ks) and the time limit on its use (Tb).
4. A transmits the ticket to B, together with the B’s nonce, the latter encrypted with the session key. The ticket provides B with the secret key that is used to de-
crypt E(Ks, Nb) to recover the nonce. The fact that B’s nonce is encrypted with the session key authenticates that the message came from A and is not a replay.
This protocol provides an effective, secure means for A and B to establish a
session with a secure session key. Furthermore, the protocol leaves A in posses-
sion of a key that can be used for subsequent authentication to B, avoiding the
need to contact the authentication server repeatedly. Suppose that A and B estab-
lish a session using the aforementioned protocol and then conclude that session.
Subsequently, but within the time limit established by the protocol, A desires a new
session with B. The following protocol ensues:
1. A S B: E(Kb, [IDA }Ks }Tb]) }Na=
2. B S A: Nb= }E(Ks, Na= )
3. A S B: E(Ks, Nb= )
When B receives the message in step 1, it verifies that the ticket has not expired.
The newly generated nonces Na = and Nb
= assure each party that there is no replay
attack.
In all the foregoing, the time specified in Tb is a time relative to B’s clock. Thus, this timestamp does not require synchronized clocks, because B checks only
self-generated timestamps.
482 CHAPTER 15 / USER AUTHENTICATION
One-Way Authentication
Using symmetric encryption, the decentralized key distribution scenario illustrated
in Figure 14.5 is impractical. This scheme requires the sender to issue a request to
the intended recipient, await a response that includes a session key, and only then
send the message.
With some refinement, the KDC strategy illustrated in Figure 14.3 is a can-
didate for encrypted electronic mail. Because we wish to avoid requiring that the
recipient (B) be on line at the same time as the sender (A), steps 4 and 5 must be
eliminated. For a message with content M, the sequence is as follows:
1. A S KDC: IDA }IDB }N1 2. KDC S A: E(Ka, [Ks }IDB }N1 }E(Kb, [Ks }IDA])]) 3. A S B: E(Kb, [Ks }IDA]) }E(Ks, M)
This approach guarantees that only the intended recipient of a message will be
able to read it. It also provides a level of authentication that the sender is A. As
specified, the protocol does not protect against replays. Some measure of defense
could be provided by including a timestamp with the message. However, because
of the potential delays in the email process, such timestamps may have limited
usefulness.
15.3 KERBEROS
Kerberos4 is an authentication service developed as part of Project Athena at MIT.
The problem that Kerberos addresses is this: Assume an open distributed environ-
ment in which users at workstations wish to access services on servers distributed
throughout the network. We would like for servers to be able to restrict access to
authorized users and to be able to authenticate requests for service. In this envi-
ronment, a workstation cannot be trusted to identify its users correctly to network
services. In particular, the following three threats exist:
1. A user may gain access to a particular workstation and pretend to be another user operating from that workstation.
2. A user may alter the network address of a workstation so that the requests sent from the altered workstation appear to come from the impersonated
workstation.
3. A user may eavesdrop on exchanges and use a replay attack to gain entrance to a server or to disrupt operations.
In any of these cases, an unauthorized user may be able to gain access to services
and data that he or she is not authorized to access. Rather than building in elaborate
4“In Greek mythology, a many headed dog, commonly three, perhaps with a serpent’s tail, the guardian of the entrance of Hades.” From Dictionary of Subjects and Symbols in Art, by James Hall, Harper & Row, 1979. Just as the Greek Kerberos has three heads, the modern Kerberos was intended to have three components to guard a network’s gate: authentication, accounting, and audit. The last two heads were never implemented.
15.3 / KERBEROS 483
authentication protocols at each server, Kerberos provides a centralized authenti-
cation server whose function is to authenticate users to servers and servers to users.
Unlike most other authentication schemes described in this book, Kerberos relies
exclusively on symmetric encryption, making no use of public-key encryption.
Two versions of Kerberos are in common use. Version 4 [MILL88, STEI88]
implementations still exist. Version 5 [KOHL94] corrects some of the security defi-
ciencies of version 4 and has been issued as a proposed Internet Standard (RFC
4120 and RFC 4121).5
We begin this section with a brief discussion of the motivation for the Kerberos
approach. Then, because of the complexity of Kerberos, it is best to start with a de-
scription of the authentication protocol used in version 4. This enables us to see the
essence of the Kerberos strategy without considering some of the details required to
handle subtle security threats. Finally, we examine version 5.
Motivation
If a set of users is provided with dedicated personal computers that have no network
connections, then a user’s resources and files can be protected by physically secur-
ing each personal computer. When these users instead are served by a centralized
time-sharing system, the time-sharing operating system must provide the security.
The operating system can enforce access-control policies based on user identity and
use the logon procedure to identify users.
Today, neither of these scenarios is typical. More common is a distributed
architecture consisting of dedicated user workstations (clients) and distributed
or centralized servers. In this environment, three approaches to security can be
envisioned.
1. Rely on each individual client workstation to assure the identity of its user or users and rely on each server to enforce a security policy based on user iden-
tification (ID).
2. Require that client systems authenticate themselves to servers, but trust the client system concerning the identity of its user.
3. Require the user to prove his or her identity for each service invoked. Also require that servers prove their identity to clients.
In a small, closed environment in which all systems are owned and operated
by a single organization, the first or perhaps the second strategy may suffice.6 But
in a more open environment in which network connections to other machines are
supported, the third approach is needed to protect user information and resources
housed at the server. Kerberos supports this third approach. Kerberos assumes a
distributed client/server architecture and employs one or more Kerberos servers to
provide an authentication service.
5Versions 1 through 3 were internal development versions. Version 4 is the “original” Kerberos. 6However, even a closed environment faces the threat of attack by a disgruntled employee.
484 CHAPTER 15 / USER AUTHENTICATION
The first published report on Kerberos [STEI88] listed the following
requirements.
■ Secure: A network eavesdropper should not be able to obtain the necessary information to impersonate a user. More generally, Kerberos should be strong
enough that a potential opponent does not find it to be the weak link.
■ Reliable: For all services that rely on Kerberos for access control, lack of availability of the Kerberos service means lack of availability of the supported
services. Hence, Kerberos should be highly reliable and should employ a
distributed server architecture with one system able to back up another.
■ Transparent: Ideally, the user should not be aware that authentication is taking place beyond the requirement to enter a password.
■ Scalable: The system should be capable of supporting large numbers of clients and servers. This suggests a modular, distributed architecture.
To support these requirements, the overall scheme of Kerberos is that of a
trusted third-party authentication service that uses a protocol based on that pro-
posed by Needham and Schroeder [NEED78], which was discussed in Section 15.2.
It is trusted in the sense that clients and servers trust Kerberos to mediate their
mutual authentication. Assuming the Kerberos protocol is well designed, then the
authentication service is secure if the Kerberos server itself is secure.7
Kerberos Version 4
Version 4 of Kerberos makes use of DES, in a rather elaborate protocol, to pro-
vide the authentication service. Viewing the protocol as a whole, it is difficult to see
the need for the many elements contained therein. Therefore, we adopt a strategy
used by Bill Bryant of Project Athena [BRYA88] and build up to the full protocol
by looking first at several hypothetical dialogues. Each successive dialogue adds
additional complexity to counter security vulnerabilities revealed in the preceding
dialogue.
After examining the protocol, we look at some other aspects of version 4.
A SIMPLE AUTHENTICATION DIALOGUE In an unprotected network environment, any client can apply to any server for service. The obvious security risk is that of im-
personation. An opponent can pretend to be another client and obtain unauthor-
ized privileges on server machines. To counter this threat, servers must be able to
confirm the identities of clients who request service. Each server can be required to
undertake this task for each client/server interaction, but in an open environment,
this places a substantial burden on each server.
7Remember that the security of the Kerberos server should not automatically be assumed but must be guarded carefully (e.g., in a locked room). It is well to remember the fate of the Greek Kerberos, whom Hercules was ordered by Eurystheus to capture as his Twelfth Labor: “Hercules found the great dog on its chain and seized it by the throat. At once the three heads tried to attack, and Kerberos lashed about with his powerful tail. Hercules hung on grimly, and Kerberos relaxed into unconsciousness. Eurystheus may have been surprised to see Hercules alive—when he saw the three slavering heads and the huge dog they belonged to he was frightened out of his wits, and leapt back into the safety of his great bronze jar.” From The Hamlyn Concise Dictionary of Greek and Roman Mythology, by Michael Stapleton, Hamlyn, 1982.
15.3 / KERBEROS 485
An alternative is to use an authentication server (AS) that knows the
passwords of all users and stores these in a centralized database. In addition, the AS
shares a unique secret key with each server. These keys have been distributed physi-
cally or in some other secure manner. Consider the following hypothetical dialogue:
(1) C S AS: IDC }PC }IDV (2) AS S C: Ticket (3) C S V: IDC }Ticket
Ticket = E(Kv, [IDC }ADC }IDV])
where
C = client AS = authentication server
V = server IDC = identifier of user on C IDV = identifier of V
PC = password of user on C ADC = network address of C
Kv = secret encryption key shared by AS and V
In this scenario, the user logs on to a workstation and requests access to server V.
The client module C in the user’s workstation requests the user’s password and then
sends a message to the AS that includes the user’s ID, the server’s ID, and the user’s
password. The AS checks its database to see if the user has supplied the proper
password for this user ID and whether this user is permitted access to server V. If
both tests are passed, the AS accepts the user as authentic and must now convince
the server that this user is authentic. To do so, the AS creates a ticket that con- tains the user’s ID and network address and the server’s ID. This ticket is encrypted
using the secret key shared by the AS and this server. This ticket is then sent back
to C. Because the ticket is encrypted, it cannot be altered by C or by an opponent.
With this ticket, C can now apply to V for service. C sends a message to V con-
taining C’s ID and the ticket. V decrypts the ticket and verifies that the user ID in
the ticket is the same as the unencrypted user ID in the message. If these two match,
the server considers the user authenticated and grants the requested service.
Each of the ingredients of message (3) is significant. The ticket is encrypted to
prevent alteration or forgery. The server’s ID (IDV) is included in the ticket so that the server can verify that it has decrypted the ticket properly. IDC is included in the ticket to indicate that this ticket has been issued on behalf of C. Finally, ADC serves to counter the following threat. An opponent could capture the ticket transmitted
in message (2), then use the name IDC and transmit a message of form (3) from another workstation. The server would receive a valid ticket that matches the user
ID and grant access to the user on that other workstation. To prevent this attack,
the AS includes in the ticket the network address from which the original request
came. Now the ticket is valid only if it is transmitted from the same workstation that
initially requested the ticket.
486 CHAPTER 15 / USER AUTHENTICATION
A MORE SECURE AUTHENTICATION DIALOGUE Although the foregoing scenario solves some of the problems of authentication in an open network environment, problems
remain. Two in particular stand out. First, we would like to minimize the number
of times that a user has to enter a password. Suppose each ticket can be used only
once. If user C logs on to a workstation in the morning and wishes to check his or her
mail at a mail server, C must supply a password to get a ticket for the mail server. If
C wishes to check the mail several times during the day, each attempt requires re-
entering the password. We can improve matters by saying that tickets are reusable.
For a single logon session, the workstation can store the mail server ticket after it is
received and use it on behalf of the user for multiple accesses to the mail server.
However, under this scheme, it remains the case that a user would need a new
ticket for every different service. If a user wished to access a print server, a mail
server, a file server, and so on, the first instance of each access would require a new
ticket and hence require the user to enter the password.
The second problem is that the earlier scenario involved a plaintext transmis-
sion of the password [message (1)]. An eavesdropper could capture the password
and use any service accessible to the victim.
To solve these additional problems, we introduce a scheme for avoiding plain-
text passwords and a new server, known as the ticket-granting server (TGS). The new (but still hypothetical) scenario is as follows.
Once per user logon session:
(1) C S AS: IDC }IDtgs (2) AS S C: E(Kc, Tickettgs)
Once per type of service:
(3) C S TGS: IDC }IDV }Tickettgs (4) TGS S C: Ticketv
Once per service session:
(5) C S V: IDC }Ticketv Tickettgs = E(Ktgs, [IDC }ADC }IDtgs }TS1 }Lifetime1])
Ticketv = E(Kv, [IDC }ADC }IDv }TS2 }Lifetime2])
The new service, TGS, issues tickets to users who have been authenticated to
AS. Thus, the user first requests a ticket-granting ticket (Tickettgs) from the AS. The client module in the user workstation saves this ticket. Each time the user requires
access to a new service, the client applies to the TGS, using the ticket to authenti-
cate itself. The TGS then grants a ticket for the particular service. The client saves
each service-granting ticket and uses it to authenticate its user to a server each time
a particular service is requested. Let us look at the details of this scheme:
1. The client requests a ticket-granting ticket on behalf of the user by sending its user’s ID to the AS, together with the TGS ID, indicating a request to use the
TGS service.
15.3 / KERBEROS 487
2. The AS responds with a ticket that is encrypted with a key that is derived from the user’s password (Kc), which is already stored at the AS. When this response arrives at the client, the client prompts the user for his or her password, gen-
erates the key, and attempts to decrypt the incoming message. If the correct
password is supplied, the ticket is successfully recovered.
Because only the correct user should know the password, only the correct user
can recover the ticket. Thus, we have used the password to obtain credentials from
Kerberos without having to transmit the password in plaintext. The ticket itself
consists of the ID and network address of the user, and the ID of the TGS. This
corresponds to the first scenario. The idea is that the client can use this ticket to
request multiple service-granting tickets. So the ticket-granting ticket is to be reus-
able. However, we do not wish an opponent to be able to capture the ticket and use
it. Consider the following scenario: An opponent captures the login ticket and waits
until the user has logged off his or her workstation. Then the opponent either gains
access to that workstation or configures his workstation with the same network ad-
dress as that of the victim. The opponent would be able to reuse the ticket to spoof
the TGS. To counter this, the ticket includes a timestamp, indicating the date and
time at which the ticket was issued, and a lifetime, indicating the length of time for
which the ticket is valid (e.g., eight hours). Thus, the client now has a reusable ticket
and need not bother the user for a password for each new service request. Finally,
note that the ticket-granting ticket is encrypted with a secret key known only to the
AS and the TGS. This prevents alteration of the ticket. The ticket is reencrypted
with a key based on the user’s password. This assures that the ticket can be recov-
ered only by the correct user, providing the authentication.
Now that the client has a ticket-granting ticket, access to any server can be
obtained with steps 3 and 4.
3. The client requests a service-granting ticket on behalf of the user. For this pur- pose, the client transmits a message to the TGS containing the user’s ID, the
ID of the desired service, and the ticket-granting ticket.
4. The TGS decrypts the incoming ticket using a key shared only by the AS and the TGS (Ktgs) and verifies the success of the decryption by the presence of its ID. It checks to make sure that the lifetime has not expired. Then it compares
the user ID and network address with the incoming information to authenti-
cate the user. If the user is permitted access to the server V, the TGS issues a
ticket to grant access to the requested service.
The service-granting ticket has the same structure as the ticket-granting ticket.
Indeed, because the TGS is a server, we would expect that the same elements are
needed to authenticate a client to the TGS and to authenticate a client to an appli-
cation server. Again, the ticket contains a timestamp and lifetime. If the user wants
access to the same service at a later time, the client can simply use the previously
acquired service-granting ticket and need not bother the user for a password. Note
that the ticket is encrypted with a secret key (Kv) known only to the TGS and the server, preventing alteration.
Finally, with a particular service-granting ticket, the client can gain access to
the corresponding service with step 5.
Hiva-Network.Com
488 CHAPTER 15 / USER AUTHENTICATION
5. The client requests access to a service on behalf of the user. For this purpose, the client transmits a message to the server containing the user’s ID and the service-
granting ticket. The server authenticates by using the contents of the ticket.
This new scenario satisfies the two requirements of only one password query
per user session and protection of the user password.
THE VERSION 4 AUTHENTICATION DIALOGUE Although the foregoing scenario en- hances security compared to the first attempt, two additional problems remain. The
heart of the first problem is the lifetime associated with the ticket-granting ticket.
If this lifetime is very short (e.g., minutes), then the user will be repeatedly asked
for a password. If the lifetime is long (e.g., hours), then an opponent has a greater
opportunity for replay. An opponent could eavesdrop on the network and capture
a copy of the ticket-granting ticket and then wait for the legitimate user to log out.
Then the opponent could forge the legitimate user’s network address and send the
message of step (3) to the TGS. This would give the opponent unlimited access to
the resources and files available to the legitimate user.
Similarly, if an opponent captures a service-granting ticket and uses it before it
expires, the opponent has access to the corresponding service.
Thus, we arrive at an additional requirement. A network service (the TGS or
an application service) must be able to prove that the person using a ticket is the
same person to whom that ticket was issued.
The second problem is that there may be a requirement for servers to authen-
ticate themselves to users. Without such authentication, an opponent could sabo-
tage the configuration so that messages to a server were directed to another loca-
tion. The false server would then be in a position to act as a real server and capture
any information from the user and deny the true service to the user.
We examine these problems in turn and refer to Table 15.1, which shows the
actual Kerberos protocol. Figure 15.2 provides a simplified overview.
(1) C S AS IDc }IDtgs }TS1 (2) AS S C E(Kc, [Kc, tgs }IDtgs }TS2 }Lifetime2 }Tickettgs])
Tickettgs = E(Ktgs, [Kc, tgs }IDC }ADC }IDtgs }TS2 }Lifetime2])
(a) Authentication Service Exchange to obtain ticket-granting ticket
(3) C S TGS IDv }Tickettgs }Authenticatorc (4) TGS S C E(Kc, tgs, [Kc, v }IDv }TS4 }Ticketv])
Tickettgs = E(Ktgs, [Kc, tgs }IDC }ADC }IDtgs }TS2 }Lifetime2]) Ticketv = E(Kv, [Kc, v }IDC }ADC }IDv }TS4 }Lifetime4])
Authenticatorc = E(Kc, tgs, [IDC }ADC }TS3])
(b) Ticket-Granting Service Exchange to obtain service-granting ticket
(5) C S V Ticketv }Authenticatorc (6) V S C E(Kc,v, [TS5 + 1]) (for mutual authentication)
Ticketv = E(Kv, [Kc, v }IDC }ADC }IDv }TS4 }Lifetime4]) Authenticatorc = E(Kc, v, [IDC }ADC }TS5])
(c) Client/Server Authentication Exchange to obtain service
Table 15.1 Summary of Kerberos Version 4 Message Exchanges
15.3 / KERBEROS 489
First, consider the problem of captured ticket-granting tickets and the need
to determine that the ticket presenter is the same as the client for whom the ticket
was issued. The threat is that an opponent will steal the ticket and use it before it
expires. To get around this problem, let us have the AS provide both the client and
the TGS with a secret piece of information in a secure manner. Then the client can
prove its identity to the TGS by revealing the secret information—again in a secure
manner. An efficient way of accomplishing this is to use an encryption key as the
secure information; this is referred to as a session key in Kerberos.
Table 15.1a shows the technique for distributing the session key. As before,
the client sends a message to the AS requesting access to the TGS. The AS re-
sponds with a message, encrypted with a key derived from the user’s password
(Kc), that contains the ticket. The encrypted message also contains a copy of the session key, Kc,tgs, where the subscripts indicate that this is a session key for C and TGS. Because this session key is inside the message encrypted with Kc, only the user’s client can read it. The same session key is included in the ticket, which can
be read only by the TGS. Thus, the session key has been securely delivered to both
C and the TGS.
Figure 15.2 Overview of Kerberos
Authentication server
Ticket- granting
server (TGS)
Host/ application
server
requ est t
icke t-
gran ting
tick et
once per user logon session
1. User logs on to workstation and requests service on host
3. Workstation prompts user for password to decrypt incoming message, and then send ticket and authenticator that contains user’s name, network address, and time to TGS.
ticke t + se
ssion key
request s ervice-
granting ticket
ticket + se ssion key
once per type of service
4. TGS decrypts ticket and authenticator, verifies request, and then creates ticket for requested application server.
Kerberos
5. Workstation sends ticket and authenticator to host.
6. Host verifies that ticket and authenticator match, and then grants access to service. If mutual authentication is required, server returns an authenticator.
request service provide server
authenticator once per service session
2. AS verifies user’s access right in database, and creates ticket-granting ticket and session key. Results are encrypted using key derived from user’s password.
490 CHAPTER 15 / USER AUTHENTICATION
Note that several additional pieces of information have been added to this
first phase of the dialogue. Message (1) includes a timestamp, so that the AS knows
that the message is timely. Message (2) includes several elements of the ticket in a
form accessible to C. This enables C to confirm that this ticket is for the TGS and to
learn its expiration time.
Armed with the ticket and the session key, C is ready to approach the TGS.
As before, C sends the TGS a message that includes the ticket plus the ID of the
requested service [message (3) in Table 15.1b]. In addition, C transmits an authentica-
tor, which includes the ID and address of C’s user and a timestamp. Unlike the ticket,
which is reusable, the authenticator is intended for use only once and has a very short
lifetime. The TGS can decrypt the ticket with the key that it shares with the AS. This
ticket indicates that user C has been provided with the session key Kc,tgs. In effect, the ticket says, “Anyone who uses Kc,tgs must be C.” The TGS uses the session key to decrypt the authenticator. The TGS can then check the name and address from the
authenticator with that of the ticket and with the network address of the incoming
message. If all match, then the TGS is assured that the sender of the ticket is indeed
the ticket’s real owner. In effect, the authenticator says, “At time TS3, I hereby use Kc,tgs.” Note that the ticket does not prove anyone’s identity but is a way to distribute keys securely. It is the authenticator that proves the client’s identity. Because the au-
thenticator can be used only once and has a short lifetime, the threat of an opponent
stealing both the ticket and the authenticator for presentation later is countered.
The reply from the TGS in message (4) follows the form of message (2). The
message is encrypted with the session key shared by the TGS and C and includes
a session key to be shared between C and the server V, the ID of V, and the time-
stamp of the ticket. The ticket itself includes the same session key.
C now has a reusable service-granting ticket for V. When C presents this ticket,
as shown in message (5), it also sends an authenticator. The server can decrypt the
ticket, recover the session key, and decrypt the authenticator.
If mutual authentication is required, the server can reply as shown in message
(6) of Table 15.1. The server returns the value of the timestamp from the authenti-
cator, incremented by 1, and encrypted in the session key. C can decrypt this mes-
sage to recover the incremented timestamp. Because the message was encrypted by
the session key, C is assured that it could have been created only by V. The contents
of the message assure C that this is not a replay of an old reply.
Finally, at the conclusion of this process, the client and server share a secret
key. This key can be used to encrypt future messages between the two or to ex-
change a new random session key for that purpose.
Figure 15.3 illustrates the Kerberos exchanges among the parties. Table 15.2
summarizes the justification for each of the elements in the Kerberos protocol.
KERBEROS REALMS AND MULTIPLE KERBERI A full-service Kerberos environment consisting of a Kerberos server, a number of clients, and a number of application
servers requires the following:
1. The Kerberos server must have the user ID and hashed passwords of all partic- ipating users in its database. All users are registered with the Kerberos server.
2. The Kerberos server must share a secret key with each server. All servers are registered with the Kerberos server.
15.3 / KERBEROS 491
Message (1) Client requests ticket-granting ticket. IDC Tells AS identity of user from this client.
IDtgs Tells AS that user requests access to TGS.
TS1 Allows AS to verify that client’s clock is synchronized with that of AS.
Message (2) AS returns ticket-granting ticket. Kc Encryption is based on user’s password, enabling AS and client to verify password, and
protecting contents of message (2).
Kc, tgs Copy of session key accessible to client created by AS to permit secure exchange between client and TGS without requiring them to share a permanent key.
IDtgs Confirms that this ticket is for the TGS.
TS2 Informs client of time this ticket was issued.
Lifetime2 Informs client of the lifetime of this ticket.
Tickettgs Ticket to be used by client to access TGS.
(a) Authentication Service Exchange
Message (3) Client requests service-granting ticket. IDV Tells TGS that user requests access to server V.
Tickettgs Assures TGS that this user has been authenticated by AS.
Authenticatorc Generated by client to validate ticket.
Table 15.2 Rationale for the Elements of the Kerberos Version 4 Protocol
Figure 15.3 Kerberos Exchanges
Client
Client authentication IDc || IDtgs || TS1
Tickettgs, server ID, and client authentication IDv || Tickettgs || Authenticatorc
Shared key and ticket E(Kc,tgs, [Kc,v || IDv || TS4 || Ticketv])
Ticketv and client authentication Ticketv || Authenticatorc
Service granted E(Kc,v, [TS5 + 1])
Shared key and ticket E(Kc, [Kc, tgs || IDtgs || TS2 ||
Lifetime2 || Tickettgs])
Authentication server (AS)
Ticket-granting server (TGS)
Service provider
(Continued)
492 CHAPTER 15 / USER AUTHENTICATION
Message (4) TGS returns service-granting ticket. Kc, tgs Key shared only by C and TGS protects contents of message (4).
Kc, v Copy of session key accessible to client created by TGS to permit secure exchange between client and server without requiring them to share a permanent key.
IDV Confirms that this ticket is for server V.
TS4 Informs client of time this ticket was issued.
TicketV Ticket to be used by client to access server V.
Tickettgs Reusable so that user does not have to reenter password.
Ktgs Ticket is encrypted with key known only to AS and TGS, to prevent tampering.
Kc, tgs Copy of session key accessible to TGS used to decrypt authenticator, thereby authenticating ticket.
IDC Indicates the rightful owner of this ticket.
ADC Prevents use of ticket from workstation other than one that initially requested the ticket.
IDtgs Assures server that it has decrypted ticket properly.
TS2 Informs TGS of time this ticket was issued.
Lifetime2 Prevents replay after ticket has expired.
Authenticatorc Assures TGS that the ticket presenter is the same as the client for whom the ticket was issued has very short lifetime to prevent replay.
Kc, tgs Authenticator is encrypted with key known only to client and TGS, to prevent tampering.
IDC Must match ID in ticket to authenticate ticket.
ADC Must match address in ticket to authenticate ticket.
TS3 Informs TGS of time this authenticator was generated.
(b) Ticket-Granting Service Exchange
Message (5) Client requests service. TicketV Assures server that this user has been authenticated by AS.
Authenticatorc Generated by client to validate ticket.
Message (6) Optional authentication of server to client. Kc, v Assures C that this message is from V.
TS5 + 1 Assures C that this is not a replay of an old reply. Ticketv Reusable so that client does not need to request a new ticket from TGS for each access to
the same server.
Kv Ticket is encrypted with key known only to TGS and server, to prevent tampering.
Kc, v Copy of session key accessible to client; used to decrypt authenticator, thereby authenticating ticket.
IDC Indicates the rightful owner of this ticket.
ADC Prevents use of ticket from workstation other than one that initially requested the ticket.
IDV Assures server that it has decrypted ticket properly.
TS4 Informs server of time this ticket was issued.
Lifetime4 Prevents replay after ticket has expired.
Authenticatorc Assures server that the ticket presenter is the same as the client for whom the ticket was issued; has very short lifetime to prevent replay.
Kc, v Authenticator is encrypted with key known only to client and server, to prevent tampering.
IDC Must match ID in ticket to authenticate ticket.
ADC Must match address in ticket to authenticate ticket.
TS5 Informs server of time this authenticator was generated.
(c) Client/Server Authentication Exchange
Table 15.2 Continued
15.3 / KERBEROS 493
Such an environment is referred to as a Kerberos realm. The concept of realm can be explained as follows. A Kerberos realm is a set of managed nodes that share the same Kerberos database. The Kerberos database resides on the
Kerberos master computer system, which should be kept in a physically secure
room. A read-only copy of the Kerberos database might also reside on other
Kerberos computer systems. However, all changes to the database must be
made on the master computer system. Changing or accessing the contents of a
Kerberos database requires the Kerberos master password. A related concept
is that of a Kerberos principal, which is a service or user that is known to the Kerberos system. Each Kerberos principal is identified by its principal name.
Principal names consist of three parts: a service or user name, an instance name,
and a realm name.
Networks of clients and servers under different administrative organizations
typically constitute different realms. That is, it generally is not practical or does
not conform to administrative policy to have users and servers in one administra-
tive domain registered with a Kerberos server elsewhere. However, users in one
realm may need access to servers in other realms, and some servers may be will-
ing to provide service to users from other realms, provided that those users are
authenticated.
Kerberos provides a mechanism for supporting such interrealm authentication.
For two realms to support interrealm authentication, a third requirement is added:
3. The Kerberos server in each interoperating realm shares a secret key with the server in the other realm. The two Kerberos servers are registered with each
other.
The scheme requires that the Kerberos server in one realm trust the Kerberos
server in the other realm to authenticate its users. Furthermore, the participating
servers in the second realm must also be willing to trust the Kerberos server in the
first realm.
With these ground rules in place, we can describe the mechanism as follows
(Figure 15.4): A user wishing service on a server in another realm needs a ticket for
that server. The user’s client follows the usual procedures to gain access to the local
TGS and then requests a ticket-granting ticket for a remote TGS (TGS in another
realm). The client can then apply to the remote TGS for a service-granting ticket for
the desired server in the realm of the remote TGS.
The details of the exchanges illustrated in Figure 15.4 are as follows (compare
Table 15.1).
(1) C S AS: IDc }IDtgs }TS1 (2) AS S C: E(Kc, [Kc, tgs }IDtgs }TS2 }Lifetime2 }Tickettgs])
(3) C S TGS: IDtgsrem }Tickettgs }Authenticatorc (4) TGS S C: E(Kc,tgs, [Kc, tgsrem }IDtgsrem }TS4 }Tickettgsrem])
(5) C S TGSrem: IDvrem }Tickettgsrem }Authenticatorc (6) TGSrem S C: E(Kc,tgsrem, [Kc, vrem }IDvrem }TS6 }Ticketvrem])
(7) C S Vrem: Ticketvrem }Authenticatorc
494 CHAPTER 15 / USER AUTHENTICATION
The ticket presented to the remote server (Vrem) indicates the realm in which the user was originally authenticated. The server chooses whether to honor the re-
mote request.
One problem presented by the foregoing approach is that it does not scale well
to many realms. If there are N realms, then there must be N(N - 1)/2 secure key exchanges so that each Kerberos realm can interoperate with all other Kerberos
realms.
Kerberos Version 5
Kerberos version 5 is specified in RFC 4120 and provides a number of improve-
ments over version 4 [KOHL94]. To begin, we provide an overview of the changes
from version 4 to version 5 and then look at the version 5 protocol.
Figure 15.4 Request for Service in Another Realm
Authentication server (AS)
Ticket- granting
server (TGS)
Kerberos
Authentication server (AS)
Ticket- granting
server (TGS)
Kerberos
Client
Realm A
Host/ application
server
Realm B
1. Req uest ti
cket fo r local
TGS
2. Tic ket fo
r loca l TGS
3. Request ticket for remoteTGS 4. Ticket for remote TGS
5. R equest ticket
for rem ote server
6. Ticket for rem ote server
7. R
eq ue
st r
em ot
e se
rv ic
e
15.3 / KERBEROS 495
DIFFERENCES BETWEEN VERSIONS 4 AND 5 Version 5 is intended to address the limita- tions of version 4 in two areas: environmental shortcomings and technical deficien-
cies. Let us briefly summarize the improvements in each area.8
Kerberos version 4 was developed for use within the Project Athena environ-
ment and, accordingly, did not fully address the need to be of general purpose. This
led to the following environmental shortcomings.
1. Encryption system dependence: Version 4 requires the use of DES. Export restriction on DES as well as doubts about the strength of DES were thus of
concern. In version 5, ciphertext is tagged with an encryption-type identifier
so that any encryption technique may be used. Encryption keys are tagged
with a type and a length, allowing the same key to be used in different al-
gorithms and allowing the specification of different variations on a given
algorithm.
2. Internet protocol dependence: Version 4 requires the use of Internet Protocol (IP) addresses. Other address types, such as the ISO network address, are not
accommodated. Version 5 network addresses are tagged with type and length,
allowing any network address type to be used.
3. Message byte ordering: In version 4, the sender of a message employs a byte ordering of its own choosing and tags the message to indicate least signifi-
cant byte in lowest address or most significant byte in lowest address. This
techniques works but does not follow established conventions. In version
5, all message structures are defined using Abstract Syntax Notation One
(ASN.1) and Basic Encoding Rules (BER), which provide an unambiguous
byte ordering.
4. Ticket lifetime: Lifetime values in version 4 are encoded in an 8-bit quantity in units of five minutes. Thus, the maximum lifetime that can be expressed is
28 * 5 = 1280 minutes (a little over 21 hours). This may be inadequate for some applications (e.g., a long-running simulation that requires valid Kerberos
credentials throughout execution). In version 5, tickets include an explicit start
time and end time, allowing tickets with arbitrary lifetimes.
5. Authentication forwarding: Version 4 does not allow credentials issued to one client to be forwarded to some other host and used by some other client. This
capability would enable a client to access a server and have that server access
another server on behalf of the client. For example, a client issues a request to
a print server that then accesses the client’s file from a file server, using the cli-
ent’s credentials for access. Version 5 provides this capability.
6. Interrealm authentication: In version 4, interoperability among N realms requires on the order of N2 Kerberos-to-Kerberos relationships, as described earlier. Version 5 supports a method that requires fewer relationships, as de-
scribed shortly.
8The following discussion follows the presentation in [KOHL94].
496 CHAPTER 15 / USER AUTHENTICATION
Apart from these environmental limitations, there are technical deficiencies in the version 4 protocol itself. Most of these deficiencies were documented in
[BELL90], and version 5 attempts to address these. The deficiencies are the
following.
1. Double encryption: Note in Table 15.1 [messages (2) and (4)] that tickets pro- vided to clients are encrypted twice—once with the secret key of the target
server and then again with a secret key known to the client. The second en-
cryption is not necessary and is computationally wasteful.
2. PCBC encryption: Encryption in version 4 makes use of a nonstandard mode of DES known as propagating cipher block chaining (PCBC).9 It has been demonstrated that this mode is vulnerable to an attack involving the inter-
change of ciphertext blocks [KOHL89]. PCBC was intended to provide an in-
tegrity check as part of the encryption operation. Version 5 provides explicit
integrity mechanisms, allowing the standard CBC mode to be used for encryp-
tion. In particular, a checksum or hash code is attached to the message prior to
encryption using CBC.
3. Session keys: Each ticket includes a session key that is used by the client to encrypt the authenticator sent to the service associated with that ticket.
In addition, the session key may subsequently be used by the client and the
server to protect messages passed during that session. However, because
the same ticket may be used repeatedly to gain service from a particular
server, there is the risk that an opponent will replay messages from an old
session to the client or the server. In version 5, it is possible for a client
and server to negotiate a subsession key, which is to be used only for that
one connection. A new access by the client would result in the use of a new
subsession key.
4. Password attacks: Both versions are vulnerable to a password attack. The mes- sage from the AS to the client includes material encrypted with a key based
on the client’s password.10 An opponent can capture this message and attempt
to decrypt it by trying various passwords. If the result of a test decryption is of
the proper form, then the opponent has discovered the client’s password and
may subsequently use it to gain authentication credentials from Kerberos. This
is the same type of password attack described in Chapter 21, with the same
kinds of countermeasures being applicable. Version 5 does provide a mecha-
nism known as preauthentication, which should make password attacks more
difficult, but it does not prevent them.
THE VERSION 5 AUTHENTICATION DIALOGUE Table 15.3 summarizes the basic ver- sion 5 dialogue. This is best explained by comparison with version 4 (Table 15.1).
First, consider the authentication service exchange. Message (1) is a client re- quest for a ticket-granting ticket. As before, it includes the ID of the user and the TGS.
The following new elements are added:
9This is described in Appendix T. 10Appendix T describes the mapping of passwords to encryption keys.
Hiva-Network.Com
15.3 / KERBEROS 497
(1) C S AS Options }IDc }Realmc }IDtgs }Times }Nonce1 (2) AS S C RealmC }IDC }Tickettgs }E(Kc, [Kc,tgs }Times }Nonce1 }Realmtgs }IDtgs])
Tickettgs = E(Ktgs, [Flags }Kc,tgs }Realmc }IDC }ADC }Times])
(a) Authentication Service Exchange to obtain ticket-granting ticket
(3) C S TGS Options }IDv }Times }Nonce2 }Tickettgs }Authenticatorc (4) TGS S C Realmc }IDC }Ticketv }E(Kc,tgs, [Kc,v }Times }Nonce2 }Realmv }IDv])
Tickettgs = E(Ktgs, [Flags }Kc,tgs }Realmc }IDC }ADC }Times]) Ticketv = E(Kv, [Flags }Kc,v }Realmc }IDC }ADC }Times])
Authenticatorc = E(Kc,tgs, [IDC }Realmc }TS1])
(b) Ticket-Granting Service Exchange to obtain service-granting ticket
(5) C S V Options }Ticketv }Authenticatorc (6) V S C E Kc,v[TS2 }Subkey }Seq #]
Ticketv = E(Kv, [Flag }Kc,v }Realmc }IDC }ADC }Times]) Authenticatorc = E(Kc,v, [IDC }Relamc }TS2 }Subkey }Seq #])
(c) Client/Server Authentication Exchange to obtain service
Table 15.3 Summary of Kerberos Version 5 Message Exchanges
■ Realm: Indicates realm of user
■ Options: Used to request that certain flags be set in the returned ticket
■ Times: Used by the client to request the following time settings in the ticket:
—from: the desired start time for the requested ticket —till: the requested expiration time for the requested ticket —rtime: requested renew-till time
■ Nonce: A random value to be repeated in message (2) to assure that the re- sponse is fresh and has not been replayed by an opponent
Message (2) returns a ticket-granting ticket, identifying information for the
client, and a block encrypted using the encryption key based on the user’s password.
This block includes the session key to be used between the client and the TGS,
times specified in message (1), the nonce from message (1), and TGS identifying
information. The ticket itself includes the session key, identifying information for
the client, the requested time values, and flags that reflect the status of this ticket
and the requested options. These flags introduce significant new functionality to
version 5. For now, we defer a discussion of these flags and concentrate on the over-
all structure of the version 5 protocol.
Let us now compare the ticket-granting service exchange for versions 4 and 5. We see that message (3) for both versions includes an authenticator, a
ticket, and the name of the requested service. In addition, version 5 includes re-
quested times and options for the ticket and a nonce—all with functions similar
to those of message (1). The authenticator itself is essentially the same as the one
used in version 4.
498 CHAPTER 15 / USER AUTHENTICATION
Message (4) has the same structure as message (2). It returns a ticket plus
information needed by the client, with the information encrypted using the session
key now shared by the client and the TGS.
Finally, for the client/server authentication exchange, several new features appear in version 5. In message (5), the client may request as an option that mutual
authentication is required. The authenticator includes several new fields:
■ Subkey: The client’s choice for an encryption key to be used to protect this specific application session. If this field is omitted, the session key from the
ticket (Kc,v) is used.
■ Sequence number: An optional field that specifies the starting sequence num- ber to be used by the server for messages sent to the client during this session.
Messages may be sequence numbered to detect replays.
If mutual authentication is required, the server responds with message (6).
This message includes the timestamp from the authenticator. Note that in version 4,
the timestamp was incremented by one. This is not necessary in version 5, because
the nature of the format of messages is such that it is not possible for an oppo-
nent to create message (6) without knowledge of the appropriate encryption keys.
The subkey field, if present, overrides the subkey field, if present, in message (5).
The optional sequence number field specifies the starting sequence number to be
used by the client.
TICKET FLAGS The flags field included in tickets in version 5 supports expanded functionality compared to that available in version 4. Table 15.4 summarizes the
flags that may be included in a ticket.
INITIAL This ticket was issued using the AS protocol and not issued based on a
ticket-granting ticket.
PRE-AUTHENT During initial authentication, the client was authenticated by the KDC
before a ticket was issued.
HW-AUTHENT The protocol employed for initial authentication required the use of hard-
ware expected to be possessed solely by the named client.
RENEWABLE Tells TGS that this ticket can be used to obtain a replacement ticket that
expires at a later date.
MAY-POSTDATE Tells TGS that a postdated ticket may be issued based on this ticket-
granting ticket.
POSTDATED Indicates that this ticket has been postdated; the end server can check the
authtime field to see when the original authentication occurred.
INVALID This ticket is invalid and must be validated by the KDC before use.
PROXIABLE Tells TGS that a new service-granting ticket with a different network
address may be issued based on the presented ticket.
PROXY Indicates that this ticket is a proxy.
FORWARDABLE Tells TGS that a new ticket-granting ticket with a different network
address may be issued based on this ticket-granting ticket.
FORWARDED Indicates that this ticket has either been forwarded or was issued based on
authentication involving a forwarded ticket-granting ticket.
Table 15.4 Kerberos Version 5 Flags
15.3 / KERBEROS 499
The INITIAL flag indicates that this ticket was issued by the AS, not by the
TGS. When a client requests a service-granting ticket from the TGS, it presents a
ticket-granting ticket obtained from the AS. In version 4, this was the only way to
obtain a service-granting ticket. Version 5 provides the additional capability that
the client can get a service-granting ticket directly from the AS. The utility of this is
as follows: A server, such as a password-changing server, may wish to know that the
client’s password was recently tested.
The PRE-AUTHENT flag, if set, indicates that when the AS received the ini-
tial request [message (1)], it authenticated the client before issuing a ticket. The
exact form of this preauthentication is left unspecified. As an example, the MIT
implementation of version 5 has encrypted timestamp preauthentication, enabled
by default. When a user wants to get a ticket, it has to send to the AS a preauthen-
tication block containing a random confounder, a version number, and a timestamp
all encrypted in the client’s password-based key. The AS decrypts the block and will
not send a ticket-granting ticket back unless the timestamp in the preauthentica-
tion block is within the allowable time skew (time interval to account for clock drift
and network delays). Another possibility is the use of a smart card that generates
continually changing passwords that are included in the preauthenticated messages.
The passwords generated by the card can be based on a user’s password but be
transformed by the card so that, in effect, arbitrary passwords are used. This pre-
vents an attack based on easily guessed passwords. If a smart card or similar device
was used, this is indicated by the HW-AUTHENT flag.
When a ticket has a long lifetime, there is the potential for it to be stolen and
used by an opponent for a considerable period. If a short lifetime is used to lessen
the threat, then overhead is involved in acquiring new tickets. In the case of a ticket-
granting ticket, the client would either have to store the user’s secret key, which is
clearly risky, or repeatedly ask the user for a password. A compromise scheme is
the use of renewable tickets. A ticket with the RENEWABLE flag set includes two
expiration times: One for this specific ticket and one that is the latest permissible
value for an expiration time. A client can have the ticket renewed by presenting it
to the TGS with a requested new expiration time. If the new time is within the limit
of the latest permissible value, the TGS can issue a new ticket with a new session
time and a later specific expiration time. The advantage of this mechanism is that
the TGS may refuse to renew a ticket reported as stolen.
A client may request that the AS provide a ticket-granting ticket with the
MAY-POSTDATE flag set. The client can then use this ticket to request a ticket
that is flagged as POSTDATED and INVALID from the TGS. Subsequently, the
client may submit the postdated ticket for validation. This scheme can be useful
for running a long batch job on a server that requires a ticket periodically. The
client can obtain a number of tickets for this session at once, with spread out time
values. All but the first ticket are initially invalid. When the execution reaches a
point in time when a new ticket is required, the client can get the appropriate ticket
validated. With this approach, the client does not have to repeatedly use its ticket-
granting ticket to obtain a service-granting ticket.
In version 5, it is possible for a server to act as a proxy on behalf of a client, in
effect adopting the credentials and privileges of the client to request a service from
another server. If a client wishes to use this mechanism, it requests a ticket-granting
500 CHAPTER 15 / USER AUTHENTICATION
ticket with the PROXIABLE flag set. When this ticket is presented to the TGS, the
TGS is permitted to issue a service-granting ticket with a different network address;
this latter ticket will have its PROXY flag set. An application receiving such a ticket
may accept it or require additional authentication to provide an audit trail.11
The proxy concept is a limited case of the more powerful forwarding procedure.
If a ticket is set with the FORWARDABLE flag, a TGS can issue to the requestor a
ticket-granting ticket with a different network address and the FORWARDED flag
set. This ticket then can be presented to a remote TGS. This capability allows a cli-
ent to gain access to a server on another realm without requiring that each Kerberos
maintain a secret key with Kerberos servers in every other realm. For example,
realms could be structured hierarchically. Then a client could walk up the tree to a
common node and then back down to reach a target realm. Each step of the walk
would involve forwarding a ticket-granting ticket to the next TGS in the path.
15.4 REMOTE USER-AUTHENTICATION USING ASYMMETRIC ENCRYPTION
Mutual Authentication
In Chapter 14, we presented one approach to the use of public-key encryption for
the purpose of session-key distribution (Figure 14.9). This protocol assumes that
each of the two parties is in possession of the current public key of the other. It may
not be practical to require this assumption.
A protocol using timestamps is provided in [DENN81]:
1. A S AS: IDA }IDB 2. AS S A: E(PRas, [IDA }PUa }T]) }E(PRas, [IDB }PUb }T]) 3. A S B: E(PRas, [IDA }PUa }T]) }E(PRas, [IDB }PUb }T]) }
E(PUb, E(PRa, [Ks }T]))
In this case, the central system is referred to as an authentication server (AS),
because it is not actually responsible for secret-key distribution. Rather, the AS pro-
vides public-key certificates. The session key is chosen and encrypted by A; hence,
there is no risk of exposure by the AS. The timestamps protect against replays of
compromised keys.
This protocol is compact but, as before, requires the synchronization of clocks.
Another approach, proposed by Woo and Lam [WOO92a], makes use of nonces.
The protocol consists of the following steps.
1. A S KDC: IDA }IDB 2. KDC S A: E(PRauth, [IDB }PUb]) 3. A S B: E(PUb, [Na }IDA]) 4. B S KDC: IDA }IDB }E(PUauth, Na) 5. KDC S B: E(PRauth, [IDA }PUa]) }E(PUb, E(PRauth, [Na }Ks }IDB]))
11For a discussion of some of the possible uses of the proxy capability, see [NEUM93b].
15.4 / REMOTE USER-AUTHENTICATION USING ASYMMETRIC ENCRYPTION 501
6. B S A: E(PUa, [E(PRauth, [(Na }Ks }IDB)]) }Nb]) 7. A S B: E(Ks, Nb)
In step 1, A informs the KDC of its intention to establish a secure connection
with B. The KDC returns to A a copy of B’s public-key certificate (step 2). Using B’s
public key, A informs B of its desire to communicate and sends a nonce Na (step 3). In step 4, B asks the KDC for A’s public-key certificate and requests a session key;
B includes A’s nonce so that the KDC can stamp the session key with that nonce.
The nonce is protected using the KDC’s public key. In step 5, the KDC returns to
B a copy of A’s public-key certificate, plus the information {Na, Ks, IDB}. This infor- mation basically says that Ks is a secret key generated by the KDC on behalf of B and tied to Na; the binding of Ks and Na will assure A that Ks is fresh. This triple is encrypted using the KDC’s private key to allow B to verify that the triple is in fact
from the KDC. It is also encrypted using B’s public key so that no other entity may
use the triple in an attempt to establish a fraudulent connection with A. In step 6,
the triple {Na, Ks, IDB}, still encrypted with the KDC’s private key, is relayed to A, together with a nonce Nb generated by B. All the foregoing are encrypted using A’s public key. A retrieves the session key Ks, uses it to encrypt Nb, and returns it to B. This last message assures B of A’s knowledge of the session key.
This seems to be a secure protocol that takes into account the various attacks.
However, the authors themselves spotted a flaw and submitted a revised version of
the algorithm in [WOO92b]:
1. A S KDC: IDA }IDB 2. KDC S A: E(PRauth, [IDB }PUb]) 3. A S B: E(PUb, [Na }IDA]) 4. B S KDC: IDA }IDB }E(PUauth, Na) 5. KDC S B: E(PRauth, [IDA }PUa]) }E(PUb, E(PRauth, [Na }Ks }IDA }IDB])) 6. B S A: E(PUa, [Nb }E(PRauth, [Na }Ks }IDA }IDB])]) 7. A S B: E(Ks, Nb)
The identifier of A, IDA, is added to the set of items encrypted with the KDC’s private key in steps 5 and 6. This binds the session key Ks to the identities of the two parties that will be engaged in the session. This inclusion of IDA accounts for the fact that the nonce value Na is considered unique only among all nonces generated by A, not among all nonces generated by all parties. Thus, it is the pair {IDA, Na} that uniquely identifies the connection request of A.
In both this example and the protocols described earlier, protocols that ap-
peared secure were revised after additional analysis. These examples highlight the
difficulty of getting things right in the area of authentication.
One-Way Authentication
We have already presented public-key encryption approaches that are suited to
electronic mail, including the straightforward encryption of the entire message for
confidentiality (Figure 12.1b), authentication (Figure 12.1c), or both (Figure 12.1d).
These approaches require that either the sender know the recipient’s public key
502 CHAPTER 15 / USER AUTHENTICATION
(confidentiality), the recipient know the sender’s public key (authentication), or
both (confidentiality plus authentication). In addition, the public-key algorithm
must be applied once or twice to what may be a long message.
If confidentiality is the primary concern, then the following may be more efficient:
A S B: E(PUb, Ks) }E(Ks, M)
In this case, the message is encrypted with a one-time secret key. A also encrypts this
one-time key with B’s public key. Only B will be able to use the corresponding private
key to recover the one-time key and then use that key to decrypt the message. This
scheme is more efficient than simply encrypting the entire message with B’s public key.
If authentication is the primary concern, then a digital signature may suffice,
as was illustrated in Figure 13.2:
A S B: M }E(PRa, H(M))
This method guarantees that A cannot later deny having sent the message.
However, this technique is open to another kind of fraud. Bob composes a mes-
sage to his boss Alice that contains an idea that will save the company money. He
appends his digital signature and sends it into the email system. Eventually, the
message will get delivered to Alice’s mailbox. But suppose that Max has heard of
Bob’s idea and gains access to the mail queue before delivery. He finds Bob’s mes-
sage, strips off his signature, appends his, and requeues the message to be delivered
to Alice. Max gets credit for Bob’s idea.
To counter such a scheme, both the message and signature can be encrypted
with the recipient’s public key:
A S B: E(PUb, [M }E(PRa, H(M))])
The latter two schemes require that B know A’s public key and be convinced
that it is timely. An effective way to provide this assurance is the digital certificate,
described in Chapter 14. Now we have
A S B: M }E(PRa, H(M)) }E(PRas, [T}IDA }PUa])
In addition to the message, A sends B the signature encrypted with A’s private
key and A’s certificate encrypted with the private key of the authentication server.
The recipient of the message first uses the certificate to obtain the sender’s public
key and verify that it is authentic and then uses the public key to verify the message
itself. If confidentiality is required, then the entire message can be encrypted with
B’s public key. Alternatively, the entire message can be encrypted with a one-time
secret key; the secret key is also transmitted, encrypted with B’s public key. This ap-
proach is explored in Chapter 19.
15.5 FEDERATED IDENTITY MANAGEMENT
Federated identity management is a relatively new concept dealing with the use of a common identity management scheme across multiple enterprises and numerous
applications and supporting many thousands, even millions, of users. We begin our
overview with a discussion of the concept of identity management and then examine
federated identity management.
15.5 / FEDERATED IDENTITY MANAGEMENT 503
Identity Management
Identity management is a centralized, automated approach to provide enterprise-
wide access to resources by employees and other authorized individuals. The focus
of identity management is defining an identity for each user (human or process),
associating attributes with the identity, and enforcing a means by which a user can
verify identity. The central concept of an identity management system is the use of
single sign-on (SSO).
SSO enables a user to access all network resources after a single authentication.
Typical services provided by a federated identity management system include
the following:
■ Point of contact: Includes authentication that a user corresponds to the user name provided, and management of user/server sessions.
■ SSO protocol services: Provides a vendor-neutral security token service for supporting a single sign on to federated services.
■ Trust services: Federation relationships require a trust relationship-based federation between business partners. A trust relationship is represented by
the combination of the security tokens used to exchange information about a
user, the cryptographic information used to protect these security tokens, and
optionally the identity mapping rules applied to the information contained
within this token.
■ Key services: Management of keys and certificates.
■ Identity services: services that provide the interface to local data stores, includ- ing user registries and databases, for identity-related information management.
■ Authorization: Granting access to specific services and/or resources based on the authentication.
■ Provisioning: Includes creating an account in each target system for the user, enrollment or registration of user in accounts, establishment of access rights or
credentials to ensure the privacy and integrity of account data.
■ Management: Services related to runtime configuration and deployment.
Note that Kerberos contains a number of the elements of an identity manage-
ment system.
Figure 15.5 illustrates entities and data flows in a generic identity manage-
ment architecture. A principal is an identity holder. Typically, this is a human user that seeks access to resources and services on the network. User devices, agent pro-
cesses, and server systems may also function as principals. Principals authenticate
themselves to an identity provider. The identity provider associates authentication information with a principal, as well as attributes and one or more identifiers.
Increasingly, digital identities incorporate attributes other than simply an iden-
tifier and authentication information (such as passwords and biometric information).
An attribute service manages the creation and maintenance of such attributes. For example, a user needs to provide a shipping address each time an order is placed at a
new Web merchant, and this information needs to be revised when the user moves.
Identity management enables the user to provide this information once, so that it
is maintained in a single place and released to data consumers in accordance with
504 CHAPTER 15 / USER AUTHENTICATION
authorization and privacy policies. Users may create some of the attributes to be
associated with their digital identity, such as an address. Administrators may also as- sign attributes to users, such as roles, access permissions, and employee information.
Data consumers are entities that obtain and employ data maintained and provided by identity and attribute providers, which are often used to support autho-
rization decisions and to collect audit information. For example, a database server
or file server is a data consumer that needs a client’s credentials so as to know what
access to provide to that client.
Identity Federation
Identity federation is, in essence, an extension of identity management to multiple
security domains. Such domains include autonomous internal business units, exter-
nal business partners, and other third-party applications and services. The goal is to
provide the sharing of digital identities so that a user can be authenticated a single
time and then access applications and resources across multiple domains. Because
these domains are relatively autonomous or independent, no centralized control is
possible. Rather, the cooperating organizations must form a federation based on
agreed standards and mutual levels of trust to securely share digital identities.
Federated identity management refers to the agreements, standards, and
technologies that enable the portability of identities, identity attributes, and entitle-
ments across multiple enterprises and numerous applications and supporting many
thousands, even millions, of users. When multiple organizations implement interop-
erable federated identity schemes, an employee in one organization can use a single
sign-on to access services across the federation with trust relationships associated
with the identity. For example, an employee may log onto her corporate intranet
and be authenticated to perform authorized functions and access authorized ser-
vices on that intranet. The employee could then access their health benefits from an
outside health-care provider without having to reauthenticate.
Figure 15.5 Generic Identity Management Architecture
Identity provider
Attribute service
Data consumer
Principal
Administrator
15.5 / FEDERATED IDENTITY MANAGEMENT 505
Beyond SSO, federated identity management provides other capabilities. One
is a standardized means of representing attributes. Increasingly, digital identities
incorporate attributes other than simply an identifier and authentication informa-
tion (such as passwords and biometric information). Examples of attributes include
account numbers, organizational roles, physical location, and file ownership. A user
may have multiple identifiers; for example, each identifier may be associated with a
unique role with its own access permissions.
Another key function of federated identity management is identity mapping.
Different security domains may represent identities and attributes differently.
Further, the amount of information associated with an individual in one domain
may be more than is necessary in another domain. The federated identity manage-
ment protocols map identities and attributes of a user in one domain to the require-
ments of another domain.
Figure 15.6 illustrates entities and data flows in a generic federated identity
management architecture.
Figure 15.6 Federated Identity Operation
User
1 Identity provider (source domain)
Service provider (destination domain)
1 End user’s browser or other application engages in an authentication dialogue with identity provider in the same domain. End user also provides attribute values associated with user’s identity.
2 Some attributes associated with an identity, such as allowable roles, may be provided by an administrator in the same domain.
3 A service provider in a remote domain, which the user wishes to access, obtains identity information, authentication information, and associated attributes from the identity provider in the source domain.
4 Service provider opens session with remote user and enforces access control restrictions based on user’s identity and attributes.
Administrator
2
3
4
Hiva-Network.Com
506 CHAPTER 15 / USER AUTHENTICATION
The identity provider acquires attribute information through dialogue and pro-
tocol exchanges with users and administrators. For example, a user needs to provide
a shipping address each time an order is placed at a new Web merchant, and this
information needs to be revised when the user moves. Identity management enables
the user to provide this information once, so that it is maintained in a single place and
released to data consumers in accordance with authorization and privacy policies.
Service providers are entities that obtain and employ data maintained and pro-
vided by identity providers, often to support authorization decisions and to collect
audit information. For example, a database server or file server is a data consumer
that needs a client’s credentials so as to know what access to provide to that client.
A service provider can be in the same domain as the user and the identity provider.
The power of this approach is for federated identity management, in which the ser-
vice provider is in a different domain (e.g., a vendor or supplier network).
STANDARDS Federated identity management uses a number of standards as the building blocks for secure identity exchange across different domains or heteroge-
neous systems. In essence, organizations issue some form of security tickets for their
users that can be processed by cooperating partners. Identity federation standards
are thus concerned with defining these tickets, in terms of content and format, pro-
viding protocols for exchanging tickets and performing a number of management
tasks. These tasks include configuring systems to perform attribute transfers and
identity mapping, and performing logging and auditing functions. The key stan-
dards are as follows:
■ The Extensible Markup Language (XML): A markup language that uses sets of embedded tags or labels to characterize text elements within a document
so as to indicate their appearance, function, meaning, or context. XML docu-
ments appear similar to HTML (Hypertext Markup Language) documents
that are visible as Web pages, but provide greater functionality. XML includes
strict definitions of the data type of each field, thus supporting database for-
mats and semantics. XML provides encoding rules for commands that are used
to transfer and update data objects.
■ The Simple Object Access Protocol (SOAP): A minimal set of conventions for invoking code using XML over HTTP. It enables applications to request
services from one another with XML-based requests and receive responses
as data formatted with XML. Thus, XML defines data objects and structures,
and SOAP provides a means of exchanging such data objects and performing
remote procedure calls related to these objects. See [ROS06] for an informa-
tive discussion.
■ WS-Security: A set of SOAP extensions for implementing message integrity and confidentiality in Web services. To provide for secure exchange of SOAP
messages among applications, WS-Security assigns security tokens to each
message for use in authentication.
■ Security Assertion Markup Language (SAML): An XML-based language for the exchange of security information between online business partners. SAML
conveys authentication information in the form of assertions about subjects.
Assertions are statements about the subject issued by an authoritative entity.
15.5 / FEDERATED IDENTITY MANAGEMENT 507
The challenge with federated identity management is to integrate multiple
technologies, standards, and services to provide a secure, user-friendly utility. The
key, as in most areas of security and networking, is the reliance on a few mature
standards widely accepted by industry. Federated identity management seems to
have reached this level of maturity.
EXAMPLES To get some feel for the functionality of identity federation, we look at three scenarios, taken from [COMP06].
In the first scenario (Figure 15.7a), Workplace.com contracts with Health.com
to provide employee health benefits. An employee uses a Web interface to sign on to
Workplace.com and goes through an authentication procedure there. This enables
the employee to access authorized services and resources at Workplace.com. When
the employee clicks on a link to access health benefits, her browser is redirected to
Health.com. At the same time, the Workplace.com software passes the user’s identi-
fier to Health.com in a secure manner. The two organizations are part of a federation
that cooperatively exchanges user identifiers. Health.com maintains user identities
Figure 15.7 Federated Identity Scenarios
User store
(a) Federation based on account linking
(c) Chained Web services
Workplace.com (employee portal)
Name Joe Jane Ravi
ID 1213 1410 1603
User store Name Joe Jane Ravi
ID 1213 1410 1603
Health.com
User store
(b) Federation based on roles
Name Joe Jane Ravi
ID 1213 1410 1603
Dept Eng Purch Purch
User store Role
Engineer Purchaser
Au the
ntic atio
n Website access
End user (employee)
User ID
Workplace.com (procurement application)
PinSupplies.com (Purchasing Web
service)
Au the
ntic atio
n
Pro cur
em ent
req ues
t
End user (employee)
SOAP message
Eship.com (shipping Web
service)
SOAP message
Workplace.com (employee portal)
PartsSupplier.com
Au the
ntic atio
n Website access
End user (employee)
Role
508 CHAPTER 15 / USER AUTHENTICATION
for every employee at Workplace.com and associates with each identity health-bene-
fits information and access rights. In this example, the linkage between the two com-
panies is based on account information and user participation is browser based.
Figure 15.7b shows a second type of browser-based scheme. PartsSupplier.
com is a regular supplier of parts to Workplace.com. In this case, a role-based
access-control (RBAC) scheme is used for access to information. An engineer of
Workplace.com authenticates at the employee portal at Workplace.com and clicks
on a link to access information at PartsSupplier.com. Because the user is authen-
ticated in the role of an engineer, he is taken to the technical documentation and
troubleshooting portion of PartsSupplier.com’s Web site without having to sign on.
Similarly, an employee in a purchasing role signs on at Workplace.com and is au-
thorized, in that role, to place purchases at PartsSupplier.com without having to
authenticate to PartsSupplier.com. For this scenario, PartsSupplier.com does not
have identity information for individual employees at Workplace.com. Rather, the
linkage between the two federated partners is in terms of roles.
The scenario illustrated in Figure 15.7c can be referred to as document based
rather than browser based. In this third example, Workplace.com has a purchasing
agreement with PinSupplies.com, and PinSupplies.com has a business relationship
with E-Ship.com. An employee of Workplace.com signs on and is authenticated to
make purchases. The employee goes to a procurement application that provides a
list of Workplace.com’s suppliers and the parts that can be ordered. The user clicks
on the PinSupplies button and is presented with a purchase order Web page (HTML
page). The employee fills out the form and clicks the submit button. The procure-
ment application generates an XML/SOAP document that it inserts into the enve-
lope body of an XML-based message. The procurement application then inserts the
user’s credentials in the envelope header of the message, together with Workplace.
com’s organizational identity. The procurement application posts the message to
the PinSupplies.com’s purchasing Web service. This service authenticates the in-
coming message and processes the request. The purchasing Web service then sends
a SOAP message to its shipping partner to fulfill the order. The message includes
a PinSupplies.com security token in the envelope header and the list of items to be
shipped as well as the end user’s shipping information in the envelope body. The
shipping Web service authenticates the request and processes the shipment order.
15.6 PERSONAL IDENTITY VERIFICATION
User authentication based on the possession of a smart card is becoming more wide-
spread. A smart card has the appearance of a credit card, has an electronic inter-
face, and may use a variety of authentication protocols.
A smart card contains within it an entire microprocessor, including processor,
memory, and I/O ports. Some versions incorporate a special co-processing circuit
for cryptographic operation to speed the task of encoding and decoding messages or
generating digital signatures to validate the information transferred. In some cards,
the I/O ports are directly accessible by a compatible reader by means of exposed
electrical contacts. Other cards rely instead on an embedded antenna for wireless
communication with the reader.
15.6 / PERSONAL IDENTITY VERIFICATION 509
A typical smart card includes three types of memory. Read-only memory
(ROM) stores data that does not change during the card’s life, such as the card
number and the cardholder’s name. Electrically erasable programmable ROM
(EEPROM) holds application data and programs, such as the protocols that the
card can execute. It also holds data that may vary with time. For example, in a tele-
phone card, the EEPROM holds the talk time remaining. Random access memory
(RAM) holds temporary data generated when applications are executed.
For the practical application of smart card authentication, a wide range of
vendors must conform to standards that cover smart card protocols, authentication
and access control formats and protocols, database entries, message formats, and so
on. An important step in this direction is FIPS 201-2 (Personal Identity Verification [PIV] of Federal Employees and Contractors, June 2012). The standard defines a reliable, government-wide PIV system for use in applications such as access to fed-
erally controlled facilities and information systems. The standard specifies a PIV
system within which common identification credentials can be created and later
used to verify a claimed identity. The standard also identifies Federal government-
wide requirements for security levels that are dependent on risks to the facility or
information being protected. The standard applies to private-sector contractors as
well, and serves as a useful guideline for any organization.
PIV System Model
Figure 15.8 illustrates the major components of FIPS 201-2 compliant systems. The
PIV front end defines the physical interface to a user who is requesting access to a
facility, which could be either physical access to a protected physical area or logical
access to an information system. The PIV front-end subsystem supports up to three- factor authentication; the number of factors used depends on the level of security
required. The front end makes use of a smart card, known as a PIV card, which
is a dual-interface contact and contactless card. The card holds a cardholder pho-
tograph, X.509 certificates, cryptographic keys, biometric data, and a cardholder
unique identifier (CHUID). Certain cardholder information may be read-protected
and require a personal identification number (PIN) for read access by the card
reader. The biometric reader, in the current version of the standard, is a fingerprint
reader or an iris scanner.
The standard defines three assurance levels for verification of the card and the
encoded data stored on the card, which in turn leads to verifying the authenticity of
the person holding the credential. A level of some confidence corresponds to use of the card reader and PIN. A level of high confidence adds a biometric comparison of a fingerprint captured and encoded on the card during the card-issuing process
and a fingerprint scanned at the physical access point. A very high confidence level requires that the process just described is completed at a control point attended by
an official observer.
The other major component of the PIV system is the PIV card issuance and management subsystem. This subsystem includes the components responsible for identity proofing and registration, card and key issuance and management, and the
various repositories and services (e.g., public key infrastructure [PKI] directory,
certificate status servers) required as part of the verification infrastructure.
510 CHAPTER 15 / USER AUTHENTICATION
The PIV system interacts with a relying subsystem, which includes compo- nents responsible for determining a particular PIV cardholder’s access to a physical
or logical resource. FIPS 201-2 standardizes data formats and protocols for interac-
tion between the PIV system and the relying system.
Unlike the typical card number/facility code encoded on most access control
cards, the FIPS 201 CHUID takes authentication to a new level, through the use of
an expiration date (a required CHUID data field) and an optional CHUID digital
signature. A digital signature can be checked to ensure that the CHUID recorded
on the card was digitally signed by a trusted source and that the CHUID data have
not been altered since the card was signed. The CHUID expiration date can be
checked to verify that the card has not expired. This is independent from whatever
expiration date is associated with cardholder privileges. Reading and verifying the
CHUID alone provides only some assurance of identity because it authenticates
the card data, not the cardholder. The PIN and biometric factors provide identity
verification of the individual.
PIV Documentation
The PIV specification is quite complex, and NIST has issued a number of docu-
ments that cover a broad range of PIV topics. These are as follows:
Figure 15.8 FIPS 201 PIV System Model
Identity profiling & registration
Card issuance & maintenance
Key management
PKI directory & certificate status
responder
Authorization data
Authorization data
Physical resource
Logical resource
PIV card issuance and management
Shapes
Relying
I&A Authorization
Physical Access Control
I&A = Identification and Authentication
Authorization
Direction of information flow
Processes
Components
Logical Access Control
Card reader /writer
PIN input device
Biometric reader
PIV card
PIV Front End
LEGEND
I&A
15.6 / PERSONAL IDENTITY VERIFICATION 511
■ FIPS 201-2—Personal Identity Verification (PIV) of Federal Employees and Contractors: Specifies the physical card characteristics, storage media, and data elements that make up the identity credentials resident on the PIV
card.
■ SP 800-73-3—Interfaces for Personal Identity Verification: Specifies the in- terfaces and card architecture for storing and retrieving identity credentials
from a smart card, and provides guidelines for the use of authentication mech-
anisms and protocols.
■ SP 800-76-2—Biometric Data Specification for Personal Identity Verification: Describes technical acquisition and formatting specifications for the biometric
credentials of the PIV system.
■ SP 800-78-3—Cryptographic Algorithms and Key Sizes for Personal Identity Verification: Identifies acceptable symmetric and asymmetric encryption algo- rithms, digital signature algorithms, and message digest algorithms, and speci-
fies mechanisms to identify the algorithms associated with PIV keys or digital
signatures.
■ SP 800-104—A Scheme for PIV Visual Card Topography: Provides additional recommendations on the PIV card color-coding for designating employee
affiliation.
■ SP 800-116—A Recommendation for the Use of PIV Credentials in Physical Access Control Systems (PACS): Describes a risk-based approach for select- ing appropriate PIV authentication mechanisms to manage physical access to
Federal government facilities and assets.
■ SP 80 0-79-1—Guidelines for the Accreditation of Personal Identity Verification Card Issuers: Provides guidelines for accrediting the reliability of issuers of PIV cards that collect, store, and disseminate personal identity
credentials and issue smart cards.
■ SP 800-96—PIV Card to Reader Interoperability Guidelines: Provides re- quirements that facilitate interoperability between any card and any reader.
In addition there are other documents that deal with conformance testing and
codes for identifiers.
PIV Credentials and Keys
The PIV card contains a number of mandatory and optional data elements that
serve as identity credentials with varying levels of strength and assurance. These
credentials are used singly or in sets to authenticate the holder of the PIV card to
achieve the level of assurance required for a particular activity or transaction. The
mandatory data elements are the following:
■ Personal Identification Number (PIN): Required to activate the card for privi- leged operation.
■ Cardholder Unique Identifier (CHUID): Includes the Federal Agency Smart Credential Number (FASC-N) and the Global Unique Identification Number
(GUID), which uniquely identify the card and the cardholder.
512 CHAPTER 15 / USER AUTHENTICATION
■ PIV Authentication Key: Asymmetric key pair and corresponding certificate for user authentication.
■ Two fingerprint templates: For biometric authentication.
■ Electronic facial image: For biometric authentication.
■ Asymmetric Card Authentication Key: Asymmetric key pair and correspond- ing certificate used for card authentication.
Optional elements include the following:
■ Digital Signature Key: Asymmetric key pair and corresponding certificate that supports document signing and signing of data elements such as the CHUID.
■ Key Management Key: Asymmetric key pair and corresponding certificate supporting key establishment and transport.
■ Symmetric Card Authentication Key: For supporting physical access applications.
■ PIV Card Application Administration Key: Symmetric key associated with the card management system.
■ One or two iris images: For biometric authentication.
Table 15.5 lists the algorithm and key size requirements for PIV key types.
Authentication
Using the electronic credentials resident on a PIV card, the card supports the fol-
lowing authentication mechanisms:
■ CHUID: The cardholder is authenticated using the signed CHUID data ele- ment on the card. The PIN is not required. This mechanism is useful in envi-
ronments where a low level of assurance is acceptable and rapid contactless
authentication is necessary.
PIV Key Type Algorithms Key Sizes (bits) Application
PIV Authentication Key
RSA 2048 Supports card and
cardholder authentication
for an interoperable
environmentECDSA 256
Card Authentication Key
3TDEA 168 Supports card authentication
for physical accessAES 128, 192, or 256
RSA 2048 Supports card
authentication for an
interoperable environmentECDSA 256
Digital Signature Key RSA 2048 or 3072 Supports document signing
and nonce signing ECDSA 256 or 384
Key Management Key RSA 2048 Supports key establishment
and transport ECDH 256 or 384
Table 15.5 PIV Algorithms and Key Sizes
15.6 / PERSONAL IDENTITY VERIFICATION 513
■ Card Authentication Key: The PIV card is authenticated using the Card Authentication Key in a challenge response protocol. The PIN is not required.
This mechanism allows contact (via card reader) or contactless (via radio
waves) authentication of the PIV card without the holder’s active participa-
tion, and provides a low level of assurance.
■ BIO: The cardholder is authenticated by matching his or her fingerprint sample(s) to the signed biometric data element in an environment without a
human attendant in view. The PIN is required to activate the card. This mecha-
nism achieves a high level of assurance and requires the cardholder’s active
participation is submitting the PIN as well as the biometric sample.
■ BIO-A: The cardholder is authenticated by matching his or her fingerprint sample(s) to the signed biometric data element in an environment with a
human attendant in view. The PIN is required to activate the card. This mecha-
nism achieves a very high level of assurance when coupled with full trust val-
idation of the biometric template retrieved from the card, and requires the
cardholder’s active participation is submitting the PIN as well as the biometric
sample.
■ PKI: The cardholder is authenticated by demonstrating control of the PIV au- thentication private key in a challenge response protocol that can be validated
using the PIV authentication certificate. The PIN is required to activate the
card. This mechanism achieves a very high level of identity assurance and re-
quires the cardholder’s knowledge of the PIN.
In each of the above use cases, except the symmetric Card Authentication Key
use case, the source and the integrity of the corresponding PIV credential are vali-
dated by verifying the digital signature on the credential, with the signature being
provided by a trusted entity.
A variety of protocols can be constructed for each of these authentication
types. SP 800-78-3 gives examples for each type. Figure 15.9 illustrates an authenti-
cation scenario that includes the use of the PIV Authentication Key. This scenario
provides a high level of assurance. This scenario would be appropriate for authenti-
cation of a user who possesses a PIV card and seeks access to a computer resource.
The computer, designated local system in the figure, includes PIV application soft- ware and communicates to the card via an application program interface that en-
ables the use of relatively high-level procedure calls. These high-level commands
are converted into PIV commands that are issued to the card through a physical
interface via a card reader or via a wireless interface. In either case, SP 800-73 refers
to the card command interface as the PIV card edge.
The process begins when the local system detects the card either through an
attached card reader or wirelessly. It then selects an application on the card for au-
thentication. The local system then requests the public-key certificate for the card’s
PIV Authentication Key. If the certificate is valid (i.e., has a valid signature, has not
expired or been revoked), authentication continues. Otherwise the card is rejected.
The next step is for the local system to request that the cardholder enter the PIN
for the card. If the submitted PIN matches the PIN stored on the card, the card
returns a positive acknowledgment; otherwise the card returns a failure message.
514 CHAPTER 15 / USER AUTHENTICATION
The local system either continues or rejects the card accordingly. The next phase is
a challenge-response protocol. The local system sends a nonce to be signed by the
PIV, and the PIV returns the signature. The local system uses the PIV authentica-
tion public key to verify the signature. If the signature is valid, the cardholder is ac-
cepted as being identified. Otherwise the local system rejects the card.
The scenario of Figure 15.9 accomplishes three types of authentication. The
combination of possession of the card and knowledge of the PIN service authenti-
cates the cardholder. The PIV Authentication Key certificate validates the card’s
credentials. The challenge-response protocol authenticates the card.
Figure 15.9 Authentication Using PIV Authentication Key
Connect
Disconnect End transaction
Verify PIN
Request card signature
PIV card app ID and Version
PIV Auth certificate returned
Signed nonce returned
PIN ACK
Read value (PIV Auth certificate)
Select application Select application
Verify PIN
Sign nonce
Begin transaction
Present card (HolderV)
PIV Application on Local System
API on Local System
PIV Card Edge
Retrieve PIV AUTH certificate
Retrieve FASC-N from the certificate
CardV = Card validation CredV = Credential validation HolderV = Cardholder validation FASC-N = Federal Agency Smart Credential Number
Validate certificate (signature, expiration, and
revocation) (CredV)
Retrieve algorithm ID and key size for signature request
Acquire PIN (HolderV)
Verify signed data—card
possesses private key (CardV)
Reject Cardholder identifier
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15.7 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 515
15.7 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
Key Terms
authentication
authentication server
claimant
credential
credential service provider
(CSP)
federated identity
management
identity management
Kerberos
Kerberos realm
mutual authentication
nonce
one-way authentication
personal identity verification
(PIV)
realm
registration authority (RA)
relying party (RP)
replay attack
subscriber
suppress-replay attack
ticket
ticket-granting server (TGS)
timestamp
verifier
Review Questions
15.1 What are the steps involved in an authentication process? 15.2 List three general approaches to dealing with replay attacks. 15.3 What is a suppress-replay attack? 15.4 What problem was Kerberos designed to address? 15.5 What are three threats associated with user authentication over a network or
Internet?
15.6 List three approaches to secure user authentication in a distributed environment. 15.7 What four requirements were defined for Kerberos? 15.8 What entities constitute a full-service Kerberos environment? 15.9 In the context of Kerberos, what is a realm? 15.10 What are the mandatory elements to authenticate a PIV card holder?
Problems
15.1 In Section 15.4, we outlined the public-key scheme proposed in [WOO92a] for the distribution of secret keys. The revised version includes IDA in steps 5 and 6. What attack, specifically, is countered by this revision?
15.2 The protocol referred to in Problem 15.1 can be reduced from seven steps to five, having the following sequence: a. A S B: b. A S KDC: c. KDC S B: d. B S A: e. A S B: Show the message transmitted at each step. Hint: The final message in this protocol is the same as the final message in the original protocol.
15.3 Reference the suppress-replay attack described in Section 15.2 to answer the following. a. Give an example of an attack when a party’s clock is ahead of that of the KDC. b. Give an example of an attack when a party’s clock is ahead of that of another
party.
516 CHAPTER 15 / USER AUTHENTICATION
15.4 There are three typical ways to use nonces as challenges. Suppose Na is a nonce gen- erated by A, A and B share key K, and f() is a function (such as an increment). The three usages are
Usage 1 Usage 2 Usage 3
(1) A S B: Na (1) A S B: E(K, Na) (1) A S B: E(K, Na) (2) B S A: E(K, Na) (2) B S A: Na (2) B S A: E(K, f(Na))
Describe situations for which each usage is appropriate.
15.5 Show that a random error in one block of ciphertext is propagated to all subsequent blocks of plaintext in PCBC mode (See Figure T.2 in Appendix T).
15.6 Suppose that, in PCBC mode, blocks Ci and Ci + 1 are interchanged during transmis- sion. Show that this affects only the decrypted blocks Pi and Pi + 1 but not subsequent blocks.
15.7 In addition to providing a standard for public-key certificate formats, X.509 specifies an authentication protocol. The original version of X.509 contains a security flaw. The essence of the protocol is as follows.
A S B: A {tA, rA, IDB} B S A: B {tB, rB, IDA, rA} A S B: A {rB}
where tA and tB are timestamps, rA and rB are nonces and the notation X{Y} indicates that the message Y is transmitted, encrypted, and signed by X.
The text of X.509 states that checking timestamps tA and tB is optional for three-way authentication. But consider the following example: Suppose A and B have used the preceding protocol on some previous occasion, and that opponent C has intercepted the preceding three messages. In addition, suppose that timestamps are not used and are all set to 0. Finally, suppose C wishes to impersonate A to B. C initially sends the first captured message to B:
C S B: A {0, rA, IDB}
B responds, thinking it is talking to A but is actually talking to C:
B S C: B {0, r B= , IDA, rA}
C meanwhile causes A to initiate authentication with C by some means. As a result, A sends C the following:
A S C: A {0, r A= , IDC}
C responds to A using the same nonce provided to C by B:
C S A: C {0, r B= , IDA, r A= }
A responds with
A S C: A {r B= }
15.7 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 517
This is exactly what C needs to convince B that it is talking to A, so C now repeats the incoming message back out to B.
C S B: A {r B= }
So B will believe it is talking to A whereas it is actually talking to C. Suggest a simple solution to this problem that does not involve the use of timestamps.
15.8 Consider a one-way authentication technique based on asymmetric encryption:
A S B: IDA B S A: R1 A S B: E(PRa, R1)
a. Explain the protocol. b. What type of attack is this protocol susceptible to?
15.9 Consider a one-way authentication technique based on asymmetric encryption:
A S B: IDA| | E(PUB,RA)
B S A: RA
a. Explain the protocol. b. What type of attack is this protocol susceptible to?
15.10 In Kerberos, when Bob receives a Ticket from Alice, how does he know it is not genuine?
15.11 In Kerberos, how does Bob know that the received token is not corresponding to Alice’s?
15.12 In Kerberos, how does Alice know that a reply to an earlier message is from Bob? 15.13 In Kerberos, what does the Ticket contain that allows Alice and Bob to talk securely?
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519
16.1 Network Access Control
Elements of a Network Access Control System
Network Access Enforcement Methods
16.2 Extensible Authentication Protocol
Authentication Methods
EAP Exchanges
16.3 IEEE 802.1X Port-Based Network Access Control
16.4 Cloud Computing
Cloud Computing Elements
Cloud Computing Reference Architecture
16.5 Cloud Security Risks and Countermeasures
16.6 Data Protection in the Cloud
16.7 Cloud Security as a Service
16.8 Addressing Cloud Computing Security Concerns
16.9 Key Terms, Review Questions, and Problems
PART SIX: NETWORK AND INTERNET SECURITY
CHAPTER
Network Access Control and Cloud Security
520 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY
This chapter begins our discussion of network security, focusing on two key topics:
network access control and cloud security. We begin with an overview of network
access control systems, summarizing the principal elements and techniques involved
in such a system. Next, we discuss the Extensible Authentication Protocol and IEEE
802.1X, two widely implemented standards that are the foundation of many network
access control systems.
The remainder of the chapter deals with cloud security. We begin with an
overview of cloud computing, and follow this with a discussion of cloud security
issues.
16.1 NETWORK ACCESS CONTROL
Network access control (NAC) is an umbrella term for managing access to a network. NAC authenticates users logging into the network and determines what
data they can access and actions they can perform. NAC also examines the health of
the user’s computer or mobile device (the endpoints).
Elements of a Network Access Control System
NAC systems deal with three categories of components:
■ Access requestor (AR): The AR is the node that is attempting to access the network and may be any device that is managed by the NAC system, including
workstations, servers, printers, cameras, and other IP-enabled devices. ARs are
also referred to as supplicants, or simply, clients.
■ Policy server: Based on the AR’s posture and an enterprise’s defined policy, the policy server determines what access should be granted. The policy server
often relies on backend systems, including antivirus, patch management, or a
user directory, to help determine the host’s condition.
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Discuss the principal elements of a network access control system.
◆ Discuss the principal network access enforcement methods.
◆ Present an overview of the Extensible Authentication Protocol.
◆ Understand the operation and role of the IEEE 802.1X Port-Based Network Access Control mechanism.
◆ Present an overview of cloud computing concepts.
◆ Understand the unique security issues related to cloud computing.
16.1 / NETWORK ACCESS CONTROL 521
■ Network access server (NAS): The NAS functions as an access control point for users in remote locations connecting to an enterprise’s internal network.
Also called a media gateway, a remote access server (RAS), or a policy server, an NAS may include its own authentication services or rely on a separate
authentication service from the policy server.
Figure 16.1 is a generic network access diagram. A variety of different ARs
seek access to an enterprise network by applying to some type of NAS. The first
step is generally to authenticate the AR. Authentication typically involves some
sort of secure protocol and the use of cryptographic keys. Authentication may be
performed by the NAS, or the NAS may mediate the authentication process. In the
latter case, authentication takes place between the supplicant and an authentication
server that is part of the policy server or that is accessed by the policy server.
The authentication process serves a number of purposes. It verifies a suppli-
cant’s claimed identity, which enables the policy server to determine what access
privileges, if any, the AR may have. The authentication exchange may result in the
Figure 16.1 Network Access Control Context
Supplicants
Network access servers
Authentication server
DHCP server
VLAN server
Policy server
Patch management
Network resources
Quarantine network
Antivirus Antispyware
Enterprise network
522 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY
establishment of session keys to enable future secure communication between the
supplicant and resources on the enterprise network.
Typically, the policy server or a supporting server will perform checks on the
AR to determine if it should be permitted interactive remote access connectivity.
These checks—sometimes called health, suitability, screening, or assessment
checks—require software on the user’s system to verify compliance with certain
requirements from the organization’s secure configuration baseline. For example,
the user’s antimalware software must be up-to-date, the operating system must
be fully patched, and the remote computer must be owned and controlled by the
organization. These checks should be performed before granting the AR access to
the enterprise network. Based on the results of these checks, the organization can
determine whether the remote computer should be permitted to use interactive
remote access. If the user has acceptable authorization credentials but the remote
computer does not pass the health check, the user and remote computer should be
denied network access or have limited access to a quarantine network so that autho-
rized personnel can fix the security deficiencies. Figure 16.1 indicates that the quar-
antine portion of the enterprise network consists of the policy server and related
AR suitability servers. There may also be application servers that do not require the
normal security threshold be met.
Once an AR has been authenticated and cleared for a certain level of access
to the enterprise network, the NAS can enable the AR to interact with resources in
the enterprise network. The NAS may mediate every exchange to enforce a security
policy for this AR, or may use other methods to limit the privileges of the AR.
Network Access Enforcement Methods
Enforcement methods are the actions that are applied to ARs to regulate access
to the enterprise network. Many vendors support multiple enforcement methods
simultaneously, allowing the customer to tailor the configuration by using one or a
combination of methods. The following are common NAC enforcement methods.
■ IEEE 802.1X: This is a link layer protocol that enforces authorization before a port is assigned an IP address. IEEE 802.1X makes use of the Extensible
Authentication Protocol for the authentication process. Sections 16.2 and 16.3
cover the Extensible Authentication Protocol and IEEE 802.1X, respectively.
■ Virtual local area networks (VLANs): In this approach, the enterprise net- work, consisting of an interconnected set of LANs, is segmented logically into
a number of virtual LANs.1 The NAC system decides to which of the network’s
VLANs it will direct an AR, based on whether the device needs security reme-
diation, Internet access only, or some level of network access to enterprise
resources. VLANs can be created dynamically and VLAN membership, of
both enterprise servers and ARs, may overlap. That is, an enterprise server or
an AR may belong to more than one VLAN.
1A VLAN is a logical subgroup within a LAN that is created via software rather than manually moving cables in the wiring closet. It combines user stations and network devices into a single unit regardless of the physical LAN segment they are attached to and allows traffic to flow more efficiently within populations of mutual interest. VLANs are implemented in port-switching hubs and LAN switches.
16.2 / EXTENSIBLE AUTHENTICATION PROTOCOL 523
■ Firewall: A firewall provides a form of NAC by allowing or denying network traffic between an enterprise host and an external user. Firewalls are discussed
in Chapter 23.
■ DHCP management: The Dynamic Host Configuration Protocol (DHCP) is an Internet protocol that enables dynamic allocation of IP addresses to hosts.
A DHCP server intercepts DHCP requests and assigns IP addresses instead.
Thus, NAC enforcement occurs at the IP layer based on subnet and IP assign-
ment. A DCHP server is easy to install and configure, but is subject to various
forms of IP spoofing, providing limited security.
There are a number of other enforcement methods available from vendors.
The ones in the preceding list are perhaps the most common, and IEEE 802.1X is by
far the most commonly implemented solution.
16.2 EXTENSIBLE AUTHENTICATION PROTOCOL
The Extensible Authentication Protocol (EAP), defined in RFC 3748, acts as a
framework for network access and authentication protocols. EAP provides a set of
protocol messages that can encapsulate various authentication methods to be used
between a client and an authentication server. EAP can operate over a variety of
network and link level facilities, including point-to-point links, LANs, and other
networks, and can accommodate the authentication needs of the various links and
networks. Figure 16.2 illustrates the protocol layers that form the context for EAP.
Authentication Methods
EAP supports multiple authentication methods. This is what is meant by referring
to EAP as extensible. EAP provides a generic transport service for the exchange of authentication information between a client system and an authentication server.
The basic EAP transport service is extended by using a specific authentication proto-
col, or method, that is installed in both the EAP client and the authentication server.
Figure 16.2 EAP Layered Context
Authentication methods
EAP layer
Data link layer
Extensible Authentication Protocol (EAP)
IEEE 802.1X EAP over LAN (EAPOL)
EAP- TLS
EAP- TTLS
EAP- PSK
EAP- IKEv2
PPP 802.3
Ethernet 802.11 WLAN
Other
Other
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524 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY
Numerous methods have been defined to work over EAP. The following are
commonly supported EAP methods:
■ EAP-TLS (EAP Transport Layer Security): EAP-TLS (RFC 5216) defines how the TLS protocol (described in Chapter 17) can be encapsulated in EAP
messages. EAP-TLS uses the handshake protocol in TLS, not its encryption
method. Client and server authenticate each other using digital certificates.
Client generates a pre-master secret key by encrypting a random number with
the server’s public key and sends it to the server. Both client and server use the
pre-master to generate the same secret key.
■ EAP-TTLS (EAP Tunneled TLS): EAP-TTLS is like EAP-TLS, except only the server has a certificate to authenticate itself to the client first. As in EAP-
TLS, a secure connection (the “tunnel”) is established with secret keys, but
that connection is used to continue the authentication process by authenti-
cating the client and possibly the server again using any EAP method or
legacy method such as PAP (Password Authentication Protocol) and CHAP
(Challenge-Handshake Authentication Protocol). EAP-TTLS is defined in
RFC 5281.
■ EAP-GPSK (EAP Generalized Pre-Shared Key): EAP-GPSK, defined in RFC 5433, is an EAP method for mutual authentication and session key deri-
vation using a Pre-Shared Key (PSK). EAP-GPSK specifies an EAP method
based on pre-shared keys and employs secret key-based cryptographic algo-
rithms. Hence, this method is efficient in terms of message flows and com-
putational costs, but requires the existence of pre-shared keys between each
peer and EAP server. The set up of these pairwise secret keys is part of the
peer registration, and thus, must satisfy the system preconditions. It provides
a protected communication channel when mutual authentication is success-
ful for both parties to communicate over and is designed for authentication
over insecure networks such as IEEE 802.11. EAP-GPSK does not require
any public-key cryptography. The EAP method protocol exchange is done in a
minimum of four messages.
■ EAP-IKEv2: It is based on the Internet Key Exchange protocol version 2 (IKEv2), which is described in Chapter 20. It supports mutual authentication
and session key establishment using a variety of methods. EAP-TLS is defined
in RFC 5106.
EAP Exchanges
Whatever method is used for authentication, the authentication information and
authentication protocol information are carried in EAP messages.
RFC 3748 defines the goal of the exchange of EAP messages to be successful
authentication. In the context of RFC 3748, successful authentication is an exchange of EAP messages, as a result of which the authenticator decides to allow access
by the peer, and the peer decides to use this access. The authenticator’s decision
typically involves both authentication and authorization aspects; the peer may
successfully authenticate to the authenticator, but access may be denied by the
authenticator due to policy reasons.
16.2 / EXTENSIBLE AUTHENTICATION PROTOCOL 525
Figure 16.3 indicates a typical arrangement in which EAP is used. The follow-
ing components are involved:
■ EAP peer: Client computer that is attempting to access a network.
■ EAP authenticator: An access point or NAS that requires EAP authentication prior to granting access to a network.
■ Authentication server: A server computer that negotiates the use of a specific EAP method with an EAP peer, validates the EAP peer’s credentials, and
authorizes access to the network. Typically, the authentication server is a
Remote Authentication Dial-In User Service (RADIUS) server.
The authentication server functions as a backend server that can authenti-
cate peers as a service to a number of EAP authenticators. The EAP authentica-
tor then makes the decision of whether to grant access. This is referred to as the
EAP pass-through mode. Less commonly, the authenticator takes over the role of the EAP server; that is, only two parties are involved in the EAP execution.
As a first step, a lower-level protocol, such as PPP (point-to-point protocol)
or IEEE 802.1X, is used to connect to the EAP authenticator. The software entity
in the EAP peer that operates at this level is referred to as the supplicant. EAP messages containing the appropriate information for a chosen EAP method are
then exchanged between the EAP peer and the authentication server.
EAP messages may include the following fields:
■ Code: Identifies the Type of EAP message. The codes are Request (1), Response (2), Success (3), and Failure (4).
■ Identifier: Used to match Responses with Requests.
■ Length: Indicates the length, in octets, of the EAP message, including the Code, Identifier, Length, and Data fields.
Figure 16.3 EAP Protocol Exchanges
Method
EAP peer/ authenticator
EAP layer
Lower layer
EAP authenticator
EAP layer
Lower layer
Method
EAP peer/ authenticator
EAP layer
Lower layer RADIUS
EAP messages
EAP messages
802.1X, PPP
EAP peer EAP authenticator Authentication server
(RADIUS)
526 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY
■ Data: Contains information related to authentication. Typically, the Data field consists of a Type subfield, indicating the type of data carried, and a Type-Data
field.
The Success and Failure messages do not include a Data field.
The EAP authentication exchange proceeds as follows. After a lower-level
exchange that established the need for an EAP exchange, the authenticator sends a
Request to the peer to request an identity, and the peer sends a Response with the
identity information. This is followed by a sequence of Requests by the authentica-
tor and Responses by the peer for the exchange of authentication information. The
information exchanged and the number of Request–Response exchanges needed
depend on the authentication method. The conversation continues until either
(1) the authenticator determines that it cannot authenticate the peer and transmits
an EAP Failure or (2) the authenticator determines that successful authentication
has occurred and transmits an EAP Success.
Figure 16.4 gives an example of an EAP exchange. Not shown in the figure is a
message or signal sent from the EAP peer to the authenticator using some protocol
other than EAP and requesting an EAP exchange to grant network access. One
protocol used for this purpose is IEEE 802.1X, discussed in the next section. The
first pair of EAP Request and Response messages is of Type identity, in which the
authenticator requests the peer’s identity, and the peer returns its claimed identity
in the Response message. This Response is passed through the authenticator to the
authentication server. Subsequent EAP messages are exchanged between the peer
and the authentication server.
Figure 16.4 EAP Message Flow in Pass-Through Mode
EAP peer
EAP-Response/Identity
EAP-Request/Identity
EAP authenticator Authentication server (RADIUS)
EAP-Response/Auth
EAP-Request/Auth
EAP-Response/Auth
EAP-Request/Auth
EAP-Success/Failure
16.3 / IEEE 802.1X PORT-BASED NETWORK ACCESS CONTROL 527
Upon receiving the identity Response message from the peer, the server
selects an EAP method and sends the first EAP message with a Type field related
to an authentication method. If the peer supports and accepts the selected EAP
method, it replies with the corresponding Response message of the same type.
Otherwise, the peer sends a NAK, and the EAP server either selects another EAP
method or aborts the EAP execution with a failure message. The selected EAP
method determines the number of Request/Response pairs. During the exchange
the appropriate authentication information, including key material, is exchanged.
The exchange ends when the server determines that authentication has succeeded
or that no further attempt can be made and authentication has failed.
16.3 IEEE 802.1X PORT-BASED NETWORK ACCESS CONTROL
IEEE 802.1X Port-Based Network Access Control was designed to provide access
control functions for LANs. Table 16.1 briefly defines key terms used in the IEEE
802.11 standard. The terms supplicant, network access point, and authentication
Authenticator An entity at one end of a point-to-point LAN segment that facilities authentication of the entity to the other
end of the link.
Authentication exchange
The two-party conversation between systems performing an authentication process.
Authentication process
The cryptographic operations and supporting data frames that perform the actual authentication.
Authentication server (AS) An entity that provides an authentication service to an authenticator. This service determines, from the
credentials provided by supplicant, whether the supplicant is authorized to access the services provided by
the system in which the authenticator resides.
Authentication transport The datagram session that actively transfers the authentication exchange between two systems.
Bridge port A port of an IEEE 802.1D or 802.1Q bridge.
Edge port A bridge port attached to a LAN that has no other bridges attached to it.
Network access port A point of attachment of a system to a LAN. It can be a physical port, such as a single LAN MAC attached to
a physical LAN segment, or a logical port, for example, an IEEE 802.11 association between a station and an
access point.
Port access entity (PAE) The protocol entity associated with a port. It can support the protocol functionality associated with the
authenticator, the supplicant, or both.
Supplicant An entity at one end of a point-to-point LAN segment that seeks to be authenticated by an authenticator
attached to the other end of that link.
Table 16.1 Terminology Related to IEEE 802.1X
528 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY
server correspond to the EAP terms peer, authenticator, and authentication server, respectively.
Until the AS authenticates a supplicant (using an authentication protocol),
the authenticator only passes control and authentication messages between the sup-
plicant and the AS; the 802.1X control channel is unblocked, but the 802.11 data
channel is blocked. Once a supplicant is authenticated and keys are provided, the
authenticator can forward data from the supplicant, subject to predefined access
control limitations for the supplicant to the network. Under these circumstances,
the data channel is unblocked.
As indicated in Figure 16.5, 802.1X uses the concepts of controlled and uncon-
trolled ports. Ports are logical entities defined within the authenticator and refer to
physical network connections. Each logical port is mapped to one of these two types
of physical ports. An uncontrolled port allows the exchange of protocol data units
(PDUs) between the supplicant and the AS, regardless of the authentication state
of the supplicant. A controlled port allows the exchange of PDUs between a sup-
plicant and other systems on the network only if the current state of the supplicant
authorizes such an exchange.
The essential element defined in 802.1X is a protocol known as EAPOL (EAP
over LAN). EAPOL operates at the network layers and makes use of an IEEE 802
LAN, such as Ethernet or Wi-Fi, at the link level. EAPOL enables a supplicant to
communicate with an authenticator and supports the exchange of EAP packets for
authentication.
Figure 16.5 802.1X Access Control
Supplicant
Network access point
Uncontrolled port
Controlled port
Authentication server
Network or Internet
16.4 / CLOUD COMPUTING 529
The most common EAPOL packets are listed in Table 16.2. When the
supplicant first connects to the LAN, it does not know the MAC address of the
authenticator. Actually it doesn’t know whether there is an authenticator present
at all. By sending an EAPOL-Start packet to a special group-multicast address reserved for IEEE 802.1X authenticators, a supplicant can determine whether an
authenticator is present and let it know that the supplicant is ready. In many cases,
the authenticator will already be notified that a new device has connected from some
hardware notification. For example, a hub knows that a cable is plugged in before
the device sends any data. In this case the authenticator may preempt the Start mes-
sage with its own message. In either case the authenticator sends an EAP-Request
Identity message encapsulated in an EAPOL-EAP packet. The EAPOL-EAP is the EAPOL frame type used for transporting EAP packets.
The authenticator uses the EAP-Key packet to send cryptographic keys to the supplicant once it has decided to admit it to the network. The EAP-Logoff packet type indicates that the supplicant wishes to be disconnected from the network.
The EAPOL packet format includes the following fields:
■ Protocol version: version of EAPOL.
■ Packet type: indicates start, EAP, key, logoff, etc.
■ Packet body length: If the packet includes a body, this field indicates the body length.
■ Packet body: The payload for this EAPOL packet. An example is an EAP packet.
Figure 16.6 shows an example of exchange using EAPOL. In Chapter 18, we
examine the use of EAP and EAPOL in the context of IEEE 802.11 wireless LAN
security.
16.4 CLOUD COMPUTING
There is an increasingly prominent trend in many organizations to move a substan-
tial portion of or even all information technology (IT) operations to an Internet-
connected infrastructure known as enterprise cloud computing. This section provides
an overview of cloud computing. For a more detailed treatment, see [STAL16].
Frame Type Definition
EAPOL-EAP Contains an encapsulated EAP packet.
EAPOL-Start A supplicant can issue this packet instead of waiting for
a challenge from the authenticator.
EAPOL-Logoff Used to return the state of the port to unauthorized when
the supplicant is finished using the network.
EAPOL-Key Used to exchange cryptographic keying information.
Table 16.2 Common EAPOL Frame Types
530 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY
Cloud Computing Elements
NIST defines cloud computing, in NIST SP-800-145 (The NIST Definition of Cloud Computing), as follows:
Figure 16.6 Example Timing Diagram for IEEE 802.1X
EAP peer
EAPOL-Start
EAPOL-EAP (EAP-Request/Identity)
EAPOL-EAP (EAP-Response/Identity)
EAP authenticator Authentication server (RADIUS)
EAPOL-Logoff
EAPOL-EAP (EAP-Response/Auth)
EAPOL-EAP (EAP-Request/Auth)
EAPOL-EAP (EAP-Response/Auth)
EAPOL-EAP (EAP-Request/Auth)
EAPOL-EAP (EAP-Success)
Cloud computing: A model for enabling ubiquitous, convenient, on-demand net- work access to a shared pool of configurable computing resources (e.g., networks,
servers, storage, applications, and services) that can be rapidly provisioned and
released with minimal management effort or service provider interaction. This
cloud model promotes availability and is composed of five essential characteris-
tics, three service models, and four deployment models.
The definition refers to various models and characteristics, whose relationship is
illustrated in Figure 16.7. The essential characteristics of cloud computing include
the following:
■ Broad network access: Capabilities are available over the network and ac- cessed through standard mechanisms that promote use by heterogeneous thin
16.4 / CLOUD COMPUTING 531
or thick client platforms (e.g., mobile phones, laptops, and PDAs) as well as
other traditional or cloud-based software services.
■ Rapid elasticity: Cloud computing gives you the ability to expand and reduce resources according to your specific service requirement. For example, you
may need a large number of server resources for the duration of a specific task.
You can then release these resources upon completion of the task.
■ Measured service: Cloud systems automatically control and optimize resource use by leveraging a metering capability at some level of abstraction appropri-
ate to the type of service (e.g., storage, processing, bandwidth, and active user
accounts). Resource usage can be monitored, controlled, and reported, provid-
ing transparency for both the provider and consumer of the utilized service.
■ On-demand self-service: A consumer can unilaterally provision computing capabilities, such as server time and network storage, as needed automati-
cally without requiring human interaction with each service provider. Because
the service is on demand, the resources are not permanent parts of your IT
infrastructure.
■ Resource pooling: The provider’s computing resources are pooled to serve multiple consumers using a multi-tenant model, with different physical and
virtual resources dynamically assigned and reassigned according to consumer
demand. There is a degree of location independence in that the customer
Figure 16.7 Cloud Computing Elements
Broad Network Access
Resource Pooling
Rapid Elasticity
E ss
en ti
al C
ha ra
ct er
is ti
cs S
er vi
ce M
od el
s D
ep lo
ym en
t M
od el
s Measured
Service On-Demand Self-Service
Public Private Hybrid Community
Software as a Service (SaaS)
Platform as a Service (PaaS)
Infrastructure as a Service (IaaS)
532 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY
generally has no control or knowledge of the exact location of the provided
resources, but may be able to specify location at a higher level of abstraction
(e.g., country, state, or data center). Examples of resources include storage,
processing, memory, network bandwidth, and virtual machines. Even private
clouds tend to pool resources between different parts of the same organization.
NIST defines three service models, which can be viewed as nested service alternatives:
■ Software as a service (SaaS): The capability provided to the consumer is to use the provider’s applications running on a cloud infrastructure. The applications
are accessible from various client devices through a thin client interface such as
a Web browser. Instead of obtaining desktop and server licenses for software
products it uses, an enterprise obtains the same functions from the cloud service.
SaaS saves the complexity of software installation, maintenance, upgrades, and
patches. Examples of services at this level are Gmail, Google’s email service,
and Salesforce.com, which helps firms keep track of their customers.
■ Platform as a service (PaaS): The capability provided to the consumer is to deploy onto the cloud infrastructure consumer-created or acquired applica-
tions created using programming languages and tools supported by the pro-
vider. PaaS often provides middleware-style services such as database and
component services for use by applications. In effect, PaaS is an operating
system in the cloud.
■ Infrastructure as a service (IaaS): The capability provided to the consumer is to provision processing, storage, networks, and other fundamental computing
resources where the consumer is able to deploy and run arbitrary software,
which can include operating systems and applications. IaaS enables custom-
ers to combine basic computing services, such as number crunching and data
storage, to build highly adaptable computer systems.
NIST defines four deployment models:
■ Public cloud: The cloud infrastructure is made available to the general public or a large industry group and is owned by an organization selling cloud ser-
vices. The cloud provider is responsible both for the cloud infrastructure and
for the control of data and operations within the cloud.
■ Private cloud: The cloud infrastructure is operated solely for an organization. It may be managed by the organization or a third party and may exist on prem-
ise or off premise. The cloud provider (CP) is responsible only for the infra-
structure and not for the control.
■ Community cloud: The cloud infrastructure is shared by several organizations and supports a specific community that has shared concerns (e.g., mission, security
requirements, policy, and compliance considerations). It may be managed by the
organizations or a third party and may exist on premise or off premise.
■ Hybrid cloud: The cloud infrastructure is a composition of two or more clouds (private, community, or public) that remain unique entities but are bound
together by standardized or proprietary technology that enables data and
application portability (e.g., cloud bursting for load balancing between clouds).
Hiva-Network.Com
16.4 / CLOUD COMPUTING 533
Figure 16.8 illustrates the typical cloud service context. An enterprise main-
tains workstations within an enterprise LAN or set of LANs, which are connected
by a router through a network or the Internet to the cloud service provider. The
cloud service provider maintains a massive collection of servers, which it man-
ages with a variety of network management, redundancy, and security tools. In the
figure, the cloud infrastructure is shown as a collection of blade servers, which is a
common architecture.
Cloud Computing Reference Architecture
NIST SP 500-292 (NIST Cloud Computing Reference Architecture) establishes a reference architecture, described as follows:
Figure 16.8 Cloud Computing Context
Router
Servers
LAN switch
Cloud service
provider Network
or Internet
Router
LAN switch
Enterprise (Cloud user)
The NIST cloud computing reference architecture focuses on the requirements
of “what” cloud services provide, not a “how to” design solution and implemen-
tation. The reference architecture is intended to facilitate the understanding of
the operational intricacies in cloud computing. It does not represent the system
architecture of a specific cloud computing system; instead it is a tool for describ-
ing, discussing, and developing a system-specific architecture using a common
framework of reference.
534 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY
NIST developed the reference architecture with the following objectives
in mind:
■ to illustrate and understand the various cloud services in the context of an
overall cloud computing conceptual model
■ to provide a technical reference for consumers to understand, discuss, catego-
rize, and compare cloud services
■ to facilitate the analysis of candidate standards for security, interoperability,
and portability and reference implementations
The reference architecture, depicted in Figure 16.9, defines five major actors
in terms of the roles and responsibilities:
■ Cloud consumer: A person or organization that maintains a business relation- ship with, and uses service from, cloud providers.
■ Cloud provider: A person, organization, or entity responsible for making a service available to interested parties.
■ Cloud auditor: A party that can conduct independent assessment of cloud services, information system operations, performance, and security of the
cloud implementation.
■ Cloud broker: An entity that manages the use, performance, and delivery of cloud services, and negotiates relationships between CPs and cloud consumers.
■ Cloud carrier: An intermediary that provides connectivity and transport of cloud services from CPs to cloud consumers.
The roles of the cloud consumer and provider have already been discussed. To
summarize, a cloud provider can provide one or more of the cloud services to meet IT and business requirements of cloud consumers. For each of the three service
Figure 16.9 NIST Cloud Computing Reference Architecture
Cloud consumer
Cloud auditor
Service intermediation
Service aggregation
Service arbitrage
Cloud broker
Cloud provider
Security audit
Performance audit
Privacy impact audit
SaaS Service layer Service orchestration Cloud
service management
PaaS
Hardware
Physical resource layer
Facility
Resource abstraction and control layer
IaaS
Business support
Provisioning/ configuration
Portability/ interoperability
S ec
ur it
y
P ri
va cy
Cloud carrier
16.5 / CLOUD SECURITY RISKS AND COUNTERMEASURES 535
models (SaaS, PaaS, IaaS), the CP provides the storage and processing facilities
needed to support that service model, together with a cloud interface for cloud
service consumers. For SaaS, the CP deploys, configures, maintains, and updates
the operation of the software applications on a cloud infrastructure so that the
services are provisioned at the expected service levels to cloud consumers. The
consumers of SaaS can be organizations that provide their members with access to
software applications, end users who directly use software applications, or software
application administrators who configure applications for end users.
For PaaS, the CP manages the computing infrastructure for the platform and
runs the cloud software that provides the components of the platform, such as run-
time software execution stack, databases, and other middleware components. Cloud
consumers of PaaS can employ the tools and execution resources provided by CPs to
develop, test, deploy, and manage the applications hosted in a cloud environment.
For IaaS, the CP acquires the physical computing resources underlying the
service, including the servers, networks, storage, and hosting infrastructure. The
IaaS cloud consumer in turn uses these computing resources, such as a virtual
computer, for their fundamental computing needs.
The cloud carrier is a networking facility that provides connectivity and trans- port of cloud services between cloud consumers and CPs. Typically, a CP will set up
service level agreements (SLAs) with a cloud carrier to provide services consistent
with the level of SLAs offered to cloud consumers, and may require the cloud carrier
to provide dedicated and secure connections between cloud consumers and CPs.
A cloud broker is useful when cloud services are too complex for a cloud con- sumer to easily manage. Three areas of support can be offered by a cloud broker:
■ Service intermediation: These are value-added services, such as identity man- agement, performance reporting, and enhanced security.
■ Service aggregation: The broker combines multiple cloud services to meet consumer needs not specifically addressed by a single CP, or to optimize per-
formance or minimize cost.
■ Service arbitrage: This is similar to service aggregation except that the services being aggregated are not fixed. Service arbitrage means a broker has the flexibil-
ity to choose services from multiple agencies. The cloud broker, for example, can
use a credit-scoring service to measure and select an agency with the best score.
A cloud auditor can evaluate the services provided by a CP in terms of secu- rity controls, privacy impact, performance, and so on. The auditor is an independent
entity that can assure that the CP conforms to a set of standards.
16.5 CLOUD SECURITY RISKS AND COUNTERMEASURES
In general terms, security controls in cloud computing are similar to the security
controls in any IT environment. However, because of the operational models and
technologies used to enable cloud service, cloud computing may present risks that
are specific to the cloud environment. The essential concept in this regard is that
the enterprise loses a substantial amount of control over resources, services, and
applications but must maintain accountability for security and privacy policies.
536 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY
The Cloud Security Alliance [CSA10] lists the following as the top cloud-
specific security threats, together with suggested countermeasures:
■ Abuse and nefarious use of cloud computing: For many CPs, it is relatively easy to register and begin using cloud services, some even offering free limited
trial periods. This enables attackers to get inside the cloud to conduct various
attacks, such as spamming, malicious code attacks, and denial of service. PaaS
providers have traditionally suffered most from this kind of attacks; however,
recent evidence shows that hackers have begun to target IaaS vendors as well.
The burden is on the CP to protect against such attacks, but cloud service cli-
ents must monitor activity with respect to their data and resources to detect
any malicious behavior.
Countermeasures include (1) stricter initial registration and valida-
tion processes; (2) enhanced credit card fraud monitoring and coordination;
(3) comprehensive introspection of customer network traffic; and (4) monitor-
ing public blacklists for one’s own network blocks.
■ Insecure interfaces and APIs: CPs expose a set of software interfaces or APIs that customers use to manage and interact with cloud services. The security
and availability of general cloud services are dependent upon the security of
these basic APIs. From authentication and access control to encryption and
activity monitoring, these interfaces must be designed to protect against both
accidental and malicious attempts to circumvent policy.
Countermeasures include (1) analyzing the security model of CP inter-
faces; (2) ensuring that strong authentication and access controls are imple-
mented in concert with encrypted transmission; and (3) understanding the
dependency chain associated with the API.
■ Malicious insiders: Under the cloud computing paradigm, an organization relinquishes direct control over many aspects of security and, in doing so, con-
fers an unprecedented level of trust onto the CP. One grave concern is the
risk of malicious insider activity. Cloud architectures necessitate certain roles
that are extremely high risk. Examples include CP system administrators and
managed security service providers.
Countermeasures include the following: (1) enforce strict supply chain
management and conduct a comprehensive supplier assessment; (2) specify
human resource requirements as part of legal contract; (3) require transpar-
ency into overall information security and management practices, as well as
compliance reporting; and (4) determine security breach notification processes.
■ Shared technology issues: IaaS vendors deliver their services in a scalable way by sharing infrastructure. Often, the underlying components that make up this
infrastructure (CPU caches, GPUs, etc.) were not designed to offer strong iso-
lation properties for a multi-tenant architecture. CPs typically approach this
risk by the use of isolated virtual machines for individual clients. This approach
is still vulnerable to attack, by both insiders and outsiders, and so can only be a
part of an overall security strategy.
Countermeasures include the following: (1) implement security best
practices for installation/configuration; (2) monitor environment for unauthor-
ized changes/activity; (3) promote strong authentication and access control
16.6 / DATA PROTECTION IN THE CLOUD 537
for administrative access and operations; (4) enforce SLAs for patching and
vulnerability remediation; and (5) conduct vulnerability scanning and
configuration audits.
■ Data loss or leakage: For many clients, the most devastating impact from a security breach is the loss or leakage of data. We address this issue in the next
subsection.
Countermeasures include the following: (1) implement strong API ac-
cess control; (2) encrypt and protect integrity of data in transit; (3) analyze
data protection at both design and run time; and (4) implement strong key
generation, storage and management, and destruction practices.
■ Account or service hijacking: Account or service hijacking, usually with stolen credentials, remains a top threat. With stolen credentials, attackers can often
access critical areas of deployed cloud computing services, allowing them to
compromise the confidentiality, integrity, and availability of those services.
Countermeasures include the following: (1) prohibit the sharing of
account credentials between users and services; (2) leverage strong two- factor
authentication techniques where possible; (3) employ proactive monitor-
ing to detect unauthorized activity; and (4) understand CP security policies
and SLAs.
■ Unknown risk profile: In using cloud infrastructures, the client necessarily cedes control to the CP on a number of issues that may affect security. Thus
the client must pay attention to and clearly define the roles and responsibili-
ties involved for managing risks. For example, employees may deploy applica-
tions and data resources at the CP without observing the normal policies and
procedures for privacy, security, and oversight.
Countermeasures include (1) disclosure of applicable logs and data;
(2) partial/full disclosure of infrastructure details (e.g., patch levels and
firewalls); and (3) monitoring and alerting on necessary information.
Similar lists have been developed by the European Network and Information
Security Agency [ENIS09] and NIST [JANS11].
16.6 DATA PROTECTION IN THE CLOUD
As can be seen from the previous section, there are numerous aspects to cloud
security and numerous approaches to providing cloud security measures.
A further example is seen in the NIST guidelines for cloud security, specified
in SP-800-14 and listed in Table 16.3. Thus, the topic of cloud security is well
beyond the scope of this chapter. In this section, we focus on one specific element
of cloud security.
There are many ways to compromise data. Deletion or alteration of records
without a backup of the original content is an obvious example. Unlinking a record
from a larger context may render it unrecoverable, as can storage on unreliable
media. Loss of an encoding key may result in effective destruction. Finally, unau-
thorized parties must be prevented from gaining access to sensitive data.
538 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY
Governance Extend organizational practices pertaining to the policies, procedures, and standards used for application
development and service provisioning in the cloud, as well as the design, implementation, testing, use, and
monitoring of deployed or engaged services.
Put in place audit mechanisms and tools to ensure organizational practices are followed throughout the
system life cycle.
Compliance Understand the various types of laws and regulations that impose security and privacy obligations on the
organization and potentially impact cloud computing initiatives, particularly those involving data location,
privacy and security controls, records management, and electronic discovery requirements.
Review and assess the cloud provider’s offerings with respect to the organizational requirements to be met
and ensure that the contract terms adequately meet the requirements.
Ensure that the cloud provider’s electronic discovery capabilities and processes do not compromise the
privacy or security of data and applications.
Trust Ensure that service arrangements have sufficient means to allow visibility into the security and privacy
controls and processes employed by the cloud provider, and their performance over time.
Establish clear, exclusive ownership rights over data.
Institute a risk management program that is flexible enough to adapt to the constantly evolving and
shifting risk landscape for the life cycle of the system.
Continuously monitor the security state of the information system to support ongoing risk management
decisions.
Architecture Understand the underlying technologies that the cloud provider uses to provision services, including the
implications that the technical controls involved have on the security and privacy of the system, over the full
system life cycle and across all system components.
Identity and access management Ensure that adequate safeguards are in place to secure authentication, authorization, and other identity and
access management functions, and are suitable for the organization.
Software isolation Understand virtualization and other logical isolation techniques that the cloud provider employs in its
multi-tenant software architecture, and assess the risks involved for the organization.
Data protection Evaluate the suitability of the cloud provider’s data management solutions for the organizational data
concerned and the ability to control access to data, to secure data while at rest, in transit, and in use, and to
sanitize data.
Take into consideration the risk of collating organizational data with those of other organizations whose
threat profiles are high or whose data collectively represent significant concentrated value.
Fully understand and weigh the risks involved in cryptographic key management with the facilities
available in the cloud environment and the processes established by the cloud provider.
Availability Understand the contract provisions and procedures for availability, data backup and recovery, and disaster
recovery, and ensure that they meet the organization’s continuity and contingency planning requirements.
Ensure that during an intermediate or prolonged disruption or a serious disaster, critical operations
can be immediately resumed, and that all operations can be eventually reinstituted in a timely and organized
manner.
Incident response Understand the contract provisions and procedures for incident response and ensure that they meet the
requirements of the organization.
Table 16.3 NIST Guidelines on Security and Privacy Issues and Recommendations
16.6 / DATA PROTECTION IN THE CLOUD 539
Ensure that the cloud provider has a transparent response process in place and sufficient mechanisms to
share information during and after an incident.
Ensure that the organization can respond to incidents in a coordinated fashion with the cloud provider in
accordance with their respective roles and responsibilities for the computing environment.
Table 16.3 Continued
The threat of data compromise increases in the cloud, due to the number of
and interactions between risks and challenges that are either unique to the cloud or
more dangerous because of the architectural or operational characteristics of the
cloud environment.
Database environments used in cloud computing can vary significantly. Some
providers support a multi-instance model, which provides a unique DBMS running on a virtual machine instance for each cloud subscriber. This gives the subscriber
complete control over role definition, user authorization, and other administrative
tasks related to security. Other providers support a multi-tenant model, which pro- vides a predefined environment for the cloud subscriber that is shared with other
tenants, typically through tagging data with a subscriber identifier. Tagging gives
the appearance of exclusive use of the instance, but relies on the CP to establish and
maintain a sound secure database environment.
Data must be secured while at rest, in transit, and in use, and access to the
data must be controlled. The client can employ encryption to protect data in transit,
though this involves key management responsibilities for the CP. The client can
enforce access control techniques but, again, the CP is involved to some extent
depending on the service model used.
For data at rest, the ideal security measure is for the client to encrypt the data-
base and only store encrypted data in the cloud, with the CP having no access to the
encryption key. So long as the key remains secure, the CP has no ability to read the
data, although corruption and other denial-of-service attacks remain a risk.
A straightforward solution to the security problem in this context is to encrypt
the entire database and not provide the encryption/decryption keys to the service
provider. This solution by itself is inflexible. The user has little ability to access
individual data items based on searches or indexing on key parameters, but rather
would have to download entire tables from the database, decrypt the tables, and
work with the results. To provide more flexibility, it must be possible to work with
the database in its encrypted form.
An example of such an approach, depicted in Figure 16.10, is reported in
[DAMI05] and [DAMI03]. A similar approach is described in [HACI02]. Four enti-
ties are involved:
■ Data owner: An organization that produces data to be made available for controlled release, either within the organization or to external users.
■ User: Human entity that presents requests (queries) to the system. The user could be an employee of the organization who is granted access to the data-
base via the server, or a user external to the organization who, after authenti-
cation, is granted access.
■ Client: Frontend that transforms user queries into queries on the encrypted data stored on the server.
540 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY
■ Server: An organization that receives the encrypted data from a data owner and makes them available for distribution to clients. The server could in fact
be owned by the data owner but, more typically, is a facility owned and main-
tained by an external provider. For our discussion, the server is a cloud server.
Before continuing this discussion, we need to define some database terms.
In relational database parlance, the basic building block is a relation, which is a flat table. Rows are referred to as tuples, and columns are referred to as attributes. A primary key is defined to be a portion of a row used to uniquely identify a row in a table; the primary key consists of one or more column names.2 For example, in
an employee table, the employee ID is sufficient to uniquely identify a row in a
particular table.
Let us first examine the simplest possible arrangement based on this scenario.
Suppose that each individual item in the database is encrypted separately, all using
the same encryption key. The encrypted database is stored at the server, but the
server does not have the encryption key. Thus, the data are secure at the server.
Even if someone were able to hack into the server’s system, all he or she would have
access to is encrypted data. The client system does have a copy of the encryption
key. A user at the client can retrieve a record from the database with the following
sequence:
1. The user issues a query for fields from one or more records with a specific value of the primary key.
2Note that a primary key has nothing to do with cryptographic keys. A primary key in a database is a means of indexing into the database.
Figure 16.10 An Encryption Scheme for a Cloud-Based Database
Query processor
1. Original query Metadata
4. Plaintext result
2. Transformed query
3. Encrypted result
Client
User Data owner
Cloud server
Encrypt/ Decrypt
Query executor
Metadata
Metadata
Encrypted database
Database
16.7 / CLOUD SECURITY AS A SERVICE 541
2. The query processor at the client encrypts the primary key, modifies the query accordingly, and transmits the query to the server.
3. The server processes the query using the encrypted value of the primary key and returns the appropriate record or records.
4. The query processor decrypts the data and returns the results.
This method is certainly straightforward but is quite limited. For example, sup-
pose the Employee table contains a salary attribute and the user wishes to retrieve
all records for salaries less than $70K. There is no obvious way to do this, because
the attribute value for salary in each record is encrypted. The set of encrypted values
does not preserve the ordering of values in the original attribute.
There are a number of ways to extend the functionality of this approach. For
example, an unencrypted index value can be associated with a given attribute and
the table can be partitioned based on these index values, enabling a user to retrieve
a certain portion of the table. The details of such schemes are beyond our scope.
See [STAL15] for more detail.
16.7 CLOUD SECURITY AS A SERVICE
The term Security as a Service (SecaaS) has generally meant a package of security services offered by a service provider that offloads much of the security respon-
sibility from an enterprise to the security service provider. Among the services
typically provided are authentication, antivirus, antimalware/-spyware, intrusion
detection, and security event management. In the context of cloud computing,
cloud security as a service, designated SecaaS, is a segment of the SaaS offering
of a CP.
The Cloud Security Alliance defines SecaaS as the provision of security
applications and services via the cloud either to cloud-based infrastructure and soft-
ware or from the cloud to the customers’ on-premise systems [CSA11b]. The Cloud
Security Alliance has identified the following SecaaS categories of service:
■ Identity and access management
■ Data loss prevention
■ Web security
■ Email security
■ Security assessments
■ Intrusion management
■ Security information and event management
■ Encryption
■ Business continuity and disaster recovery
■ Network security
In this section, we examine these categories with a focus on security of the
cloud-based infrastructure and services (Figure 16.11).
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542 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY
Identity and access management (IAM) includes people, processes, and systems that are used to manage access to enterprise resources by assuring that the
identity of an entity is verified, and then granting the correct level of access based
on this assured identity. One aspect of identity management is identity provision-
ing, which has to do with providing access to identified users and subsequently
deprovisioning, or deny access, to users when the client enterprise designates such
users as no longer having access to enterprise resources in the cloud. Another aspect
of identity management is for the cloud to participate in the federated identity man-
agement scheme (see Chapter 15) scheme used by the client enterprise. Among
other requirements, the cloud service provider (CSP) must be able to exchange
identity attributes with the enterprise’s chosen identity provider.
The access management portion of IAM involves authentication and access
control services. For example, the CSP must be able to authenticate users in a
trustworthy manner. The access control requirements in SPI environments include
establishing trusted user profile and policy information, using it to control access
within the cloud service, and doing this in an auditable way.
Data loss prevention (DLP) is the monitoring, protecting, and verifying the security of data at rest, in motion, and in use. Much of DLP can be implemented by
Figure 16.11 Elements of Cloud Security as a Service
Cloud service clients and adversaries
Identity and access management Network security
Data loss prevention
Web security Intrusion management
Encryption
Email security
Security assessments Security information and event management Business continuity and disaster recovery
16.7 / CLOUD SECURITY AS A SERVICE 543
the cloud client, such as discussed in Section 16.6. The CSP can also provide DLP
services, such as implementing rules about what functions can be performed on data
in various contexts.
Web security is real-time protection offered either on premise through soft- ware/appliance installation or via the cloud by proxying or redirecting Web traffic
to the CP. This provides an added layer of protection on top of things like antivi-
ruses to prevent malware from entering the enterprise via activities such as Web
browsing. In addition to protecting against malware, a cloud-based Web security
service might include usage policy enforcement, data backup, traffic control, and
Web access control.
A CSP may provide a Web-based email service, for which security measures
are needed. Email security provides control over inbound and outbound email, protecting the organization from phishing, malicious attachments, enforcing corpo-
rate polices such as acceptable use and spam prevention. The CSP may also incor-
porate digital signatures on all email clients and provide optional email encryption.
Security assessments are third-part audits of cloud services. While this service is outside the province of the CSP, the CSP can provide tools and access points to
facilitate various assessment activities.
Intrusion management encompasses intrusion detection, prevention, and response. The core of this service is the implementation of intrusion detection sys-
tems (IDSs) and intrusion prevention systems (IPSs) at entry points to the cloud
and on servers in the cloud. An IDS is a set of automated tools designed to detect
unauthorized access to a host system. We discuss this in Chapter 21. An IPS incor-
porates IDS functionality but also includes mechanisms designed to block traffic
from intruders.
Security information and event management (SIEM) aggregates (via push or pull mechanisms) log and event data from virtual and real networks, applications,
and systems. This information is then correlated and analyzed to provide real-time
reporting and alerting on information/events that may require intervention or other
type of response. The CSP typically provides an integrated service that can put
together information from a variety of sources both within the cloud and within the
client enterprise network.
Encryption is a pervasive service that can be provided for data at rest in the cloud, email traffic, client-specific network management information, and identity
information. Encryption services provided by the CSP involve a range of complex
issues, including key management, how to implement virtual private network (VPN)
services in the cloud, application encryption, and data content access.
Business continuity and disaster recovery comprise measures and mechanisms to ensure operational resiliency in the event of any service interruptions. This is
an area where the CSP, because of economies of scale, can offer obvious benefits
to a cloud service client [WOOD10]. The CSP can provide backup at multiple
locations, with reliable failover and disaster recovery facilities. This service must
include a flexible infrastructure, redundancy of functions and hardware, monitored
operations, geographically distributed data centers, and network survivability.
Network security consists of security services that allocate access, distribute, monitor, and protect the underlying resource services. Services include perimeter
and server firewalls and denial-of-service protection. Many of the other services
544 CHAPTER 16 / NETWORK ACCESS CONTROL AND CLOUD SECURITY
listed in this section, including intrusion management, identity and access man-
agement, data loss protection, and Web security, also contribute to the network
security service.
16.8 ADDRESSING CLOUD COMPUTING SECURITY CONCERNS
Numerous documents have been developed to guide businesses thinking about the
security issues associated with cloud computing. In addition to SP 800-144, which
provides overall guidance, NIST has issued SP 800-146 (Cloud Computing Synopsis and Recommendations, May 2012). NIST’s recommendations systematically con- sider each of the major types of cloud services consumed by businesses including
Software as a Service (SaaS), Infrastructure as a Service (IaaS), and Platform as
a Service (PaaS). While security issues vary somewhat depending on the type of
cloud service, there are multiple NIST recommendations that are independent of
service type. Not surprisingly, NIST recommends selecting cloud providers that
support strong encryption, have appropriate redundancy mechanisms in place,
employ authentication mechanisms, and offer subscribers sufficient visibility about
mechanisms used to protect subscribers from other subscribers and the provider.
SP 800-146 also lists the overall security controls that are relevant in a cloud com-
puting environment and that must be assigned to the different cloud actors. These
are shown in Table 16.4.
As more businesses incorporate cloud services into their enterprise net-
work infrastructures, cloud computing security will persist as an important issue.
Examples of cloud computing security failures have the potential to have a chilling
effect on business interest in cloud services and this is inspiring service providers
to be serious about incorporating security mechanisms that will allay concerns of
potential subscribers. Some service providers have moved their operations to Tier 4
data centers to address user concerns about availability and redundancy. Because so
many businesses remain reluctant to embrace cloud computing in a big way, cloud
service providers will have to continue to work hard to convince potential customers
that computing support for core business processes and mission critical applications
can be moved safely and securely to the cloud.
Technical Operational Management
Access Control
Audit and Accountability
Identification and Authentication
System and Communication
Protection
Awareness and Training
Configuration and Management
Contingency Planning
Incident Response
Maintenance
Media Protection
Physical and Environmental
Protection
Personnel Security System and
Information Integrity
Certification, Accreditation, and
Security Assessment
Planning Risk Assessment
System and Services Acquisition
Table 16.4 Control Functions and Classes
16.9 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 545
Key Terms
access requestor (AR)
authentication server
cloud
cloud auditor
cloud broker
cloud carrier
cloud computing
cloud consumer
cloud provider
community cloud
Dynamic Host Configuration
Protocol (DHCP)
EAP authenticator
EAP-GPSK
EAP-IKEv2
EAP over LAN (EAPOL)
EAP method
EAP pass-through mode
EAP peer
EAP-TLS
EAP-TTLS
Extensible Authentication
Protocol (EAP)
firewall
IEEE 802.1X
media gateway
Network Access Control
(NAC)
Network Access Server
(NAS)
Platform as a Service (PaaS)
policy server
private cloud
public cloud
Remote Access Server (RAS)
Security as a Service (SecaaS)
Software as a Service (SaaS)
supplicant
Virtual Local Area Network
(VLAN)
16.9 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
Review Questions 16.1 Provide a brief definition of network access control. 16.2 What is an EAP? 16.3 List and briefly define four EAP authentication methods. 16.4 What is DHCP? How useful is it to help achieve security of IP addresses? 16.5 Why is EAPOL an essential element of IEEE 802.1X? 16.6 What are the essential characteristics of cloud computing? 16.7 List and briefly define the deployment models of cloud computing. 16.8 What is the cloud computing reference architecture? 16.9 Describe some of the main cloud-specific security threats.
Problems 16.1 Investigate the network access control scheme used at your school or place of
employment. Draw a diagram and describe the principal components.
16.2 Figure 16.3 suggests that EAP can be described in the context of a four-layer model. Indicate the functions and formats of each of the four layers. You may need to refer to RFC 3748.
16.3 List some commonly used cloud-based data services. Explore and compare these services based on their use of encryption, flexibility, efficiency, speed, and ease of use. Study security breaches on these services in recent past. What changes were made by the services after these attacks?
546546
Transport-Level Security 17.1 Web Security Considerations
Web Security Threats
Web Traffic Security Approaches
17.2 Transport Layer Security
TLS Architecture
TLS Record Protocol
Change Cipher Spec Protocol
Alert Protocol
Handshake Protocol
Cryptographic Computations
Heartbeat Protocol
SSL/TLS Attacks
TLSv1.3
17.3 HTTPS
Connection Initiation
Connection Closure
17.4 Secure Shell (SSH)
Transport Layer Protocol
User Authentication Protocol
Connection Protocol
17.5 Key Terms, Review Questions, and Problems
CHAPTER
17.1 / WEB SECURITY CONSIDERATIONS 547
Virtually all businesses, most government agencies, and many individuals now have
Web sites. The number of individuals and companies with Internet access is expanding
rapidly and all of these have graphical Web browsers. As a result, businesses are enthu-
siastic about setting up facilities on the Web for electronic commerce. But the reality
is that the Internet and the Web are extremely vulnerable to compromises of various
sorts. As businesses wake up to this reality, the demand for secure Web services grows.
The topic of Web security is a broad one and can easily fill a book. In this chap-
ter, we begin with a discussion of the general requirements for Web security and then
focus on three standardized schemes that are becoming increasingly important as part
of Web commerce and that focus on security at the transport layer: SSL/TLS, HTTPS,
and SSH.
17.1 WEB SECURITY CONSIDERATIONS
The World Wide Web is fundamentally a client/server application running over the
Internet and TCP/IP intranets. As such, the security tools and approaches discussed
so far in this book are relevant to the issue of Web security. However, the following
characteristics of Web usage suggest the need for tailored security tools:
■ Although Web browsers are very easy to use, Web servers are relatively easy
to configure and manage, and Web content is increasingly easy to develop, the
underlying software is extraordinarily complex. This complex software may
hide many potential security flaws. The short history of the Web is filled with
examples of new and upgraded systems, properly installed, that are vulnerable
to a variety of security attacks.
■ A Web server can be exploited as a launching pad into the corporation’s or
agency’s entire computer complex. Once the Web server is subverted, an
attacker may be able to gain access to data and systems not part of the Web
itself but connected to the server at the local site.
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Summarize Web security threats and Web traffic security approaches.
◆ Present an overview of Transport Layer Security (TLS).
◆ Understand the differences between Secure Sockets Layer and Transport Layer Security.
◆ Compare the pseudorandom function used in Transport Layer Security with those discussed earlier in the book.
◆ Present an overview of HTTPS (HTTP over SSL).
◆ Present an overview of Secure Shell (SSH).
548 CHAPTER 17 / TRANSPORT-LEVEL SECURITY
■ Casual and untrained (in security matters) users are common clients for Web-
based services. Such users are not necessarily aware of the security risks that
exist and do not have the tools or knowledge to take effective countermeasures.
Web Security Threats
Table 17.1 provides a summary of the types of security threats faced when using the
Web. One way to group these threats is in terms of passive and active attacks. Passive
attacks include eavesdropping on network traffic between browser and server and
gaining access to information on a Web site that is supposed to be restricted. Active
attacks include impersonating another user, altering messages in transit between
client and server, and altering information on a Web site.
Another way to classify Web security threats is in terms of the location of the
threat: Web server, Web browser, and network traffic between browser and server.
Issues of server and browser security fall into the category of computer system secu-
rity; Part Six of this book addresses the issue of system security in general but is also
applicable to Web system security. Issues of traffic security fall into the category of
network security and are addressed in this chapter.
Web Traffic Security Approaches
A number of approaches to providing Web security are possible. The various
approaches that have been considered are similar in the services they provide and,
to some extent, in the mechanisms that they use, but they differ with respect to their
scope of applicability and their relative location within the TCP/IP protocol stack.
Threats Consequences Countermeasures
Integrity Modification of user data Trojan horse browser
Modification of memory
Modification of message
traffic in transit
Loss of information
Compromise of machine
Vulnerability to all other
threats
Cryptographic
checksums
Confidentiality Eavesdropping on the net Theft of info from server
Theft of data from client
Info about network
configuration
Info about which client talks
to server
Loss of information
Loss of privacy
Encryption, Web
proxies
Denial of Service
Killing of user threads
Flooding machine with bogus
requests
Filling up disk or memory
Isolating machine by DNS
attacks
Disruptive
Annoying
Prevent user from getting work
done
Difficult to prevent
Authentication Impersonation of legitimate users
Data forgery
Misrepresentation of user
Belief that false information
is valid
Cryptographic
techniques
Table 17.1 A Comparison of Threats on the Web
17.2 / TRANSPORT LAYER SECURITY 549
Figure 17.1 illustrates this difference. One way to provide Web security is
to use IP security (IPsec) (Figure 17.1a). The advantage of using IPsec is that it is
transparent to end users and applications and provides a general-purpose solution.
Furthermore, IPsec includes a filtering capability so that only selected traffic need
incur the overhead of IPsec processing.
Another relatively general-purpose solution is to implement security just
above TCP (Figure 17.1b). The foremost example of this approach is the Secure
Sockets Layer (SSL) and the follow-on Internet standard known as Transport
Layer Security (TLS). At this level, there are two implementation choices. For full
generality, SSL (or TLS) could be provided as part of the underlying protocol suite
and therefore be transparent to applications. Alternatively, TLS can be embedded
in specific packages. For example, virtually all browsers come equipped with TLS,
and most Web servers have implemented the protocol.
Application-specific security services are embedded within the particular
application. Figure 17.1c shows examples of this architecture. The advantage of this
approach is that the service can be tailored to the specific needs of a given application.
17.2 TRANSPORT LAYER SECURITY
One of the most widely used security services is Transport Layer Security (TSL); the current version is Version 1.2, defined in RFC 5246. TLS is an Internet stan-
dard that evolved from a commercial protocol known as Secure Sockets Layer (SSL). Although SSL implementations are still around, it has been deprecated by IETF and is disabled by most corporations offering TLS software. TLS is a general-
purpose service implemented as a set of protocols that rely on TCP. At this level,
there are two implementation choices. For full generality, TLS could be provided
as part of the underlying protocol suite and therefore be transparent to applica-
tions. Alternatively, TLS can be embedded in specific packages. For example, most
browsers come equipped with TLS, and most Web servers have implemented the
protocol.
TLS Architecture
TLS is designed to make use of TCP to provide a reliable end-to-end secure ser-
vice. TLS is not a single protocol but rather two layers of protocols, as illustrated in
Figure 17.2.
Figure 17.1 Relative Location of Security Facilities in the TCP/IP Protocol Stack
SMTPHTTP
TCP
IP/IPSec
(a) Network level
FTP
SMTPHTTP
TCP
SSL or TLS
IP
(b) Transport level
FTP
IP
S/MIME
HTTPKerberos
UDP
SMTP
(c) Application level
TCP
550 CHAPTER 17 / TRANSPORT-LEVEL SECURITY
The TLS Record Protocol provides basic security services to various higher-
layer protocols. In particular, the Hypertext Transfer Protocol (HTTP), which provides the transfer service for Web client/server interaction, can operate on top
of TLS. Three higher-layer protocols are defined as part of TLS: the Handshake
Protocol; the Change Cipher Spec Protocol; and the Alert Protocol. These TLS-
specific protocols are used in the management of TLS exchanges and are examined
later in this section. A fourth protocol, the Heartbeat Protocol, is defined in a sepa-
rate RFC and is also discussed subsequently in this section.
Two important TLS concepts are the TLS session and the TLS connection,
which are defined in the specification as follows:
■ Connection: A connection is a transport (in the OSI layering model definition) that provides a suitable type of service. For TLS, such connections are peer-to-
peer relationships. The connections are transient. Every connection is associ-
ated with one session.
■ Session: A TLS session is an association between a client and a server. Sessions are created by the Handshake Protocol. Sessions define a set of cryptographic
security parameters, which can be shared among multiple connections. Sessions
are used to avoid the expensive negotiation of new security parameters for
each connection.
Between any pair of parties (applications such as HTTP on client and server),
there may be multiple secure connections. In theory, there may also be multiple
simultaneous sessions between parties, but this feature is not used in practice.
There are a number of states associated with each session. Once a session is
established, there is a current operating state for both read and write (i.e., receive
and send). In addition, during the Handshake Protocol, pending read and write
states are created. Upon successful conclusion of the Handshake Protocol, the
pending states become the current states.
A session state is defined by the following parameters:
■ Session identifier: An arbitrary byte sequence chosen by the server to identify an active or resumable session state.
■ Peer certificate: An X509.v3 certificate of the peer. This element of the state may be null.
Figure 17.2 TLS Protocol Stack
IP
TCP
Record protocol
Handshake protocol
Change cipher spec
protocol
Alert protocol
HTTP Heartbeat protocol
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17.2 / TRANSPORT LAYER SECURITY 551
■ Compression method: The algorithm used to compress data prior to encryption.
■ Cipher spec: Specifies the bulk data encryption algorithm (such as null, AES, etc.) and a hash algorithm (such as MD5 or SHA-1) used for MAC calculation.
It also defines cryptographic attributes such as the hash_size.
■ Master secret: 48-byte secret shared between the client and server.
■ Is resumable: A flag indicating whether the session can be used to initiate new connections.
A connection state is defined by the following parameters:
■ Server and client random: Byte sequences that are chosen by the server and client for each connection.
■ Server write MAC secret: The secret key used in MAC operations on data sent by the server.
■ Client write MAC secret: The symmetric key used in MAC operations on data sent by the client.
■ Server write key: The symmetric encryption key for data encrypted by the server and decrypted by the client.
■ Client write key: The symmetric encryption key for data encrypted by the client and decrypted by the server.
■ Initialization vectors: When a block cipher in CBC mode is used, an initial- ization vector (IV) is maintained for each key. This field is first initialized by
the TLS Handshake Protocol. Thereafter, the final ciphertext block from each
record is preserved for use as the IV with the following record.
■ Sequence numbers: Each party maintains separate sequence numbers for transmitted and received messages for each connection. When a party sends or
receives a “change cipher spec message,” the appropriate sequence number is
set to zero. Sequence numbers may not exceed 264 - 1.
TLS Record Protocol
The TLS Record Protocol provides two services for TLS connections:
■ Confidentiality: The Handshake Protocol defines a shared secret key that is used for conventional encryption of TLS payloads.
■ Message Integrity: The Handshake Protocol also defines a shared secret key that is used to form a message authentication code (MAC).
Figure 17.3 indicates the overall operation of the TLS Record Protocol. The
Record Protocol takes an application message to be transmitted, fragments the data
into manageable blocks, optionally compresses the data, applies a MAC, encrypts,
adds a header, and transmits the resulting unit in a TCP segment. Received data
are decrypted, verified, decompressed, and reassembled before being delivered to
higher-level users.
The first step is fragmentation. Each upper-layer message is fragmented into blocks of 214 bytes (16,384 bytes) or less. Next, compression is optionally applied. Compression must be lossless and may not increase the content length by more than
552 CHAPTER 17 / TRANSPORT-LEVEL SECURITY
1024 bytes.1 In TLSv2, no compression algorithm is specified, so the default com-
pression algorithm is null.
The next step in processing is to compute a message authentication code over the compressed data. TLS makes use of the HMAC algorithm defined in RFC 2104.
Recall from Chapter 12 that HMAC is defined as
HMACK(M) = H[(K + ⊕ opad) ‘ H[(K+ ⊕ ipad) ‘ M]]
where
H = embedded hash function (for TLS, either MD5 or SHA-1) M = message input to HMAC K+ = secret key padded with zeros on the left so that the result is equal to
the block length of the hash code (for MD5 and SHA-1, block
length = 512 bits) ipad = 00110110 (36 in hexadecimal) repeated 64 times (512 bits) opad = 01011100 (5C in hexadecimal) repeated 64 times (512 bits)
For TLS, the MAC calculation encompasses the fields indicated in the
following expression:
HMAC_hash(MAC_write_secret, seq_num ‘ TLSCompressed.type ‘ TLSCompressed.version ‘ TLSCompressed.length ‘ TLSCompressed.fragment)
The MAC calculation covers all of the fields XXX, plus the field
TLSCompressed.version, which is the version of the protocol being employed. Next, the compressed message plus the MAC are encrypted using symmetric
encryption. Encryption may not increase the content length by more than 1024 bytes,
Figure 17.3 TLS Record Protocol Operation
Application data
Fragment
Compress
Add MAC
Encrypt
Append TLS record header
1Of course, one hopes that compression shrinks rather than expands the data. However, for very short blocks, it is possible, because of formatting conventions, that the compression algorithm will actually pro- vide output that is longer than the input.
17.2 / TRANSPORT LAYER SECURITY 553
so that the total length may not exceed 214 + 2048. The following encryption algo- rithms are permitted:
Block Cipher Stream Cipher
Algorithm Key Size Algorithm Key Size
AES
3DES
128, 256
168
RC4-128 128
For stream encryption, the compressed message plus the MAC are encrypted.
Note that the MAC is computed before encryption takes place and that the MAC is
then encrypted along with the plaintext or compressed plaintext.
For block encryption, padding may be added after the MAC prior to encryp-
tion. The padding is in the form of a number of padding bytes followed by a one-
byte indication of the length of the padding. The padding can be any amount that
results in a total that is a multiple of the cipher’s block length, up to a maximum
of 255 bytes. For example, if the cipher block length is 16 bytes (e.g., AES) and if
the plaintext (or compressed text if compression is used) plus MAC plus padding
length byte is 79 bytes long, then the padding length (in bytes) can be 1, 17, 33, and
so on, up to 161. At a padding length of 161, the total length is 79 + 161 = 240. A variable padding length may be used to frustrate attacks based on an analysis of
the lengths of exchanged messages.
The final step of TLS Record Protocol processing is to prepend a header con-
sisting of the following fields:
■ Content Type (8 bits): The higher-layer protocol used to process the enclosed fragment.
■ Major Version (8 bits): Indicates major version of TLS in use. For TLSv2, the value is 3.
■ Minor Version (8 bits): Indicates minor version in use. For TLSv2, the value is 1.
■ Compressed Length (16 bits): The length in bytes of the plaintext fragment (or compressed fragment if compression is used). The maximum value is
214 + 2048.
The content types that have been defined are change_cipher_spec, alert, handshake, and application_data. The first three are the TLS- specific protocols, discussed next. Note that no distinction is made among the vari-
ous applications (e.g., HTTP) that might use TLS; the content of the data created by
such applications is opaque to TLS.
Figure 17.4 illustrates the TLS record format.
Change Cipher Spec Protocol
The Change Cipher Spec Protocol is one of the four TLS-specific protocols that use
the TLS Record Protocol, and it is the simplest. This protocol consists of a single
message (Figure 17.5a), which consists of a single byte with the value 1. The sole
purpose of this message is to cause the pending state to be copied into the current
state, which updates the cipher suite to be used on this connection.
554 CHAPTER 17 / TRANSPORT-LEVEL SECURITY
Alert Protocol
The Alert Protocol is used to convey TLS-related alerts to the peer entity. As with
other applications that use TLS, alert messages are compressed and encrypted, as
specified by the current state.
Each message in this protocol consists of two bytes (Figure 17.5b). The first
byte takes the value warning (1) or fatal (2) to convey the severity of the message.
If the level is fatal, TLS immediately terminates the connection. Other connections
on the same session may continue, but no new connections on this session may
be established. The second byte contains a code that indicates the specific alert.
The following alerts are always fatal:
■ unexpected_message: An inappropriate message was received.
■ bad_record_mac: An incorrect MAC was received.
■ decompression_failure: The decompression function received improper input (e.g., unable to decompress or decompress to greater than maximum allowable
length).
■ handshake_failure: Sender was unable to negotiate an acceptable set of secu- rity parameters given the options available.
■ illegal_parameter: A field in a handshake message was out of range or incon- sistent with other fields.
Figure 17.5 TLS Record Protocol Payload
1
(a) Change Cipher Spec Protocol
1 byte
Type
(c) Handshake Protocol
1 byte
Length
3 bytes
Content
Ú 0 bytes
(d) Other Upper-Layer Protocol (e.g., HTTP)
Opaque content
Ú 1 byte
Level
(b) Alert Protocol
1 byte 1 byte
Alert
Figure 17.4 TLS Record Format
Content type
Major version
Minor version
Compressed length
Plaintext (optionally
compressed)
MAC (0, 16, or 20 bytes)
E nc
ry pt
ed
17.2 / TRANSPORT LAYER SECURITY 555
■ decryption_failed: A ciphertext decrypted in an invalid way; either it was not an even multiple of the block length or its padding values, when checked, were
incorrect.
■ record_overflow: A TLS record was received with a payload (ciphertext) whose length exceeds 214 + 2048 bytes, or the ciphertext decrypted to a length of greater than 214 + 1024 bytes.
■ unknown_ca: A valid certificate chain or partial chain was received, but the certificate was not accepted because the CA certificate could not be located or
could not be matched with a known, trusted CA.
■ access_denied: A valid certificate was received, but when access control was applied, the sender decided not to proceed with the negotiation.
■ decode_error: A message could not be decoded, because either a field was out of its specified range or the length of the message was incorrect.
■ export_restriction: A negotiation not in compliance with export restrictions on key length was detected.
■ protocol_version: The protocol version the client attempted to negotiate is recognized but not supported.
■ insufficient_security: Returned instead of handshake_failure when a negotia- tion has failed specifically because the server requires ciphers more secure
than those supported by the client.
■ internal_error: An internal error unrelated to the peer or the correctness of the protocol makes it impossible to continue.
The remaining alerts are the following.
■ close_notify: Notifies the recipient that the sender will not send any more mes- sages on this connection. Each party is required to send a close_notify alert
before closing the write side of a connection.
■ bad_certificate: A received certificate was corrupt (e.g., contained a signature that did not verify).
■ unsupported_certificate: The type of the received certificate is not supported.
■ certificate_revoked: A certificate has been revoked by its signer.
■ certificate_expired: A certificate has expired.
■ certificate_unknown: Some other unspecified issue arose in processing the certificate, rendering it unacceptable.
■ decrypt_error: A handshake cryptographic operation failed, including being unable to verify a signature, decrypt a key exchange, or validate a finished
message.
■ user_canceled: This handshake is being canceled for some reason unrelated to a protocol failure.
■ no_renegotiation: Sent by a client in response to a hello request or by the server in response to a client hello after initial handshaking. Either of these
messages would normally result in renegotiation, but this alert indicates that
the sender is not able to renegotiate. This message is always a warning.
556 CHAPTER 17 / TRANSPORT-LEVEL SECURITY
Handshake Protocol
The most complex part of TLS is the Handshake Protocol. This protocol allows the server and client to authenticate each other and to negotiate an encryption and
MAC algorithm and cryptographic keys to be used to protect data sent in a TLS
record. The Handshake Protocol is used before any application data is transmitted.
The Handshake Protocol consists of a series of messages exchanged by client
and server. All of these have the format shown in Figure 17.5c . Each message has
three fields:
■ Type (1 byte): Indicates one of 10 messages. Table 17.2 lists the defined mes- sage types.
■ Length (3 bytes): The length of the message in bytes.
■ Content (# 0 bytes): The parameters associated with this message; these are listed in Table 17.2.
Figure 17.6 shows the initial exchange needed to establish a logical connection
between client and server. The exchange can be viewed as having four phases.
PHASE 1. ESTABLISH SECURITY CAPABILITIES Phase 1 initiates a logical connection and establishes the security capabilities that will be associated with it. The exchange
is initiated by the client, which sends a client_hello message with the following parameters:
■ Version: The highest TLS version understood by the client.
■ Random: A client-generated random structure consisting of a 32-bit time- stamp and 28 bytes generated by a secure random number generator. These
values serve as nonces and are used during key exchange to prevent replay
attacks.
■ Session ID: A variable-length session identifier. A nonzero value indicates that the client wishes to update the parameters of an existing connection or to cre-
ate a new connection on this session. A zero value indicates that the client
wishes to establish a new connection on a new session.
Message Type Parameters
hello_request null
client_hello version, random, session id, cipher suite, compression method
server_hello version, random, session id, cipher suite, compression method
certificate chain of X.509v3 certificates
server_key_exchange parameters, signature
certificate_request type, authorities
server_done null
certificate_verify signature
client_key_exchange parameters, signature
finished hash value
Table 17.2 TLS Handshake Protocol Message Types
17.2 / TRANSPORT LAYER SECURITY 557
■ CipherSuite: This is a list that contains the combinations of cryptographic algorithms supported by the client, in decreasing order of preference. Each
element of the list (each cipher suite) defines both a key exchange algorithm
and a CipherSpec; these are discussed subsequently.
■ Compression Method: This is a list of the compression methods the client supports.
After sending the client_hello message, the client waits for the server_ hello message, which contains the same parameters as the client_hello
Figure 17.6 Handshake Protocol Action
Client Server
Phase 1 Establish security capabilities, including protocol version, session ID, cipher suite, compression method, and initial random numbers.
Phase 2 Server may send certificate, key exchange, and request certificate. Server signals end of hello message phase.
Phase 3 Client sends certificate if requested. Client sends key exchange. Client may send certificate verification.
Phase 4 Change cipher suite and finish handshake protocol.
Note: Shaded transfers are optional or situation-dependent messages that are not always sent.
finished
change_c ipher_sp
ec
finished
change_cipher_spec
certificate_verify
client_key_exchange
certificate
server_he llo_done
certificat e_reques
t
server_ke y_exchan
ge
certificat e
server_he llo
client_hello T
im e
558 CHAPTER 17 / TRANSPORT-LEVEL SECURITY
message. For the server_hello message, the following conventions apply. The Version field contains the lowest of the version suggested by the client and the highest
supported by the server. The Random field is generated by the server and is indepen-
dent of the client’s Random field. If the SessionID field of the client was nonzero, the
same value is used by the server; otherwise the server’s SessionID field contains the
value for a new session. The CipherSuite field contains the single cipher suite selected
by the server from those proposed by the client. The Compression field contains the
compression method selected by the server from those proposed by the client.
The first element of the Ciphersuite parameter is the key exchange method
(i.e., the means by which the cryptographic keys for conventional encryption and
MAC are exchanged). The following key exchange methods are supported.
■ RSA: The secret key is encrypted with the receiver’s RSA public key. A public- key certificate for the receiver’s key must be made available.
■ Fixed Diffie–Hellman: This is a Diffie–Hellman key exchange in which the server’s certificate contains the Diffie–Hellman public parameters signed by
the certificate authority (CA). That is, the public-key certificate contains the
Diffie–Hellman public-key parameters. The client provides its Diffie–Hellman
public-key parameters either in a certificate, if client authentication is re-
quired, or in a key exchange message. This method results in a fixed secret key
between two peers based on the Diffie–Hellman calculation using the fixed
public keys.
■ Ephemeral Diffie-Hellman: This technique is used to create ephemeral (tem- porary, one-time) secret keys. In this case, the Diffie–Hellman public keys are
exchanged and signed using the sender’s private RSA or DSS key. The receiver
can use the corresponding public key to verify the signature. Certificates are used
to authenticate the public keys. This would appear to be the most secure of the
three Diffie–Hellman options because it results in a temporary, authenticated key.
■ Anonymous Diffie–Hellman: The base Diffie–Hellman algorithm is used with no authentication. That is, each side sends its public Diffie–Hellman pa-
rameters to the other with no authentication. This approach is vulnerable to
man-in-the-middle attacks, in which the attacker conducts anonymous Diffie–
Hellman with both parties.
Following the definition of a key exchange method is the CipherSpec, which
includes the following fields:
■ CipherAlgorithm: Any of the algorithms mentioned earlier: RC4, RC2, DES, 3DES, DES40, or IDEA
■ MACAlgorithm: MD5 or SHA-1
■ CipherType: Stream or Block
■ IsExportable: True or False
■ HashSize: 0, 16 (for MD5), or 20 (for SHA-1) bytes
■ Key Material: A sequence of bytes that contain data used in generating the write keys
■ IV Size: The size of the Initialization Value for Cipher Block Chaining (CBC) encryption
17.2 / TRANSPORT LAYER SECURITY 559
PHASE 2. SERVER AUTHENTICATION AND KEY EXCHANGE The server begins this phase by sending its certificate if it needs to be authenticated; the message con-
tains one or a chain of X.509 certificates. The certificate message is required for any agreed-on key exchange method except anonymous Diffie–Hellman. Note
that if fixed Diffie–Hellman is used, this certificate message functions as the serv-
er’s key exchange message because it contains the server’s public Diffie–Hellman
parameters.
Next, a server_key_exchange message may be sent if it is required. It is not required in two instances: (1) The server has sent a certificate with fixed Diffie–
Hellman parameters; or (2) RSA key exchange is to be used. The server_key_
exchange message is needed for the following:
■ Anonymous Diffie–Hellman: The message content consists of the two global Diffie–Hellman values (a prime number and a primitive root of that number)
plus the server’s public Diffie–Hellman key (see Figure 10.1).
■ Ephemeral Diffie–Hellman: The message content includes the three Diffie– Hellman parameters provided for anonymous Diffie–Hellman plus a signature
of those parameters.
■ RSA key exchange (in which the server is using RSA but has a signature-only RSA key): Accordingly, the client cannot simply send a secret key encrypted with the server’s public key. Instead, the server must create a temporary RSA
public/private key pair and use the server_key_exchange message to send the
public key. The message content includes the two parameters of the temporary
RSA public key (exponent and modulus; see Figure 9.5) plus a signature of
those parameters.
Some further details about the signatures are warranted. As usual, a signature
is created by taking the hash of a message and encrypting it with the sender’s private
key. In this case, the hash is defined as
hash(ClientHello.random ‘ ServerHello.random ‘ ServerParams)
So the hash covers not only the Diffie–Hellman or RSA parameters but also the
two nonces from the initial hello messages. This ensures against replay attacks and
misrepresentation. In the case of a DSS signature, the hash is performed using the
SHA-1 algorithm. In the case of an RSA signature, both an MD5 and an SHA-1
hash are calculated, and the concatenation of the two hashes (36 bytes) is encrypted
with the server’s private key.
Next, a nonanonymous server (server not using anonymous Diffie–Hellman)
can request a certificate from the client. The certificate_request message includes two parameters: certificate_type and certificate_authorities. The certificate type in-
dicates the public-key algorithm and its use:
■ RSA, signature only
■ DSS, signature only
■ RSA for fixed Diffie–Hellman; in this case the signature is used only for
authentication, by sending a certificate signed with RSA
■ DSS for fixed Diffie–Hellman; again, used only for authentication
Hiva-Network.Com
560 CHAPTER 17 / TRANSPORT-LEVEL SECURITY
The second parameter in the certificate_request message is a list of the distin-
guished names of acceptable certificate authorities.
The final message in phase 2, and one that is always required, is the server_ done message, which is sent by the server to indicate the end of the server hello and associated messages. After sending this message, the server will wait for a client
response. This message has no parameters.
PHASE 3. CLIENT AUTHENTICATION AND KEY EXCHANGE Upon receipt of the server_done message, the client should verify that the server provided a valid certificate (if required) and check that the server_hello parameters are accept- able. If all is satisfactory, the client sends one or more messages back to the server.
If the server has requested a certificate, the client begins this phase by send-
ing a certificate message. If no suitable certificate is available, the client sends a no_certificate alert instead.
Next is the client_key_exchange message, which must be sent in this phase. The content of the message depends on the type of key exchange, as follows:
■ RSA: The client generates a 48-byte pre-master secret and encrypts with the public key from the server’s certificate or temporary RSA key from a server_
key_exchange message. Its use to compute a master secret is explained later.
■ Ephemeral or Anonymous Diffie–Hellman: The client’s public Diffie–Hellman parameters are sent.
■ Fixed Diffie–Hellman: The client’s public Diffie–Hellman parameters were sent in a certificate message, so the content of this message is null.
Finally, in this phase, the client may send a certificate_verify message to pro- vide explicit verification of a client certificate. This message is only sent following
any client certificate that has signing capability (i.e., all certificates except those
containing fixed Diffie–Hellman parameters). This message signs a hash code based
on the preceding messages, defined as
CertificateVerify.signature.md5_hash
MD5(handshake_messages);
Certificate.signature.sha_hash
SHA(handshake_messages);
where handshake_messages refers to all Handshake Protocol messages sent or received starting at client_hello but not including this message. If the user’s private key is DSS, then it is used to encrypt the SHA-1 hash. If the user’s private
key is RSA, it is used to encrypt the concatenation of the MD5 and SHA-1 hashes.
In either case, the purpose is to verify the client’s ownership of the private key for
the client certificate. Even if someone is misusing the client’s certificate, he or she
would be unable to send this message.
PHASE 4. FINISH Phase 4 completes the setting up of a secure connection. The client sends a change_cipher_spec message and copies the pending CipherSpec into the current CipherSpec. Note that this message is not considered part of the Handshake
Protocol but is sent using the Change Cipher Spec Protocol. The client then imme-
diately sends the finished message under the new algorithms, keys, and secrets.
17.2 / TRANSPORT LAYER SECURITY 561
The finished message verifies that the key exchange and authentication processes
were successful. The content of the finished message is:
PRF(master_secret, finished_label, MD5(handshake_messages) ‘ SHA@1 (handshake_messages))
where finished_label is the string “client finished” for the client and “server finished” for the server.
In response to these two messages, the server sends its own change_ cipher_ spec message, transfers the pending to the current CipherSpec, and sends its fin- ished message. At this point, the handshake is complete and the client and server
may begin to exchange application-layer data.
Cryptographic Computations
Two further items are of interest: (1) the creation of a shared master secret by
means of the key exchange; and (2) the generation of cryptographic parameters
from the master secret.
MASTER SECRET CREATION The shared master secret is a one-time 48-byte value (384 bits) generated for this session by means of secure key exchange. The creation
is in two stages. First, a pre_master_secret is exchanged. Second, the master_ secret is calculated by both parties. For pre_master_secret exchange, there are two possibilities.
■ RSA: A 48-byte pre_master_secret is generated by the client, encrypted with the server’s public RSA key, and sent to the server. The server decrypts the
ciphertext using its private key to recover the pre_master_secret.
■ Diffie–Hellman: Both client and server generate a Diffie–Hellman public key. After these are exchanged, each side performs the Diffie–Hellman calculation
to create the shared pre_master_secret.
Both sides now compute the master_secret as
master_secret = PRF(pre_master_secret, “master secret”, ClientHello.random ‘ ServerHello .random)
where ClientHello.random and ServerHello.random are the two nonce values exchanged in the initial hello messages.
The algorithm is performed until 48 bytes of pseudorandom output are pro-
duced. The calculation of the key block material (MAC secret keys, session encryp-
tion keys, and IVs) is defined as
key_block = PRF(SecurityParameters.master_secret, “key expansion”,
SecurityParameters.server_random ‘ SecurityParameters.client_random)
until enough output has been generated.
562 CHAPTER 17 / TRANSPORT-LEVEL SECURITY
GENERATION OF CRYPTOGRAPHIC PARAMETERS CipherSpecs require a client write MAC secret, a server write MAC secret, a client write key, a server write key, a
client write IV, and a server write IV, which are generated from the master secret
in that order. These parameters are generated from the master secret by hashing
the master secret into a sequence of secure bytes of sufficient length for all needed
parameters.
The generation of the key material from the master secret uses the same for-
mat for generation of the master secret from the pre-master secret as
key_block = MD5(master_secret ‘ SHA(=A> ‘ master_secret ‘ ServerHello.random ‘ ClientHello.random)) ‘
MD5(master_secret ‘ SHA(=BB> ‘ master_secret ‘ ServerHello.random ‘ ClientHello.random)) ‘
MD5(master_secret ‘ SHA(=CCC> ‘ master_secret ‘ ServerHello.random ‘ ClientHello.random)) ‘ c
until enough output has been generated. The result of this algorithmic structure is a
pseudorandom function. We can view the master_secret as the pseudorandom seed value to the function. The client and server random numbers can be viewed as
salt values to complicate cryptanalysis (see Chapter 21 for a discussion of the use of
salt values).
PSEUDORANDOM FUNCTION TLS makes use of a pseudorandom function referred to as PRF to expand secrets into blocks of data for purposes of key generation or
validation. The objective is to make use of a relatively small, shared secret value but
to generate longer blocks of data in a way that is secure from the kinds of attacks
made on hash functions and MACs. The PRF is based on the data expansion func-
tion (Figure 17.7) given as
P_hash(secret, seed) = HMAC_hash(secret, A(1) ‘ seed) ‘ HMAC_hash(secret, A(2) ‘ seed) ‘ HMAC_hash(secret, A(3) ‘ seed) ‘
where A() is defined as
A(0) = seed A(i) = HMAC_hash(secret, A(i - 1))
The data expansion function makes use of the HMAC algorithm with either MD5
or SHA-1 as the underlying hash function. As can be seen, P_hash can be iterated as many times as necessary to produce the required quantity of data. For example, if
P_SHA256 was used to generate 80 bytes of data, it would have to be iterated three times (through A(3)), producing 96 bytes of data of which the last 16 would be dis-
carded. In this case, P_MD5 would have to be iterated four times, producing exactly 64 bytes of data. Note that each iteration involves two executions of HMAC, each
of which in turn involves two executions of the underlying hash algorithm.
17.2 / TRANSPORT LAYER SECURITY 563
To make PRF as secure as possible, it uses two hash algorithms in a way that
should guarantee its security if either algorithm remains secure. PRF is defined as
PRF(secret, label, seed) = P_ 6 hash 7 (secret, label ‘ seed)
PRF takes as input a secret value, an identifying label, and a seed value and
produces an output of arbitrary length.
Heartbeat Protocol
In the context of computer networks, a heartbeat is a periodic signal generated by
hardware or software to indicate normal operation or to synchronize other parts of
a system. A heartbeat protocol is typically used to monitor the availability of a pro-
tocol entity. In the specific case of TLS, a Heartbeat protocol was defined in 2012 in
RFC 6250 (Transport Layer Security (TLS) and Datagram Transport Layer Security (DTLS) Heartbeat Extension).
Figure 17.7 TLS Function P_hash(secret, seed)
Secret
Seed
Seed
A(1) HMAC
Secret
Secret
Length = hash size
Secret
Seed
A(2) HMAC
HMAC Secret
Seed
A(3) HMAC
HMAC
Secret HMAC
564 CHAPTER 17 / TRANSPORT-LEVEL SECURITY
The Heartbeat protocol runs on top of the TLS Record Protocol and con-
sists of two message types: heartbeat_request and heartbeat_response. The use of the Heartbeat protocol is established during Phase 1 of the Handshake
protocol (Figure 17.6). Each peer indicates whether it supports heartbeats. If heart-
beats are supported, the peer indicates whether it is willing to receive heartbeat_ request messages and respond with heartbeat_response messages or only willing to send heartbeat_request messages.
A heartbeat_request message can be sent at any time. Whenever a re- quest message is received, it should be answered promptly with a corresponding
heartbeat_response message. The heartbeat_request message includes payload length, payload, and padding fields. The payload is a random content
between 16 bytes and 64 Kbytes in length. The corresponding heartbeat_ response message must include an exact copy of the received payload. The pad- ding is also random content. The padding enables the sender to perform a path
MTU (maximum transfer unit) discovery operation, by sending requests with in-
creasing padding until there is no answer anymore, because one of the hosts on
the path cannot handle the message.
The heartbeat serves two purposes. First, it assures the sender that the recipi-
ent is still alive, even though there may not have been any activity over the under-
lying TCP connection for a while. Second, the heartbeat generates activity across
the connection during idle periods, which avoids closure by a firewall that does not
tolerate idle connections.
The requirement for the exchange of a payload was designed into the Heartbeat
protocol to support its use in a connectionless version of TLS known as Datagram
Transport Layer Security (DTLS). Because a connectionless service is subject
to packet loss, the payload enables the requestor to match response messages to
request messages. For simplicity, the same version of the Heartbeat protocol is used
with both TLS and DTLS. Thus, the payload is required for both TLS and DTLS.
SSL/TLS ATTACKS
Since the first introduction of SSL in 1994, and the subsequent standardization of
TLS, numerous attacks have been devised against these protocols. The appearance
of each attack has necessitated changes in the protocol, the encryption tools used, or
some aspect of the implementation of SSL and TLS to counter these threats.
ATTACK CATEGORIES We can group the attacks into four general categories:
■ Attacks on the handshake protocol: As early as 1998, an approach to com- promising the handshake protocol based on exploiting the formatting and
implementation of the RSA encryption scheme was presented [BLEI98]. As
countermeasures were implemented the attack was refined and adjusted to not
only thwart the countermeasures but also speed up the attack [e.g., BARD12].
■ Attacks on the record and application data protocols: A number of vulnerabili- ties have been discovered in these protocols, leading to patches to counter the
new threats. As a recent example, in 2011, researchers Thai Duong and Juliano
Rizzo demonstrated a proof of concept called BEAST (Browser Exploit Against
SSL/TLS) that turned what had been considered only a theoretical vulnerability
17.2 / TRANSPORT LAYER SECURITY 565
into a practical attack [GOOD11]. BEAST leverages a type of cryptographic
attack called a chosen-plaintext attack. The attacker mounts the attack by
choosing a guess for the plaintext that is associated with a known ciphertext. The
researchers developed a practical algorithm for launching successful attacks.
Subsequent patches were able to thwart this attack. The authors of the BEAST
attack are also the creators of the 2012 CRIME (Compression Ratio Info-leak
Made Easy) attack, which can allow an attacker to recover the content of web
cookies when data compression is used along with TLS [GOOD12]. When used
to recover the content of secret authentication cookies, it allows an attacker to
perform session hijacking on an authenticated web session.
■ Attacks on the PKI: Checking the validity of X.509 certificates is an activity subject to a variety of attacks, both in the context of SSL/TLS and elsewhere.
For example, [GEOR12] demonstrated that commonly used libraries for
SSL/TLS suffer from vulnerable certificate validation implementations. The
authors revealed weaknesses in the source code of OpenSSL, GnuTLS, JSSE,
ApacheHttpClient, Weberknecht, cURL, PHP, Python and applications built
upon or with these products.
■ Other attacks: [MEYE13] lists a number of attacks that do not fit into any of the preceding categories. One example is an attack announced in 2011 by the
German hacker group The Hackers Choice, which is a DoS attack [KUMA11].
The attack creates a heavy processing load on a server by overwhelming the
target with SSL/TLS handshake requests. Boosting system load is done by
establishing new connections or using renegotiation. Assuming that the major-
ity of computation during a handshake is done by the server, the attack creates
more system load on the server than on the source device, leading to a DoS.
The server is forced to continuously recompute random numbers and keys.
The history of attacks and countermeasures for SSL/TLS is representative of
that for other Internet-based protocols. A “perfect” protocol and a “perfect” imple-
mentation strategy are never achieved. A constant back-and-forth between threats
and countermeasures determines the evolution of Internet-based protocols.
TLSv1.3
In 2014, the IETF TLS working group began work on a version 1.3 of TLS. The
primary aim is to improve the security of TLS. As of this writing, TLSv1.3 is still
in a draft stage, but the final standard is likely to be very close to the current draft.
Among the significant changes from version 1.2 are the following:
■ TLSv1.3 removes support for a number of options and functions. Remov-
ing code that implements functions no longer needed reduces the chances
of potentially dangerous coding errors and reduces the attack surface. The
deleted items include:
–Compression
–Ciphers that do not offer authenticated encryption
–Static RSA and DH key exchange
–32-bit timestamp as part of the Random parameter in the client_hello
message
566 CHAPTER 17 / TRANSPORT-LEVEL SECURITY
–Renegotiation
–Change Cipher Spec Protocol
–RC4
–Use of MD5 and SHA-224 hashes with signatures
■ TLSv1.3 uses Diffie–Hellman or Elliptic Curve Diffie–Hellman for key
exchange and does not permit RSA. The danger with RSA is that if the private
key is compromised, all handshakes using these cipher suites will be compro-
mised. With DH or ECDH, a new key is negotiated for each handshake.
■ TLSv1.3 allows for a “1 round trip time” handshake by changing the order of
message sent with establishing a secure connection. The client sends a Client
Key Exchange message containing its cryptographic parameters for key estab-
lishment before a cipher suite has been negotiated. This enables a server
to calculate keys for encryption and authentication before sending its first
response. Reducing the number of packets sent during this handshake phase
speeds up the process and reduces the attack surface.
These changes should improve the efficiency and security of TLS.
17.3 HTTPS
HTTPS (HTTP over SSL) refers to the combination of HTTP and SSL to imple-
ment secure communication between a Web browser and a Web server. The HTTPS
capability is built into all modern Web browsers. Its use depends on the Web server
supporting HTTPS communication. For example, some search engines do not sup-
port HTTPS.
The principal difference seen by a user of a Web browser is that URL (uniform
resource locator) addresses begin with https:// rather than http://. A normal HTTP
connection uses port 80. If HTTPS is specified, port 443 is used, which invokes SSL.
When HTTPS is used, the following elements of the communication are
encrypted:
■ URL of the requested document
■ Contents of the document
■ Contents of browser forms (filled in by browser user)
■ Cookies sent from browser to server and from server to browser
■ Contents of HTTP header
HTTPS is documented in RFC 2818, HTTP Over TLS. There is no fundamen- tal change in using HTTP over either SSL or TLS, and both implementations are
referred to as HTTPS.
Connection Initiation
For HTTPS, the agent acting as the HTTP client also acts as the TLS client. The
client initiates a connection to the server on the appropriate port and then sends
the TLS ClientHello to begin the TLS handshake. When the TLS handshake has
17.4 / SECURE SHELL (SSH) 567
finished, the client may then initiate the first HTTP request. All HTTP data is to be
sent as TLS application data. Normal HTTP behavior, including retained connec-
tions, should be followed.
There are three levels of awareness of a connection in HTTPS. At the HTTP
level, an HTTP client requests a connection to an HTTP server by sending a con-
nection request to the next lowest layer. Typically, the next lowest layer is TCP,
but it also may be TLS/SSL. At the level of TLS, a session is established between a
TLS client and a TLS server. This session can support one or more connections at
any time. As we have seen, a TLS request to establish a connection begins with the
establishment of a TCP connection between the TCP entity on the client side and
the TCP entity on the server side.
Connection Closure
An HTTP client or server can indicate the closing of a connection by including the
following line in an HTTP record: Connection: close. This indicates that the connection will be closed after this record is delivered.
The closure of an HTTPS connection requires that TLS close the connec-
tion with the peer TLS entity on the remote side, which will involve closing the
underlying TCP connection. At the TLS level, the proper way to close a connec-
tion is for each side to use the TLS alert protocol to send a close_notify alert. TLS implementations must initiate an exchange of closure alerts before closing a
connection. A TLS implementation may, after sending a closure alert, close the
connection without waiting for the peer to send its closure alert, generating an
“incomplete close”. Note that an implementation that does this may choose to
reuse the session. This should only be done when the application knows (typically
through detecting HTTP message boundaries) that it has received all the message
data that it cares about.
HTTP clients also must be able to cope with a situation in which the underlying
TCP connection is terminated without a prior close_notify alert and without a Connection: close indicator. Such a situation could be due to a programming error on the server or a communication error that causes the TCP connection to drop.
However, the unannounced TCP closure could be evidence of some sort of attack. So
the HTTPS client should issue some sort of security warning when this occurs.
17.4 SECURE SHELL (SSH)
Secure Shell (SSH) is a protocol for secure network communications designed to
be relatively simple and inexpensive to implement. The initial version, SSH1 was
focused on providing a secure remote logon facility to replace TELNET and other
remote logon schemes that provided no security. SSH also provides a more general
client/server capability and can be used for such network functions as file transfer and
email. A new version, SSH2, fixes a number of security flaws in the original scheme.
SSH2 is documented as a proposed standard in IETF RFCs 4250 through 4256.
SSH client and server applications are widely available for most operating
systems. It has become the method of choice for remote login and X tunneling and
568 CHAPTER 17 / TRANSPORT-LEVEL SECURITY
is rapidly becoming one of the most pervasive applications for encryption technol-
ogy outside of embedded systems.
SSH is organized as three protocols that typically run on top of TCP
(Figure 17.8):
■ Transport Layer Protocol: Provides server authentication, data confidentiality, and data integrity with forward secrecy (i.e., if a key is compromised during
one session, the knowledge does not affect the security of earlier sessions). The
transport layer may optionally provide compression.
■ User Authentication Protocol: Authenticates the user to the server.
■ Connection Protocol: Multiplexes multiple logical communications channels over a single, underlying SSH connection.
Transport Layer Protocol
HOST KEYS Server authentication occurs at the transport layer, based on the server possessing a public/private key pair. A server may have multiple host keys using
multiple different asymmetric encryption algorithms. Multiple hosts may share
the same host key. In any case, the server host key is used during key exchange to
authenticate the identity of the host. For this to be possible, the client must have a
priori knowledge of the server’s public host key. RFC 4251 dictates two alternative
trust models that can be used:
1. The client has a local database that associates each host name (as typed by the user) with the corresponding public host key. This method requires no centrally
administered infrastructure and no third-party coordination. The downside is that
the database of name-to-key associations may become burdensome to maintain.
Figure 17.8 SSH Protocol Stack
SSH User Authentication Protocol
SSH Transport Layer Protocol
TCP
IP Internet protocol provides datagram delivery across multiple networks.
Transmission control protocol provides reliable, connection- oriented end-to-end delivery.
Provides server authentication, confidentiality, and integrity. It may optionally also provide compression.
Authenticates the client-side user to the server.
SSH Connection Protocol
Multiplexes the encrypted tunnel into several logical channels.
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17.4 / SECURE SHELL (SSH) 569
2. The host name-to-key association is certified by a trusted certification author- ity (CA). The client only knows the CA root key and can verify the validity of
all host keys certified by accepted CAs. This alternative eases the maintenance
problem, since ideally, only a single CA key needs to be securely stored on the
client. On the other hand, each host key must be appropriately certified by a
central authority before authorization is possible.
PACKET EXCHANGE Figure 17.9 illustrates the sequence of events in the SSH Transport Layer Protocol. First, the client establishes a TCP connection to the
server. This is done via the TCP protocol and is not part of the Transport Layer
Protocol. Once the connection is established, the client and server exchange data,
referred to as packets, in the data field of a TCP segment. Each packet is in the
following format (Figure 17.10).
■ Packet length: Length of the packet in bytes, not including the packet length and MAC fields.
■ Padding length: Length of the random padding field.
■ Payload: Useful contents of the packet. Prior to algorithm negotiation, this field is uncompressed. If compression is negotiated, then in subsequent
packets, this field is compressed.
Figure 17.9 SSH Transport Layer Protocol Packet Exchanges
Client Server
SSH-protoversion-softwareversion Identification string
exchange
Algorithm negotiation
End of key exchange
Service request
SSH-protoversion-softwareversion
SSH_MSG_KEXINIT
SSH_MSG_KEXINIT
SSH_MSG_NEWKEYS
SSH_MSG_NEWKEYS
SSH_MSG_SERVICE_REQUEST
Establish TCP Connection
Key Exchange
570 CHAPTER 17 / TRANSPORT-LEVEL SECURITY
■ Random padding: Once an encryption algorithm has been negotiated, this field is added. It contains random bytes of padding so that the total length of
the packet (excluding the MAC field) is a multiple of the cipher block size, or
8 bytes for a stream cipher.
■ Message authentication code (MAC): If message authentication has been negotiated, this field contains the MAC value. The MAC value is computed
over the entire packet plus a sequence number, excluding the MAC field. The
sequence number is an implicit 32-bit packet sequence that is initialized to
zero for the first packet and incremented for every packet. The sequence num-
ber is not included in the packet sent over the TCP connection.
Once an encryption algorithm has been negotiated, the entire packet
(excluding the MAC field) is encrypted after the MAC value is calculated.
The SSH Transport Layer packet exchange consists of a sequence of steps
(Figure 17.9). The first step, the identification string exchange, begins with the client sending a packet with an identification string of the form:
SSH-protoversion-softwareversion SP comments CR LF
Figure 17.10 SSH Transport Layer Protocol Packet Formation
pdlpktl
pktl = packet length pdl = padding length
gniddaP# qes
Payload
SSH Packet
Compressed payload
Ciphertext
COMPRESS
ENCRYPT MAC
17.4 / SECURE SHELL (SSH) 571
where SP,CR, and LF are space character, carriage return, and line feed, respec- tively. An example of a valid string is SSH-2.0-billsSSH_3.6.3q3<CR><LF>. The server responds with its own identification string. These strings are used in the
Diffie–Hellman key exchange.
Next comes algorithm negotiation. Each side sends an SSH_MSG_KEXINIT containing lists of supported algorithms in the order of preference to the sender.
There is one list for each type of cryptographic algorithm. The algorithms include
key exchange, encryption, MAC algorithm, and compression algorithm. Table 17.3
shows the allowable options for encryption, MAC, and compression. For each cat-
egory, the algorithm chosen is the first algorithm on the client’s list that is also sup-
ported by the server.
The next step is key exchange. The specification allows for alternative meth- ods of key exchange, but at present, only two versions of Diffie–Hellman key
exchange are specified. Both versions are defined in RFC 2409 and require only one
packet in each direction. The following steps are involved in the exchange. In this,
C is the client; S is the server; p is a large safe prime; g is a generator for a subgroup of GF(p); q is the order of the subgroup; V_S is S’s identification string; V_C is
Table 17.3 SSH Transport Layer Cryptographic Algorithms
MAC algorithm
hmac-sha1* HMAC-SHA1; digest length = key length = 20
hmac-sha1-96** First 96 bits of HMAC- SHA1; digest length = 12; key length = 20
hmac-md5 HMAC-MD5; digest length = key length = 16
hmac-md5-96 First 96 bits of HMAC-MD5;
digest length = 12; key length = 16
Compression algorithm
none* No compression
zlib Defined in RFC 1950 and RFC 1951
Cipher
3des-cbc* Three-key 3DES in CBC mode
blowfish-cbc Blowfish in CBC mode
twofish256-cbc Twofish in CBC mode with a 256-bit key
twofish192-cbc Twofish with a 192-bit key
twofish128-cbc Twofish with a 128-bit key
aes256-cbc AES in CBC mode with a 256-bit key
aes192-cbc AES with a 192-bit key
aes128-cbc** AES with a 128-bit key
Serpent256-cbc Serpent in CBC mode with a 256-bit key
Serpent192-cbc Serpent with a 192-bit key
Serpent128-cbc Serpent with a 128-bit key
arcfour RC4 with a 128-bit key
cast128-cbc CAST-128 in CBC mode
* = Required ** = Recommended
572 CHAPTER 17 / TRANSPORT-LEVEL SECURITY
C’s identification string; K_S is S’s public host key; I_C is C’s SSH_MSG_KEXINIT message and I_S is S’s SSH_MSG_KEXINIT message that have been exchanged before this part begins. The values of p, g, and q are known to both client and server as a result of the algorithm selection negotiation. The hash function hash() is also decided during algorithm negotiation.
1. C generates a random number x(1 6 x 6 q) and computes e = gx mod p. C sends e to S.
2. S generates a random number y(0 6 y 6 q) and computes f = gy mod p. S receives e. It computes K = ey mod p, H = hash(V_C ‘ V_S ‘ I_C ‘ I_S ‘ K_S ‘ e ‘ f ‘ K), and signature s on H with its private host key. S sends (K_S ‘ f ‘ s) to C. The signing operation may involve a second hashing operation.
3. C verifies that K_S really is the host key for S (e.g., using certificates or a local database). C is also allowed to accept the key without verification; however,
doing so will render the protocol insecure against active attacks (but may be
desirable for practical reasons in the short term in many environments). C then
computes K = f x mod p, H = hash(V_C ‘ V_S ‘ I_C ‘ I_S ‘ K_S ‘ e ‘ f ‘ K), and verifies the signature s on H.
As a result of these steps, the two sides now share a master key K. In addition, the server has been authenticated to the client, because the server has used its pri-
vate key to sign its half of the Diffie-Hellman exchange. Finally, the hash value H serves as a session identifier for this connection. Once computed, the session identi-
fier is not changed, even if the key exchange is performed again for this connection
to obtain fresh keys.
The end of key exchange is signaled by the exchange of SSH_MSG_NEWKEYS packets. At this point, both sides may start using the keys generated from K, as dis- cussed subsequently.
The final step is service request. The client sends an SSH_MSG_SERVICE_ REQUEST packet to request either the User Authentication or the Connection Protocol. Subsequent to this, all data is exchanged as the payload of an SSH
Transport Layer packet, protected by encryption and MAC.
KEY GENERATION The keys used for encryption and MAC (and any needed IVs) are generated from the shared secret key K, the hash value from the key exchange H, and the session identifier, which is equal to H unless there has been a subsequent key exchange after the initial key exchange. The values are computed as follows.
■ Initial IV client to server: HASH(K ‘ H ‘ ;A< ‘ session_id) ■ Initial IV server to client: HASH(K ‘ H ‘ ;B< ‘ session_id) ■ Encryption key client to server: HASH(K ‘ H ‘ ;C< ‘ session_id) ■ Encryption key server to client: HASH(K ‘ H ‘ ;D< ‘ session_id) ■ Integrity key client to server: HASH(K ‘ H ‘ ;E< ‘ session_id) ■ Integrity key server to client: HASH(K ‘ H ‘ ;F< ‘ session_id)
where HASH() is the hash function determined during algorithm negotiation.
17.4 / SECURE SHELL (SSH) 573
User Authentication Protocol
The User Authentication Protocol provides the means by which the client is
authenticated to the server.
MESSAGE TYPES AND FORMATS Three types of messages are always used in the User Authentication Protocol. Authentication requests from the client have the format:
byte SSH_MSG_USERAUTH_REQUEST (50)
string user name
string service name
string method name
. . . method specific fields
where user name is the authorization identity the client is claiming, service
name is the facility to which the client is requesting access (typically the SSH
Connection Protocol), and method name is the authentication method being
used in this request. The first byte has decimal value 50, which is interpreted as
SSH_MSG_USERAUTH_REQUEST. If the server either (1) rejects the authentication request or (2) accepts the
request but requires one or more additional authentication methods, the server
sends a message with the format:
byte SSH_MSG_USERAUTH_FAILURE (51)
name-list authentications that can continue
boolean partial success
where the name-list is a list of methods that may productively continue the dialog.
If the server accepts authentication, it sends a single byte message: SSH_MSG_ USERAUTH_SUCCESS (52).
MESSAGE EXCHANGE The message exchange involves the following steps.
1. The client sends a SSH_MSG_USERAUTH_REQUEST with a requested method of none.
2. The server checks to determine if the user name is valid. If not, the server returns SSH_MSG_USERAUTH_FAILURE with the partial success value of false. If the user name is valid, the server proceeds to step 3.
3. The server returns SSH_MSG_USERAUTH_FAILURE with a list of one or more authentication methods to be used.
4. The client selects one of the acceptable authentication methods and sends a SSH_MSG_USERAUTH_REQUEST with that method name and the required method-specific fields. At this point, there may be a sequence of exchanges to
perform the method.
574 CHAPTER 17 / TRANSPORT-LEVEL SECURITY
5. If the authentication succeeds and more authentication methods are required, the server proceeds to step 3, using a partial success value of true. If the
authentication fails, the server proceeds to step 3, using a partial success value
of false.
6. When all required authentication methods succeed, the server sends a SSH_MSG_USERAUTH_SUCCESS message, and the Authentication Protocol is over.
AUTHENTICATION METHODS The server may require one or more of the following authentication methods.
■ publickey: The details of this method depend on the public-key algorithm chosen. In essence, the client sends a message to the server that contains
the client’s public key, with the message signed by the client’s private key.
When the server receives this message, it checks whether the supplied key
is acceptable for authentication and, if so, it checks whether the signature is
correct.
■ password: The client sends a message containing a plaintext password, which is protected by encryption by the Transport Layer Protocol.
■ hostbased: Authentication is performed on the client’s host rather than the client itself. Thus, a host that supports multiple clients would provide authen-
tication for all its clients. This method works by having the client send a signa-
ture created with the private key of the client host. Thus, rather than directly
verifying the user’s identity, the SSH server verifies the identity of the client
host—and then believes the host when it says the user has already authenti-
cated on the client side.
Connection Protocol
The SSH Connection Protocol runs on top of the SSH Transport Layer Protocol
and assumes that a secure authentication connection is in use.2 That secure authen-
tication connection, referred to as a tunnel, is used by the Connection Protocol to multiplex a number of logical channels.
CHANNEL MECHANISM All types of communication using SSH, such as a terminal session, are supported using separate channels. Either side may open a channel.
For each channel, each side associates a unique channel number, which need not be
the same on both ends. Channels are flow controlled using a window mechanism.
No data may be sent to a channel until a message is received to indicate that window
space is available.
2RFC 4254, The Secure Shell (SSH) Connection Protocol, states that the Connection Protocol runs on top of the Transport Layer Protocol and the User Authentication Protocol. RFC 4251, SSH Protocol Architecture, states that the Connection Protocol runs over the User Authentication Protocol. In fact, the Connection Protocol runs over the Transport Layer Protocol, but assumes that the User Authentication Protocol has been previously invoked.
17.4 / SECURE SHELL (SSH) 575
The life of a channel progresses through three stages: opening a channel, data
transfer, and closing a channel.
When either side wishes to open a new channel, it allocates a local number for the channel and then sends a message of the form:
byte SSH_MSG_CHANNEL_OPEN
string channel type
uint32 sender channel
uint32 initial window size
uint32 maximum packet size
.... channel type specific data follows
where uint32 means unsigned 32-bit integer. The channel type identifies the appli-
cation for this channel, as described subsequently. The sender channel is the local
channel number. The initial window size specifies how many bytes of channel data
can be sent to the sender of this message without adjusting the window. The maxi-
mum packet size specifies the maximum size of an individual data packet that can
be sent to the sender. For example, one might want to use smaller packets for inter-
active connections to get better interactive response on slow links.
If the remote side is able to open the channel, it returns a SSH_MSG_CHANNEL_ OPEN_CONFIRMATION message, which includes the sender channel number, the recipient channel number, and window and packet size values for incoming traffic.
Otherwise, the remote side returns a SSH_MSG_CHANNEL_OPEN_FAILURE message with a reason code indicating the reason for failure.
Once a channel is open, data transfer is performed using a SSH_MSG_ CHANNEL_DATA message, which includes the recipient channel number and a block of data. These messages, in both directions, may continue as long as the channel
is open.
When either side wishes to close a channel, it sends a SSH_MSG_CHANNEL_ CLOSE message, which includes the recipient channel number.
Figure 17.11 provides an example of Connection Protocol Message Exchange.
CHANNEL TYPES Four channel types are recognized in the SSH Connection Protocol specification.
■ session: The remote execution of a program. The program may be a shell, an application such as file transfer or email, a system command, or some built-in
subsystem. Once a session channel is opened, subsequent requests are used to
start the remote program.
■ x11: This refers to the X Window System, a computer software system and network protocol that provides a graphical user interface (GUI) for net-
worked computers. X allows applications to run on a network server but to be
displayed on a desktop machine.
576 CHAPTER 17 / TRANSPORT-LEVEL SECURITY
■ forwarded-tcpip: This is remote port forwarding, as explained in the next subsection.
■ direct-tcpip: This is local port forwarding, as explained in the next subsection.
PORT FORWARDING One of the most useful features of SSH is port forwarding. In essence, port forwarding provides the ability to convert any insecure TCP connec-
tion into a secure SSH connection. This is also referred to as SSH tunneling. We
need to know what a port is in this context. A port is an identifier of a user of TCP. So, any application that runs on top of TCP has a port number. Incoming TCP
traffic is delivered to the appropriate application on the basis of the port number.
An application may employ multiple port numbers. For example, for the Simple
Mail Transfer Protocol (SMTP), the server side generally listens on port 25, so an
Figure 17.11 Example of SSH Connection Protocol Message Exchange
Client Server
SSH_MSG_CHANNEL_OPEN Open a channel
Data transfer
Close a channel
SSH_MSG_CHANNEL_OPEN_CONFIRMATION
SSH_MSG_CHANNEL_DATA
SSH_MSG_CHANNEL_DATA
SSH_MSG_CHANNEL_DATA
SSH_MSG_CHANNEL_DATA
SSH_MSG_CHANNEL_CLOSE
Establish Authenticated Transport Layer Connection
17.4 / SECURE SHELL (SSH) 577
incoming SMTP request uses TCP and addresses the data to destination port 25.
TCP recognizes that this is the SMTP server address and routes the data to the
SMTP server application.
Figure 17.12 illustrates the basic concept behind port forwarding. We have
a client application that is identified by port number x and a server application identified by port number y. At some point, the client application invokes the local TCP entity and requests a connection to the remote server on port y. The local TCP entity negotiates a TCP connection with the remote TCP entity, such that the
connection links local port x to remote port y. To secure this connection, SSH is configured so that the SSH Transport Layer
Protocol establishes a TCP connection between the SSH client and server entities,
with TCP port numbers a and b, respectively. A secure SSH tunnel is established
Figure 17.12 SSH Transport Layer Packet Exchanges
Client Server
Client application
Unsecure TCP connection
(a) Connection via TCP
TCP entity
x y
Server application
TCP entity
Client application
Secure SSH tunnel
(b) Connection via SSH tunnel
SSH entity
x y
Server application
SSH entity
Unsecure TCP connectionTCP entity
a b TCP entity
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578 CHAPTER 17 / TRANSPORT-LEVEL SECURITY
over this TCP connection. Traffic from the client at port x is redirected to the local SSH entity and travels through the tunnel where the remote SSH entity delivers the
data to the server application on port y. Traffic in the other direction is similarly redirected.
SSH supports two types of port forwarding: local forwarding and remote for-
warding. Local forwarding allows the client to set up a “hijacker” process. This will intercept selected application-level traffic and redirect it from an unsecured TCP
connection to a secure SSH tunnel. SSH is configured to listen on selected ports.
SSH grabs all traffic using a selected port and sends it through an SSH tunnel. On
the other end, the SSH server sends the incoming traffic to the destination port dic-
tated by the client application.
The following example should help clarify local forwarding. Suppose you have
an email client on your desktop and use it to get email from your mail server via the
Post Office Protocol (POP). The assigned port number for POP3 is port 110. We
can secure this traffic in the following way:
1. The SSH client sets up a connection to the remote server.
2. Select an unused local port number, say 9999, and configure SSH to accept traffic from this port destined for port 110 on the server.
3. The SSH client informs the SSH server to create a connection to the destina- tion, in this case mailserver port 110.
4. The client takes any bits sent to local port 9999 and sends them to the server inside the encrypted SSH session. The SSH server decrypts the incoming bits
and sends the plaintext to port 110.
5. In the other direction, the SSH server takes any bits received on port 110 and sends them inside the SSH session back to the client, who decrypts and sends
them to the process connected to port 9999.
With remote forwarding, the user’s SSH client acts on the server’s behalf. The client receives traffic with a given destination port number, places the traf-
fic on the correct port and sends it to the destination the user chooses. A typical
example of remote forwarding is the following. You wish to access a server at
work from your home computer. Because the work server is behind a firewall, it
will not accept an SSH request from your home computer. However, from work
you can set up an SSH tunnel using remote forwarding. This involves the follow-
ing steps.
1. From the work computer, set up an SSH connection to your home computer. The firewall will allow this, because it is a protected outgoing connection.
2. Configure the SSH server to listen on a local port, say 22, and to deliver data across the SSH connection addressed to remote port, say 2222.
3. You can now go to your home computer, and configure SSH to accept traffic on port 2222.
4. You now have an SSH tunnel that can be used for remote logon to the work server.
17.5 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 579
17.5 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
Key Terms
Alert protocol
Change Cipher Spec protocol
Handshake protocol
HTTPS (HTTP over SSL)
Master Secret
Secure Shell (SSH)
Secure Socket Layer (SSL)
Transport Layer Security
(TLS)
Review Questions
17.1 What are the advantages of each of the three approaches shown in Figure 17.1? 17.2 What protocols comprise TLS? 17.3 What is the difference between a TLS connection and a TLS session? 17.4 List and briefly define the parameters that define a TLS session state. 17.5 List and briefly define the parameters that define a TLS session connection. 17.6 What services are provided by the TLS Record Protocol? 17.7 What steps are involved in the TLS Record Protocol transmission? 17.8 Give brief details about different level of awareness of a connection in HTTPS. 17.9 Which protocol was replaced by SSH and why? Which version is currently in the pro-
cess of being standardized?
17.10 List and briefly define the SSH protocols.
Problems
17.1 In SSL and TLS, why is there a separate Change Cipher Spec Protocol rather than including a change_cipher_spec message in the Handshake Protocol?
17.2 What purpose does the MAC serve during the change cipher spec TLS exchange? 17.3 Consider the following threats to Web security and describe how each is countered by
a particular feature of TLS. a. Brute-Force Cryptanalytic Attack: An exhaustive search of the key space for a
conventional encryption algorithm. b. Known Plaintext Dictionary Attack: Many messages will contain predictable
plaintext, such as the HTTP GET command. An attacker constructs a diction- ary containing every possible encryption of the known-plaintext message. When an encrypted message is intercepted, the attacker takes the portion containing the encrypted known plaintext and looks up the ciphertext in the dictionary. The ciphertext should match against an entry that was encrypted with the same secret key. If there are several matches, each of these can be tried against the full cipher- text to determine the right one. This attack is especially effective against small key sizes (e.g., 40-bit keys).
c. Replay Attack: Earlier TLS handshake messages are replayed. d. Man-in-the-Middle Attack: An attacker interposes during key exchange, acting as
the client to the server and as the server to the client. e. Password Sniffing: Passwords in HTTP or other application traffic are eaves-
dropped. f. IP Spoofing: Uses forged IP addresses to fool a host into accepting bogus data.
580 CHAPTER 17 / TRANSPORT-LEVEL SECURITY
g. IP Hijacking: An active, authenticated connection between two hosts is disrupted and the attacker takes the place of one of the hosts.
h. SYN Flooding: An attacker sends TCP SYN messages to request a connection but does not respond to the final message to establish the connection fully. The attacked TCP module typically leaves the “half-open connection” around for a few minutes. Repeated SYN messages can clog the TCP module.
17.4 Based on what you have learned in this chapter, is it possible in TLS for the receiver to reorder TLS record blocks that arrive out of order? If so, explain how it can be done. If not, why not?
17.5 For SSH packets, what is the advantage, if any, of not including the MAC in the scope of the packet encryption?
581
Wireless Network Security 18.1 Wireless Security
Wireless Network Threats
Wireless Security Measures
18.2 Mobile Device Security
Security Threats
Mobile Device Security Strategy
18.3 IEEE 802.11 Wireless LAN Overview
The Wi-Fi Alliance
IEEE 802 Protocol Architecture
IEEE 802.11 Network Components and Architectural Model
IEEE 802.11 Services
18.4 IEEE 802.11i Wireless LAN Security
IEEE 802.11i Services
IEEE 802.11i Phases of Operation
Discovery Phase
Authentication Phase
Key Management Phase
Protected Data Transfer Phase
The IEEE 802.11i Pseudorandom Function
18.5 Key Terms, Review Questions, and Problems
CHAPTER
582 CHAPTER 18 / WIRELESS NETWORK SECURITY
This chapter begins with a general overview of wireless security issues. We then focus
on the relatively new area of mobile device security, examining threats and counter-
measures for mobile devices used in the enterprise. Then, we look at the IEEE 802.11i
standard for wireless LAN security. This standard is part of IEEE 802.11, also referred
to as Wi-Fi. We begin the discussion with an overview of IEEE 802.11, and then we
look in some detail at IEEE 802.11i.
18.1 WIRELESS SECURITY
Wireless networks, and the wireless devices that use them, introduce a host of secu-
rity problems over and above those found in wired networks. Some of the key fac-
tors contributing to the higher security risk of wireless networks compared to wired
networks include the following [MA10]:
■ Channel: Wireless networking typically involves broadcast communications, which is far more susceptible to eavesdropping and jamming than wired
networks. Wireless networks are also more vulnerable to active attacks that
exploit vulnerabilities in communications protocols.
■ Mobility: Wireless devices are, in principal and usually in practice, far more portable and mobile than wired devices. This mobility results in a number of
risks, described subsequently.
■ Resources: Some wireless devices, such as smartphones and tablets, have sophisticated operating systems but limited memory and processing resources
with which to counter threats, including denial of service and malware.
■ Accessibility: Some wireless devices, such as sensors and robots, may be left unattended in remote and/or hostile locations. This greatly increases their
vulnerability to physical attacks.
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Present an overview of security threats and countermeasures for wireless networks.
◆ Understand the unique security threats posed by the use of mobile devices with enterprise networks.
◆ Describe the principal elements in a mobile device security strategy.
◆ Understand the essential elements of the IEEE 802.11 wireless LAN standard.
◆ Summarize the various components of the IEEE 802.11i wireless LAN security architecture.
18.1 / WIRELESS SECURITY 583
In simple terms, the wireless environment consists of three components that
provide point of attack (Figure 18.1). The wireless client can be a cell phone, a
Wi-Fi–enabled laptop or tablet, a wireless sensor, a Bluetooth device, and so on.
The wireless access point provides a connection to the network or service. Examples
of access points are cell towers, Wi-Fi hotspots, and wireless access points to wired
local or wide area networks. The transmission medium, which carries the radio
waves for data transfer, is also a source of vulnerability.
Wireless Network Threats
[CHOI08] lists the following security threats to wireless networks:
■ Accidental association: Company wireless LANs or wireless access points to wired LANs in close proximity (e.g., in the same or neighboring buildings)
may create overlapping transmission ranges. A user intending to connect to
one LAN may unintentionally lock on to a wireless access point from a neigh-
boring network. Although the security breach is accidental, it nevertheless
exposes resources of one LAN to the accidental user.
■ Malicious association: In this situation, a wireless device is configured to appear to be a legitimate access point, enabling the operator to steal pass-
words from legitimate users and then penetrate a wired network through a
legitimate wireless access point.
■ Ad hoc networks: These are peer-to-peer networks between wireless comput- ers with no access point between them. Such networks can pose a security
threat due to a lack of a central point of control.
■ Nontraditional networks: Nontraditional networks and links, such as personal network Bluetooth devices, barcode readers, and handheld PDAs, pose a secu-
rity risk in terms of both eavesdropping and spoofing.
■ Identity theft (MAC spoofing): This occurs when an attacker is able to eaves- drop on network traffic and identify the MAC address of a computer with
network privileges.
■ Man-in-the middle attacks: This type of attack is described in Chapter 10 in the context of the Diffie–Hellman key exchange protocol. In a broader sense,
this attack involves persuading a user and an access point to believe that they
are talking to each other when in fact the communication is going through an
intermediate attacking device. Wireless networks are particularly vulnerable
to such attacks.
Figure 18.1 Wireless Networking Components
Endpoint Wireless medium Access point
584 CHAPTER 18 / WIRELESS NETWORK SECURITY
■ Denial of service (DoS): This type of attack is discussed in detail in Chapter 21. In the context of a wireless network, a DoS attack occurs when an attacker
continually bombards a wireless access point or some other accessible wireless
port with various protocol messages designed to consume system resources.
The wireless environment lends itself to this type of attack, because it is so
easy for the attacker to direct multiple wireless messages at the target.
■ Network injection: A network injection attack targets wireless access points that are exposed to nonfiltered network traffic, such as routing protocol mes-
sages or network management messages. An example of such an attack is
one in which bogus reconfiguration commands are used to affect routers and
switches to degrade network performance.
Wireless Security Measures
Following [CHOI08], we can group wireless security measures into those dealing
with wireless transmissions, wireless access points, and wireless networks (consist-
ing of wireless routers and endpoints).
SECURING WIRELESS TRANSMISSIONS The principal threats to wireless transmission are eavesdropping, altering or inserting messages, and disruption. To deal with
eavesdropping, two types of countermeasures are appropriate:
■ Signal-hiding techniques: Organizations can take a number of measures to make it more difficult for an attacker to locate their wireless access points,
including turning off service set identifier (SSID) broadcasting by wireless
access points; assigning cryptic names to SSIDs; reducing signal strength to the
lowest level that still provides requisite coverage; and locating wireless access
points in the interior of the building, away from windows and exterior walls.
Greater security can be achieved by the use of directional antennas and of
signal-shielding techniques.
■ Encryption: Encryption of all wireless transmission is effective against eaves- dropping to the extent that the encryption keys are secured.
The use of encryption and authentication protocols is the standard method of
countering attempts to alter or insert transmissions.
The methods discussed in Chapter 21 for dealing with DoS apply to wireless
transmissions. Organizations can also reduce the risk of unintentional DoS attacks.
Site surveys can detect the existence of other devices using the same frequency
range, to help determine where to locate wireless access points. Signal strengths can
be adjusted and shielding used in an attempt to isolate a wireless environment from
competing nearby transmissions.
SECURING WIRELESS ACCESS POINTS The main threat involving wireless access points is unauthorized access to the network. The principal approach for preventing
such access is the IEEE 802.1X standard for port-based network access control. The
standard provides an authentication mechanism for devices wishing to attach to a
LAN or wireless network. The use of 802.1X can prevent rogue access points and
other unauthorized devices from becoming insecure backdoors.
Section 16.3 provides an introduction to 802.1X.
18.2 / MOBILE DEVICE SECURITY 585
SECURING WIRELESS NETWORKS [CHOI08] recommends the following techniques for wireless network security:
1. Use encryption. Wireless routers are typically equipped with built-in encryp- tion mechanisms for router-to-router traffic.
2. Use antivirus and antispyware software, and a firewall. These facilities should be enabled on all wireless network endpoints.
3. Turn off identifier broadcasting. Wireless routers are typically configured to broadcast an identifying signal so that any device within range can learn of
the router’s existence. If a network is configured so that authorized devices
know the identity of routers, this capability can be disabled, so as to thwart
attackers.
4. Change the identifier on your router from the default. Again, this measure thwarts attackers who will attempt to gain access to a wireless network using
default router identifiers.
5. Change your router’s pre-set password for administration. This is another prudent step.
6. Allow only specific computers to access your wireless network. A router can be configured to only communicate with approved MAC addresses. Of course,
MAC addresses can be spoofed, so this is just one element of a security strategy.
18.2 MOBILE DEVICE SECURITY
Prior to the widespread use of smartphones, the dominant paradigm for computer
and network security in organizations was as follows. Corporate IT was tightly con-
trolled. User devices were typically limited to Windows PCs. Business applications
were controlled by IT and either run locally on endpoints or on physical servers
in data centers. Network security was based upon clearly defined perimeters that
separated trusted internal networks from the untrusted Internet. Today, there have
been massive changes in each of these assumptions. An organization’s networks
must accommodate the following:
■ Growing use of new devices: Organizations are experiencing significant growth in employee use of mobile devices. In many cases, employees are allowed to
use a combination of endpoint devices as part of their day-to-day activities.
■ Cloud-based applications: Applications no longer run solely on physical servers in corporate data centers. Quite the opposite, applications can run
anywhere—on traditional physical servers, on mobile virtual servers, or in the
cloud. Additionally, end users can now take advantage of a wide variety of
cloud-based applications and IT services for personal and professional use.
Facebook can be used for an employee’s personal profiles or as a component
of a corporate marketing campaign. Employees depend upon Skype to speak
with friends abroad or for legitimate business video conferencing. Dropbox
and Box can be used to distribute documents between corporate and personal
devices for mobility and user productivity.
586 CHAPTER 18 / WIRELESS NETWORK SECURITY
■ De-perimeterization: Given new device proliferation, application mobility, and cloud-based consumer and corporate services, the notion of a static net-
work perimeter is all but gone. Now there are a multitude of network perim-
eters around devices, applications, users, and data. These perimeters have also
become quite dynamic as they must adapt to various environmental conditions
such as user role, device type, server virtualization mobility, network location,
and time-of-day.
■ External business requirements: The enterprise must also provide guests, third-party contractors, and business partners network access using various
devices from a multitude of locations.
The central element in all of these changes is the mobile computing device.
Mobile devices have become an essential element for organizations as part of the
overall network infrastructure. Mobile devices such as smartphones, tablets, and
memory sticks provide increased convenience for individuals as well as the poten-
tial for increased productivity in the workplace. Because of their widespread use
and unique characteristics, security for mobile devices is a pressing and complex
issue. In essence, an organization needs to implement a security policy through a
combination of security features built into the mobile devices and additional secu-
rity controls provided by network components that regulate the use of the mobile
devices.
Security Threats
Mobile devices need additional, specialized protection measures beyond those
implemented for other client devices, such as desktop and laptop devices that are
used only within the organization’s facilities and on the organization’s networks.
SP 800-14 (Guidelines for Managing and Securing Mobile Devices in the Enterprise, July 2012) lists seven major security concerns for mobile devices. We examine each
of these in turn.
LACK OF PHYSICAL SECURITY CONTROLS Mobile devices are typically under the com- plete control of the user, and are used and kept in a variety of locations outside the
organization’s control, including off premises. Even if a device is required to remain
on premises, the user may move the device within the organization between secure
and nonsecured locations. Thus, theft and tampering are realistic threats.
The security policy for mobile devices must be based on the assumption that
any mobile device may be stolen or at least accessed by a malicious party. The threat
is twofold: A malicious party may attempt to recover sensitive data from the device
itself, or may use the device to gain access to the organization’s resources.
USE OF UNTRUSTED MOBILE DEVICES In addition to company-issued and company- controlled mobile devices, virtually all employees will have personal smartphones
and/or tablets. The organization must assume that these devices are not trustworthy.
That is, the devices may not employ encryption and either the user or a third party
may have installed a bypass to the built-in restrictions on security, operating system
use, and so on.
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18.2 / MOBILE DEVICE SECURITY 587
USE OF UNTRUSTED NETWORKS If a mobile device is used on premises, it can connect to organization resources over the organization’s own in-house wireless networks.
However, for off-premises use, the user will typically access organizational resources
via Wi-Fi or cellular access to the Internet and from the Internet to the organiza-
tion. Thus, traffic that includes an off-premises segment is potentially susceptible to
eavesdropping or man-in-the-middle types of attacks. Thus, the security policy must
be based on the assumption that the networks between the mobile device and the
organization are not trustworthy.
USE OF APPLICATIONS CREATED BY UNKNOWN PARTIES By design, it is easy to find and install third-party applications on mobile devices. This poses the obvious risk of
installing malicious software. An organization has several options for dealing with
this threat, as described subsequently.
INTERACTION WITH OTHER SYSTEMS A common feature found on smartphones and tablets is the ability to automatically synchronize data, apps, contacts, photos, and
so on with other computing devices and with cloud-based storage. Unless an orga-
nization has control of all the devices involved in synchronization, there is consider-
able risk of the organization’s data being stored in an unsecured location, plus the
risk of the introduction of malware.
USE OF UNTRUSTED CONTENT Mobile devices may access and use content that other computing devices do not encounter. An example is the Quick Response (QR)
code, which is a two-dimensional barcode. QR codes are designed to be captured
by a mobile device camera and used by the mobile device. The QR code translates
to a URL, so that a malicious QR code could direct the mobile device to malicious
Web sites.
USE OF LOCATION SERVICES The GPS capability on mobile devices can be used to maintain a knowledge of the physical location of the device. While this feature
might be useful to an organization as part of a presence service, it creates security
risks. An attacker can use the location information to determine where the device
and user are located, which may be of use to the attacker.
Mobile Device Security Strategy
With the threats listed in the preceding discussion in mind, we outline the principal
elements of a mobile device security strategy. They fall into three categories: device
security, client/server traffic security, and barrier security (Figure 18.2).
DEVICE SECURITY A number of organizations will supply mobile devices for employee use and preconfigure those devices to conform to the enterprise secu-
rity policy. However, many organizations will find it convenient or even necessary
to adopt a bring-your-own-device (BYOD) policy that allows the personal mobile
devices of employees to have access to corporate resources. IT managers should be
able to inspect each device before allowing network access. IT will want to estab-
lish configuration guidelines for operating systems and applications. For example,
“rooted” or “jail-broken” devices are not permitted on the network, and mobile
588 CHAPTER 18 / WIRELESS NETWORK SECURITY
devices cannot store corporate contacts on local storage. Whether a device is owned
by the organization or BYOD, the organization should configure the device with
security controls, including the following:
■ Enable auto-lock, which causes the device to lock if it has not been used for a
given amount of time, requiring the user to re-enter a four-digit PIN or a pass-
word to re-activate the device.
■ Enable password or PIN protection. The PIN or password is needed to unlock
the device. In addition, it can be configured so that email and other data on the
device are encrypted using the PIN or password and can only be retrieved with
the PIN or password.
■ Avoid using auto-complete features that remember user names or passwords.
■ Enable remote wipe.
■ Ensure that SSL protection is enabled, if available.
■ Make sure that software, including operating systems and applications, is up
to date.
■ Install antivirus software as it becomes available.
Figure 18.2 Mobile Device Security Elements
Firewall
Firewall limits scope of data and application access
Authentication and access control protocols used to verify device and user and establish limits on access
Mobile device is configured with security mechanisms and parameters to conform to organization security policy
Traffic is encrypted; uses SSL or IPsec VPN tunnel
Authentication/ access control server
Mobile device configuration server
Application/ database server
18.3 / IEEE 802.11 WIRELESS LAN OVERVIEW 589
■ Either sensitive data should be prohibited from storage on the mobile device
or it should be encrypted.
■ IT staff should also have the ability to remotely access devices, wipe the device
of all data, and then disable the device in the event of loss or theft.
■ The organization may prohibit all installation of third-party applications,
implement whitelisting to prohibit installation of all unapproved applications,
or implement a secure sandbox that isolates the organization’s data and appli-
cations from all other data and applications on the mobile device. Any applica-
tion that is on an approved list should be accompanied by a digital signature
and a public-key certificate from an approved authority.
■ The organization can implement and enforce restrictions on what devices can
synchronize and on the use of cloud-based storage.
■ To deal with the threat of untrusted content, security responses can include
training of personnel on the risks inherent in untrusted content and disabling
camera use on corporate mobile devices.
■ To counter the threat of malicious use of location services, the security policy
can dictate that such service is disabled on all mobile devices.
TRAFFIC SECURITY Traffic security is based on the usual mechanisms for encryption and authentication. All traffic should be encrypted and travel by secure means, such
as SSL or IPv6. Virtual private networks (VPNs) can be configured so that all traffic
between the mobile device and the organization’s network is via a VPN.
A strong authentication protocol should be used to limit the access from the
device to the resources of the organization. Often, a mobile device has a single
device-specific authenticator, because it is assumed that the device has only one
user. A preferable strategy is to have a two-layer authentication mechanism, which
involves authenticating the device and then authenticating the user of the device.
BARRIER SECURITY The organization should have security mechanisms to protect the network from unauthorized access. The security strategy can also include fire-
wall policies specific to mobile device traffic. Firewall policies can limit the scope
of data and application access for all mobile devices. Similarly, intrusion detection
and intrusion prevention systems can be configured to have tighter rules for mobile
device traffic.
18.3 IEEE 802.11 WIRELESS LAN OVERVIEW
IEEE 802 is a committee that has developed standards for a wide range of local area
networks (LANs). In 1990, the IEEE 802 Committee formed a new working group,
IEEE 802.11, with a charter to develop a protocol and transmission specifications
for wireless LANs (WLANs). Since that time, the demand for WLANs at different
frequencies and data rates has exploded. Keeping pace with this demand, the IEEE
802.11 working group has issued an ever-expanding list of standards. Table 18.1
briefly defines key terms used in the IEEE 802.11 standard.
590 CHAPTER 18 / WIRELESS NETWORK SECURITY
The Wi-Fi Alliance
The first 802.11 standard to gain broad industry acceptance was 802.11b. Although
802.11b products are all based on the same standard, there is always a concern
whether products from different vendors will successfully interoperate. To meet
this concern, the Wireless Ethernet Compatibility Alliance (WECA), an indus-
try consortium, was formed in 1999. This organization, subsequently renamed the
Wi-Fi (Wireless Fidelity) Alliance, created a test suite to certify interoperability for
802.11b products. The term used for certified 802.11b products is Wi-Fi. Wi-Fi certi- fication has been extended to 802.11g products. The Wi-Fi Alliance has also devel-
oped a certification process for 802.11a products, called Wi-Fi5. The Wi-Fi Alliance is concerned with a range of market areas for WLANs, including enterprise, home,
and hot spots.
More recently, the Wi-Fi Alliance has developed certification procedures for
IEEE 802.11 security standards, referred to as Wi-Fi Protected Access (WPA). The
most recent version of WPA, known as WPA2, incorporates all of the features of
the IEEE 802.11i WLAN security specification.
IEEE 802 Protocol Architecture
Before proceeding, we need to briefly preview the IEEE 802 protocol architecture.
IEEE 802.11 standards are defined within the structure of a layered set of protocols.
This structure, used for all IEEE 802 standards, is illustrated in Figure 18.3.
PHYSICAL LAYER The lowest layer of the IEEE 802 reference model is the physical layer, which includes such functions as encoding/decoding of signals and bit trans- mission/reception. In addition, the physical layer includes a specification of the
transmission medium. In the case of IEEE 802.11, the physical layer also defines
frequency bands and antenna characteristics.
Access point (AP) Any entity that has station functionality and provides access to the
distribution system via the wireless medium for associated stations.
Basic service set (BSS) A set of stations controlled by a single coordination function.
Coordination function The logical function that determines when a station operating within a BSS
is permitted to transmit and may be able to receive PDUs.
Distribution system (DS) A system used to interconnect a set of BSSs and integrated LANs to create
an ESS.
Extended service set (ESS) A set of one or more interconnected BSSs and integrated LANs that
appear as a single BSS to the LLC layer at any station associated with one
of these BSSs.
MAC protocol data unit
(MPDU)
The unit of data exchanged between two peer MAC entities using the
services of the physical layer.
MAC service data unit
(MSDU)
Information that is delivered as a unit between MAC users.
Station Any device that contains an IEEE 802.11 conformant MAC and physical
layer.
Table 18.1 IEEE 802.11 Terminology
18.3 / IEEE 802.11 WIRELESS LAN OVERVIEW 591
MEDIA ACCESS CONTROL All LANs consist of collections of devices that share the network’s transmission capacity. Some means of controlling access to the transmis-
sion medium is needed to provide an orderly and efficient use of that capacity. This
is the function of a media access control (MAC) layer. The MAC layer receives data from a higher-layer protocol, typically the Logical Link Control (LLC) layer, in the
form of a block of data known as the MAC service data unit (MSDU). In general, the MAC layer performs the following functions:
■ On transmission, assemble data into a frame, known as a MAC protocol data unit (MPDU) with address and error-detection fields.
■ On reception, disassemble frame, and perform address recognition and error
detection.
■ Govern access to the LAN transmission medium.
The exact format of the MPDU differs somewhat for the various MAC proto-
cols in use. In general, all of the MPDUs have a format similar to that of Figure 18.4.
The fields of this frame are as follows.
■ MAC Control: This field contains any protocol control information needed for the functioning of the MAC protocol. For example, a priority level could be
indicated here.
■ Destination MAC Address: The destination physical address on the LAN for this MPDU.
■ Source MAC Address: The source physical address on the LAN for this MPDU.
Figure 18.3 IEEE 802.11 Protocol Stack
Logical Link Control
Medium Access Control
Physical Encoding/decoding of signals Bit transmission/reception Transmission medium
Assemble data into frame Addressing Error detection Medium access
Flow control Error control
General IEEE 802 functions
Specific IEEE 802.11 functions
Frequency band definition Wireless signal encoding
Reliable data delivery Wireless access control protocols
592 CHAPTER 18 / WIRELESS NETWORK SECURITY
■ MAC Service Data Unit: The data from the next higher layer.
■ CRC: The cyclic redundancy check field; also known as the Frame Check Sequence (FCS) field. This is an error-detecting code, such as that which is
used in other data-link control protocols. The CRC is calculated based on the
bits in the entire MPDU. The sender calculates the CRC and adds it to the
frame. The receiver performs the same calculation on the incoming MPDU
and compares that calculation to the CRC field in that incoming MPDU. If
the two values don’t match, then one or more bits have been altered in transit.
The fields preceding the MSDU field are referred to as the MAC header, and the field following the MSDU field is referred to as the MAC trailer. The header and trailer contain control information that accompany the data field and that are
used by the MAC protocol.
LOGICAL LINK CONTROL In most data-link control protocols, the data-link protocol entity is responsible not only for detecting errors using the CRC, but for recovering
from those errors by retransmitting damaged frames. In the LAN protocol archi-
tecture, these two functions are split between the MAC and LLC layers. The MAC
layer is responsible for detecting errors and discarding any frames that contain er-
rors. The LLC layer optionally keeps track of which frames have been successfully
received and retransmits unsuccessful frames.
IEEE 802.11 Network Components and Architectural Model
Figure 18.5 illustrates the model developed by the 802.11 working group. The small-
est building block of a wireless LAN is a basic service set (BSS), which consists of wireless stations executing the same MAC protocol and competing for access to the
same shared wireless medium. A BSS may be isolated, or it may connect to a back-
bone distribution system (DS) through an access point (AP). The AP functions as a bridge and a relay point. In a BSS, client stations do not communicate directly with
one another. Rather, if one station in the BSS wants to communicate with another
station in the same BSS, the MAC frame is first sent from the originating station to
the AP and then from the AP to the destination station. Similarly, a MAC frame
from a station in the BSS to a remote station is sent from the local station to the AP
and then relayed by the AP over the DS on its way to the destination station. The
BSS generally corresponds to what is referred to as a cell in the literature. The DS
can be a switch, a wired network, or a wireless network.
When all the stations in the BSS are mobile stations that communicate directly
with one another (not using an AP), the BSS is called an independent BSS (IBSS). An IBSS is typically an ad hoc network. In an IBSS, the stations all communicate
directly, and no AP is involved.
Figure 18.4 General IEEE 802 MPDU Format
MAC Control
reliart CAMredaeh CAM
Destination MAC Address
Source MAC Address MAC Service Data Unit (MSDU) CRC
18.3 / IEEE 802.11 WIRELESS LAN OVERVIEW 593
A simple configuration is shown in Figure 18.5, in which each station belongs
to a single BSS; that is, each station is within wireless range only of other stations
within the same BSS. It is also possible for two BSSs to overlap geographically, so
that a single station could participate in more than one BSS. Furthermore, the asso-
ciation between a station and a BSS is dynamic. Stations may turn off, come within
range, and go out of range.
An extended service set (ESS) consists of two or more basic service sets interconnected by a distribution system. The extended service set appears as a sin-
gle logical LAN to the logical link control (LLC) level.
IEEE 802.11 Services
IEEE 802.11 defines nine services that need to be provided by the wireless LAN to
achieve functionality equivalent to that which is inherent to wired LANs. Table 18.2
lists the services and indicates two ways of categorizing them.
1. The service provider can be either the station or the DS. Station services are implemented in every 802.11 station, including AP stations. Distribution ser-
vices are provided between BSSs; these services may be implemented in an AP
or in another special-purpose device attached to the distribution system.
2. Three of the services are used to control IEEE 802.11 LAN access and confi- dentiality. Six of the services are used to support delivery of MSDUs between
stations. If the MSDU is too large to be transmitted in a single MPDU, it may
be fragmented and transmitted in a series of MPDUs.
Figure 18.5 IEEE 802.11 Extended Service Set
STA 2
STA 3
STA4
STA 1
STA 6 STA 7
STA 8
AP 2
AP 1
Basic Service Set (BSS)
Basic Service Set (BSS)
Distribution System
594 CHAPTER 18 / WIRELESS NETWORK SECURITY
Following the IEEE 802.11 document, we next discuss the services in an order
designed to clarify the operation of an IEEE 802.11 ESS network. MSDU delivery, which is the basic service, already has been mentioned. Services related to security
are introduced in Section 18.4.
DISTRIBUTION OF MESSAGES WITHIN A DS The two services involved with the dis- tribution of messages within a DS are distribution and integration. Distribution is the primary service used by stations to exchange MPDUs when the MPDUs must
traverse the DS to get from a station in one BSS to a station in another BSS. For
example, suppose a frame is to be sent from station 2 (STA 2) to station 7 (STA 7)
in Figure 18.5. The frame is sent from STA 2 to AP 1, which is the AP for this BSS.
The AP gives the frame to the DS, which has the job of directing the frame to the
AP associated with STA 7 in the target BSS. AP 2 receives the frame and forwards
it to STA 7. How the message is transported through the DS is beyond the scope of
the IEEE 802.11 standard.
If the two stations that are communicating are within the same BSS, then the
distribution service logically goes through the single AP of that BSS.
The integration service enables transfer of data between a station on an IEEE 802.11 LAN and a station on an integrated IEEE 802.x LAN. The term integrated refers to a wired LAN that is physically connected to the DS and whose stations
may be logically connected to an IEEE 802.11 LAN via the integration service. The
integration service takes care of any address translation and media conversion logic
required for the exchange of data.
ASSOCIATION-RELATED SERVICES The primary purpose of the MAC layer is to transfer MSDUs between MAC entities; this purpose is fulfilled by the distribu-
tion service. For that service to function, it requires information about stations
within the ESS that is provided by the association-related services. Before the
distribution service can deliver data to or accept data from a station, that sta-
tion must be associated. Before looking at the concept of association, we need
Service Provider Used to support
Association Distribution system MSDU delivery
Authentication Station LAN access and security
Deauthentication Station LAN access and security
Disassociation Distribution system MSDU delivery
Distribution Distribution system MSDU delivery
Integration Distribution system MSDU delivery
MSDU delivery Station MSDU delivery
Privacy Station LAN access and security
Reassociation Distribution system MSDU delivery
Table 18.2 IEEE 802.11 Services
18.4 / IEEE 802.11i WIRELESS LAN SECURITY 595
to describe the concept of mobility. The standard defines three transition types,
based on mobility:
■ No transition: A station of this type is either stationary or moves only within the direct communication range of the communicating stations of a single BSS.
■ BSS transition: This is defined as a station movement from one BSS to another BSS within the same ESS. In this case, delivery of data to the station requires that
the addressing capability be able to recognize the new location of the station.
■ ESS transition: This is defined as a station movement from a BSS in one ESS to a BSS within another ESS. This case is supported only in the sense that
the station can move. Maintenance of upper-layer connections supported by
802.11 cannot be guaranteed. In fact, disruption of service is likely to occur.
To deliver a message within a DS, the distribution service needs to know where
the destination station is located. Specifically, the DS needs to know the identity of
the AP to which the message should be delivered in order for that message to reach
the destination station. To meet this requirement, a station must maintain an asso-
ciation with the AP within its current BSS. Three services relate to this requirement:
■ Association: Establishes an initial association between a station and an AP. Before a station can transmit or receive frames on a wireless LAN, its iden-
tity and address must be known. For this purpose, a station must establish an
association with an AP within a particular BSS. The AP can then communicate
this information to other APs within the ESS to facilitate routing and delivery
of addressed frames.
■ Reassociation: Enables an established association to be transferred from one AP to another, allowing a mobile station to move from one BSS to another.
■ Disassociation: A notification from either a station or an AP that an existing association is terminated. A station should give this notification before leaving
an ESS or shutting down. However, the MAC management facility protects
itself against stations that disappear without notification.
18.4 IEEE 802.11i WIRELESS LAN SECURITY
There are two characteristics of a wired LAN that are not inherent in a wireless LAN.
1. In order to transmit over a wired LAN, a station must be physically connected to the LAN. On the other hand, with a wireless LAN, any station within radio
range of the other devices on the LAN can transmit. In a sense, there is a form
of authentication with a wired LAN in that it requires some positive and pre-
sumably observable action to connect a station to a wired LAN.
2. Similarly, in order to receive a transmission from a station that is part of a wired LAN, the receiving station also must be attached to the wired LAN.
On the other hand, with a wireless LAN, any station within radio range can
receive. Thus, a wired LAN provides a degree of privacy, limiting reception of
data to stations connected to the LAN.
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596 CHAPTER 18 / WIRELESS NETWORK SECURITY
These differences between wired and wireless LANs suggest the increased
need for robust security services and mechanisms for wireless LANs. The original
802.11 specification included a set of security features for privacy and authenti-
cation that were quite weak. For privacy, 802.11 defined the Wired Equivalent Privacy (WEP) algorithm. The privacy portion of the 802.11 standard contained major weaknesses. Subsequent to the development of WEP, the 802.11i task
group has developed a set of capabilities to address the WLAN security issues.
In order to accelerate the introduction of strong security into WLANs, the Wi-Fi
Alliance promulgated Wi-Fi Protected Access (WPA) as a Wi-Fi standard. WPA is a set of security mechanisms that eliminates most 802.11 security issues and
was based on the current state of the 802.11i standard. The final form of the
802.11i standard is referred to as Robust Security Network (RSN). The Wi-Fi Alliance certifies vendors in compliance with the full 802.11i specification under
the WPA2 program.
The RSN specification is quite complex, and occupies 145 pages of the 2012
IEEE 802.11 standard. In this section, we provide an overview.
IEEE 802.11i Services
The 802.11i RSN security specification defines the following services.
■ Authentication: A protocol is used to define an exchange between a user and an AS that provides mutual authentication and generates temporary keys to
be used between the client and the AP over the wireless link.
■ Access control:1 This function enforces the use of the authentication function, routes the messages properly, and facilitates key exchange. It can work with a
variety of authentication protocols.
■ Privacy with message integrity: MAC-level data (e.g., an LLC PDU) are encrypted along with a message integrity code that ensures that the data have
not been altered.
Figure 18.6a indicates the security protocols used to support these services,
while Figure 18.6b lists the cryptographic algorithms used for these services.
IEEE 802.11i Phases of Operation
The operation of an IEEE 802.11i RSN can be broken down into five distinct phases
of operation. The exact nature of the phases will depend on the configuration and
the end points of the communication. Possibilities include (see Figure 18.5):
1. Two wireless stations in the same BSS communicating via the access point (AP) for that BSS.
2. Two wireless stations (STAs) in the same ad hoc IBSS communicating directly with each other.
1In this context, we are discussing access control as a security function. This is a different function than media access control (MAC) as described in Section 18.3. Unfortunately, the literature and the standards use the term access control in both contexts.
18.4 / IEEE 802.11i WIRELESS LAN SECURITY 597
3. Two wireless stations in different BSSs communicating via their respective APs across a distribution system.
4. A wireless station communicating with an end station on a wired network via its AP and the distribution system.
IEEE 802.11i security is concerned only with secure communication between
the STA and its AP. In case 1 in the preceding list, secure communication is assured
if each STA establishes secure communications with the AP. Case 2 is similar, with
the AP functionality residing in the STA. For case 3, security is not provided across
the distribution system at the level of IEEE 802.11, but only within each BSS. End-
to-end security (if required) must be provided at a higher layer. Similarly, in case 4,
security is only provided between the STA and its AP.
Figure 18.6 Elements of IEEE 802.11i
Access Control S
er vi
ce s
P ro
to co
ls S
er vi
ce s
A lg
or it
hm s
IEEE 802.1 Port-based
Access Control
Extensible Authentication Protocol (EAP)
Authentication and Key
Generation
(a) Services and protocols
Confidentiality, Data Origin Authentication
and Integrity and Replay Protection
TKIP CCMP
Robust Security Network (RSN)
Confidentiality
TKIP (Michael
MIC)
CCM (AES- CBC- MAC)
CCM (AES- CTR)
NIST Key
Wrap
HMAC- MD5
HMAC- SHA-1
Integrity and Data Origin
Authentication
(b) Cryptographic algorithms
Key Generation
TKIP (RC4)
Robust Security Network (RSN)
HMAC- SHA-1
RFC 1750
CBC-MAC = Cipher Block Chaining Message Authentication Code (MAC) CCM = Counter Mode with Cipher Block Chaining Message Authentication Code CCMP = Counter Mode with Cipher Block Chaining MAC Protocol TKIP = Temporal Key Integrity Protocol
598 CHAPTER 18 / WIRELESS NETWORK SECURITY
With these considerations in mind, Figure 18.7 depicts the five phases of op-
eration for an RSN and maps them to the network components involved. One new
component is the authentication server (AS). The rectangles indicate the exchange
of sequences of MPDUs. The five phases are defined as follows.
■ Discovery: An AP uses messages called Beacons and Probe Responses to ad- vertise its IEEE 802.11i security policy. The STA uses these to identify an AP
for a WLAN with which it wishes to communicate. The STA associates with
the AP, which it uses to select the cipher suite and authentication mechanism
when the Beacons and Probe Responses present a choice.
■ Authentication: During this phase, the STA and AS prove their identities to each other. The AP blocks non-authentication traffic between the STA and AS
until the authentication transaction is successful. The AP does not participate
in the authentication transaction other than forwarding traffic between the
STA and AS.
■ Key generation and distribution: The AP and the STA perform several opera- tions that cause cryptographic keys to be generated and placed on the AP and
the STA. Frames are exchanged between the AP and STA only.
■ Protected data transfer: Frames are exchanged between the STA and the end station through the AP. As denoted by the shading and the encryption module
icon, secure data transfer occurs between the STA and the AP only; security is
not provided end-to-end.
Figure 18.7 IEEE 802.11i Phases of Operation
Phase 1 - Discovery
STA AP AS End Station
Phase 5 - Connection Termination
Phase 3 - Key Management
Phase 4 - Protected Data Transfer
Phase 2 - Authentication
18.4 / IEEE 802.11i WIRELESS LAN SECURITY 599
■ Connection termination: The AP and STA exchange frames. During this phase, the secure connection is torn down and the connection is restored to the origi-
nal state.
Discovery Phase
We now look in more detail at the RSN phases of operation, beginning with the
discovery phase, which is illustrated in the upper portion of Figure 18.8. The pur-
pose of this phase is for an STA and an AP to recognize each other, agree on a set
of security capabilities, and establish an association for future communication using
those security capabilities.
Figure 18.8 IEEE 802.11i Phases of Operation: Capability Discovery, Authentication, and Association
STA AP AS
Probe requestStation sends a request to join network AP sends possible
security parameter (security capabilities set per the security policy)
AP performs null authentication
AP sends the associated security parameters
Station sends a request to perform
null authentication
Station sends a request to associate with AP with
security parameters
Station sets selected security parameters
Open system authentication request
Probe response
802.1X EAP request
Access request (EAP request)
802.1X EAP response
Accept/EAP-success key material
802.1X EAP success
Association request
Association response
Open system authentication response
802.1X-controlled port blocked
802.1X-controlled port blocked
Extensible Authentication Protocol Exchange
600 CHAPTER 18 / WIRELESS NETWORK SECURITY
SECURITY CAPABILITIES During this phase, the STA and AP decide on specific tech- niques in the following areas:
■ Confidentiality and MPDU integrity protocols for protecting unicast traffic
(traffic only between this STA and AP)
■ Authentication method
■ Cryptography key management approach
Confidentiality and integrity protocols for protecting multicast/broadcast traf-
fic are dictated by the AP, since all STAs in a multicast group must use the same
protocols and ciphers. The specification of a protocol, along with the chosen key
length (if variable) is known as a cipher suite. The options for the confidentiality and integrity cipher suite are
■ WEP, with either a 40-bit or 104-bit key, which allows backward compatibility
with older IEEE 802.11 implementations
■ TKIP
■ CCMP
■ Vendor-specific methods
The other negotiable suite is the authentication and key management (AKM)
suite, which defines (1) the means by which the AP and STA perform mutual au-
thentication and (2) the means for deriving a root key from which other keys may
be generated. The possible AKM suites are
■ IEEE 802.1X
■ Pre-shared key (no explicit authentication takes place and mutual authentica-
tion is implied if the STA and AP share a unique secret key)
■ Vendor-specific methods
MPDU EXCHANGE The discovery phase consists of three exchanges.
■ Network and security capability discovery: During this exchange, STAs dis- cover the existence of a network with which to communicate. The AP either
periodically broadcasts its security capabilities (not shown in figure), indicated
by RSN IE (Robust Security Network Information Element), in a specific
channel through the Beacon frame; or responds to a station’s Probe Request
through a Probe Response frame. A wireless station may discover available
access points and corresponding security capabilities by either passively moni-
toring the Beacon frames or actively probing every channel.
■ Open system authentication: The purpose of this frame sequence, which pro- vides no security, is simply to maintain backward compatibility with the IEEE
802.11 state machine, as implemented in existing IEEE 802.11 hardware. In
essence, the two devices (STA and AP) simply exchange identifiers.
■ Association: The purpose of this stage is to agree on a set of security capa- bilities to be used. The STA then sends an Association Request frame to
the AP. In this frame, the STA specifies one set of matching capabilities
18.4 / IEEE 802.11i WIRELESS LAN SECURITY 601
(one authentication and key management suite, one pairwise cipher suite,
and one group-key cipher suite) from among those advertised by the AP.
If there is no match in capabilities between the AP and the STA, the AP
refuses the Association Request. The STA blocks it too, in case it has associ-
ated with a rogue AP or someone is inserting frames illicitly on its channel.
As shown in Figure 18.8, the IEEE 802.1X controlled ports are blocked, and
no user traffic goes beyond the AP. The concept of blocked ports is explained
subsequently.
Authentication Phase
As was mentioned, the authentication phase enables mutual authentication between
an STA and an authentication server (AS) located in the DS. Authentication is
designed to allow only authorized stations to use the network and to provide the
STA with assurance that it is communicating with a legitimate network.
IEEE 802.1X ACCESS CONTROL APPROACH IEEE 802.11i makes use of another stan- dard that was designed to provide access control functions for LANs. The standard
is IEEE 802.1X, Port-Based Network Access Control. The authentication proto-
col that is used, the Extensible Authentication Protocol (EAP), is defined in the
IEEE 802.1X standard. IEEE 802.1X uses the terms supplicant, authenticator, and authentication server (AS). In the context of an 802.11 WLAN, the first two terms correspond to the wireless station and the AP. The AS is typically a separate device
on the wired side of the network (i.e., accessible over the DS) but could also reside
directly on the authenticator.
Before a supplicant is authenticated by the AS using an authentication proto-
col, the authenticator only passes control or authentication messages between the
supplicant and the AS; the 802.1X control channel is unblocked, but the 802.11 data
channel is blocked. Once a supplicant is authenticated and keys are provided, the
authenticator can forward data from the supplicant, subject to predefined access
control limitations for the supplicant to the network. Under these circumstances,
the data channel is unblocked.
As indicated in Figure 16.5, 802.1X uses the concepts of controlled and uncon-
trolled ports. Ports are logical entities defined within the authenticator and refer to
physical network connections. For a WLAN, the authenticator (the AP) may have
only two physical ports: one connecting to the DS and one for wireless communica-
tion within its BSS. Each logical port is mapped to one of these two physical ports.
An uncontrolled port allows the exchange of PDUs between the supplicant and the
other AS, regardless of the authentication state of the supplicant. A controlled port
allows the exchange of PDUs between a supplicant and other systems on the LAN
only if the current state of the supplicant authorizes such an exchange. IEEE 802.1X
is covered in more detail in Chapter 16.
The 802.1X framework, with an upper-layer authentication protocol, fits
nicely with a BSS architecture that includes a number of wireless stations and an
AP. However, for an IBSS, there is no AP. For an IBSS, 802.11i provides a more
complex solution that, in essence, involves pairwise authentication between stations
on the IBSS.
602 CHAPTER 18 / WIRELESS NETWORK SECURITY
MPDU EXCHANGE The lower part of Figure 18.8 shows the MPDU exchange dic- tated by IEEE 802.11 for the authentication phase. We can think of authentication
phase as consisting of the following three phases.
■ Connect to AS: The STA sends a request to its AP (the one with which it has an association) for connection to the AS. The AP acknowledges this request
and sends an access request to the AS.
■ EAP exchange: This exchange authenticates the STA and AS to each other. A number of alternative exchanges are possible, as explained subsequently.
■ Secure key delivery: Once authentication is established, the AS generates a master session key (MSK), also known as the Authentication, Authorization,
and Accounting (AAA) key and sends it to the STA. As explained subse-
quently, all the cryptographic keys needed by the STA for secure communi-
cation with its AP are generated from this MSK. IEEE 802.11i does not pre-
scribe a method for secure delivery of the MSK but relies on EAP for this.
Whatever method is used, it involves the transmission of an MPDU containing
an encrypted MSK from the AS, via the AP, to the AS.
EAP EXCHANGE As mentioned, there are a number of possible EAP exchanges that can be used during the authentication phase. Typically, the message flow between
STA and AP employs the EAP over LAN (EAPOL) protocol, and the message
flow between the AP and AS uses the Remote Authentication Dial In User Service
(RADIUS) protocol, although other options are available for both STA-to-AP and
AP-to-AS exchanges. [FRAN07] provides the following summary of the authenti-
cation exchange using EAPOL and RADIUS.
1. The EAP exchange begins with the AP issuing an EAP-Request/Identity frame to the STA.
2. The STA replies with an EAP-Response/Identity frame, which the AP receives over the uncontrolled port. The packet is then encapsulated in RADIUS over
EAP and passed on to the RADIUS server as a RADIUS-Access-Request packet.
3. The AAA server replies with a RADIUS-Access-Challenge packet, which is passed on to the STA as an EAP-Request. This request is of the appropriate
authentication type and contains relevant challenge information.
4. The STA formulates an EAP-Response message and sends it to the AS. The response is translated by the AP into a Radius-Access-Request with the re-
sponse to the challenge as a data field. Steps 3 and 4 may be repeated multiple
times, depending on the EAP method in use. For TLS tunneling methods, it is
common for authentication to require 10 to 20 round trips.
5. The AAA server grants access with a Radius-Access-Accept packet. The AP issues an EAP-Success frame. (Some protocols require confirmation of the
EAP success inside the TLS tunnel for authenticity validation.) The controlled
port is authorized, and the user may begin to access the network.
Note from Figure 18.8 that the AP controlled port is still blocked to general
user traffic. Although the authentication is successful, the ports remain blocked
18.4 / IEEE 802.11i WIRELESS LAN SECURITY 603
until the temporal keys are installed in the STA and AP, which occurs during the
4-Way Handshake.
Key Management Phase
During the key management phase, a variety of cryptographic keys are generated
and distributed to STAs. There are two types of keys: pairwise keys used for com-
munication between an STA and an AP and group keys used for multicast com-
munication. Figure 18.9, based on [FRAN07], shows the two key hierarchies, and
Table 18.3 defines the individual keys.
Figure 18.9 IEEE 802.11i Key Hierarchies
Out-of-band path EAP method path
Pre-shared key
EAPOL key confirmation key EAPOL key encryption key Temporal key
PSK
256 bits
384 bits (CCMP) 512 bits (TKIP)
128 bits (CCMP) 256 bits (TKIP)
40 bits, 104 bits (WEP) 128 bits (CCMP) 256 bits (TKIP)
256 bits
128 bits
No modification Legend
Possible truncation PRF (pseudo random function) using HMAC-SHA-1
128 bits
User-defined cryptoid
EAP authentication
Following EAP authentication or PSK
During 4-way handshake
These keys are components of the PTK
≥ 256 bits
PMK
KCK
PTK
KTKEK
AAAK or MSK
Pairwise master key
(b) Group key hierarchy
(a) Pairwise key hierarchy
AAA key
Pairwise transient key
256 bits Changes periodically or if compromised
Changes based on policy (dissociation, deauthentication)
GMK (generated by AS)
GTK
Group master key
Group temporal key
604 CHAPTER 18 / WIRELESS NETWORK SECURITY
Abbreviation Name Description / Purpose Size (bits) Type
AAA Key Authentication,
Accounting, and
Authorization Key
Used to derive the PMK.
Used with the IEEE
802.1X authentication
and key management
approach. Same as
MMSK.
Ú 256 Key generation key, root key
PSK Pre-shared Key Becomes the PMK
in pre-shared key
environments.
256 Key generation key,
root key
PMK Pairwise Master Key Used with other inputs to
derive the PTK.
256 Key generation key
GMK Group Master Key Used with other inputs to
derive the GTK.
128 Key generation key
PTK Pair-wise Transient
Key
Derived from the PMK.
Comprises the EAPOL-
KCK, EAPOL-KEK, and
TK and (for TKIP) the
MIC key.
512 (TKIP)
384 (CCMP)
Composite key
TK Temporal Key Used with TKIP or
CCMP to provide
confidentiality and
integrity protection for
unicast user traffic.
256 (TKIP)
128 (CCMP)
Traffic key
GTK Group Temporal Key Derived from the
GMK. Used to provide
confidentiality and
integrity protection for
multicast/broadcast user
traffic.
256 (TKIP)
128 (CCMP)
40,104 (WEP)
Traffic key
MIC Key Message Integrity
Code Key
Used by TKIP’s Michael
MIC to provide integrity
protection of messages.
64 Message integrity key
EAPOL-KCK EAPOL-Key
Confirmation Key
Used to provide integrity
protection for key
material distributed
during the 4-Way
Handshake.
128 Message integrity key
EAPOL-KEK EAPOL-Key
Encryption Key
Used to ensure the
confidentiality of the
GTK and other key
material in the 4-Way
Handshake.
128 Traffic key / key
encryption key
WEP Key Wired Equivalent
Privacy Key
Used with WEP. 40,104 Traffic key
Table 18.3 IEEE 802.11i Keys for Data Confidentiality and Integrity Protocols
Hiva-Network.Com
18.4 / IEEE 802.11i WIRELESS LAN SECURITY 605
PAIRWISE KEYS Pairwise keys are used for communication between a pair of de- vices, typically between an STA and an AP. These keys form a hierarchy beginning
with a master key from which other keys are derived dynamically and used for a
limited period of time.
At the top level of the hierarchy are two possibilities. A pre-shared key (PSK) is a secret key shared by the AP and a STA and installed in some fashion outside
the scope of IEEE 802.11i. The other alternative is the master session key (MSK), also known as the AAAK, which is generated using the IEEE 802.1X protocol dur-
ing the authentication phase, as described previously. The actual method of key
generation depends on the details of the authentication protocol used. In either case
(PSK or MSK), there is a unique key shared by the AP with each STA with which
it communicates. All the other keys derived from this master key are also unique
between an AP and an STA. Thus, each STA, at any time, has one set of keys, as
depicted in the hierarchy of Figure 18.9a, while the AP has one set of such keys for
each of its STAs.
The pairwise master key (PMK) is derived from the master key. If a PSK is used, then the PSK is used as the PMK; if a MSK is used, then the PMK is derived
from the MSK by truncation (if necessary). By the end of the authentication phase,
marked by the 802.1X EAP Success message (Figure 18.8), both the AP and the
STA have a copy of their shared PMK.
The PMK is used to generate the pairwise transient key (PTK), which in fact consists of three keys to be used for communication between an STA and AP after
they have been mutually authenticated. To derive the PTK, the HMAC-SHA-1
function is applied to the PMK, the MAC addresses of the STA and AP, and nonces
generated when needed. Using the STA and AP addresses in the generation of the
PTK provides protection against session hijacking and impersonation; using nonces
provides additional random keying material.
The three parts of the PTK are as follows.
■ EAP Over LAN (EAPOL) Key Confirmation Key (EAPOL-KCK): Supports the integrity and data origin authenticity of STA-to-AP control frames during
operational setup of an RSN. It also performs an access control function:
proof-of-possession of the PMK. An entity that possesses the PMK is autho-
rized to use the link.
■ EAPOL Key Encryption Key (EAPOL-KEK): Protects the confidentiality of keys and other data during some RSN association procedures.
■ Temporal Key (TK): Provides the actual protection for user traffic.
GROUP KEYS Group keys are used for multicast communication in which one STA sends MPDU’s to multiple STAs. At the top level of the group key hierarchy is the
group master key (GMK). The GMK is a key-generating key used with other inputs to derive the group temporal key (GTK). Unlike the PTK, which is generated using material from both AP and STA, the GTK is generated by the AP and transmitted
606 CHAPTER 18 / WIRELESS NETWORK SECURITY
to its associated STAs. Exactly how this GTK is generated is undefined. IEEE
802.11i, however, requires that its value is computationally indistinguishable from
random. The GTK is distributed securely using the pairwise keys that are already
established. The GTK is changed every time a device leaves the network.
PAIRWISE KEY DISTRIBUTION The upper part of Figure 18.10 shows the MPDU exchange for distributing pairwise keys. This exchange is known as the 4-way handshake. The STA and AP use this handshake to confirm the existence of the
Figure 18.10 IEEE 802.11i Phases of Operation: Four-Way Handshake and Group Key Handshake
STA AP
Message 1 delivers a nonce to the STA so that it can generate the PTK.
Message 1 delivers a new GTK to the STA. The GTK is encrypted before it is sent and the entire message is integrity protected.
The AP installs the GTK.
Message 3 demonstrates to the STA that the authenticator is alive, ensures that the PTK is fresh (new) and that there is no man-in-the-middle.
Message 2 delivers another nonce to the AP so that it can also generate the PTK. It demonstrates to the AP that the STA is alive, ensures that the PTK is fresh (new) and that there is no man-in-the-middle.
The STA decrypts the GTK and installs it for use.
Message 2 is delivered to the AP. This frame serves only as an acknowledgment to the AP.
Message 4 serves as an acknowledgment to Message 3. It serves no cryptographic function. This message also ensures the reliable start of the group key handshake.
Message 2 EAPOL-key (Snonce,
Unicast, MIC)
Message 1 EAPOL-key (Anonce, Unicast)
Message 1 EAPOL-key (GTK, MIC)
Message 4 EAPOL-key (Unicast, MIC)
Message 2 EAPOL-key (MIC)
Message 3 EAPOL-key (Install PTK,
Unicast, MIC)
AP’s 802.1X-controlled port blocked
AP’s 802.1X-controlled port unblocked for unicast traffic
18.4 / IEEE 802.11i WIRELESS LAN SECURITY 607
PMK, verify the selection of the cipher suite, and derive a fresh PTK for the follow-
ing data session. The four parts of the exchange are as follows.
■ AP S STA: Message includes the MAC address of the AP and a nonce (Anonce)
■ STA S AP: The STA generates its own nonce (Snonce) and uses both nonces and both MAC addresses, plus the PMK, to generate a PTK. The STA then
sends a message containing its MAC address and Snonce, enabling the AP to
generate the same PTK. This message includes a message integrity code
(MIC)2 using HMAC-MD5 or HMAC-SHA-1-128. The key used with the MIC
is KCK.
■ AP S STA: The AP is now able to generate the PTK. The AP then sends a message to the STA, containing the same information as in the first message,
but this time including a MIC.
■ STA S AP: This is merely an acknowledgment message, again protected by a MIC.
GROUP KEY DISTRIBUTION For group key distribution, the AP generates a GTK and distributes it to each STA in a multicast group. The two-message exchange with
each STA consists of the following:
■ AP S STA: This message includes the GTK, encrypted either with RC4 or with AES. The key used for encryption is KEK, using a key wrapping algo-
rithm (as discussed in Chapter 12). A MIC value is appended.
■ STA S AP: The STA acknowledges receipt of the GTK. This message includes a MIC value.
Protected Data Transfer Phase
IEEE 802.11i defines two schemes for protecting data transmitted in 802.11 MPDUs:
the Temporal Key Integrity Protocol (TKIP), and the Counter Mode-CBC MAC
Protocol (CCMP).
TKIP TKIP is designed to require only software changes to devices that are imple- mented with the older wireless LAN security approach called Wired Equivalent
Privacy (WEP). TKIP provides two services:
■ Message integrity: TKIP adds a message integrity code (MIC) to the 802.11 MAC frame after the data field. The MIC is generated by an algorithm, called
Michael, that computes a 64-bit value using as input the source and destination
MAC address values and the Data field, plus key material.
■ Data confidentiality: Data confidentiality is provided by encrypting the MPDU plus MIC value using RC4.
2 While MAC is commonly used in cryptography to refer to a Message Authentication Code, the term MIC is used instead in connection with 802.11i because MAC has another standard meaning, Media Access Control, in networking.
608 CHAPTER 18 / WIRELESS NETWORK SECURITY
The 256-bit TK (Figure 18.9) is employed as follows. Two 64-bit keys are used
with the Michael message digest algorithm to produce a message integrity code.
One key is used to protect STA-to-AP messages, and the other key is used to pro-
tect AP-to-STA messages. The remaining 128 bits are truncated to generate the
RC4 key used to encrypt the transmitted data.
For additional protection, a monotonically increasing TKIP sequence counter
(TSC) is assigned to each frame. The TSC serves two purposes. First, the TSC is
included with each MPDU and is protected by the MIC to protect against replay
attacks. Second, the TSC is combined with the session TK to produce a dynamic en-
cryption key that changes with each transmitted MPDU, thus making cryptanalysis
more difficult.
CCMP CCMP is intended for newer IEEE 802.11 devices that are equipped with the hardware to support this scheme. As with TKIP, CCMP provides two services:
■ Message integrity: CCMP uses the cipher block chaining message authentica- tion code (CBC-MAC), described in Chapter 12.
■ Data confidentiality: CCMP uses the CTR block cipher mode of operation with AES for encryption. CTR is described in Chapter 7.
The same 128-bit AES key is used for both integrity and confidentiality. The
scheme uses a 48-bit packet number to construct a nonce to prevent replay attacks.
The IEEE 802.11i Pseudorandom Function
At a number of places in the IEEE 802.11i scheme, a pseudorandom function (PRF) is
used. For example, it is used to generate nonces, to expand pairwise keys, and to gen-
erate the GTK. Best security practice dictates that different pseudorandom number
streams be used for these different purposes. However, for implementation efficiency,
we would like to rely on a single pseudorandom number generator function.
The PRF is built on the use of HMAC-SHA-1 to generate a pseudorandom
bit stream. Recall that HMAC-SHA-1 takes a message (block of data) and a key of
length at least 160 bits and produces a 160-bit hash value. SHA-1 has the property
that the change of a single bit of the input produces a new hash value with no appar-
ent connection to the preceding hash value. This property is the basis for pseudo-
random number generation.
The IEEE 802.11i PRF takes four parameters as input and produces the de-
sired number of random bits. The function is of the form PRF(K, A, B, Len), where
K = a secret key A = a text string specific to the application (e.g., nonce generation or pairwise
key expansion)
B = some data specific to each case Len = desired number of pseudorandom bits
For example, for the pairwise transient key for CCMP:
PTK = PRF (PMK, “Pairwise key expansion”, min (AP- Addr, STA-Addr) || max (AP-Addr, STA-Addr) || min (Anonce, Snonce) || max (Anonce, Snonce), 384)
18.4 / IEEE 802.11i WIRELESS LAN SECURITY 609
So, in this case, the parameters are
K = PMK A = the text string “Pairwise key expansion” B = a sequence of bytes formed by concatenating the two MAC addresses
and the two nonces
Len = 384 bits
Similarly, a nonce is generated by
Nonce = PRF (Random Number, “InitCounter”, MAC || Time, 256)
where Time is a measure of the network time known to the nonce generator. The group temporal key is generated by
GTK = PRF (GMK, “Group key expansion”, MAC || Gnonce, 256)
Figure 18.11 illustrates the function PRF(K, A, B, Len). The parameter K serves as the key input to HMAC. The message input consists of four items concat-
enated together: the parameter A, a byte with value 0, the parameter B, and a coun- ter i. The counter is initialized to 0. The HMAC algorithm is run once, producing a 160-bit hash value. If more bits are required, HMAC is run again with the same
inputs, except that i is incremented each time until the necessary number of bits is generated. We can express the logic as
PRF (K, A, B, Len) R
S
null string for i
S
0 to ((Len + 159)/160 − 1) do R
S
R || HMAC-SHA-1 (K, A || 0 || B || i) Return Truncate-to-Len (R, Len)
Figure 18.11 IEEE 802.11i Pseudorandom Function
HMAC-SHA-1
| |
K
A 0 B i
R = HMAC-SHA-1(K, A || 0 || B || i)
+ 1
610 CHAPTER 18 / WIRELESS NETWORK SECURITY
18.5 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
Key Terms
4-way handshake
access point (AP)
basic service set (BSS)
Counter Mode-CBC MAC
Protocol (CCMP)
distribution system (DS)
extended service set (ESS)
group keys
IEEE 802.1X
IEEE 802.11
IEEE 802.11i
independent BSS (IBSS)
logical link control (LLC)
media access control (MAC)
MAC protocol data unit
(MPDU)
MAC service data unit
(MSDU)
message integrity code
(MIC)
Michael
pairwise keys
pseudorandom function
Robust Security Network
(RSN)
Temporal Key Integrity
Protocol (TKIP)
Wi-Fi
Wi-Fi Protected Access
(WPA)
Wired Equivalent Privacy
(WEP)
Wireless LAN (WLAN)
Review Questions
18.1 What is the basic building block of an 802.11 WLAN? 18.2 List and briefly define threats to a wireless network. 18.3 List and briefly define IEEE 802.11 services. 18.4 List some security threats related to mobile devices. 18.5 How is the concept of an association related to that of mobility? 18.6 What security areas are addressed by IEEE 802.11i? 18.7 Briefly describe the five IEEE 802.11i phases of operation. 18.8 What is the difference between TKIP and CCMP?
Problems
18.1 In IEEE 802.11, open system authentication simply consists of two communications. An authentication is requested by the client, which contains the station ID (typically the MAC address). This is followed by an authentication response from the AP/router containing a success or failure message. An example of when a failure may occur is if the client’s MAC address is explicitly excluded in the AP/router configuration. a. What are the benefits of this authentication scheme? b. What are the security vulnerabilities of this authentication scheme?
18.2 Prior to the introduction of IEEE 802.11i, the security scheme for IEEE 802.11 was Wired Equivalent Privacy (WEP). WEP assumed all devices in the network share a secret key. The purpose of the authentication scenario is for the STA to prove that it possesses the secret key. Authentication proceeds as shown in Figure 18.12. The STA sends a message to the AP requesting authentication. The AP issues a chal- lenge, which is a sequence of 128 random bytes sent as plaintext. The STA encrypts the challenge with the shared key and returns it to the AP. The AP decrypts the incoming value and compares it to the challenge that it sent. If there is a match, the AP confirms that authentication has succeeded. a. What are the benefits of this authentication scheme? b. This authentication scheme is incomplete. What is missing and why is this impor-
tant? Hint: The addition of one or two messages would fix the problem. c. What is a cryptographic weakness of this scheme?
18.5 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 611
18.3 For WEP, data integrity and data confidentiality are achieved using the RC4 stream encryption algorithm. The transmitter of an MPDU performs the following steps, referred to as encapsulation: 1. The transmitter selects an initial vector (IV) value. 2. The IV value is concatenated with the WEP key shared by transmitter and receiver
to form the seed, or key input, to RC4. 3. A 32-bit cyclic redundancy check (CRC) is computed over all the bits of the MAC
data field and appended to the data field. The CRC is a common error-detection code used in data link control protocols. In this case, the CRC serves as a integrity check value (ICV).
4. The result of step 3 is encrypted using RC4 to form the ciphertext block. 5. The plaintext IV is prepended to the ciphertext block to form the encapsulated
MPDU for transmission. a. Draw a block diagram that illustrates the encapsulation process. b. Describe the steps at the receiver end to recover the plaintext and perform the
integrity check. c. Draw a block diagram that illustrates part b.
18.4 A potential weakness of the CRC as an integrity check is that it is a linear function. This means that you can predict which bits of the CRC are changed if a single bit of the message is changed. Furthermore, it is possible to determine which combination of bits could be flipped in the message so that the net result is no change in the CRC. Thus, there are a number of combinations of bit flippings of the plaintext message that leave the CRC unchanged, so message integrity is defeated. However, in WEP, if an attacker does not know the encryption key, the attacker does not have access to the plaintext, only to the ciphertext block. Does this mean that the ICV is protected from the bit flipping attack? Explain.
Figure 18.12 WEP Authentication; refer to Problem 18.2
STA AP
RequestStation sends a request for authentication
AP sends challenge message containing 128-bit random number
AP decrypts challenge response. If match, send authentication success message
Station responds with encrypted version
of challenge number
Response
Challenge
Success
612612
CHAPTER
Electronic Mail Security 19.1 Internet Mail Architecture
Email Components
Email Protocols
19.2 Email Formats
RFC 5322
Multipurpose Internet Mail Extensions
19.3 Email Threats and Comprehensive Email Security
19.4 S/MIME
Operational Description
S/MIME Message Content Types
Approved Cryptographic Algorithms
S/MIME Messages
S/MIME Certificate Processing
Enhanced Security Services
19.5 Pretty Good Privacy
19.6 DNSSEC
Domain Name System
DNS Security Extensions
19.7 DNS-Based Authentication of Named Entities
TLSA Record
Use of DANE for SMTP
Use of DNSSEC for S/MIME
19.8 Sender Policy Framework
SPF on the Sender Side
SPF on the Receiver Side
19.9 DomainKeys Identified Mail
Email Threats
DKIM Strategy
DKIM Functional Flow
19.1 / INTERNET MAIL ARCHITECTURE 613
19.1 INTERNET MAIL ARCHITECTURE
For an understanding of the topics in this chapter, it is useful to have a basic grasp of
the Internet mail architecture, which is currently defined in RFC 5598 (Internet Mail Architecture, July 2009). This section provides an overview of the basic concepts.
19.10 Domain-Based Message Authentication, Reporting, and Conformance
Identifier Alignment
DMARC on the Sender Side
DMARC on the Receiver Side
DMARC Reports
19.11 Key Terms, Review Questions, and Problems
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Summarize the key functional components of the Internet mail architecture.
◆ Explain the basic functionality of SMTP, POP3, and IMAP.
◆ Explain the need for MIME as an enhancement to ordinary email.
◆ Describe the key elements of MIME.
◆ Understand the functionality of S/MIME and the security threats it addresses.
◆ Understand the basic mechanisms of STARTTLS and its role in email security.
◆ Understand the basic mechanisms of DANE and its role in email security.
◆ Understand the basic mechanisms of SPF and its role in email security.
◆ Understand the basic mechanisms of DKIM and its role in email security.
◆ Understand the basic mechanisms of DMARC and its role in email security.
In virtually all distributed environments, electronic mail is the most heavily used
network-based application. Users expect to be able to, and do, send email to others
who are connected directly or indirectly to the Internet, regardless of host operat-
ing system or communications suite. With the explosively growing reliance on email,
there grows a demand for authentication and confidentiality services. Two schemes
stand out as approaches that enjoy widespread use: Pretty Good Privacy (PGP) and
S/MIME. Both are examined in this chapter. This chapter concludes with a discussion
of DomainKeys Identified Mail.
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614 CHAPTER 19 / ELECTRONIC MAIL SECURITY
Email Components
At its most fundamental level, the Internet mail architecture consists of a user world
in the form of Message User Agents (MUA), and the transfer world, in the form
of the Message Handling Service (MHS), which is composed of Message Transfer Agents (MTA). The MHS accepts a message from one user and delivers it to one
or more other users, creating a virtual MUA-to-MUA exchange environment. This
architecture involves three types of interoperability. One is directly between users:
messages must be formatted by the MUA on behalf of the message author so that
the message can be displayed to the message recipient by the destination MUA.
There are also interoperability requirements between the MUA and the MHS—
first when a message is posted from an MUA to the MHS and later when it is deliv-
ered from the MHS to the destination MUA. Interoperability is required among the
MTA components along the transfer path through the MHS.
Figure 19.1 illustrates the key components of the Internet mail architecture,
which include the following.
■ Message User Agent (MUA): Operates on behalf of user actors and user applications. It is their representative within the email service. Typically, this
function is housed in the user’s computer and is referred to as a client email
Figure 19.1 Function Modules and Standardized Protocols Used between them in the Internet Mail Architecture
Message user agent (MUA)
Message author
Message recipient
ESMTP (Submission)
SMTP
SMTP SMTP
ESMTP (Submission)
(SMTP, local)
(IMAP, POP, local)
Mail submission agent (MSA)
Message transfer agent (MTA)
Message transfer agent (MTA)
MESSAGE HANDLING SYSTEM (MHS)
Message transfer agent (MTA)
Mail delivery agent (MDA)
Message store (MS)
Message user agent (MUA)
19.1 / INTERNET MAIL ARCHITECTURE 615
program or a local network email server. The author MUA formats a message
and performs initial submission into the MHS via a MSA. The recipient MUA
processes received mail for storage and/or display to the recipient user.
■ Mail Submission Agent (MSA): Accepts the message submitted by an MUA and enforces the policies of the hosting domain and the requirements of
Internet standards. This function may be located together with the MUA or
as a separate functional model. In the latter case, the Simple Mail Transfer
Protocol (SMTP) is used between the MUA and the MSA.
■ Message Transfer Agent (MTA): Relays mail for one application-level hop. It is like a packet switch or IP router in that its job is to make routing assessments
and to move the message closer to the recipients. Relaying is performed by a
sequence of MTAs until the message reaches a destination MDA. An MTA
also adds trace information to the message header. SMTP is used between
MTAs and between an MTA and an MSA or MDA.
■ Mail Delivery Agent (MDA): Responsible for transferring the message from the MHS to the MS.
■ Message Store (MS): An MUA can employ a long-term MS. An MS can be located on a remote server or on the same machine as the MUA. Typically,
an MUA retrieves messages from a remote server using POP (Post Office
Protocol) or IMAP (Internet Message Access Protocol).
Two other concepts need to be defined. An administrative management domain (ADMD) is an Internet email provider. Examples include a department that operates a local mail relay (MTA), an IT department that operates an enterprise
mail relay, and an ISP that operates a public shared email service. Each ADMD
can have different operating policies and trust-based decision making. One obvi-
ous example is the distinction between mail that is exchanged within an organiza-
tion and mail that is exchanged between independent organizations. The rules for
handling the two types of traffic tend to be quite different.
The Domain Name System (DNS) is a directory lookup service that provides a mapping between the name of a host on the Internet and its numerical address.
DNS is discussed subsequently in this chapter.
Email Protocols
Two types of protocols are used for transferring email. The first type is used to move
messages through the Internet from source to destination. The protocol used for
this purpose is SMTP, with various extensions and in some cases restrictions. The
second type consists of protocols used to transfer messages between mail servers, of
which IMAP and POP are the most commonly used.
SIMPLE MAIL TRANSFER PROTOCOL SMTP encapsulates an email message in an envelope and is used to relay the encapsulated messages from source to destination
through multiple MTAs. SMTP was originally specified in 1982 as RFC 821 and has
undergone several revisions, the most current being RFC 5321 (October 2008). These
revisions have added additional commands and introduced extensions. The term
Extended SMTP (ESMTP) is often used to refer to these later versions of SMTP.
616 CHAPTER 19 / ELECTRONIC MAIL SECURITY
SMTP is a text-based client-server protocol where the client (email sender)
contacts the server (next-hop recipient) and issues a set of commands to tell the
server about the message to be sent, then sending the message itself. The majority
of these commands are ASCII text messages sent by the client and a resulting return
code (and additional ASCII text) returned by the server.
The transfer of a message from a source to its ultimate destination can occur
over a single SMTP client/server conversation over a single TCP connection.
Alternatively, an SMTP server may be an intermediate relay that assumes the role
of an SMTP client after receiving a message and then forwards that message to an
SMTP server along a route to the ultimate destination.
The operation of SMTP consists of a series of commands and responses
exchanged between the SMTP sender and receiver. The initiative is with the SMTP
sender, who establishes the TCP connection. Once the connection is established,
the SMTP sender sends commands over the connection to the receiver. Each com-
mand consists of a single line of text, beginning with a four-letter command code
followed in some cases by an argument field. Each command generates exactly one
reply from the SMTP receiver. Most replies are a single-line, although multiple-line
replies are possible. Each reply begins with a three-digit code and may be followed
by additional information.
Figure 19.2 illustrates the SMTP exchange between a client (C) and server (S).
The interchange begins with the client establishing a TCP connection to TCP port
25 on the server (not shown in figure). This causes the server to activate SMTP
S: 220 foo.com Simple Mail Transfer Service Ready
C: HELO bar.com
S: 250 OK
C: MAIL FROM:<[email protected]>
S: 250 OK
C: RCPT TO:<[email protected]>
S: 250 OK
C: RCPT TO:<[email protected]>
S: 550 No such user here
C: RCPT TO:<[email protected]>
S: 250 OK
C: DATA
S: 354 Start mail input; end with <crlf>.<crlf>
C: Blah blah blah . . .
C: . . . etc. etc. etc.
C: <crlf><crlf>
S: 250 OK
C: QUIT
S: 221 foo.com Service closing transmission channel
Figure 19.2 Example SMTP Transaction Scenario
19.2 / EMAIL FORMATS 617
and send a 220 reply to the client. The HELO command identifies the sending
domain, which the server acknowledges and accepts with a 250 reply. The SMTP
sender is transmitting mail that originates with the user [email protected]. The MAIL
command identifies the originator of the message. The message is addressed to
three users on machine foo.com, namely, Jones, Green, and Brown. The client iden-
tifies each of these in a separate RCPT command. The SMTP receiver indicates
that it has mailboxes for Jones and Brown but does not have information on Green.
Because at least one of the intended recipients has been verified, the client proceeds
to send the text message, by first sending a DATA command to ensure the server
is ready for the data. After the server acknowledges receipt of all the data, it issues
a 250 OK message. Then the client issues a QUIT command and the server closes
the connection.
A significant security-related extension for SMTP, called STARTTLS, is
defined in RFC 3207 (SMTP Service Extension for Secure SMTP over Transport Layer Security, February 2002). STARTTLS enables the addition of confidentiality and authentication in the exchange between SMTP agents. This gives SMTP agents
the ability to protect some or all of their communications from eavesdroppers
and attackers. If the client does initiate the connection over a TLS-enabled port
(e.g., port 465 was previously used for SMTP over SSL), the server may prompt with
a message indicating that the STARTTLS option is available. The client can then
issue the STARTTLS command in the SMTP command stream, and the two parties
proceed to establish a secure TLS connection. An advantage of using STARTTLS
is that the server can offer SMTP service on a single port, rather than requiring
separate port numbers for secure and cleartext operations. Similar mechanisms are
available for running TLS over IMAP and POP protocols.
Historically, MUA/MSA message transfers have used SMTP. The standard
currently preferred is SUBMISSION, defined in RFC 6409 (Message Submission for Mail, November 2011). Although SUBMISSION derives from SMTP, it uses a separate TCP port and imposes distinct requirements, such as access authorization.
MAIL ACCESS PROTOCOLS (POP3, IMAP) Post Office Protocol (POP3) allows an email client (user agent) to download an email from an email server (MTA). POP3
user agents connect via TCP to the server (typically port 110). The user agent enters
a username and password (either stored internally for convenience or entered each
time by the user for stronger security). After authorization, the UA can issue POP3
commands to retrieve and delete mail.
As with POP3, Internet Mail Access Protocol (IMAP) also enables an email
client to access mail on an email server. IMAP also uses TCP, with server TCP port
143. IMAP is more complex than POP3. IMAP provides stronger authentication
than POP3 and provides other functions not supported by POP3.
19.2 EMAIL FORMATS
To understand S/MIME, we need first to have a general understanding of the
underlying email format that it uses, namely, MIME. But to understand the sig-
nificance of MIME, we need to go back to the traditional email format standard,
618 CHAPTER 19 / ELECTRONIC MAIL SECURITY
RFC 822, which is still in common use. The most recent version of this format speci-
fication is RFC 5322 (Internet Message Format, October 2008). Accordingly, this section first provides an introduction to these two earlier standards and then moves
on to a discussion of S/MIME.
RFC 5322
RFC 5322 defines a format for text messages that are sent using electronic mail. It
has been the standard for Internet-based text mail messages and remains in com-
mon use. In the RFC 5322 context, messages are viewed as having an envelope and
contents. The envelope contains whatever information is needed to accomplish
transmission and delivery. The contents compose the object to be delivered to the
recipient. The RFC 5322 standard applies only to the contents. However, the con-
tent standard includes a set of header fields that may be used by the mail system to
create the envelope, and the standard is intended to facilitate the acquisition of such
information by programs.
The overall structure of a message that conforms to RFC 5322 is very simple.
A message consists of some number of header lines (the header) followed by unrestricted text (the body). The header is separated from the body by a blank line. Put differently, a message is ASCII text, and all lines up to the first blank line are
assumed to be header lines used by the user agent part of the mail system.
A header line usually consists of a keyword, followed by a colon, followed by
the keyword’s arguments; the format allows a long line to be broken up into several
lines. The most frequently used keywords are From, To, Subject, and Date. Here is an example message:
Date: October 8, 2009 2:15:49 PM EDT
From: “William Stallings” <[email protected]>
Subject: The Syntax in RFC 5322
Hello. This section begins the actual
message body, which is delimited from the
message heading by a blank line.
Another field that is commonly found in RFC 5322 headers is Message-ID. This field contains a unique identifier associated with this message.
Multipurpose Internet Mail Extensions
Multipurpose Internet Mail Extension (MIME) is an extension to the RFC 5322
framework that is intended to address some of the problems and limitations of the
use of Simple Mail Transfer Protocol (SMTP) or some other mail transfer protocol
and RFC 5322 for electronic mail. RFCs 2045 through 2049 define MIME, and there
have been a number of updating documents since then.
19.2 / EMAIL FORMATS 619
As justification for the use of MIME, [PARZ06] lists the following limitations
of the SMTP/5322 scheme.
1. SMTP cannot transmit executable files or other binary objects. A number of schemes are in use for converting binary files into a text form that can be used
by SMTP mail systems, including the popular UNIX UUencode/UUdecode
scheme. However, none of these is a standard or even a de facto standard.
2. SMTP cannot transmit text data that includes national language characters, because these are represented by 8-bit codes with values of 128 decimal or
higher, and SMTP is limited to 7-bit ASCII.
3. SMTP servers may reject mail message over a certain size.
4. SMTP gateways that translate between ASCII and the character code EBCDIC do not use a consistent set of mappings, resulting in translation problems.
5. SMTP gateways to X.400 electronic mail networks cannot handle nontextual data included in X.400 messages.
6. Some SMTP implementations do not adhere completely to the SMTP standards defined in RFC 821. Common problems include:
—Deletion, addition, or reordering of carriage return and linefeed
—Truncating or wrapping lines longer than 76 characters
—Removal of trailing white space (tab and space characters)
—Padding of lines in a message to the same length
—Conversion of tab characters into multiple space characters
MIME is intended to resolve these problems in a manner that is compatible
with existing RFC 5322 implementations.
OVERVIEW The MIME specification includes the following elements.
1. Five new message header fields are defined, which may be included in an RFC 5322 header. These fields provide information about the body of the
message.
2. A number of content formats are defined, thus standardizing representations that support multimedia electronic mail.
3. Transfer encodings are defined that enable the conversion of any content format into a form that is protected from alteration by the mail system.
In this subsection, we introduce the five message header fields. The next two
subsections deal with content formats and transfer encodings.
The five header fields defined in MIME are as follows:
■ MIME-Version: Must have the parameter value 1.0. This field indicates that the message conforms to RFCs 2045 and 2046.
■ Content-Type: Describes the data contained in the body with sufficient detail that the receiving user agent can pick an appropriate agent or mechanism to
represent the data to the user or otherwise deal with the data in an appropriate
manner.
620 CHAPTER 19 / ELECTRONIC MAIL SECURITY
■ Content-Transfer-Encoding: Indicates the type of transformation that has been used to represent the body of the message in a way that is acceptable for
mail transport.
■ Content-ID: Used to identify MIME entities uniquely in multiple contexts.
■ Content-Description: A text description of the object with the body; this is useful when the object is not readable (e.g., audio data).
Any or all of these fields may appear in a normal RFC 5322 header. A compli-
ant implementation must support the MIME-Version, Content-Type, and Content-
Transfer-Encoding fields; the Content-ID and Content-Description fields are
optional and may be ignored by the recipient implementation.
MIME CONTENT TYPES The bulk of the MIME specification is concerned with the definition of a variety of content types. This reflects the need to provide stan-
dardized ways of dealing with a wide variety of information representations in a
multimedia environment.
Table 19.1 lists the content types specified in RFC 2046. There are seven dif-
ferent major types of content and a total of 15 subtypes. In general, a content type
declares the general type of data, and the subtype specifies a particular format for
that type of data.
Type Subtype Description
Text Plain Unformatted text; may be ASCII or ISO 8859.
Enriched Provides greater format flexibility.
Multipart Mixed The different parts are independent but are to be transmitted
together. They should be presented to the receiver in the order
that they appear in the mail message.
Parallel Differs from Mixed only in that no order is defined for delivering
the parts to the receiver.
Alternative The different parts are alternative versions of the same
information. They are ordered in increasing faithfulness to the
original, and the recipient’s mail system should display the “best”
version to the user.
Digest Similar to Mixed, but the default type/subtype of each part is
message/rfc822.
Message rfc822 The body is itself an encapsulated message that conforms to RFC 822.
Partial Used to allow fragmentation of large mail items, in a way that is
transparent to the recipient.
External-body Contains a pointer to an object that exists elsewhere.
Image jpeg The image is in JPEG format, JFIF encoding.
gif The image is in GIF format.
Video mpeg MPEG format.
Audio Basic Single-channel 8-bit ISDN m-law encoding at a sample rate of
8 kHz.
Application PostScript Adobe Postscript format.
octet-stream General binary data consisting of 8-bit bytes.
Table 19.1 MIME Content Types
19.2 / EMAIL FORMATS 621
For the text type of body, no special software is required to get the full meaning of the text aside from support of the indicated character set. The primary subtype is
plain text, which is simply a string of ASCII characters or ISO 8859 characters. The enriched subtype allows greater formatting flexibility.
The multipart type indicates that the body contains multiple, independent parts. The Content-Type header field includes a parameter (called boundary) that
defines the delimiter between body parts. This boundary should not appear in
any parts of the message. Each boundary starts on a new line and consists of two
hyphens followed by the boundary value. The final boundary, which indicates the
end of the last part, also has a suffix of two hyphens. Within each part, there may be
an optional ordinary MIME header.
Here is a simple example of a multipart message containing two parts—both
consisting of simple text (taken from RFC 2046):
From: Nathaniel Borenstein <[email protected]>
To: Ned Freed <[email protected]>
Subject: Sample message
MIME-Version: 1.0
Content-type: multipart/mixed; boundary=“simple boundary”
This is the preamble. It is to be ignored, though it is a handy place for mail composers to include an explanatory note to non-MIME conformant readers.
—simple boundary
This is implicitly typed plain ASCII text. It does NOT end with a linebreak.
—simple boundary
Content-type: text/plain; charset=us-ascii
This is explicitly typed plain ASCII text. It DOES end with a linebreak.
—simple boundary—
This is the epilogue. It is also to be ignored.
There are four subtypes of the multipart type, all of which have the same
overall syntax. The multipart/mixed subtype is used when there are multiple inde- pendent body parts that need to be bundled in a particular order. For the multipart/ parallel subtype, the order of the parts is not significant. If the recipient’s system is appropriate, the multiple parts can be presented in parallel. For example, a picture
or text part could be accompanied by a voice commentary that is played while the
picture or text is displayed.
For the multipart/alternative subtype, the various parts are different represen- tations of the same information. The following is an example:
From: Nathaniel Borenstein <[email protected]> To: Ned Freed <[email protected]> Subject: Formatted text mail
622 CHAPTER 19 / ELECTRONIC MAIL SECURITY
MIME-Version: 1.0
Content-Type: multipart/alternative;
boundary=boundary42
—boundary42
Content-Type: text/plain; charset=us-ascii
. . . plain text version of message goes here. . . .
—boundary42
Content-Type: text/enriched
. . . RFC 1896 text/enriched version of same message goes here . . .
—boundary42—
In this subtype, the body parts are ordered in terms of increasing preference.
For this example, if the recipient system is capable of displaying the message in the
text/enriched format, this is done; otherwise, the plain text format is used.
The multipart/digest subtype is used when each of the body parts is inter- preted as an RFC 5322 message with headers. This subtype enables the construction
of a message whose parts are individual messages. For example, the moderator of a
group might collect email messages from participants, bundle these messages, and
send them out in one encapsulating MIME message.
The message type provides a number of important capabilities in MIME. The message/rfc822 subtype indicates that the body is an entire message, including header and body. Despite the name of this subtype, the encapsulated message may
be not only a simple RFC 5322 message, but also any MIME message.
The message/partial subtype enables fragmentation of a large message into a number of parts, which must be reassembled at the destination. For this subtype,
three parameters are specified in the Content-Type: Message/Partial field: an id common to all fragments of the same message, a sequence number unique to each fragment, and the total number of fragments.
The message/external-body subtype indicates that the actual data to be con- veyed in this message are not contained in the body. Instead, the body contains the
information needed to access the data. As with the other message types, the mes-
sage/external-body subtype has an outer header and an encapsulated message with
its own header. The only necessary field in the outer header is the Content-Type
field, which identifies this as a message/external-body subtype. The inner header is
the message header for the encapsulated message. The Content-Type field in the
outer header must include an access-type parameter, which indicates the method of
access, such as FTP (file transfer protocol).
The application type refers to other kinds of data, typically either uninter- preted binary data or information to be processed by a mail-based application.
MIME TRANSFER ENCODINGS The other major component of the MIME specifica- tion, in addition to content type specification, is a definition of transfer encodings
for message bodies. The objective is to provide reliable delivery across the largest
range of environments.
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19.2 / EMAIL FORMATS 623
The MIME standard defines two methods of encoding data. The Content-
Transfer-Encoding field can actually take on six values, as listed in Table 19.2.
However, three of these values (7-bit, 8-bit, and binary) indicate that no encod-
ing has been done but provide some information about the nature of the data. For
SMTP transfer, it is safe to use the 7-bit form. The 8-bit and binary forms may be
usable in other mail transport contexts. Another Content-Transfer-Encoding value
is x-token, which indicates that some other encoding scheme is used for which
a name is to be supplied. This could be a vendor-specific or application-specific
scheme. The two actual encoding schemes defined are quoted-printable and base64.
Two schemes are defined to provide a choice between a transfer technique that is
essentially human readable and one that is safe for all types of data in a way that is
reasonably compact.
The quoted-printable transfer encoding is useful when the data consists largely of octets that correspond to printable ASCII characters. In essence, it represents
nonsafe characters by the hexadecimal representation of their code and introduces
reversible (soft) line breaks to limit message lines to 76 characters.
The base64 transfer encoding, also known as radix-64 encoding, is a common one for encoding arbitrary binary data in such a way as to be invulnerable to the
processing by mail-transport programs. It is also used in PGP and is described in
Appendix X.
A MULTIPART EXAMPLE Figure 19.3, taken from RFC 2045, is the outline of a com- plex multipart message. The message has five parts to be displayed serially: two
introductory plain text parts, an embedded multipart message, a richtext part, and
a closing encapsulated text message in a non-ASCII character set. The embedded
multipart message has two parts to be displayed in parallel: a picture and an audio
fragment.
CANONICAL FORM An important concept in MIME and S/MIME is that of canonical form. Canonical form is a format, appropriate to the content type, that is standard-
ized for use between systems. This is in contrast to native form, which is a format that
may be peculiar to a particular system. RFC 2049 defines these two forms as follows:
■ Native form: The body to be transmitted is created in the system’s native for- mat. The native character set is used and, where appropriate, local end-of-line
conventions are used as well. The body may be any format that corresponds to
7 bit The data are all represented by short lines of ASCII characters.
8 bit The lines are short, but there may be non-ASCII characters (octets with the
high-order bit set).
binary Not only may non-ASCII characters be present but the lines are not necessarily
short enough for SMTP transport.
quoted-printable Encodes the data in such a way that if the data being encoded are mostly ASCII
text, the encoded form of the data remains largely recognizable by humans.
base64 Encodes data by mapping 6-bit blocks of input to 8-bit blocks of output, all of
which are printable ASCII characters.
x-token A named nonstandard encoding.
Table 19.2 MIME Transfer Encodings
624 CHAPTER 19 / ELECTRONIC MAIL SECURITY
MIME-Version: 1.0
From: Nathaniel Borenstein <[email protected]>
To: Ned Freed <[email protected]>
Subject: A multipart example
Content-Type: multipart/mixed;
boundary=unique-boundary-1
This is the preamble area of a multipart message. Mail readers that understand multipart format should ignore this preamble. If you are reading this text, you might want to consider changing to a mail reader that understands how to properly display multipart messages.
—unique-boundary-1
. . . Some text appears here . . .
[Note that the preceding blank line means no header fields were given and this is text, with charset US ASCII. It could have been done with explicit typing as in the next part.]
—unique-boundary-1
Content-type: text/plain; charset=US-ASCII
This could have been part of the previous part, but illustrates explicit versus implicit typing of body parts.
—unique-boundary-1
Content-Type: multipart/parallel; boundary=unique-boundary-2
—unique-boundary-2
Content-Type: audio/basic
Content-Transfer-Encoding: base64
. . . base64-encoded 8000 Hz single-channel mu-law-format audio data goes here . . . .
—unique-boundary-2
Content-Type: image/jpeg
Content-Transfer-Encoding: base64
. . . base64-encoded image data goes here . . . .
—unique-boundary-2—
—unique-boundary-1
Content-type: text/enriched
This is richtext. as defined in RFC 1896
Isn’t it cool?
—unique-boundary-1
Content-Type: message/rfc822
From: (mailbox in US-ASCII)
To: (address in US-ASCII)
Subject: (subject in US-ASCII)
Content-Type: Text/plain; charset=ISO-8859-1
Content-Transfer-Encoding: Quoted-printable
. . . Additional text in ISO-8859-1 goes here . . .
—unique-boundary-1—
Figure 19.3 Example MIME Message Structure
19.3 / EMAIL THREATS AND COMPREHENSIVE EMAIL SECURITY 625
the local model for the representation of some form of information. Examples
include a UNIX-style text file, or a Sun raster image, or a VMS indexed file, and
audio data in a system-dependent format stored only in memory. In essence,
the data are created in the native form that corresponds to the type specified
by the media type.
■ Canonical form: The entire body, including out-of-band information such as record lengths and possibly file attribute information, is converted to a univer-
sal canonical form. The specific media type of the body as well as its associated
attributes dictates the nature of the canonical form that is used. Conversion to
the proper canonical form may involve character set conversion, transforma-
tion of audio data, compression, or various other operations specific to the
various media types.
19.3 EMAIL THREATS AND COMPREHENSIVE EMAIL SECURITY
For both organizations and individuals, email is both pervasive and especially vul-
nerable to a wide range of security threats. In general terms, email security threats
can be classified as follows:
■ Authenticity-related threats: Could result in unauthorized access to an enter- prise’s email system.
■ Integrity-related threats: Could result in unauthorized modification of email content.
■ Confidentiality-related threats: Could result in unauthorized disclosure of sensitive information.
■ Availability-related threats: Could prevent end users from being able to send or receive email.
A useful list of specific email threats, together with approaches to mitigation,
is provided in NIST SP 800-177 (Trustworthy Email, September 2015) and is shown in Table 19.3.
SP 800-177 recommends use of a variety of standardized protocols as a means
for countering these threats. These include:
■ STARTTLS: An SMTP security extension that provides authentication, integ- rity, non-repudiation (via digital signatures) and confidentiality (via encryp-
tion) for the entire SMTP message by running SMTP over TLS.
■ S/MIME: Provides authentication, integrity, non-repudiation (via digital signatures) and confidentiality (via encryption) of the message body carried
in SMTP messages.
■ DNS Security Extensions (DNSSEC): Provides authentication and integ- rity protection of DNS data, and is an underlying tool used by various email
security protocols.
■ DNS-based Authentication of Named Entities (DANE): Is designed to over- come problems in the certificate authority (CA) system by providing an
alternative channel for authenticating public keys based on DNSSEC, with the
626 CHAPTER 19 / ELECTRONIC MAIL SECURITY
Threat Impact on Purported
Sender Impact on Receiver Mitigation
Email sent by
unauthorized MTA in
enterprise (e.g., malware
botnet)
Loss of reputation, valid
email from enterprise
may be blocked as
possible spam/phishing
attack.
UBE and/or email
containing malicious
links may be delivered
into user inboxes.
Deployment of domain-
based authentication
techniques. Use of
digital signatures over
email.
Email message sent
using spoofed or
unregistered sending
domain
Loss of reputation, valid
email from enterprise
may be blocked as
possible spam/phishing
attack.
UBE and/or email
containing malicious
links may be delivered
into user inboxes.
Deployment of domain-
based authentication
techniques. Use of
digital signatures over
email.
Email message sent
using forged sending
address or email address
(i.e., phishing, spear
phishing)
Loss of reputation, valid
email from enterprise
may be blocked as
possible spam/phishing
attack.
UBE and/or email
containing malicious
links may be delivered.
Users may inadvertently
divulge sensitive
information or PII.
Deployment of domain-
based authentication
techniques. Use of
digital signatures over
email.
Email modified in transit Leak of sensitive
information or PII.
Leak of sensitive
information, altered
message may contain
malicious information.
Use of TLS to encrypt
email transfer between
servers. Use of end-to-
end email encryption.
Disclosure of sensitive
information (e.g., PII)
via monitoring and
capturing of email traffic
Leak of sensitive
information or PII.
Leak of sensitive
information, altered
message may contain
malicious information.
Use of TLS to encrypt
email transfer between
servers. Use of end-to-
end email encryption.
Unsolicited Bulk Email
(UBE) (i.e., spam)
None, unless purported
sender is spoofed.
UBE and/or email
containing malicious
links may be delivered
into user inboxes.
Techniques to address
UBE.
DoS/DDoS attack
against an enterprises’
email servers
Inability to send email. Inability to receive
email.
Multiple mail servers,
use of cloud-based email
providers.
Table 19.3 Email Threats and Mitigations
result that the same trust relationships used to certify IP addresses are used to
certify servers operating on those addresses.
■ Sender Policy Framework (SPF): Uses the Domain Name System (DNS) to allow domain owners to create records that associate the domain name with a
specific IP address range of authorized message senders. It is a simple matter
for receivers to check the SPF TXT record in the DNS to confirm that the pur-
ported sender of a message is permitted to use that source address and reject
mail that does not come from an authorized IP address.
■ DomainKeys Identified Mail (DKIM): Enables an MTA to sign selected headers and the body of a message. This validates the source domain of the
mail and provides message body integrity.
■ Domain-based Message Authentication, Reporting, and Conformance (DMARC): Lets senders know the proportionate effectiveness of their SPF and DKIM policies, and signals to receivers what action should be taken in
various individual and bulk attack scenarios.
19.4 / S/MIME 627
Figure 19.4 shows how these components interact to provide message authen-
ticity and integrity. Not shown, for simplicity, is that S/MIME also provides message
confidentiality by encrypting messages.
19.4 S/MIME
Secure/Multipurpose Internet Mail Extension (S/MIME) is a security enhancement
to the MIME Internet email format standard based on technology from RSA Data
Security. S/MIME is a complex capability that is defined in a number of documents.
The most important documents relevant to S/MIME include the following:
■ RFC 5750, S/MIME Version 3.2 Certificate Handling: Specifies conventions for X.509 certificate usage by (S/MIME) v3.2.
Figure 19.4 The Interrelationship of DNSSEC, SPF, DKIM, DMARC, DANE, and S/MIME for Assuring Message Authenticity and Integrity
msg
msg
sig
msg
sig
msg
sig
Sender MUA
Sender’s S/MIME signing key
(private key)
DKIM signature
DKIM TXT RR provides
sending MTA’s public key
to receiving MTA
DM ARC TXT tells receiving
M TA that sender uses
DKIM and SPF
DA NE
TL SA
RR
spe cifi
es S MT
P
TL S ce
rtif icat
e
Receiver MUA verifies S/MIME
signature
DNSSEC secured
DNSSEC secured
MTA’s DKIM signing key
DANE = DNS-based Authentication of Named Entities DKIM = DomainKeys Identified Mail DMARC = Domain-based Message Authentication, Reporting, and Conformance DNSSEC = Domain Name System Security Extensions SPF = Sender Policy Framework S/MIME = Secure Multi-Purpose Internet Mail Extensions TLSA RR = Transport Layer Security Authentication Resource Record
SP F
TX T
sp ec
fie s
se nd
er ’s
IP ad
dr es
s
Sender DNS
Receiver DNS
Receiver MUA
Sending MTA
Receiving MTA
628 CHAPTER 19 / ELECTRONIC MAIL SECURITY
■ RFC 5751, S/MIME) Version 3.2 Message Specification: The principal defining document for S/MIME message creation and processing.
■ RFC 4134, Examples of S/MIME Messages: Gives examples of message bodies formatted using S/MIME.
■ RFC 2634, Enhanced Security Services for S/MIME: Describes four optional security service extensions for S/MIME.
■ RFC 5652, Cryptographic Message Syntax (CMS): Describes the Crypto- graphic Message Syntax (CMS). This syntax is used to digitally sign, digest,
authenticate, or encrypt arbitrary message content.
■ RFC 3370, CMS Algorithms: Describes the conventions for using several cryptographic algorithms with the CMS.
■ RFC 5752, Multiple Signatures in CMS: Describes the use of multiple, parallel signatures for a message.
■ RFC 1847, Security Multiparts for MIME—Multipart/Signed and Multipart/ Encrypted: Defines a framework within which security services may be applied to MIME body parts. The use of a digital signature is relevant to S/MIME, as
explained subsequently.
Operational Description
S/MIME provides for four message-related services: authentication, confidential-
ity, compression, and email compatibility (Table 19.4). This subsection provides
an overview. We then look in more detail at this capability by examining message
formats and message preparation.
AUTHENTICATION Authentication is provided by means of a digital signature, using the general scheme discussed in Chapter 13 and illustrated in Figure 13.1. Most
commonly RSA with SHA-256 is used. The sequence is as follows:
1. The sender creates a message.
2. SHA-256 is used to generate a 256-bit message digest of the message.
Function Typical Algorithm Typical Action
Digital signature RSA/SHA-256 A hash code of a message is created using SHA-256.
This message digest is encrypted using SHA-256
with the sender’s private key and included with
the message.
Message encryption AES-128 with CBC A message is encrypted using AES-128 with CBC
with a one-time session key generated by the
sender. The session key is encrypted using RSA
with the recipient’s public key and included with
the message.
Compression unspecified A message may be compressed for storage or
transmission.
Email compatibility Radix-64 conversion To provide transparency for email applications, an
encrypted message may be converted to an ASCII
string using radix-64 conversion.
Table 19.4 Summary of S/MIME Services
19.4 / S/MIME 629
3. The message digest is encrypted with RSA using the sender’s private key, and the result is appended to the message. Also appended is identifying information
for the signer, which will enable the receiver to retrieve the signer’s public key.
4. The receiver uses RSA with the sender’s public key to decrypt and recover the message digest.
5. The receiver generates a new message digest for the message and compares it with the decrypted hash code. If the two match, the message is accepted as
authentic.
The combination of SHA-256 and RSA provides an effective digital signature
scheme. Because of the strength of RSA, the recipient is assured that only the pos-
sessor of the matching private key can generate the signature. Because of the strength
of SHA-256, the recipient is assured that no one else could generate a new message
that matches the hash code and, hence, the signature of the original message.
Although signatures normally are found attached to the message or file that
they sign, this is not always the case: Detached signatures are supported. A detached
signature may be stored and transmitted separately from the message it signs. This
is useful in several contexts. A user may wish to maintain a separate signature log
of all messages sent or received. A detached signature of an executable program
can detect subsequent virus infection. Finally, detached signatures can be used
when more than one party must sign a document, such as a legal contract. Each
person’s signature is independent and therefore is applied only to the document.
Otherwise, signatures would have to be nested, with the second signer signing both
the document and the first signature, and so on.
CONFIDENTIALITY S/MIME provides confidentiality by encrypting messages. Most commonly AES with a 128-bit key is used, with the cipher block chaining (CBC)
mode. The key itself is also encrypted, typically with RSA, as explained below.
As always, one must address the problem of key distribution. In S/MIME,
each symmetric key, referred to as a content-encryption key, is used only once. That
is, a new key is generated as a random number for each message. Because it is to be
used only once, the content-encryption key is bound to the message and transmit-
ted with it. To protect the key, it is encrypted with the receiver’s public key. The
sequence can be described as follows:
1. The sender generates a message and a random 128-bit number to be used as a content-encryption key for this message only.
2. The message is encrypted using the content-encryption key.
3. The content-encryption key is encrypted with RSA using the recipient’s public key and is attached to the message.
4. The receiver uses RSA with its private key to decrypt and recover the content-encryption key.
5. The content-encryption key is used to decrypt the message.
Several observations may be made. First, to reduce encryption time, the com-
bination of symmetric and public-key encryption is used in preference to simply
using public-key encryption to encrypt the message directly: Symmetric algorithms
630 CHAPTER 19 / ELECTRONIC MAIL SECURITY
are substantially faster than asymmetric ones for a large block of content. Second,
the use of the public-key algorithm solves the session-key distribution problem,
because only the recipient is able to recover the session key that is bound to the
message. Note that we do not need a session-key exchange protocol of the type
discussed in Chapter 14, because we are not beginning an ongoing session. Rather,
each message is a one-time independent event with its own key. Furthermore, given
the store-and-forward nature of electronic mail, the use of handshaking to assure
that both sides have the same session key is not practical. Finally, the use of one-
time symmetric keys strengthens what is already a strong symmetric encryption
approach. Only a small amount of plaintext is encrypted with each key, and there is
no relationship among the keys. Thus, to the extent that the public-key algorithm is
secure, the entire scheme is secure.
CONFIDENTIALITY AND AUTHENTICATION As Figure 19.5 illustrates, both confi- dentiality and encryption may be used for the same message. The figure shows a
sequence in which a signature is generated for the plaintext message and appended
to the message. Then the plaintext message and signature are encrypted as a single
block using symmetric encryption and the symmetric encryption key is encrypted
using public-key encryption.
S/MIME allows the signing and message encryption operations to be per-
formed in either order. If signing is done first, the identity of the signer is hidden
by the encryption. Plus, it is generally more convenient to store a signature with a
plaintext version of a message. Furthermore, for purposes of third-party verifica-
tion, if the signature is performed first, a third party need not be concerned with the
symmetric key when verifying the signature.
If encryption is done first, it is possible to verify a signature without exposing
the message content. This can be useful in a context in which automatic signature
verification is desired, as no private key material is required to verify a signature.
However, in this case the recipient cannot determine any relationship between the
signer and the unencrypted content of the message.
EMAIL COMPATIBILITY When S/MIME is used, at least part of the block to be trans- mitted is encrypted. If only the signature service is used, then the message digest is
encrypted (with the sender’s private key). If the confidentiality service is used, the
message plus signature (if present) are encrypted (with a one-time symmetric key).
Thus, part or all of the resulting block consists of a stream of arbitrary 8-bit octets.
However, many electronic mail systems only permit the use of blocks consisting
of ASCII text. To accommodate this restriction, S/MIME provides the service of
converting the raw 8-bit binary stream to a stream of printable ASCII characters,
a process referred to as 7-bit encoding.
The scheme typically used for this purpose is Base64 conversion. Each group
of three octets of binary data is mapped into four ASCII characters. See Appendix
X for a description.
One noteworthy aspect of the Base64 algorithm is that it blindly converts the
input stream to Base64 format regardless of content, even if the input happens to
be ASCII text. Thus, if a message is signed but not encrypted and the conversion
is applied to the entire block, the output will be unreadable to the casual observer,
which provides a certain level of confidentiality.
19.4 / S/MIME 631
RFC 5751 also recommends that even if outer 7-bit encoding is not used, the
original MIME content should be 7-bit encoded. The reason for this is that it allows
the MIME entity to be handled in any environment without changing it. For exam-
ple, a trusted gateway might remove the encryption, but not the signature, of a mes-
sage, and then forward the signed message on to the end recipient so that they can
verify the signatures directly. If the transport internal to the site is not 8-bit clean,
such as on a wide area network with a single mail gateway, verifying the signature
will not be possible unless the original MIME entity was only 7-bit data.
COMPRESSION S/MIME also offers the ability to compress a message. This has the benefit of saving space both for email transmission and for file storage. Compression
Figure 19.5 Simplified S/MIME Functional Flow
Sign (e.g., RSA/ SHA-256)
Sender’s private key
(a) Sender signs, then encrypts message
(b) Receiver decrypts message, then verifies sender’s signature
One-time secret key
Encrypt (e.g,
AES-128/ CBC
Encrypt (e.g., RSA)
msg msg
sig sig
sig
msg
sig
Receiver’s public key
Sender’s public key
Decrypt (e.g., RSA)
Receiver’s private key
Secret key generated by
sender
Decrypt (e.g,
AES-128/ CBC
Verify signature
(e.g., RSA/ SHA-256)
msg
msg
Hiva-Network.Com
632 CHAPTER 19 / ELECTRONIC MAIL SECURITY
can be applied in any order with respect to the signing and message encryption
operations. RFC 5751 provides the following guidelines:
■ Compression of binary encoded encrypted data is discouraged, since it will not
yield significant compression. Base64 encrypted data could very well benefit,
however.
■ If a lossy compression algorithm is used with signing, you will need to compress
first, then sign.
S/MIME Message Content Types
S/MIME uses the following message content types, which are defined in RFC 5652,
Cryptographic Message Syntax:
■ Data: Refers to the inner MIME-encoded message content, which may then be encapsulated in a SignedData, EnvelopedData, or CompressedData con-
tent type.
■ SignedData: Used to apply a digital signature to a message.
■ EnvelopedData: This consists of encrypted content of any type and encrypted- content encryption keys for one or more recipients.
■ CompressedData: Used to apply data compression to a message.
The Data content type is also used for a procedure known as clear signing.
For clear signing, a digital signature is calculated for a MIME-encoded message and
the two parts, the message and signature, form a multipart MIME message. Unlike
SignedData, which involves encapsulating the message and signature in a special
format, clear-signed messages can be read and their signatures verified by email
entities that do not implement S/MIME.
Approved Cryptographic Algorithms
Table 19.5 summarizes the cryptographic algorithms used in S/MIME. S/MIME
uses the following terminology taken from RFC 2119 (Key Words for use in RFCs to Indicate Requirement Levels, March 1997) to specify the requirement level:
■ MUST: The definition is an absolute requirement of the specification. An implementation must include this feature or function to be in conformance
with the specification.
■ SHOULD: There may exist valid reasons in particular circumstances to ignore this feature or function, but it is recommended that an implementation include
the feature or function.
The S/MIME specification includes a discussion of the procedure for deciding
which content encryption algorithm to use. In essence, a sending agent has two deci-
sions to make. First, the sending agent must determine if the receiving agent is capable
of decrypting using a given encryption algorithm. Second, if the receiving agent is only
capable of accepting weakly encrypted content, the sending agent must decide if it is
acceptable to send using weak encryption. To support this decision process, a sending
agent may announce its decrypting capabilities in order of preference for any message
that it sends out. A receiving agent may store that information for future use.
19.4 / S/MIME 633
The following rules, in the following order, should be followed by a sending agent.
1. If the sending agent has a list of preferred decrypting capabilities from an intended recipient, it SHOULD choose the first (highest preference) capabil-
ity on the list that it is capable of using.
2. If the sending agent has no such list of capabilities from an intended recipient but has received one or more messages from the recipient, then the outgoing
message SHOULD use the same encryption algorithm as was used on the last
signed and encrypted message received from that intended recipient.
3. If the sending agent has no knowledge about the decryption capabilities of the intended recipient and is willing to risk that the recipient may not be able to
decrypt the message, then the sending agent SHOULD use triple DES.
4. If the sending agent has no knowledge about the decryption capabilities of the intended recipient and is not willing to risk that the recipient may not be able
to decrypt the message, then the sending agent MUST use RC2/40.
If a message is to be sent to multiple recipients and a common encryption
algorithm cannot be selected for all, then the sending agent will need to send two
messages. However, in that case, it is important to note that the security of the
message is made vulnerable by the transmission of one copy with lower security.
S/MIME Messages
S/MIME makes use of a number of new MIME content types. All of the new applica-
tion types use the designation PKCS. This refers to a set of public-key cryptography
specifications issued by RSA Laboratories and made available for the S/MIME effort.
Function Requirement
Create a message digest to be used in
forming a digital signature.
MUST support SHA-256
SHOULD support SHA-1
Receiver SHOULD support MD5 for backward compatibility
Use message digest to form a digital
signature.
MUST support RSA with SHA-256
SHOULD support
—DSA with SHA-256
—RSASSA-PSS with SHA-256
—RSA with SHA-1
—DSA with SHA-1
—RSA with MD5
Encrypt session key for transmission with
a message.
MUST support RSA encryption
SHOULD support
—RSAES-OAEP
—Diffie–Hellman ephemeral-static mode
Encrypt message for transmission with a
one-time session key.
MUST support AES-128 with CBC
SHOULD support
—AES-192 CBC and AES-256 CBC
—Triple DES CBC
Table 19.5 Cryptographic Algorithms Used in S/MIME
634 CHAPTER 19 / ELECTRONIC MAIL SECURITY
We examine each of these in turn after first looking at the general procedures
for S/MIME message preparation.
SECURING A MIME ENTITY S/MIME secures a MIME entity with a signature, encryption, or both. A MIME entity may be an entire message (except for the RFC
5322 headers), or if the MIME content type is multipart, then a MIME entity is one
or more of the subparts of the message. The MIME entity is prepared according
to the normal rules for MIME message preparation. Then the MIME entity plus
some security-related data, such as algorithm identifiers and certificates, are pro-
cessed by S/MIME to produce what is known as a PKCS object. A PKCS object is
then treated as message content and wrapped in MIME (provided with appropriate
MIME headers). This process should become clear as we look at specific objects
and provide examples.
In all cases, the message to be sent is converted to canonical form. In par-
ticular, for a given type and subtype, the appropriate canonical form is used for the
message content. For a multipart message, the appropriate canonical form is used
for each subpart.
The use of transfer encoding requires special attention. For most cases, the
result of applying the security algorithm will be to produce an object that is partially
or totally represented in arbitrary binary data. This will then be wrapped in an outer
MIME message and transfer encoding can be applied at that point, typically base64.
However, in the case of a multipart signed message (described in more detail later),
the message content in one of the subparts is unchanged by the security process.
Unless that content is 7 bit, it should be transfer encoded using base64 or quoted-
printable so that there is no danger of altering the content to which the signature
was applied.
We now look at each of the S/MIME content types.
ENVELOPEDDATA An application/pkcs7-mime subtype is used for one of four cat- egories of S/MIME processing, each with a unique smime-type parameter. In all
cases, the resulting entity, (referred to as an object) is represented in a form known as Basic Encoding Rules (BER), which is defined in ITU-T Recommendation
X.209. The BER format consists of arbitrary octet strings and is therefore binary
data. Such an object should be transfer encoded with base64 in the outer MIME
message. We first look at envelopedData.
The steps for preparing an envelopedData MIME entity are:
1. Generate a pseudorandom session key for a particular symmetric encryption algorithm (RC2/40 or triple DES).
2. For each recipient, encrypt the session key with the recipient’s public RSA key.
3. For each recipient, prepare a block known as RecipientInfo that contains an identifier of the recipient’s public-key certificate,1 an identifier of the
algorithm used to encrypt the session key, and the encrypted session key.
4. Encrypt the message content with the session key.
1This is an X.509 certificate, discussed later in this section.
19.4 / S/MIME 635
The RecipientInfo blocks followed by the encrypted content constitute the
envelopedData. This information is then encoded into base64. A sample message (excluding the RFC 5322 headers) is given below.
Content-Type: application/pkcs7-mime; smime-type=enveloped-
data; name=smime.p7m
Content-Transfer-Encoding: base64
Content-Disposition: attachment; filename=smime.p7m
rfvbnj756tbBghyHhHUujhJhjH77n8HHGT9HG4VQpfyF467GhIGfHfYT6
7n8HHGghyHhHUujhJh4VQpfyF467GhIGfHfYGTrfvbnjT6jH7756tbB9H
f8HHGTrfvhJhjH776tbB9HG4VQbnj7567GhIGfHfYT6ghyHhHUujpfyF4
0GhIGfHfQbnj756YT64V
To recover the encrypted message, the recipient first strips off the base64
encoding. Then the recipient’s private key is used to recover the session key. Finally,
the message content is decrypted with the session key.
SIGNEDDATA The signedData smime-type can be used with one or more signers. For clarity, we confine our description to the case of a single digital signature. The
steps for preparing a signedData MIME entity are as follows.
1. Select a message digest algorithm (SHA or MD5).
2. Compute the message digest (hash function) of the content to be signed.
3. Encrypt the message digest with the signer’s private key.
4. Prepare a block known as SignerInfo that contains the signer’s public-key certificate, an identifier of the message digest algorithm, an identifier of the
algorithm used to encrypt the message digest, and the encrypted message
digest.
The signedData entity consists of a series of blocks, including a message digest algorithm identifier, the message being signed, and SignerInfo. The signedData entity may also include a set of public-key certificates sufficient to constitute a chain from a recognized root or top-level certification authority to the
signer. This information is then encoded into base64. A sample message (excluding
the RFC 5322 headers) is the following.
Content-Type: application/pkcs7-mime; smime-type=signed-
data; name=smime.p7m
Content-Transfer-Encoding: base64
Content-Disposition: attachment; filename=smime.p7m
567GhIGfHfYT6ghyHhHUujpfyF4f8HHGTrfvhJhjH776tbB9HG4VQbnj7
77n8HHGT9HG4VQpfyF467GhIGfHfYT6rfvbnj756tbBghyHhHUujhJhjH
HUujhJh4VQpfyF467GhIGfHfYGTrfvbnjT6jH7756tbB9H7n8HHGghyHh
6YT64V0GhIGfHfQbnj75
636 CHAPTER 19 / ELECTRONIC MAIL SECURITY
To recover the signed message and verify the signature, the recipient first strips
off the base64 encoding. Then the signer’s public key is used to decrypt the message
digest. The recipient independently computes the message digest and compares it to
the decrypted message digest to verify the signature.
CLEAR SIGNING Clear signing is achieved using the multipart content type with a signed subtype. As was mentioned, this signing process does not involve trans-
forming the message to be signed, so that the message is sent “in the clear.” Thus,
recipients with MIME capability but not S/MIME capability are able to read the
incoming message.
A multipart/signed message has two parts. The first part can be any MIME
type but must be prepared so that it will not be altered during transfer from source
to destination. This means that if the first part is not 7 bit, then it needs to be encoded
using base64 or quoted-printable. Then this part is processed in the same manner
as signedData, but in this case an object with signedData format is created that has an empty message content field. This object is a detached signature. It is then
transfer encoded using base64 to become the second part of the multipart/signed
message. This second part has a MIME content type of application and a subtype of
pkcs7-signature. Here is a sample message:
Content-Type: multipart/signed;
protocol=”application/pkcs7-signature”;
micalg=sha1; boundary=boundary42
—boundary42
Content-Type: text/plain
This is a clear-signed message.
—boundary42
Content-Type: application/pkcs7-signature; name=smime.p7s
Content-Transfer-Encoding: base64
Content-Disposition: attachment; filename=smime.p7s
ghyHhHUujhJhjH77n8HHGTrfvbnj756tbB9HG4VQpfyF467GhIGfHfYT6
4VQpfyF467GhIGfHfYT6jH77n8HHGghyHhHUujhJh756tbB9HGTrfvbnj
n8HHGTrfvhJhjH776tbB9HG4VQbnj7567GhIGfHfYT6ghyHhHUujpfyF4
7GhIGfHfYT64VQbnj756
—boundary42—
The protocol parameter indicates that this is a two-part clear-signed entity.
The micalg parameter indicates the type of message digest used. The receiver can verify the signature by taking the message digest of the first part and comparing this
to the message digest recovered from the signature in the second part.
REGISTRATION REQUEST Typically, an application or user will apply to a certi- fication authority for a public-key certificate. The application/pkcs10 S/MIME
19.4 / S/MIME 637
entity is used to transfer a certification request. The certification request
includes certificationRequestInfo block, followed by an identifier of the public-key encryption algorithm, followed by the signature of the
certificationRequestInfo block, made using the sender’s private key. The certificationRequestInfo block includes a name of the certificate subject (the entity whose public key is to be certified) and a bit-string representation of the
user’s public key.
CERTIFICATES-ONLY MESSAGE A message containing only certificates or a certificate revocation list (CRL) can be sent in response to a registration request. The message
is an application/pkcs7-mime type/subtype with an smime-type parameter of degen-
erate. The steps involved are the same as those for creating a signedData message, except that there is no message content and the signerInfo field is empty.
S/MIME Certificate Processing
S/MIME uses public-key certificates that conform to version 3 of X.509 (see
Chapter 14). S/MIME managers and/or users must configure each client with a list of
trusted keys and with certificate revocation lists. That is, the responsibility is local for
maintaining the certificates needed to verify incoming signatures and to encrypt outgo-
ing messages. On the other hand, the certificates are signed by certification authorities.
USER AGENT ROLE An S/MIME user has several key management functions to perform.
■ Key generation: The user of some related administrative utility (e.g., one associated with LAN management) MUST be capable of generating separate
Diffie–Hellman and DSS key pairs and SHOULD be capable of generating
RSA key pairs. Each key pair MUST be generated from a good source of
nondeterministic random input and be protected in a secure fashion. A user
agent SHOULD generate RSA key pairs with a length in the range of 768 to
1024 bits and MUST NOT generate a length of less than 512 bits.
■ Registration: A user’s public key must be registered with a certification authority in order to receive an X.509 public-key certificate.
■ Certificate storage and retrieval: A user requires access to a local list of certifi- cates in order to verify incoming signatures and to encrypt outgoing messages.
Such a list could be maintained by the user or by some local administrative
entity on behalf of a number of users.
Enhanced Security Services
RFC 2634 defines four enhanced security services for S/MIME:
■ Signed receipts: A signed receipt may be requested in a SignedData object. Returning a signed receipt provides proof of delivery to the originator of a
message and allows the originator to demonstrate to a third party that the
recipient received the message. In essence, the recipient signs the entire
original message plus the original (sender’s) signature and appends the new
signature to form a new S/MIME message.
638 CHAPTER 19 / ELECTRONIC MAIL SECURITY
■ Security labels: A security label may be included in the authenticated attributes of a SignedData object. A security label is a set of security information regarding the sensitivity of the content that is protected by S/MIME encapsu-
lation. The labels may be used for access control, by indicating which users are
permitted access to an object. Other uses include priority (secret, confidential,
restricted, and so on) or role based, describing which kind of people can see
the information (e.g., patient’s health-care team, medical billing agents).
■ Secure mailing lists: When a user sends a message to multiple recipients, a certain amount of per-recipient processing is required, including the use of
each recipient’s public key. The user can be relieved of this work by employ-
ing the services of an S/MIME Mail List Agent (MLA). An MLA can take a
single incoming message, perform the recipient-specific encryption for each
recipient, and forward the message. The originator of a message need only
send the message to the MLA with encryption performed using the MLA’s
public key.
■ Signing certificates: This service is used to securely bind a sender’s certificate to their signature through a signing certificate attribute.
19.5 PRETTY GOOD PRIVACY
An alternative email security protocol is Pretty Good Privacy (PGP), which has
essentially the same functionality as S/MIME. PGP was created by Phil Zimmerman
and implemented as a product first released in 1991. It was made available free of
charge and became quite popular for personal use. The initial PGP protocol was
proprietary and used some encryption algorithms with intellectual property restric-
tions. In 1996, version 5.x of PGP was defined in IETF RFC 1991, PGP Message Exchange Formats. Subsequently, OpenPGP was developed as a new standard protocol based on PGP version 5.x. OpenPGP is defined in RFC 4880 (OpenPGP Message Format, November 2007) and RFC 3156 (MIME Security with OpenPGP, August 2001).
There are two significant differences between S/MIME and OpenPGP:
■ Key Certification: S/MIME uses X.509 certificates that are issued by Certificate Authorities (or local agencies that have been delegated authority by a CA to
issue certificates). In OpenPGP, users generate their own OpenPGP public
and private keys and then solicit signatures for their public keys from individu-
als or organizations to which they are known. Whereas X.509 certificates are
trusted if there is a valid PKIX chain to a trusted root, an OpenPGP public key
is trusted if it is signed by another OpenPGP public key that is trusted by the
recipient. This is called the Web-of-Trust.
■ Key Distribution: OpenPGP does not include the sender’s public key with each message, so it is necessary for recipients of OpenPGP messages to sepa-
rately obtain the sender’s public key in order to verify the message. Many
organizations post OpenPGP keys on TLS-protected websites: People who
wish to verify digital signatures or send these organizations encrypted mail
19.6 / DNSSEC 639
need to manually download these keys and add them to their OpenPGP
clients. Keys may also be registered with the OpenPGP public key servers,
which are servers that maintain a database of PGP public keys organized by
email address. Anyone may post a public key to the OpenPGP key servers,
and that public key may contain any email address. There is no vetting of
OpenPGP keys, so users must use the Web-of-Trust to decide whether to trust
a given public key.
NIST 800-177 recommends the use of S/MIME rather than PGP because of
the greater confidence in the CA system of verifying public keys.
Appendix P provides an overview of PGP.
19.6 DNSSEC
DNS Security Extensions (DNSSEC) are used by several protocols that provide
email security. This section provides a brief overview of the Domain Name System
(DNS) and then looks at DNSSEC.
Domain Name System
DNS is a directory lookup service that provides a mapping between the name of a
host on the Internet and its numeric IP address. DNS is essential to the functioning
of the Internet. The DNS is used by MUAs and MTAs to find the address of the
next hop server for mail delivery. Sending MTAs query DNS for the Mail Exchange
Resource Record (MX RR) of the recipient’s domain (the right hand side of the
“@” symbol) in order to find the receiving MTA to contact.
Four elements comprise the DNS:
■ Domain name space: DNS uses a tree-structured name space to identify resources on the Internet.
■ DNS database: Conceptually, each node and leaf in the name space tree struc- ture names a set of information (e.g., IP address, name server for this domain
name) that is contained in resource record. The collection of all RRs is orga-
nized into a distributed database.
■ Name servers: These are server programs that hold information about a por- tion of the domain name tree structure and the associated RRs.
■ Resolvers: These are programs that extract information from name servers in response to client requests. A typical client request is for an IP address corre-
sponding to a given domain name.
THE DNS DATABASE DNS is based on a hierarchical database containing resource records (RRs) that include the name, IP address, and other information about hosts. The key features of the database are as follows:
■ Variable-depth hierarchy for names: DNS allows essentially unlimited levels and uses the period (.) as the level delimiter in printed names, as described
earlier.
640 CHAPTER 19 / ELECTRONIC MAIL SECURITY
■ Distributed database: The database resides in DNS servers scattered through- out the Internet.
■ Distribution controlled by the database: The DNS database is divided into thousands of separately managed zones, which are managed by separate
administrators. Distribution and update of records is controlled by the database
software.
Using this database, DNS servers provide a name-to-address directory service
for network applications that need to locate specific servers. For example, every
time an email message is sent or a Web page is accessed, there must be a DNS name
lookup to determine the IP address of the email server or Web server.
Table 19.6 lists the various types of resource records.
DNS OPERATION DNS operation typically includes the following steps (Figure 19.6):
1. A user program requests an IP address for a domain name.
2. A resolver module in the local host or local ISP queries a local name server in the same domain as the resolver.
3. The local name server checks to see if the name is in its local database or cache, and, if so, returns the IP address to the requestor. Otherwise, the name server
queries other available name servers, if necessary going to the root server, as
explained subsequently.
4. When a response is received at the local name server, it stores the name/ address mapping in its local cache and may maintain this entry for the amount
of time specified in the time-to-live field of the retrieved RR.
5. The user program is given the IP address or an error message.
Type Description
A A host address. This RR type maps the name of a system to its IPv4 address. Some
systems (e.g., routers) have multiple addresses, and there is a separate RR for each.
AAAA Similar to A type, but for IPv6 addresses.
CNAME Canonical name. Specifies an alias name for a host and maps this to the canonical
(true) name.
HINFO Host information. Designates the processor and operating system used by the host.
MINFO Mailbox or mail list information. Maps a mailbox or mail list name to a host name.
MX Mail exchange. Identifies the system(s) via which mail to the queried domain name
should be relayed.
NS Authoritative name server for this domain.
PTR Domain name pointer. Points to another part of the domain name space.
SOA Start of a zone of authority (which part of naming hierarchy is implemented). Includes
parameters related to this zone.
SRV For a given service provides name of server or servers in domain that provide that service.
TXT Arbitrary text. Provides a way to add text comments to the database.
WKS Well-known services. May list the application services available at this host.
Table 19.6 Resource Record Types
Hiva-Network.Com
19.6 / DNSSEC 641
Figure 19.6 DNS Name Resolution
User program
User system
Internet user
query query
qu ery
user response
response
res po
nse Name
resolver
Cache
Name server
Cache
Database
Database
Foreign name server
Cache
The distributed DNS database that supports the DNS functionality must be
updated frequently because of the rapid and continued growth of the Internet.
Further, the DNS must cope with dynamic assignment of IP addresses, such as is
done for home DSL users by their ISP. Accordingly, dynamic updating functions
for DNS have been defined. In essence, DNS name servers automatically send out
updates to other relevant name servers as conditions warrant.
DNS Security Extensions
DNSSEC provides end-to-end protection through the use of digital signatures that
are created by responding zone administrators and verified by a recipient’s resolver
software. In particular, DNSSEC avoids the need to trust intermediate name servers
and resolvers that cache or route the DNS records originating from the responding
zone administrator before they reach the source of the query. DNSSEC consists of
a set of new resource record types and modifications to the existing DNS protocol,
and is defined in the following documents:
■ RFC 4033, DNS Security Introduction and Requirements: Introduces the DNS security extensions and describes their capabilities and limitations. The
document also discusses the services that the DNS security extensions do and
do not provide.
■ RFC 4034, Resource Records for the DNS Security Extensions: Defines four new resource records that provide security for DNS.
642 CHAPTER 19 / ELECTRONIC MAIL SECURITY
■ RFC 4035, Protocol Modifications for the DNS Security Extensions: Defines the concept of a signed zone, along with the requirements for serving and
resolving by using DNSSEC. These techniques allow a security-aware resolver
to authenticate both DNS resource records and authoritative DNS error
indications.
DNSSEC OPERATION In essence, DNSSEC is designed to protect DNS clients from accepting forged or altered DNS resource records. It does this by using digital
signatures to provide:
■ Data origin authentication: Ensures that data has originated from the correct source.
■ Data integrity verification: Ensures that the content of a RR has not been modified.
The DNS zone administrator digitally signs every Resource Record set
(RRset) in the zone, and publishes this collection of digital signatures, along with
the zone administrator’s public key, in the DNS itself. In DNSSEC, trust in the pub-
lic key (for signature verification) of the source is established not by going to a third
party or a chain of third parties (as in public key infrastructure [PKI] chaining), but
by starting from a trusted zone (such as the root zone) and establishing the chain of
trust down to the current source of response through successive verifications of sig-
nature of the public key of a child by its parent. The public key of the trusted zone
is called the trust anchor.
RESOURCE RECORDS FOR DNSSEC RFC 4034 defines four new DNS resource records:
■ DNSKEY: Contains a public key.
■ RRSIG: A resource record digital signature.
■ NSEC: Authenticated denial of existence record.
■ DS: Delegation signer.
An RRSIG is associated with each RRset, where an RRset is the set of
resource records that have the same label, class, and type. When a client requests
data, an RRset is returned, together with the associated digital signature in an
RRSIG record. The client obtains the relevant DNSKEY public key and verifies
the signature for this RRset.
DNSSEC depends on establishing the authenticity of the DNS hierarchy lead-
ing to the domain name in question, and thus its operation depends on beginning
the use of cryptographic digital signatures in the root zone. The DS resource record
facilitates key signing and authentication between DNS zones to create an authen-
tication chain, or trusted sequence of signed data, from the root of the DNS tree
down to a specific domain name. To secure all DNS lookups, including those for
non-existent domain names and record types, DNSSEC uses the NSEC resource
record to authenticate negative responses to queries. NSEC is used to identify the
19.7 / DNS-BASED AUTHENTICATION OF NAMED ENTITIES 643
range of DNS names or resource record types that do not exist among the sequence
of domain names in a zone.
19.7 DNS-BASED AUTHENTICATION OF NAMED ENTITIES
DANE is a protocol to allow X.509 certificates, commonly used for Transport Layer
Security (TLS), to be bound to DNS names using DNSSEC. It is proposed in RFC
6698 as a way to authenticate TLS client and server entities without a certificate
authority (CA).
The rationale for DANE is the vulnerability of the use of CAs in a global PKI
system. Every browser developer and operating system supplier maintains a list of
CA root certificates as trust anchors. These are called the software’s root certifi-
cates and are stored in its root certificate store. The PKIX procedure allows a cer-
tificate recipient to trace a certificate back to the root. So long as the root certificate
remains trustworthy, and the authentication concludes successfully, the client can
proceed with the connection.
However, if any of the hundreds of CAs operating on the Internet is compro-
mised, the effects can be widespread. The attacker can obtain the CA’s private key,
get issued certificates under a false name, or introduce new bogus root certificates
into a root certificate store. There is no limitation of scope for the global PKI and
a compromise of a single CA damages the integrity of the entire PKI system. In
addition, some CAs have engaged in poor security practices. For example, some
CAs have issued wildcard certificates that allow the holder to issue sub-certificates
for any domain or entity, anywhere in the world.
The purpose of DANE is to replace reliance on the security of the CA system
with reliance on the security provided by DNSSEC. Given that the DNS adminis-
trator for a domain name is authorized to give identifying information about the
zone, it makes sense to allow that administrator to also make an authoritative bind-
ing between the domain name and a certificate that might be used by a host at that
domain name.
TLSA Record
DANE defines a new DNS record type, TLSA, that can be used for a secure method
of authenticating SSL/TLS certificates. The TLSA provides for:
■ Specifying constraints on which CA can vouch for a certificate, or which
specific PKIX end-entity certificate is valid.
■ Specifying that a service certificate or a CA can be directly authenticated in
the DNS itself.
The TLSA RR enables certificate issue and delivery to be tied to a given
domain. A server domain owner creates a TLSA resource record that identifies the
certificate and its public key. When a client receives an X.509 certificate in the TLS
negotiation, it looks up the TLSA RR for that domain and matches the TLSA data
against the certificate as part of the client’s certificate validation procedure.
644 CHAPTER 19 / ELECTRONIC MAIL SECURITY
Figure 19.7 shows the format of a TLSA RR as it is transmitted to a request-
ing entity. It contains four fields. The Certificate Usage field defines four different usage models, to accommodate users who require different forms of authentication.
The usage models are:
■ PKIX-TA (CA constraint): Specifies which CA should be trusted to authen- ticate the certificate for the service. This usage model limits which CA can be
used to issue certificates for a given service on a host. The server certificate
chain must pass PKIX validation that terminates with a trusted root certificate
stored in the client.
■ PKIX-EE (service certificate constraint): Defines which specific end entity service certificate should be trusted for the service. This usage model limits
which end entity certificate can be used by a given service on a host. The server
certificate chain must pass PKIX validation that terminates with a trusted root
certificate stored in the client.
■ DANE-TA (trust anchor assertion): Specifies a domain-operated CA to be used as a trust anchor. This usage model allows a domain name administrator
to specify a new trust anchor—for example, if the domain issues its own certifi-
cates under its own CA that is not expected to be in the end users’ collection
of trust anchors. The server certificate chain is self-issued and does not need to
verify against a trusted root stored in the client.
■ DANE-EE (domain-issued certificate): Specifies a domain-operated CA to be used as a trust anchor. This certificate usage allows a domain name admin-
istrator to issue certificates for a domain without involving a third-party CA.
The server certificate chain is self-issued and does not need to verify against a
trusted root stored in the client.
The first two usage models are designed to co-exist with and strengthen
the public CA system. The final two usage models operate without the use of
public CAs.
The Selector field indicates whether the full certificate will be matched or just the value of the public key. The match is made between the certificate presented
in TLS negotiation and the certificate in the TLSA RR. The Matching Type field indicates how the match of the certificate is made. The options are exact match,
SHA-256 hash match, or SHA-512 hash match. The Certificate Association Data is the raw certificate data in hex format.
Figure 19.7 TLSA RR Transmission Format
Certificate usage Selector Matching type
Certificate association data
0Bit: 318 16 24
19.8 / SENDER POLICY FRAMEWORK 645
Use of DANE for SMTP
DANE can be used in conjunction with SMTP over TLS, as provided by STARTTLS,
to more fully secure email delivery. DANE can authenticate the certificate of the
SMTP submission server that the user’s mail client (MUA) communicates with. It
can also authenticate the TLS connections between SMTP servers (MTAs). The
use of DANE with SMTP is documented in an Internet Draft (SMTP Security via Opportunistic DANE TLS, draft-ietf-dane-smtp-with-dane-19, May 29, 2015).
As discussed in Section 19.1, SMTP can use the STARTTLS extension to
run SMTP over TLS, so that the entire email message plus SMTP envelope are
encrypted. This is done opportunistically, that is, if both sides support STARTTLS.
Even when TLS is used to provide confidentiality, it is vulnerable to attack in the
following ways:
■ Attackers can strip away the TLS capability advertisement and downgrade the
connection to not use TLS.
■ TLS connections are often unauthenticated (e.g., the use of self-signed certifi-
cates as well as mismatched certificates is common).
DANE can address both these vulnerabilities. A domain can use the presence
of the TLSA RR as an indicator that encryption must be performed, thus prevent-
ing malicious downgrade. A domain can authenticate the certificate used in the TLS
connection setup using a DNSSEC-signed TLSA RR.
Use of DNSSEC for S/MIME
DNSSEC can be used in conjunction with S/MIME to more fully secure email
delivery, in a manner similar to the DANE functionality. This use is documented in
an Internet Draft (Using Secure DNS to Associate Certificates with Domain Names for S/MIME, draft-ietf-dane-smime-09, August 27, 2015), which proposes a new SMIMEA DNS RR. The purpose of the SMIMEA RR is to associate certificates
with DNS domain names.
As discussed in Section 19.4, S/MIME messages often contain certificates
that can assist in authenticating the message sender and can be used in encrypt-
ing messages sent in reply. This feature requires that the receiving MUA validate
the certificate associated with the purported sender. SMIMEA RRs can provide a
secure means of doing this validation.
In essence, the SMIMEA RR will have the same format and content as the
TLSA RR, with the same functionality. The difference is that it is geared to the
needs of MUAs in dealing with domain names as specified in email addresses in the
message body, rather than domain names specified in the outer SMTP envelope.
19.8 SENDER POLICY FRAMEWORK
SPF is the standardized way for a sending domain to identify and assert the mail
senders for a given domain. The problem that SPF addresses is the following: With
the current email infrastructure, any host can use any domain name for each of the
646 CHAPTER 19 / ELECTRONIC MAIL SECURITY
various identifiers in the mail header, not just the domain name where the host is
located. Two major drawbacks of this freedom are:
■ It is a major obstacle to reducing unsolicited bulk email (UBE), also known as
spam. It makes it difficult for mail handlers to filter out emails on the basis of
known UBE sources.
■ ADMDs (see Section 19.1) are understandably concerned about the ease with
which other entities can make use of their domain names, often with malicious
intent.
RFC 7208 defines the SPF. It provides a protocol by which ADMDs can
authorize hosts to use their domain names in the “MAIL FROM” or “HELO”
identities. Compliant ADMDs publish Sender Policy Framework (SPF) records in
the DNS specifying which hosts are permitted to use their names, and compliant
mail receivers use the published SPF records to test the authorization of sending
Mail Transfer Agents (MTAs) using a given “HELO” or “MAIL FROM” identity
during a mail transaction.
SPF works by checking a sender’s IP address against the policy encoded in any
SPF record found at the sending domain. The sending domain is the domain used
in the SMTP connection, not the domain indicated in the message header as dis-
played in the MUA. This means that SPF checks can be applied before the message
content is received from the sender.
Figure 19.8 is an example in which SPF would come into play. Assume that the
sender’s IP address is 192.168.0.1. The message arrives from the MTA with domain
mta.example.net. The sender uses the MAIL FROM tag of [email protected],
indicating that the message originates in the example.org domain. But the message
header specifies [email protected]. The receiver uses SPF to query for the
SPF RR that corresponds to example.com to check if the IP address 192.168.0.1 is
S: 220 foo.com Simple Mail Transfer Service Ready
C: HELO mta.example.net
S: 250 OK
C: MAIL FROM:<[email protected]>
S: 250 OK
C: RCPT TO:<[email protected]>
S: 250 OK
C: DATA
S: 354 Start mail input; end with <crlf>.<crlf>
C: To: [email protected]
C: From: [email protected]
C: Date: Today
C: Subject: Meeting Today
. . .
Figure 19.8 Example in which SMTP Envelope Header Does Not Match Message Header
19.8 / SENDER POLICY FRAMEWORK 647
Tag Description
ip4 Specifies an IPv4 address or range of addresses that are authorized senders for
a domain.
ip6 Specifies an IPv6 address or range of addresses that are authorized senders for
a domain.
mx Asserts that the listed hosts for the Mail Exchange RRs are also valid senders for
the domain.
include Lists another domain where the receiver should look for an SPF RR for further
senders. This can be useful for large organizations with many domains or
sub-domains that have a single set of shared senders. The include mechanism is
recursive, in that the SPF check in the record found is tested in its entirety before
proceeding. It is not simply a concatenation of the checks.
all Matches every IP address that has not otherwise been matched.
(a) SPF Mechanisms
Modifier Description
+ The given mechanism check must pass. This is the default mechanism and does not need to be explicitly listed.
- The given mechanism is not allowed to send email on behalf of the domain.
∼ The given mechanism is in transition and if an email is seen from the listed host/IP address, then it should be accepted but marked for closer inspection.
? The SPF RR explicitly states nothing about the mechanism. In this case, the default
behavior is to accept the email. (This makes it equivalent to = + > unless some sort of discrete or aggregate message review is conducted.)
(b) SPF Mechanism Modifiers
Table 19.7 Common SPF Mechanisms and Modifiers
listed as a valid sender, and then takes appropriate action based on the results of
checking the RR.
SPF on the Sender Side
A sending domain needs to identify all the senders for a given domain and add
that information into the DNS as a separate resource record. Next, the sending
domain encodes the appropriate policy for each sender using the SPF syntax. The
encoding is done in a TXT DNS resource record as a list of mechanisms and mod-
ifiers. Mechanisms are used to define an IP address or range of addresses to be
matched, and modifiers indicate the policy for a given match. Table 19.7 lists the
most important mechanisms and modifiers used in SPF.
The SPF syntax is fairly complex and can express complex relationships
between senders. For more detail, see RFC 7208.
SPF on the Receiver Side
If SPF is implemented at a receiver, the SPF entity uses the SMTP envelope MAIL
FROM: address domain and the IP address of the sender to query an SPF TXT RR.
The SPF checks can be started before the body of the email message is received,
648 CHAPTER 19 / ELECTRONIC MAIL SECURITY
which may result in blocking the transmission of the email content. Alternatively,
the entire message can be absorbed and buffered until all the checks are finished.
In either case, checks must be completed before the mail message is sent to the end
user’s inbox.
The checking involves the following rules:
1. If no SPF TXT RR is returned, the default behavior is to accept the message.
2. If the SPF TXT RR has formatting errors, the default behavior is to accept the message.
3. Otherwise the mechanisms and modifiers in the RR are used to determine disposition of the email message.
Figure 19.9 illustrates SPF operation.
19.9 DOMAINKEYS IDENTIFIED MAIL
DomainKeys Identified Mail (DKIM) is a specification for cryptographically
signing email messages, permitting a signing domain to claim responsibility for a
message in the mail stream. Message recipients (or agents acting in their behalf)
can verify the signature by querying the signer’s domain directly to retrieve the
appropriate public key and thereby can confirm that the message was attested to
by a party in possession of the private key for the signing domain. DKIM is an
Internet Standard (RFC 6376: DomainKeys Identified Mail (DKIM) Signatures). DKIM has been widely adopted by a range of email providers, including
corporations, government agencies, gmail, Yahoo!, and many Internet Service
Providers (ISPs).
Figure 19.9 Sender Policy Framework Operation
Sender Inbound
mail server
SPF record lookup
Authorization pass/fail Further
policy checks
Inbox
Junk email
Quarantine
Block/delete
DNS
Internet
19.9 / DOMAINKEYS IDENTIFIED MAIL 649
Email Threats
RFC 4686 (Analysis of Threats Motivating DomainKeys Identified Mail) describes the threats being addressed by DKIM in terms of the characteristics, capabilities,
and location of potential attackers.
CHARACTERISTICS RFC 4686 characterizes the range of attackers on a spectrum of three levels of threat.
1. At the low end are attackers who simply want to send email that a recipient does not want to receive. The attacker can use one of a number of commercially
available tools that allow the sender to falsify the origin address of messages.
This makes it difficult for the receiver to filter spam on the basis of originating
address or domain.
2. At the next level are professional senders of bulk spam mail. These attackers often operate as commercial enterprises and send messages on behalf of third
parties. They employ more comprehensive tools for attack, including Mail
Transfer Agents (MTAs) and registered domains and networks of compro-
mised computers (zombies), to send messages and (in some cases) to harvest
addresses to which to send.
3. The most sophisticated and financially motivated senders of messages are those who stand to receive substantial financial benefit, such as from an email-
based fraud scheme. These attackers can be expected to employ all of the
above mechanisms and additionally may attack the Internet infrastructure
itself, including DNS cache-poisoning attacks and IP routing attacks.
CAPABILITIES RFC 4686 lists the following as capabilities that an attacker might have.
1. Submit messages to MTAs and Message Submission Agents (MSAs) at multiple locations in the Internet.
2. Construct arbitrary Message Header fields, including those claiming to be mailing lists, resenders, and other mail agents.
3. Sign messages on behalf of domains under their control.
4. Generate substantial numbers of either unsigned or apparently signed messages that might be used to attempt a denial-of-service attack.
5. Resend messages that may have been previously signed by the domain.
6. Transmit messages using any envelope information desired.
7. Act as an authorized submitter for messages from a compromised computer.
8. Manipulation of IP routing. This could be used to submit messages from specific IP addresses or difficult-to-trace addresses, or to cause diversion of
messages to a specific domain.
9. Limited influence over portions of DNS using mechanisms such as cache poisoning. This might be used to influence message routing or to falsify adver-
tisements of DNS-based keys or signing practices.
Hiva-Network.Com
650 CHAPTER 19 / ELECTRONIC MAIL SECURITY
10. Access to significant computing resources, for example, through the conscrip- tion of worm-infected “zombie” computers. This could allow the “bad actor” to
perform various types of brute-force attacks.
11. Ability to eavesdrop on existing traffic, perhaps from a wireless network.
LOCATION DKIM focuses primarily on attackers located outside of the administra- tive units of the claimed originator and the recipient. These administrative units
frequently correspond to the protected portions of the network adjacent to the orig-
inator and recipient. It is in this area that the trust relationships required for authen-
ticated message submission do not exist and do not scale adequately to be practical.
Conversely, within these administrative units, there are other mechanisms (such as
authenticated message submission) that are easier to deploy and more likely to be
used than DKIM. External bad actors are usually attempting to exploit the “any-to-
any” nature of email that motivates most recipient MTAs to accept messages from
anywhere for delivery to their local domain. They may generate messages without
signatures, with incorrect signatures, or with correct signatures from domains with
little traceability. They may also pose as mailing lists, greeting cards, or other agents
that legitimately send or resend messages on behalf of others.
DKIM Strategy
DKIM is designed to provide an email authentication technique that is transparent
to the end user. In essence, a user’s email message is signed by a private key of the
administrative domain from which the email originates. The signature covers all of
the content of the message and some of the RFC 5322 message headers. At the
receiving end, the MDA can access the corresponding public key via a DNS and
verify the signature, thus authenticating that the message comes from the claimed
administrative domain. Thus, mail that originates from somewhere else but claims
to come from a given domain will not pass the authentication test and can be
rejected. This approach differs from that of S/MIME and PGP, which use the origi-
nator’s private key to sign the content of the message. The motivation for DKIM is
based on the following reasoning:2
1. S/MIME depends on both the sending and receiving users employing S/MIME. For almost all users, the bulk of incoming mail does not use S/MIME, and the
bulk of the mail the user wants to send is to recipients not using S/MIME.
2. S/MIME signs only the message content. Thus, RFC 5322 header information concerning origin can be compromised.
3. DKIM is not implemented in client programs (MUAs) and is therefore trans- parent to the user; the user need not take any action.
4. DKIM applies to all mail from cooperating domains.
5. DKIM allows good senders to prove that they did send a particular message and to prevent forgers from masquerading as good senders.
2 The reasoning is expressed in terms of the use of S/MIME. The same argument applies to PGP.
19.9 / DOMAINKEYS IDENTIFIED MAIL 651
Figure 19.10 Simple Example of DKIM Deployment
Mail origination network
Mail delivery network
DNS Public key query/response
DNS = Domain Name System MDA = Mail Delivery Agent MSA = Mail Submission Agent MTA = Message Transfer Agent MUA = Message User Agent
SMTP
MUA
MUA
SMTP
SMTP
Signer Verifier
SMTP POP, IMAP
M T
A M
S A
M T
A M
D A
D N
S
Figure 19.10 is a simple example of the operation of DKIM. We begin with a
message generated by a user and transmitted into the MHS to an MSA that is within
the user’s administrative domain. An email message is generated by an email client
program. The content of the message, plus selected RFC 5322 headers, is signed by
the email provider using the provider’s private key. The signer is associated with a
domain, which could be a corporate local network, an ISP, or a public email facility
such as gmail. The signed message then passes through the Internet via a sequence
of MTAs. At the destination, the MDA retrieves the public key for the incoming
signature and verifies the signature before passing the message on to the destination
email client. The default signing algorithm is RSA with SHA-256. RSA with SHA-1
also may be used.
DKIM Functional Flow
Figure 19.11 provides a more detailed look at the elements of DKIM operation.
Basic message processing is divided between a signing Administrative Management
Domain (ADMD) and a verifying ADMD. At its simplest, this is between the origi-
nating ADMD and the delivering ADMD, but it can involve other ADMDs in the
handling path.
Signing is performed by an authorized module within the signing ADMD
and uses private information from a Key Store. Within the originating ADMD,
652 CHAPTER 19 / ELECTRONIC MAIL SECURITY
this might be performed by the MUA, MSA, or an MTA. Verifying is performed
by an authorized module within the verifying ADMD. Within a delivering
ADMD, verifying might be performed by an MTA, MDA or MUA. The mod-
ule verifies the signature or determines whether a particular signature was
required. Verifying the signature uses public information from the Key Store.
If the signature passes, reputation information is used to assess the signer and
that information is passed to the message filtering system. If the signature fails
or there is no signature using the author’s domain, information about signing
practices related to the author can be retrieved remotely and/or locally, and that
information is passed to the message filtering system. For example, if the sender
(e.g., gmail) uses DKIM but no DKIM signature is present, then the message
may be considered fraudulent.
Figure 19.11 DKIM Functional Flow
Originating or relaying ADMD: Sign message with SDID
RFC 5322 message
yes
pass fail
no
Relaying or delivering ADMD: Message signed?
Verify signature
Private key
store
(paired)
Public key
store
Remote sender
practices
Local info on sender practices
Reputation/ accreditation information
Assessments
Message filtering engine
Check signing
practices
Internet
19.9 / DOMAINKEYS IDENTIFIED MAIL 653
The signature is inserted into the RFC 5322 message as an additional header
entry, starting with the keyword Dkim-Signature. You can view examples from your
own incoming mail by using the View Long Headers (or similar wording) option for
an incoming message. Here is an example:
Dkim-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=gamma; h=domainkey- signature:mime-version:received:date: message-id:subject :from:to:content-type: content-transfer-encoding; bh=5mZvQDyCRuyLb1Y28K4zgS2MPOemFToDBgvbJ 7GO90s=; b=PcUvPSDygb4ya5Dyj1rbZGp/VyRiScuaz7TTG
J5qW5slM+klzv6kcfYdGDHzEVJW+Z FetuPfF1ETOVhELtwH0zjSccOyPkEiblOf6gILO bm3DDRm3Ys1/FVrbhVOlA+/jH9Aei uIIw/5iFnRbSH6qPDVv/beDQqAWQfA/wF7O5k=
Before a message is signed, a process known as canonicalization is performed
on both the header and body of the RFC 5322 message. Canonicalization is necessary
to deal with the possibility of minor changes in the message made en route, includ-
ing character encoding, treatment of trailing white space in message lines, and the
“folding” and “unfolding” of header lines. The intent of canonicalization is to make a
minimal transformation of the message (for the purpose of signing; the message itself
is not changed, so the canonicalization must be performed again by the verifier) that
will give it its best chance of producing the same canonical value at the receiving end.
DKIM defines two header canonicalization algorithms (“simple” and “relaxed”) and
two for the body (with the same names). The simple algorithm tolerates almost no
modification, while the relaxed algorithm tolerates common modifications.
The signature includes a number of fields. Each field begins with a tag consist-
ing of a tag code followed by an equals sign and ends with a semicolon. The fields
include the following:
■ v= DKIM version/
■ a= Algorithm used to generate the signature; must be either rsa-sha1 or rsa-sha256
■ c= Canonicalization method used on the header and the body.
■ d= A domain name used as an identifier to refer to the identity of a responsible person or organization. In DKIM, this identifier is called the Signing Domain
IDentifier (SDID). In our example, this field indicates that the sender is using
a gmail address.
■ s= In order that different keys may be used in different circumstances for the same signing domain (allowing expiration of old keys, separate departmen-
tal signing, or the like), DKIM defines a selector (a name associated with a
key) that is used by the verifier to retrieve the proper key during signature
verification.
654 CHAPTER 19 / ELECTRONIC MAIL SECURITY
■ h= Signed Header fields. A colon-separated list of header field names that identify the header fields presented to the signing algorithm. Note that in our
example above, the signature covers the domainkey-signature field. This refers
to an older algorithm (since replaced by DKIM) that is still in use.
■ bh= The hash of the canonicalized body part of the message. This provides additional information for diagnosing signature verification failures.
■ b= The signature data in base64 format; this is the encrypted hash code.
19.10 DOMAIN-BASED MESSAGE AUTHENTICATION, REPORTING, AND CONFORMANCE
Domain-Based Message Authentication, Reporting, and Conformance (DMARC)
allows email senders to specify policy on how their mail should be handled, the
types of reports that receivers can send back, and the frequency those reports
should be sent. It is defined in RFC 7489 (Domain-based Message Authentication, Reporting, and Conformance, March 2015).
DMARC works with SPF and DKIM. SPF and DKM enable senders to advise
receivers, via DNS, whether mail purporting to come from the sender is valid, and
whether it should be delivered, flagged, or discarded. However, neither SPF nor
DKIM include a mechanism to tell receivers if SPF or DKIM are in use, nor do they
have feedback mechanism to inform senders of the effectiveness of the anti-spam
techniques. For example, if a message arrives at a receiver without a DKIM signa-
ture, DKIM provides no mechanism to allow the receiver to learn if the message is
authentic but was sent from a sender that did not implement DKIM, or if the mes-
sage is a spoof. DMARC addresses these issues essentially by standardizing how
email receivers perform email authentication using SPF and DKIM mechanisms.
Identifier Alignment
DKIM, SPF, and DMARC authenticate various aspects of an individual mes-
sage. DKIM authenticates the domain that affixed a signature to the message. SPF
focuses on the SMTP envelope, defined in RFC 5321. It can authenticate either the
domain that appears in the MAIL FROM portion of the SMTP envelope or the
HELO domain, or both. These may be different domains, and they are typically not
visible to the end user.
DMARC authentication deals with the From domain in the message header,
as defined in RFC 5322. This field is used as the central identity of the DMARC
mechanism because it is a required message header field and therefore guaranteed
to be present in compliant messages, and most MUAs represent the RFC 5322 From
field as the originator of the message and render some or all of this header field’s
content to end users. The email address in this field is the one used by end users to
identify the source of the message and therefore is a prime target for abuse.
DMARC requires that From address match (be aligned with) an Authenticated
Identifier from DKIM or SPF. In the case of DKIM, the match is made between
the DKIM signing domain and the From domain. In the case of SPF, the match is
between the SPF-authenticated domain and the From domain.
19.10 / DOMAIN-BASED MESSAGE AUTHENTICATION 655
DMARC on the Sender Side
A mail sender that uses DMARC must also use SPF or DKIM, or both. The sender
posts a DMARC policy in the DNS that advises receivers on how to treat messages
that purport to originate from the sender’s domain. The policy is in the form of
a DNS TXT resource record. The sender also needs to establish email addresses
to receive aggregate and forensic reports. As these email addresses are published
unencrypted in the DNS TXT RR, they are easily discovered, leaving the poster
subject to unsolicited bulk email. Thus, the poster of the DNS TXT RR needs to
employ some kind of abuse countermeasures.
Similar to SPF and DKIM, the DMARC policy in the TXT RR is encoded
in a series of tag=value pairs separated by semicolons. Table 19.8 describes the common tags.
Once the DMARC RR is posted, messages from the sender are typically
processed as follows:
1. The domain owner constructs an SPF policy and publishes it in its DNS database. The domain owner also configures its system for DKIM signing.
Finally, the domain owner publishes via the DNS a DMARC message- handling
policy.
2. The author generates a message and hands the message to the domain owner’s designated mail submission service.
3. The submission service passes relevant details to the DKIM signing module in order to generate a DKIM signature to be applied to the message.
4. The submission service relays the now-signed message to its designated trans- port service for routing to its intended recipient(s).
DMARC on the Receiver Side
A message generated on the sender side may pass through other relays but even-
tually arrives at a receiver’s transport service. The typical processing order for
DMARC on the receiving side is the following:
1. The receiver performs standard validation tests, such as checking against IP blocklists and domain reputation lists, as well as enforcing rate limits from a
particular source.
2. The receiver extracts the RFC 5322 From address from the message. This must contain a single, valid address or else the mail is refused as an error.
3. The receiver queries for the DMARC DNS record based on the sending do- main. If none exists, terminate DMARC processing.
4. The receiver performs DKIM signature checks. If more than one DKIM signa- ture exists in the message, one must verify.
5. The receiver queries for the sending domain’s SPF record and performs SPF validation checks.
6. The receiver conducts Identifier Alignment checks between the RFC 5321 From and the results of the SPF and DKIM records (if present).
656 CHAPTER 19 / ELECTRONIC MAIL SECURITY
Tag (Name) Description
v= (Version) Version field that must be present as the first element. By default the value is always DMARC1.
p= (Policy) Mandatory policy field. May take values none or quarantine or reject. This allows for a gradually tightening policy where the sender domain recommends
no specific action on mail that fails DMARC checks (p= none), through treating failed mail as suspicious (p= quarantine), to rejecting all failed mail (p= reject), preferably at the SMTP transaction stage.
aspf= (SPF Policy) Values are r (default) for relaxed and s for strict SPF domain enforcement. Strict alignment requires an exact match between the From address domain and the
(passing) SPF check must exactly match the MailFrom address (HELO address).
Relaxed requires that only the From and MailFrom address domains be in
alignment. For example, the MailFrom address domain smtp.example.org and the From address [email protected] are in alignment, but not a strict match.
adkim= (DKIM Policy)
Optional. Values are r (default) for relaxed and s for strict DKIM domain enforcement. Strict alignment requires an exact match between the From
domain in the message header and the DKIM domain presented in the
(d= DKIM), tag. Relaxed requires only that the domain part is in alignment (as in aspf).
fo= (Failure reporting options)
Optional. Ignore if a ruf argument is not also present. Value 0 indicates the receiver should generate a DMARC failure report if all underlying mechanisms
fail to produce an aligned pass result. Value 1 means generate a DMARC failure report if any underlying mechanism produces something other than an aligned
pass result. Other possible values are d (generate a DKIM failure report if a signature failed evaluation), and s (generate an SPF failure report if the message failed SPF evaluation). These values are not exclusive and may be combined.
ruf= Optional, but requires the fo argument to be present. Lists a series of URIs (currently just mailto:<emailaddress>) that list where to send forensic feedback reports. This is for reports on message-specific failures.
rua= Optional list of URIs (like in ruf= , using the mailto: URI) listing where to send aggregate feedback back to the sender. These reports are sent based on the
interval requested using the ri= option with a default of 86400 seconds if not listed.
ri= (Reporting interval) Optional with the default value of 86400 seconds. The value listed is the reporting interval desired by the sender.
pct= (Percent) Optional with the default value of 100. Expresses the percentage of a sender’s mail that should be subject to the given DMARC policy. This allows senders to
ramp up their policy enforcement gradually and prevent having to commit to a
rigorous policy before getting feedback on their existing policy.
sp= (Receiver Policy) Optional with a default value of none. Other values include the same range of values as the p= argument. This is the policy to be applied to mail from all identified subdomains of the given DMARC RR.
Table 19.8 DMARC Tag and Value Descriptions
7. The results of these steps are passed to the DMARC module along with the Author’s domain. The DMARC module attempts to retrieve a policy from the
DNS for that domain. If none is found, the DMARC module determines the
organizational domain and repeats the attempt to retrieve a policy from the DNS.
8. If a policy is found, it is combined with the Author’s domain and the SPF and DKIM results to produce a DMARC policy result (a “pass” or “fail”) and can
optionally cause one of two kinds of reports to be generated.
19.10 / DOMAIN-BASED MESSAGE AUTHENTICATION, 657
Figure 19.12 DMARC Functional Flow
DKIM
DKIM
SPF SPF
Failure report
Block
Pass
Sender Receiver
Fail
Quaran- tine
Author composes and sends email
Standard processing (including antispam)
Sending mail server attaches DKIM signature
Standard validation tests at receiver (including IP
blocklists, reputation, rate
limits, etc)
Retrieve verified DKIM domains
Retrieve “envelope from”
via SPF
Update periodic aggregate report
to be sent to sender
Apply DMARC
policy
9. Recipient transport service either delivers the message to the recipient inbox or takes other local policy action based on the DMARC result.
10. When requested, Recipient transport service collects data from the message delivery session to be used in providing feedback.
Figure 19.12, based on one at DMARC.org, summarizes the sending and
receiving functional flow.
658 CHAPTER 19 / ELECTRONIC MAIL SECURITY
DMARC Reports
DMARC reporting provides the sender’s feedback on their SPF, DKIM, Identifier
Alignment, and message disposition policies, which enable the sender to make
these policies more effective. Two types of reports are sent: aggregate reports and
forensic reports.
Aggregate reports are sent by receivers periodically and include aggregate
figures for successful and unsuccessful message authentications, including:
■ The sender’s DMARC policy for that interval.
■ The message disposition by the receiver (i.e., delivered, quarantined, rejected).
■ SPF result for a given SPF identifier.
■ DKIM result for a given DKIM identifier.
■ Whether identifiers are in alignment or not.
■ Results classified by sender subdomain.
■ The sending and receiving domain pair.
■ The policy applied, and whether this is different from the policy requested.
■ The number of successful authentications.
■ Totals for all messages received.
This information enables the sender to identify gaps in email infrastruc-
ture and policy. SP 800-177 recommends that a sending domain begin by setting
a DMARC policy of p= none, so that the ultimate disposition of a message that fails some check is determined by the receiver’s local policy. As DMARC aggregate
reports are collected, the sender will have a quantitatively better assessment of the
extent to which the sender’s email is authenticated by outside receivers, and will
be able to set a policy of p=reject, indicating that any message that fails the SPF, DKIM, and alignment checks really should be rejected. From their own traffic anal-
ysis, receivers can develop a determination of whether a sender’s p=reject policy is sufficiently trustworthy to act on.
A forensic report helps the sender refine the component SPF and DKIM
mechanisms as well as alerting the sender that their domain is being used as part
of a phishing/spam campaign. Forensic reports are similar in format to aggregation
reports, with these changes:
■ Receivers include as much of the message and message header as is reason-
able to allow the domain to investigate the failure. Add an Identity-Alignment
field, with DKIM and SPF DMARC-method fields as appropriate.
■ Optionally add a Delivery-Result field.
■ Add DKIM Domain, DKIM Identity, and DKIM selector fields, if the message
was DKIM signed. Optionally also add DKIM Canonical header and body
fields.
■ Add an additional DMARC authentication failure type, for use when some
authentication mechanisms fail to produce aligned identifiers.
Hiva-Network.Com
19.11 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 659
19.11 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
Key Terms
administrative management
domain (ADMD)
base64
Cryptographic Message
Syntax (CMS)
detached signature
DNS-based Authentication of
Named Entities (DANE)
DNS Security Extensions
(DNSSEC)
Domain-based Message
Authentication, Reporting,
and Conformance
(DMARC)
Domain Name System (DNS)
DomainKeys Identified Mail
(DKIM)
electronic mail
Internet Mail Access Protocol
(IMAP)
Mail Delivery Agent (MDA)
Mail Submission Agent
(MSA)
Message Handling Service
(MHS)
Message Store
Message Transfer Agents
(MTA)
Message User Agent (MUA)
Multipurpose Internet Mail
Extensions (MIME)
Post Office Protocol (POP3)
Pretty Good Privacy (PGP)
Sender Policy Framework
(SPF)
session key
Simple Mail Transfer Protocol
(SMTP)
STARTTLS
SUBMISSION
S/MIME
trust
Review Questions 19.1 What types of interoperability issues are involved in internet mail architecture and
how are they handled?
19.2 What are the SMTP and MIME standards? 19.3 What is the difference between a MIME content type and a MIME transfer encoding? 19.4 Briefly explain base64 encoding. 19.5 Why is base64 conversion useful for an email application? 19.6 What is S/MIME? 19.7 What are the four principal services provided by S/MIME? 19.8 What is the utility of a detached signature? 19.9 What is DKIM?
Problems 19.1 The character sequence “<CR><LF>.<CR><LF>” indicates the end of mail data to a
SMTP-server. What happens if the mail data itself contains that character sequence?
19.2 What are POP3 and IMAP? 19.3 If a lossless compression algorithm, such as ZIP, is used with S/MIME, why is it pref-
erable to generate a signature before applying compression?
19.4 Before the deployment of the Domain Name System, a simple text file (HOSTS. TXT) centrally maintained at the SRI Network Information Center was used to enable mapping between host names and addresses. Each host connected to the Internet had to have an updated local copy of it to be able to use host names instead of having to cope directly with their IP addresses. Discuss the main advantages of the DNS over the old centralized HOSTS.TXT system.
19.5 For this problem and the next few, consult Appendix P. In Figure P.2, each entry in the public-key ring contains an Owner Trust field that indicates the degree of trust associated with this public-key owner. Why is that not enough? That is, if this owner is trusted and this is supposed to be the owner’s public key, why is that trust not enough to permit PGP to use this public key?
660 CHAPTER 19 / ELECTRONIC MAIL SECURITY
19.6 What is the basic difference between X.509 and PGP in terms of key hierarchies and key trust?
19.7 In PGP, what is the expected number of session keys generated before a previously created key is produced?
19.8 A PGP user may have multiple public keys. So that a recipient knows which public key is being used by a sender, a key ID, consisting of the least significant 64 bits of the public key, is sent with the message. What is the probability that a user with N public keys will have at least one duplicate key ID?
19.9 The first 16 bits of the message digest in a PGP signature are translated in the clear. This enables the recipient to determine if the correct public key was used to decrypt the message digest by comparing this plaintext copy of the first two octets with the first two octets of the decrypted digest. a. To what extent does this compromise the security of the hash algorithm? b. To what extent does it in fact perform its intended function, namely, to help deter-
mine if the correct RSA key was used to decrypt the digest?
19.10 Consider base64 conversion as a form of encryption. In this case, there is no key. But suppose that an opponent knew only that some form of substitution algorithm was being used to encrypt English text and did not guess that it was base64. How effective would this algorithm be against cryptanalysis?
19.11 Encode the text “ciphertext” using the following techniques. Assume characters are stored in 8-bit ASCII with zero parity. a. base64 b. Quoted-printable
19.12 Use a 2 * 2 matrix to categorize the properties of the four certificate usage models in DANE.
661
CHAPTER
20.1 IP Security Overview
Applications of IPsec
Benefits of IPsec
Routing Applications
IPsec Documents
IPsec Services
Transport and Tunnel Modes
20.2 IP Security Policy
Security Associations
Security Association Database
Security Policy Database
IP Traffic Processing
20.3 Encapsulating Security Payload
ESP Format
Encryption and Authentication Algorithms
Padding
Anti-Replay Service
Transport and Tunnel Modes
20.4 Combining Security Associations
Authentication Plus Confidentiality
Basic Combinations of Security Associations
20.5 Internet Key Exchange
Key Determination Protocol
Header and Payload Formats
20.6 Cryptographic Suites
20.7 Key Terms, Review Questions, and Problems
IP Security
662 CHAPTER 20 / IP SECURITY
There are application-specific security mechanisms for a number of application
areas, including electronic mail (S/MIME, PGP), client/server (Kerberos), Web ac-
cess (Secure Sockets Layer), and others. However, users have security concerns that
cut across protocol layers. For example, an enterprise can run a secure, private IP
network by disallowing links to untrusted sites, encrypting packets that leave the
premises, and authenticating packets that enter the premises. By implementing se-
curity at the IP level, an organization can ensure secure networking not only for
applications that have security mechanisms but also for the many security-ignorant
applications.
IP-level security encompasses three functional areas: authentication, confiden-
tiality, and key management. The authentication mechanism assures that a received
packet was, in fact, transmitted by the party identified as the source in the packet
header. In addition, this mechanism assures that the packet has not been altered in
transit. The confidentiality facility enables communicating nodes to encrypt messages
to prevent eavesdropping by third parties. The key management facility is concerned
with the secure exchange of keys.
We begin this chapter with an overview of IP security (IPsec) and an introduc-
tion to the IPsec architecture. We then look at each of the three functional areas in
detail. Appendix L reviews Internet protocols.
20.1 IP SECURITY OVERVIEW
In 1994, the Internet Architecture Board (IAB) issued a report titled “Security in
the Internet Architecture” (RFC 1636). The report identified key areas for security
mechanisms. Among these were the need to secure the network infrastructure from
LEARNING OBJECTIVES
After studying this chapter, you should be able to:
◆ Present an overview of IP security (IPsec).
◆ Explain the difference between transport mode and tunnel mode.
◆ Understand the concept of security association.
◆ Explain the difference between the security association database and the security policy database.
◆ Summarize the traffic processing functions performed by IPsec for out- bound packets and for inbound packets.
◆ Present an overview of Encapsulating Security Payload.
◆ Discuss the alternatives for combining security associations.
◆ Present an overview of Internet Key Exchange.
◆ Summarize the alternative cryptographic suites approved for use with IPsec.
20.1 / IP SECURITY OVERVIEW 663
unauthorized monitoring and control of network traffic and the need to secure end-
user-to-end-user traffic using authentication and encryption mechanisms.
To provide security, the IAB included authentication and encryption as nec-
essary security features in the next-generation IP, which has been issued as IPv6.
Fortunately, these security capabilities were designed to be usable both with the
current IPv4 and the future IPv6. This means that vendors can begin offering these
features now, and many vendors now do have some IPsec capability in their prod-
ucts. The IPsec specification now exists as a set of Internet standards.
Applications of IPsec
IPsec provides the capability to secure communications across a LAN, across pri-
vate and public WANs, and across the Internet. Examples of its use include:
■ Secure branch office connectivity over the Internet: A company can build a secure virtual private network over the Internet or over a public WAN. This
enables a business to rely heavily on the Internet and reduce its need for pri-
vate networks, saving costs and network management overhead.
■ Secure remote access over the Internet: An end user whose system is equipped with IP security protocols can make a local call to an Internet Service Provider
(ISP) and gain secure access to a company network. This reduces the cost of
toll charges for traveling employees and telecommuters.
■ Establishing extranet and intranet connectivity with partners: IPsec can be used to secure communication with other organizations, ensuring authentica-
tion and confidentiality and providing a key exchange mechanism.
■ Enhancing electronic commerce security: Even though some Web and elec- tronic commerce applications have built-in security protocols, the use of IPsec
enhances that security. IPsec guarantees that all traffic designated by the net-
work administrator is both encrypted and authenticated, adding an additional
layer of security to whatever is provided at the application layer.
The principal feature of IPsec that enables it to support these varied applica-
tions is that it can encrypt and/or authenticate all traffic at the IP level. Thus, all dis- tributed applications (including remote logon, client/server, email, file transfer, Web
access, and so on) can be secured. Figure 20.1a shows a simplified packet format for
an IPsec option known as tunnel mode, described subsequently. Tunnel mode makes
use of an IPsec function, a combined authentication/encryption function called
Encapsulating Security Payload (ESP), and a key exchange function. For VPNs,
both authentication and encryption are generally desired, because it is important
both to (1) assure that unauthorized users do not penetrate the VPN, and (2) assure
that eavesdroppers on the Internet cannot read messages sent over the VPN.
Figure 20.1b is a typical scenario of IPsec usage. An organization maintains
LANs at dispersed locations. Nonsecure IP traffic is conducted on each LAN. For
traffic offsite, through some sort of private or public WAN, IPsec protocols are used.
These protocols operate in networking devices, such as a router or firewall, that
connect each LAN to the outside world. The IPsec networking device will typically
encrypt all traffic going into the WAN and decrypt traffic coming from the WAN;
these operations are transparent to workstations and servers on the LAN. Secure
664 CHAPTER 20 / IP SECURITY
transmission is also possible with individual users who dial into the WAN. Such user
workstations must implement the IPsec protocols to provide security.
Benefits of IPsec
Some of the benefits of IPsec:
■ When IPsec is implemented in a firewall or router, it provides strong security
that can be applied to all traffic crossing the perimeter. Traffic within a com-
pany or workgroup does not incur the overhead of security-related processing.
Figure 20.1 An IPSec VPN Scenario
Networking device with IPSec
Ethernet switch
Unprotected IP traffic
Legend:
User system with IPSec
(a) Tunnel-mode format
(b) Example configuration
Public (Internet) or private network
authenticated
encrypted
ESP auth
orig IP hdr
IP payload ESP trlr
ESP hdr
IP traffic protected by IPSec
Virtual tunnel: protected by IPSec
New IP hdr
20.1 / IP SECURITY OVERVIEW 665
■ IPsec in a firewall is resistant to bypass if all traffic from the outside must use
IP and the firewall is the only means of entrance from the Internet into the
organization.
■ IPsec is below the transport layer (TCP, UDP) and so is transparent to appli-
cations. There is no need to change software on a user or server system when
IPsec is implemented in the firewall or router. Even if IPsec is implemented in
end systems, upper-layer software, including applications, is not affected.
■ IPsec can be transparent to end users. There is no need to train users on secu-
rity mechanisms, issue keying material on a per-user basis, or revoke keying
material when users leave the organization.
■ IPsec can provide security for individual users if needed. This is useful for off-
site workers and for setting up a secure virtual subnetwork within an organiza-
tion for sensitive applications.
Routing Applications
In addition to supporting end users and protecting premises systems and networks,
IPsec can play a vital role in the routing architecture required for internetworking.
[HUIT98] lists the following examples of the use of IPsec. IPsec can assure that
■ A router advertisement (a new router advertises its presence) comes from an
authorized router.
■ A neighbor advertisement (a router seeks to establish or maintain a neighbor
relationship with a router in another routing domain) comes from an autho-
rized router.
■ A redirect message comes from the router to which the initial IP packet was sent.
■ A routing update is not forged.
Without such security measures, an opponent can disrupt communications
or divert some traffic. Routing protocols such as Open Shortest Path First (OSPF)
should be run on top of security associations between routers that are defined by
IPsec.
IPsec Documents
IPsec encompasses three functional areas: authentication, confidentiality, and key
management. The totality of the IPsec specification is scattered across dozens of
RFCs and draft IETF documents, making this the most complex and difficult to
grasp of all IETF specifications. The best way to grasp the scope of IPsec is to
consult the latest version of the IPsec document roadmap, which as of this writ-
ing is RFC 6071 [IP Security (IPsec) and Internet Key Exchange (IKE) Document Roadmap, February 2011]. The documents can be categorized into the following groups.
■ Architecture: Covers the general concepts, security requirements, definitions, and mechanisms defining IPsec technology. The current specification is RFC
4301, Security Architecture for the Internet Protocol.
666 CHAPTER 20 / IP SECURITY
■ Authentication Header (AH): AH is an extension header to provide mes- sage authentication. The current specification is RFC 4302, IP Authentication Header. Because message authentication is provided by ESP, the use of AH is deprecated. It is included in IPsecv3 for backward compatibility
but should not be used in new applications. We do not discuss AH in this
chapter.
■ Encapsulating Security Payload (ESP): ESP consists of an encapsulat- ing header and trailer used to provide encryption or combined encryption/
authentication. The current specification is RFC 4303, IP Encapsulating Security Payload (ESP).
■ Internet Key Exchange (IKE): This is a collection of documents describing the key management schemes for use with IPsec. The main specification is
RFC 7296, Internet Key Exchange (IKEv2) Protocol, but there are a number of related RFCs.
■ Cryptographic algorithms: This category encompasses a large set of docu- ments that define and describe cryptographic algorithms for encryption, mes-
sage authentication, pseudorandom functions (PRFs), and cryptographic key
exchange.
■ Other: There are a variety of other IPsec-related RFCs, including those deal- ing with security policy and management information base (MIB) content.
IPsec Services
IPsec provides security services at the IP layer by enabling a system to select
required security protocols, determine the algorithm(s) to use for the service(s),
and put in place any cryptographic keys required to provide the requested services.
Two protocols are used to provide security: an authentication protocol designated
by the header of the protocol, Authentication Header (AH); and a combined
encryption/authentication protocol designated by the format of the packet for
that protocol, Encapsulating Security Payload (ESP). RFC 4301 lists the following
services:
■ Access control
■ Connectionless integrity
■ Data origin authentication
■ Rejection of replayed packets (a form of partial sequence integrity)
■ Confidentiality (encryption)
■ Limited traffic flow confidentiality
Transport and Tunnel Modes
Both AH and ESP support two modes of use: transport and tunnel mode. The oper-
ation of these two modes is best understood in the context of a description of ESP,
which is covered in Section 20.3. Here we provide a brief overview.
20.1 / IP SECURITY OVERVIEW 667
TRANSPORT MODE Transport mode provides protection primarily for upper-layer protocols. That is, transport mode protection extends to the payload of an IP
packet.1 Examples include a TCP or UDP segment or an ICMP packet, all of which
operate directly above IP in a host protocol stack. Typically, transport mode is used
for end-to-end communication between two hosts (e.g., a client and a server, or two
workstations). When a host runs AH or ESP over IPv4, the payload is the data that
normally follow the IP header. For IPv6, the payload is the data that normally fol-
low both the IP header and any IPv6 extensions headers that are present, with the
possible exception of the destination options header, which may be included in the
protection.
ESP in transport mode encrypts and optionally authenticates the IP payload
but not the IP header. AH in transport mode authenticates the IP payload and
selected portions of the IP header.
TUNNEL MODE Tunnel mode provides protection to the entire IP packet. To achieve this, after the AH or ESP fields are added to the IP packet, the entire packet plus
security fields is treated as the payload of new outer IP packet with a new outer
IP header. The entire original, inner, packet travels through a tunnel from one
point of an IP network to another; no routers along the way are able to examine
the inner IP header. Because the original packet is encapsulated, the new, larger
packet may have totally different source and destination addresses, adding to the
security. Tunnel mode is used when one or both ends of a security association (SA)
are a security gateway, such as a firewall or router that implements IPsec. With tun-
nel mode, a number of hosts on networks behind firewalls may engage in secure
communications without implementing IPsec. The unprotected packets generated
by such hosts are tunneled through external networks by tunnel mode SAs set up
by the IPsec software in the firewall or secure router at the boundary of the local
network.
Here is an example of how tunnel mode IPsec operates. Host A on a network
generates an IP packet with the destination address of host B on another network.
This packet is routed from the originating host to a firewall or secure router at the
boundary of A’s network. The firewall filters all outgoing packets to determine the
need for IPsec processing. If this packet from A to B requires IPsec, the firewall
performs IPsec processing and encapsulates the packet with an outer IP header.
The source IP address of this outer IP packet is this firewall, and the destination
address may be a firewall that forms the boundary to B’s local network. This packet
is now routed to B’s firewall, with intermediate routers examining only the outer IP
header. At B’s firewall, the outer IP header is stripped off, and the inner packet is
delivered to B.
ESP in tunnel mode encrypts and optionally authenticates the entire inner IP
packet, including the inner IP header. AH in tunnel mode authenticates the entire
inner IP packet and selected portions of the outer IP header.
Table 20.1 summarizes transport and tunnel mode functionality.
1In this chapter, the term IP packet refers to either an IPv4 datagram or an IPv6 packet.
Hiva-Network.Com
668 CHAPTER 20 / IP SECURITY
20.2 IP SECURITY POLICY
Fundamental to the operation of IPsec is the concept of a security policy applied
to each IP packet that transits from a source to a destination. IPsec policy is
determined primarily by the interaction of two databases, the security association database (SAD) and the security policy database (SPD). This section provides an overview of these two databases and then summarizes their use during IPsec opera-
tion. Figure 20.2 illustrates the relevant relationships.
Security Associations
A key concept that appears in both the authentication and confidentiality mecha-
nisms for IP is the security association (SA). An association is a one-way logical
connection between a sender and a receiver that affords security services to the traf-
fic carried on it. If a peer relationship is needed for two-way secure exchange, then
two security associations are required.
A security association is uniquely identified by three parameters.
■ Security Parameters Index (SPI): A 32-bit unsigned integer assigned to this SA and having local significance only. The SPI is carried in AH and ESP head-
ers to enable the receiving system to select the SA under which a received
packet will be processed.
■ IP Destination Address: This is the address of the destination endpoint of the SA, which may be an end-user system or a network system such as a firewall
or router.
■ Security Protocol Identifier: This field from the outer IP header indicates whether the association is an AH or ESP security association.
Hence, in any IP packet, the security association is uniquely identified by the
Destination Address in the IPv4 or IPv6 header and the SPI in the enclosed exten-
sion header (AH or ESP).
Transport Mode SA Tunnel Mode SA
AH Authenticates IP payload and selected
portions of IP header and IPv6
extension headers.
Authenticates entire inner IP packet (inner
header plus IP payload) plus selected portions
of outer IP header and outer IPv6 extension
headers.
ESP Encrypts IP payload and any IPv6
extension headers following the ESP
header.
Encrypts entire inner IP packet.
ESP with
Authentication
Encrypts IP payload and any IPv6
extension headers following the ESP
header. Authenticates IP payload but
not IP header.
Encrypts entire inner IP packet. Authenticates
inner IP packet.
Table 20.1 Tunnel Mode and Transport Mode Functionality
20.2 / IP SECURITY POLICY 669
Security Association Database
In each IPsec implementation, there is a nominal2 Security Association Database
that defines the parameters associated with each SA. A security association is nor-
mally defined by the following parameters in an SAD entry.
■ Security Parameter Index: A 32-bit value selected by the receiving end of an SA to uniquely identify the SA. In an SAD entry for an outbound SA, the SPI
is used to construct the packet’s AH or ESP header. In an SAD entry for an
inbound SA, the SPI is used to map traffic to the appropriate SA.
■ Sequence Number Counter: A 32-bit value used to generate the Sequence Number field in AH or ESP headers, described in Section 20.3 (required for all
implementations).
■ Sequence Counter Overflow: A flag indicating whether overflow of the Sequence Number Counter should generate an auditable event and prevent
further transmission of packets on this SA (required for all implementations).
■ Anti-Replay Window: Used to determine whether an inbound AH or ESP packet is a replay, described in Section 20.3 (required for all implementations).
■ AH Information: Authentication algorithm, keys, key lifetimes, and related parameters being used with AH (required for AH implementations).
■ ESP Information: Encryption and authentication algorithm, keys, initialization values, key lifetimes, and related parameters being used with ESP (required
for ESP implementations).
■ Lifetime of this Security Association: A time interval or byte count after which an SA must be replaced with a new SA (and new SPI) or terminated,
plus an indication of which of these actions should occur (required for all
implementations).
2Nominal in the sense that the functionality provided by a Security Association Database must be present in any IPsec implementation, but the way in which that functionality is provided is up to the implementer.
Figure 20.2 IPsec Architecture
SPD SPD
SAD
IKEv2 IKEv2
IPsecv3 IPsecv3
Security association database
Key exchange
IKE SA
IPsec SA Pair
ESP protects data
Security association database
Security policy
database
Security policy
database
SAD
670 CHAPTER 20 / IP SECURITY
■ IPsec Protocol Mode: Tunnel, transport, or wildcard.
■ Path MTU: Any observed path maximum transmission unit (maximum size of a packet that can be transmitted without fragmentation) and aging variables
(required for all implementations).
The key management mechanism that is used to distribute keys is coupled to
the authentication and privacy mechanisms only by way of the Security Parameters
Index (SPI). Hence, authentication and privacy have been specified independent of
any specific key management mechanism.
IPsec provides the user with considerable flexibility in the way in which IPsec
services are applied to IP traffic. As we will see later, SAs can be combined in a
number of ways to yield the desired user configuration. Furthermore, IPsec pro-
vides a high degree of granularity in discriminating between traffic that is afforded
IPsec protection and traffic that is allowed to bypass IPsec, as in the former case
relating IP traffic to specific SAs.
Security Policy Database
The means by which IP traffic is related to specific SAs (or no SA in the case of traffic
allowed to bypass IPsec) is the nominal Security Policy Database (SPD). In its simplest
form, an SPD contains entries, each of which defines a subset of IP traffic and points
to an SA for that traffic. In more complex environments, there may be multiple entries
that potentially relate to a single SA or multiple SAs associated with a single SPD
entry. The reader is referred to the relevant IPsec documents for a full discussion.
Each SPD entry is defined by a set of IP and upper-layer protocol field values,
called selectors. In effect, these selectors are used to filter outgoing traffic in order to map it into a particular SA. Outbound processing obeys the following general
sequence for each IP packet.
1. Compare the values of the appropriate fields in the packet (the selector fields) against the SPD to find a matching SPD entry, which will point to zero or more SAs.
2. Determine the SA if any for this packet and its associated SPI.
3. Do the required IPsec processing (i.e., AH or ESP processing).
The following selectors determine an SPD entry:
■ Remote IP Address: This may be a single IP address, an enumerated list or range of addresses, or a wildcard (mask) address. The latter two are required to
support more than one destination system sharing the same SA (e.g., behind
a firewall).
■ Local IP Address: This may be a single IP address, an enumerated list or range of addresses, or a wildcard (mask) address. The latter two are required to sup-
port more than one source system sharing the same SA (e.g., behind a firewall).
■ Next Layer Protocol: The IP protocol header (IPv4, IPv6, or IPv6 Extension) includes a field (Protocol for IPv4, Next Header for IPv6 or IPv6 Extension)
that designates the protocol operating over IP. This is an individual protocol
number, ANY, or for IPv6 only, OPAQUE. If AH or ESP is used, then this IP
protocol header immediately precedes the AH or ESP header in the packet.
20.2 / IP SECURITY POLICY 671
■ Name: A user identifier from the operating system. This is not a field in the IP or upper-layer headers but is available if IPsec is running on the same operat-
ing system as the user.
■ Local and Remote Ports: These may be individual TCP or UDP port values, an enumerated list of ports, or a wildcard port.
Table 20.2 provides an example of an SPD on a host system (as opposed to
a network system such as a firewall or router). This table reflects the following
configuration: A local network configuration consists of two networks. The basic
corporate network configuration has the IP network number 1.2.3.0/24. The local
configuration also includes a secure LAN, often known as a DMZ, that is identified
as 1.2.4.0/24. The DMZ is protected from both the outside world and the rest of the
corporate LAN by firewalls. The host in this example has the IP address 1.2.3.10,
and it is authorized to connect to the server 1.2.4.10 in the DMZ.
The entries in the SPD should be self-explanatory. For example, UDP port
500 is the designated port for IKE. Any traffic from the local host to a remote host
for purposes of an IKE exchange bypasses the IPsec processing.
IP Traffic Processing
IPsec is executed on a packet-by-packet basis. When IPsec is implemented, each
outbound IP packet is processed by the IPsec logic before transmission, and each
inbound packet is processed by the IPsec logic after reception and before passing
the packet contents on to the next higher layer (e.g., TCP or UDP). We look at the
logic of these two situations in turn.
OUTBOUND PACKETS Figure 20.3 highlights the main elements of IPsec processing for outbound traffic. A block of data from a higher layer, such as TCP, is passed
down to the IP layer and an IP packet is formed, consisting of an IP header and an
IP body. Then the following steps occur:
1. IPsec searches the SPD for a match to this packet.
2. If no match is found, then the packet is discarded and an error message is generated.
Protocol Local IP Port Remote IP Port Action Comment
UDP 1.2.3.101 500 * 500 BYPASS IKE
ICMP 1.2.3.101 * * * BYPASS Error messages
* 1.2.3.101 * 1.2.3.0/24 * PROTECT: ESP
intransport-mode
Encrypt intranet traffic
TCP 1.2.3.101 * 1.2.4.10 80 PROTECT: ESP
intransport-mode
Encrypt to server
TCP 1.2.3.101 * 1.2.4.10 443 BYPASS TLS: avoid double encryption
* 1.2.3.101 * 1.2.4.0/24 * DISCARD Others in DMZ
* 1.2.3.101 * * * BYPASS Internet
Table 20.2 Host SPD Example
672 CHAPTER 20 / IP SECURITY
3. If a match is found, further processing is determined by the first matching entry in the SPD. If the policy for this packet is DISCARD, then the packet is
discarded. If the policy is BYPASS, then there is no further IPsec processing;
the packet is forwarded to the network for transmission.
4. If the policy is PROTECT, then a search is made of the SAD for a matching entry. If no entry is found, then IKE is invoked to create an SA with the ap-
propriate keys and an entry is made in the SA.
5. The matching entry in the SAD determines the processing for this packet. Either encryption, authentication, or both can be performed, and either trans-
port or tunnel mode can be used. The packet is then forwarded to the network
for transmission.
INBOUND PACKETS Figure 20.4 highlights the main elements of IPsec processing for inbound traffic. An incoming IP packet triggers the IPsec processing. The following
steps occur:
1. IPsec determines whether this is an unsecured IP packet or one that has ESP or AH headers/trailers, by examining the IP Protocol field (IPv4) or Next
Header field (IPv6).
Figure 20.3 Processing Model for Outbound Packets
Search security policy
database
Search security association
database
Determine policy
Outbound IP packet (e.g., from TCP or UDP)
Discard packet
No match found
No match found
Match found
Match found
DISCARD PROTECT
BYPASS
Forward packet via
IP
Internet key
exchange
Process (AH/ESP)
20.3 / ENCAPSULATING SECURITY PAYLOAD 673
2. If the packet is unsecured, IPsec searches the SPD for a match to this packet. If the first matching entry has a policy of BYPASS, the IP header is processed
and stripped off and the packet body is delivered to the next higher layer, such
as TCP. If the first matching entry has a policy of PROTECT or DISCARD, or
if there is no matching entry, the packet is discarded.
3. For a secured packet, IPsec searches the SAD. If no match is found, the packet is discarded. Otherwise, IPsec applies the appropriate ESP or AH processing.
Then, the IP header is processed and stripped off and the packet body is deliv-
ered to the next higher layer, such as TCP.
20.3 ENCAPSULATING SECURITY PAYLOAD
ESP can be used to provide confidentiality, data origin authentication, connection-
less integrity, an anti-replay service (a form of partial sequence integrity), and (lim-
ited) traffic flow confidentiality. The set of services provided depends on options
selected at the time of Security Association (SA) establishment and on the location
of the implementation in a network topology.
ESP can work with a variety of encryption and authentication algorithms, in-
cluding authenticated encryption algorithms such as GCM.
Figure 20.4 Processing Model for Inbound Packets
Search security policy
database
Search security association
database
Packet type
Inbound IP packet (from Internet)
Discard packet
No match found
cesPIPI
Not BYPASS
Match foundBYPASS
Deliver packet to higher layer
(e.g., TCP, UDP)
Process (AH/ESP)
674 CHAPTER 20 / IP SECURITY
ESP Format
Figure 20.5a shows the top-level format of an ESP packet. It contains the following fields.
■ Security Parameters Index (32 bits): Identifies a security association.
■ Sequence Number (32 bits): A monotonically increasing counter value; this provides an anti-replay function, as discussed for AH.
■ Payload Data (variable): This is a transport-level segment (transport mode) or IP packet (tunnel mode) that is protected by encryption.
■ Padding (0–255 bytes): The purpose of this field is discussed later.
■ Pad Length (8 bits): Indicates the number of pad bytes immediately preceding this field.
■ Next Header (8 bits): Identifies the type of data contained in the payload data field by identifying the first header in that payload (e.g., an extension header
in IPv6, or an upper-layer protocol such as TCP).
■ Integrity Check Value (variable): A variable-length field (must be an integral number of 32-bit words) that contains the Integrity Check Value computed
over the ESP packet minus the Authentication Data field.
Figure 20.5 ESP Packet Format
Security parameters index (SPI)
32 bits
Sequence number
Padding (0–255 bytes) Pad length Next header
Payload data (variable)
Integrity check value - ICV (variable)
IC V
c ov
er ag
e
E nc
ry pt
ed E
nc ry
pt ed
(a) Top-level format of an ESP Packet
(b) Substructure of payload data
Security parameters index (SPI) Sequence number
Initialization value - IV (optional)
Padding (0–255 bytes) TFC padding (optional, variable)
Pad length Next header
Rest of payload data (variable)
Integrity check value - ICV (variable)
IC V
c ov
er ag
e
P ay
lo ad
20.3 / ENCAPSULATING SECURITY PAYLOAD 675
When any combined mode algorithm is employed, the algorithm itself is ex-
pected to return both decrypted plaintext and a pass/fail indication for the integrity
check. For combined mode algorithms, the ICV that would normally appear at the
end of the ESP packet (when integrity is selected) may be omitted. When the ICV
is omitted and integrity is selected, it is the responsibility of the combined mode
algorithm to encode within the Payload Data an ICV-equivalent means of verifying
the integrity of the packet.
Two additional fields may be present in the payload (Figure 20.5b).
An initialization value (IV), or nonce, is present if this is required by the encryption or authenticated encryption algorithm used for ESP. If tunnel mode is being used,
then the IPsec implementation may add traffic flow confidentiality (TFC) padding after the Payload Data and before the Padding field, as explained subsequently.
Encryption and Authentication Algorithms
The Payload Data, Padding, Pad Length, and Next Header fields are encrypted by
the ESP service. If the algorithm used to encrypt the payload requires cryptographic
synchronization data, such as an initialization vector (IV), then these data may be
carried explicitly at the beginning of the Payload Data field. If included, an IV is
usually not encrypted, although it is often referred to as being part of the ciphertext.
The ICV field is optional. It is present only if the integrity service is selected
and is provided by either a separate integrity algorithm or a combined mode algo-
rithm that uses an ICV. The ICV is computed after the encryption is performed.
This order of processing facilitates rapid detection and rejection of replayed or
bogus packets by the receiver prior to decrypting the packet, hence potentially re-
ducing the impact of denial of service (DoS) attacks. It also allows for the possibility
of parallel processing of packets at the receiver that is decryption can take place in
parallel with integrity checking. Note that because the ICV is not protected by en-
cryption, a keyed integrity algorithm must be employed to compute the ICV.
Padding
The Padding field serves several purposes:
■ If an encryption algorithm requires the plaintext to be a multiple of some
number of bytes (e.g., the multiple of a single block for a block cipher), the
Padding field is used to expand the plaintext (consisting of the Payload Data,
Padding, Pad Length, and Next Header fields) to the required length.
■ The ESP format requires that the Pad Length and Next Header fields be right
aligned within a 32-bit word. Equivalently, the ciphertext must be an integer
multiple of 32 bits. The Padding field is used to assure this alignment.
■ Additional padding may be added to provide partial traffic-flow confidential-
ity by concealing the actual length of the payload.
Anti-Replay Service
A replay attack is one in which an attacker obtains a copy of an authenticated packet and later transmits it to the intended destination. The receipt of duplicate,
authenticated IP packets may disrupt service in some way or may have some other
676 CHAPTER 20 / IP SECURITY
undesired consequence. The Sequence Number field is designed to thwart such at-
tacks. First, we discuss sequence number generation by the sender, and then we
look at how it is processed by the recipient.
When a new SA is established, the sender initializes a sequence number counter to 0. Each time that a packet is sent on this SA, the sender increments the
counter and places the value in the Sequence Number field. Thus, the first value to
be used is 1. If anti-replay is enabled (the default), the sender must not allow the
sequence number to cycle past 232 - 1 back to zero. Otherwise, there would be mul- tiple valid packets with the same sequence number. If the limit of 232 - 1 is reached, the sender should terminate this SA and negotiate a new SA with a new key.
Because IP is a connectionless, unreliable service, the protocol does not guar-
antee that packets will be delivered in order and does not guarantee that all packets
will be delivered. Therefore, the IPsec authentication document dictates that the
receiver should implement a window of size W, with a default of W = 64. The right edge of the window represents the highest sequence number, N, so far received for a valid packet. For any packet with a sequence number in the range from N - W + 1 to N that has been correctly received (i.e., properly authenticated), the correspond- ing slot in the window is marked (Figure 20.6). Inbound processing proceeds as fol-
lows when a packet is received:
1. If the received packet falls within the window and is new, the MAC is checked. If the packet is authenticated, the corresponding slot in the window is marked.
2. If the received packet is to the right of the window and is new, the MAC is checked. If the packet is authenticated, the window is advanced so that this
sequence number is the right edge of the window, and the corresponding slot
in the window is marked.
3. If the received packet is to the left of the window or if authentication fails, the packet is discarded; this is an auditable event.
Transport and Tunnel Modes
Figure 20.7 shows two ways in which the IPsec ESP service can be used. In the upper
part of the figure, encryption (and optionally authentication) is provided directly be-
tween two hosts. Figure 20.7b shows how tunnel mode operation can be used to set up
Figure 20.6 Anti-replay Mechanism
Fixed window size W
N
N + 1N – W
Marked if valid packet received
Unmarked if valid packet not yet received
Advance window if valid packet to the
right is received
Hiva-Network.Com
20.3 / ENCAPSULATING SECURITY PAYLOAD 677
a virtual private network. In this example, an organization has four private networks interconnected across the Internet. Hosts on the internal networks use the Internet
for transport of data but do not interact with other Internet-based hosts. By terminat-
ing the tunnels at the security gateway to each internal network, the configuration al-
lows the hosts to avoid implementing the security capability. The former technique is
supported by a transport mode SA, while the latter technique uses a tunnel mode SA.
In this section, we look at the scope of ESP for the two modes. The consid-
erations are somewhat different for IPv4 and IPv6. We use the packet formats of
Figure 20.8a as a starting point.
TRANSPORT MODE ESP Transport mode ESP is used to encrypt and optionally au- thenticate the data carried by IP (e.g., a TCP segment), as shown in Figure 20.8b.
For this mode using IPv4, the ESP header is inserted into the IP packet immedi-
ately prior to the transport-layer header (e.g., TCP, UDP, ICMP), and an ESP
trailer (Padding, Pad Length, and Next Header fields) is placed after the IP packet.
If authentication is selected, the ESP Authentication Data field is added after the
ESP trailer. The entire transport-level segment plus the ESP trailer are encrypted.
Authentication covers all of the ciphertext plus the ESP header.
In the context of IPv6, ESP is viewed as an end-to-end payload; that is, it is
not examined or processed by intermediate routers. Therefore, the ESP header ap-
pears after the IPv6 base header and the hop-by-hop, routing, and fragment exten-
sion headers. The destination options extension header could appear before or after
the ESP header, depending on the semantics desired. For IPv6, encryption covers
Figure 20.7 Transport-Mode versus Tunnel-Mode Encryptionx
Internal Network
External Network
Encrypted TCP Session
(a) Transport-level security
Internet
Corporate network
Corporate network
Corporate network
Corporate network
(b) A virtual private network via tunnel mode
Encrypted tunnels carrying IP traffic
678 CHAPTER 20 / IP SECURITY
the entire transport-level segment plus the ESP trailer plus the destination options
extension header if it occurs after the ESP header. Again, authentication covers the
ciphertext plus the ESP header.
Transport mode operation may be summarized as follows.
1. At the source, the block of data consisting of the ESP trailer plus the entire transport-layer segment is encrypted and the plaintext of this block is replaced
Figure 20.8 Scope of ESP Encryption and Authentication
Orig IP hdr
Hop-by-hop, dest, routing, fragmentIPv6
Orig IP hdrIPv4
New IP hdrIPv4
(b) Transport Mode
New IP hdr
Ext headersIPv6
authenticated encrypted
authenticated encrypted
authenticated encrypted
authenticated encrypted
(c) Tunnel Mode
Orig IP hdr
Ext headers TCP Data
ESP trlr
ESP auth
ESP hdr
ESP auth
Orig IP hdr TCP Data
ESP trlr
ESP auth
ESP hdr
Dest TCP Data
TCP Data
ESP trlr
ESP auth
ESP trlr
ESP hdr
ESP hdr
Orig IP hdr
Extension headers (if present) TCP DataIPv6
Orig IP hdr
TCP DataIPv4
(a) Before Applying ESP
20.3 / ENCAPSULATING SECURITY PAYLOAD 679
with its ciphertext to form the IP packet for transmission. Authentication is
added if this option is selected.
2. The packet is then routed to the destination. Each intermediate router needs to examine and process the IP header plus any plaintext IP extension headers
but does not need to examine the ciphertext.
3. The destination node examines and processes the IP header plus any plaintext IP extension headers. Then, on the basis of the SPI in the ESP header, the
destination node decrypts the remainder of the packet to recover the plaintext
transport-layer segment.
Transport mode operation provides confidentiality for any application that
uses it, thus avoiding the need to implement confidentiality in every individual ap-
plication. One drawback to this mode is that it is possible to do traffic analysis on
the transmitted packets.
TUNNEL MODE ESP Tunnel mode ESP is used to encrypt an entire IP packet (Figure 20.8c). For this mode, the ESP header is prefixed to the packet and then the packet
plus the ESP trailer is encrypted. This method can be used to counter traffic analysis.
Because the IP header contains the destination address and possibly source
routing directives and hop-by-hop option information, it is not possible simply to
transmit the encrypted IP packet prefixed by the ESP header. Intermediate routers
would be unable to process such a packet. Therefore, it is necessary to encapsulate
the entire block (ESP header plus ciphertext plus Authentication Data, if present)
with a new IP header that will contain sufficient information for routing but not for
traffic analysis.
Whereas the transport mode is suitable for protecting connections between
hosts that support the ESP feature, the tunnel mode is useful in a configuration that
includes a firewall or other sort of security gateway that protects a trusted network
from external networks. In this latter case, encryption occurs only between an exter-
nal host and the security gateway or between two security gateways. This relieves
hosts on the internal network of the processing burden of encryption and simplifies
the key distribution task by reducing the number of needed keys. Further, it thwarts
traffic analysis based on ultimate destination.
Consider a case in which an external host wishes to communicate with a host
on an internal network protected by a firewall, and in which ESP is implemented
in the external host and the firewalls. The following steps occur for transfer of a
transport-layer segment from the external host to the internal host.
1. The source prepares an inner IP packet with a destination address of the target internal host. This packet is prefixed by an ESP header; then the packet and
ESP trailer are encrypted and Authentication Data may be added. The result-
ing block is encapsulated with a new IP header (base header plus optional ex-
tensions such as routing and hop-by-hop options for IPv6) whose destination
address is the firewall; this forms the outer IP packet.
2. The outer packet is routed to the destination firewall. Each intermediate router needs to examine and process the outer IP header plus any outer IP
extension headers but does not need to examine the ciphertext.
680 CHAPTER 20 / IP SECURITY
3. The destination firewall examines and processes the outer IP header plus any outer IP extension headers. Then, on the basis of the SPI in the ESP header, the
destination node decrypts the remainder of the packet to recover the plaintext
inner IP packet. This packet is then transmitted in the internal network.
4. The inner packet is routed through zero or more routers in the internal net- work to the destination host.
Figure 20.9 shows the protocol architecture for the two modes.
Figure 20.9 Protocol Operation for ESP
Data
Data
TCP hdr
Data
TCP hdr
DataOrig IP hdr
TCP hdr
Data ESP trlr
ESP hdr
Orig IP hdr
ESP auth
New IP hdr
TCP hdr
Data ESP trlr
ESP hdr
Orig IP hdr
ESP auth
TCP hdr
Data
Orig IP hdr
TCP hdr
Data
Orig IP hdr
TCP hdr
Data
(a) Transport mode
(b) Tunnel mode
ESP trlr
ESP hdr
ESP auth
Application
TCP
IP
IPsec
Application
TCP
IP
IPsec
IP
20.4 / COMBINING SECURITY ASSOCIATIONS 681
20.4 COMBINING SECURITY ASSOCIATIONS
An individual SA can implement either the AH or ESP protocol but not both.
Sometimes a particular traffic flow will call for the services provided by both AH
and ESP. Further, a particular traffic flow may require IPsec services between hosts
and, for that same flow, separate services between security gateways, such as fire-
walls. In all of these cases, multiple SAs must be employed for the same traffic flow
to achieve the desired IPsec services. The term security association bundle refers to a sequence of SAs through which traffic must be processed to provide a desired set
of IPsec services. The SAs in a bundle may terminate at different endpoints or at
the same endpoints.
Security associations may be combined into bundles in two ways:
■ Transport adjacency: Refers to applying more than one security protocol to the same IP packet without invoking tunneling. This approach to combining
AH and ESP allows for only one level of combination; further nesting yields
no added benefit since the processing is performed at one IPsec instance: the
(ultimate) destination.
■ Iterated tunneling: Refers to the application of multiple layers of security pro- tocols effected through IP tunneling. This approach allows for multiple levels
of nesting, since each tunnel can originate or terminate at a different IPsec site
along the path.
The two approaches can be combined, for example, by having a transport SA be-
tween hosts travel part of the way through a tunnel SA between security gateways.
One interesting issue that arises when considering SA bundles is the order in
which authentication and encryption may be applied between a given pair of end-
points and the ways of doing so. We examine that issue next. Then we look at com-
binations of SAs that involve at least one tunnel.
Authentication Plus Confidentiality
Encryption and authentication can be combined in order to transmit an IP packet
that has both confidentiality and authentication between hosts. We look at several
approaches.
ESP WITH AUTHENTICATION OPTION This approach is illustrated in Figure 20.8. In this approach, the user first applies ESP to the data to be protected and then
appends the authentication data field. There are actually two subcases:
■ Transport mode ESP: Authentication and encryption apply to the IP payload delivered to the host, but the IP header is not protected.
■ Tunnel mode ESP: Authentication applies to the entire IP packet delivered to the outer IP destination address (e.g., a firewall), and authentication is per-
formed at that destination. The entire inner IP packet is protected by the pri-
vacy mechanism for delivery to the inner IP destination.
For both cases, authentication applies to the ciphertext rather than the plaintext.
682 CHAPTER 20 / IP SECURITY
TRANSPORT ADJACENCY Another way to apply authentication after encryption is to use two bundled transport SAs, with the inner being an ESP SA and the outer being
an AH SA. In this case, ESP is used without its authentication option. Because the
inner SA is a transport SA, encryption is applied to the IP payload. The resulting
packet consists of an IP header (and possibly IPv6 header extensions) followed by
an ESP. AH is then applied in transport mode, so that authentication covers the
ESP plus the original IP header (and extensions) except for mutable fields. The
advantage of this approach over simply using a single ESP SA with the ESP authen-
tication option is that the authentication covers more fields, including the source
and destination IP addresses. The disadvantage is the overhead of two SAs versus
one SA.
TRANSPORT-TUNNEL BUNDLE The use of authentication prior to encryption might be preferable for several reasons. First, because the authentication data are pro-
tected by encryption, it is impossible for anyone to intercept the message and alter
the authentication data without detection. Second, it may be desirable to store the
authentication information with the message at the destination for later reference.
It is more convenient to do this if the authentication information applies to the un-
encrypted message; otherwise the message would have to be reencrypted to verify
the authentication information.
One approach to applying authentication before encryption between two hosts
is to use a bundle consisting of an inner AH transport SA and an outer ESP tunnel
SA. In this case, authentication is applied to the IP payload plus the IP header (and
extensions) except for mutable fields. The resulting IP packet is then processed in
tunnel mode by ESP; the result is that the entire, authenticated inner packet is en-
crypted and a new outer IP header (and extensions) is added.
Basic Combinations of Security Associations
The IPsec Architecture document lists four examples of combinations of SAs that
must be supported by compliant IPsec hosts (e.g., workstation, server) or security
gateways (e.g., firewall, router). These are illustrated in Figure 20.10. The lower
part of each case in the figure represents the physical connectivity of the elements;
the upper part represents logical connectivity via one or more nested SAs. Each SA
can be either AH or ESP. For host-to-host SAs, the mode may be either transport
or tunnel; otherwise it must be tunnel mode.
Case 1. All security is provided between end systems that implement IPsec. For any two end systems to communicate via an SA, they must share the appropri-
ate secret keys. Among the possible combinations are
a. AH in transport mode
b. ESP in transport mode
c. ESP followed by AH in transport mode (an ESP SA inside an AH SA)
d. Any one of a, b, or c inside an AH or ESP in tunnel mode
We have already discussed how these various combinations can be used to
support authentication, encryption, authentication before encryption, and authenti-
cation after encryption.
20.4 / COMBINING SECURITY ASSOCIATIONS 683
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684 CHAPTER 20 / IP SECURITY
Case 2. Security is provided only between gateways (routers, firewalls, etc.) and no hosts implement IPsec. This case illustrates simple virtual private network
support. The security architecture document specifies that only a single tunnel SA is
needed for this case. The tunnel could support AH, ESP, or ESP with the authenti-
cation option. Nested tunnels are not required, because the IPsec services apply to
the entire inner packet.
Case 3. This builds on case 2 by adding end-to-end security. The same combi- nations discussed for cases 1 and 2 are allowed here. The gateway-to-gateway tun-
nel provides either authentication, confidentiality, or both for all traffic between
end systems. When the gateway-to-gateway tunnel is ESP, it also provides a limited
form of traffic confidentiality. Individual hosts can implement any additional IPsec
services required for given applications or given users by means of end-to-end SAs.
Case 4. This provides support for a remote host that uses the Internet to reach an organization’s firewall and then to gain access to some server or workstation
behind the firewall. Only tunnel mode is required between the remote host and the
firewall. As in case 1, one or two SAs may be used between the remote host and the
local host.
20.5 INTERNET KEY EXCHANGE
The key management portion of IPsec involves the determination and distribution
of secret keys. A typical requirement is four keys for communication between two
applications: transmit and receive pairs for both integrity and confidentiality. The
IPsec Architecture document mandates support for two types of key management:
■ Manual: A system administrator manually configures each system with its own keys and with the keys of other communicating systems. This is practical for
small, relatively static environments.
■ Automated: An automated system enables the on-demand creation of keys for SAs and facilitates the use of keys in a large distributed system with an evolv-
ing configuration.
The default automated key management protocol for IPsec is referred to as
ISAKMP/Oakley and consists of the following elements:
■ Oakley Key Determination Protocol: Oakley is a key exchange protocol based on the Diffie–Hellman algorithm but providing added security. Oakley is ge-
neric in that it does not dictate specific formats.
■ Internet Security Association and Key Management Protocol (ISAKMP): ISAKMP provides a framework for Internet key management and provides
the specific protocol support, including formats, for negotiation of security
attributes.
ISAKMP by itself does not dictate a specific key exchange algorithm; rather,
ISAKMP consists of a set of message types that enable the use of a variety of key
exchange algorithms. Oakley is the specific key exchange algorithm mandated for
use with the initial version of ISAKMP.
20.5 / INTERNET KEY EXCHANGE 685
In IKEv2, the terms Oakley and ISAKMP are no longer used, and there
are significant differences from the use of Oakley and ISAKMP in IKEv1.
Nevertheless, the basic functionality is the same. In this section, we describe the
IKEv2 specification.
Key Determination Protocol
IKE key determination is a refinement of the Diffie–Hellman key exchange algo-
rithm. Recall that Diffie–Hellman involves the following interaction between users
A and B. There is prior agreement on two global parameters: q, a large prime num- ber; and a, a primitive root of q. A selects a random integer XA as its private key and transmits to B its public key ΥA = a
XA mod q. Similarly, B selects a random integer XB as its private key and transmits to A its public key ΥB = a
XB mod q. Each side can now compute the secret session key:
K = (ΥB) XA mod q = (ΥA)
XB mod q = aXAXB mod q
The Diffie–Hellman algorithm has two attractive features:
■ Secret keys are created only when needed. There is no need to store secret
keys for a long period of time, exposing them to increased vulnerability.
■ The exchange requires no pre-existing infrastructure other than an agreement
on the global parameters.
However, there are a number of weaknesses to Diffie–Hellman, as pointed out in
[HUIT98].
■ It does not provide any information about the identities of the parties.
■ It is subject to a man-in-the-middle attack, in which a third party C imperson-
ates B while communicating with A and impersonates A while communicating
with B. Both A and B end up negotiating a key with C, which can then listen to
and pass on traffic. The man-in-the-middle attack proceeds as
1. B sends his public key YB in a message addressed to A (see Figure 10.2).
2. The enemy (E) intercepts this message. E saves B’s public key and sends a message to A that has B’s User ID but E’s public key YE. This message is sent in such a way that it appears as though it was sent from B’s host system.
A receives E’s message and stores E’s public key with B’s User ID. Similarly,
E sends a message to B with E’s public key, purporting to come from A.
3. B computes a secret key K1 based on B’s private key and YE. A computes a secret key K2 based on A’s private key and YE. E computes K1 using E’s secret key XE and YB and computers K2 using XE and YA.
4. From now on, E is able to relay messages from A to B and from B to A, appropriately changing their encipherment en route in such a way that nei-
ther A nor B will know that they share their communication with E.
■ It is computationally intensive. As a result, it is vulnerable to a clogging attack,
in which an opponent requests a high number of keys. The victim spends con-
siderable computing resources doing useless modular exponentiation rather
than real work.
Hiva-Network.Com
686 CHAPTER 20 / IP SECURITY
IKE key determination is designed to retain the advantages of Diffie–Hellman,
while countering its weaknesses.
FEATURES OF IKE KEY DETERMINATION The IKE key determination algorithm is characterized by five important features:
1. It employs a mechanism known as cookies to thwart clogging attacks.
2. It enables the two parties to negotiate a group; this, in essence, specifies the global parameters of the Diffie–Hellman key exchange.
3. It uses nonces to ensure against replay attacks.
4. It enables the exchange of Diffie–Hellman public key values.
5. It authenticates the Diffie–Hellman exchange to thwart man-in-the-middle attacks.
We have already discussed Diffie–Hellman. Let us look at the remainder of
these elements in turn. First, consider the problem of clogging attacks. In this at-
tack, an opponent forges the source address of a legitimate user and sends a public
Diffie–Hellman key to the victim. The victim then performs a modular exponentia-
tion to compute the secret key. Repeated messages of this type can clog the vic- tim’s system with useless work. The cookie exchange requires that each side send a pseudorandom number, the cookie, in the initial message, which the other side
acknowledges. This acknowledgment must be repeated in the first message of the
Diffie–Hellman key exchange. If the source address was forged, the opponent gets
no answer. Thus, an opponent can only force a user to generate acknowledgments
and not to perform the Diffie–Hellman calculation.
IKE mandates that cookie generation satisfy three basic requirements:
1. The cookie must depend on the specific parties. This prevents an attacker from obtaining a cookie using a real IP address and UDP port and then using it to
swamp the victim with requests from randomly chosen IP addresses or ports.
2. It must not be possible for anyone other than the issuing entity to generate cookies that will be accepted by that entity. This implies that the issuing entity
will use local secret information in the generation and subsequent verification
of a cookie. It must not be possible to deduce this secret information from any
particular cookie. The point of this requirement is that the issuing entity need
not save copies of its cookies, which are then more vulnerable to discovery, but
can verify an incoming cookie acknowledgment when it needs to.
3. The cookie generation and verification methods must be fast to thwart attacks intended to sabotage processor resources.
The recommended method for creating the cookie is to perform a fast hash
(e.g., MD5) over the IP Source and Destination addresses, the UDP Source and
Destination ports, and a locally generated secret value.
IKE key determination supports the use of different groups for the Diffie–
Hellman key exchange. Each group includes the definition of the two global pa-
rameters and the identity of the algorithm. The current specification includes the
following groups.
20.5 / INTERNET KEY EXCHANGE 687
■ Modular exponentiation with a 768-bit modulus
q = 2768 - 2704 - 1 + 264 * (:2638 * p; + 149686) a = 2
■ Modular exponentiation with a 1024-bit modulus
q = 21024 - 2960 - 1 + 264 * (:2894 * p; + 129093) a = 2
■ Modular exponentiation with a 1536-bit modulus
■ Parameters to be determined
■ Elliptic curve group over 2155
■ Generator (hexadecimal): X = 7B, Y = 1C8 ■ Elliptic curve parameters (hexadecimal): A = 0, Y = 7338F
■ Elliptic curve group over 2185
■ Generator (hexadecimal): X = 18, Y = D ■ Elliptic curve parameters (hexadecimal): A = 0, Y = 1EE9
The first three groups are the classic Diffie–Hellman algorithm using modular
exponentiation. The last two groups use the elliptic curve analog to Diffie–Hellman,
which was described in Chapter 10.
IKE key determination employs nonces to ensure against replay attacks. Each nonce is a locally generated pseudorandom number. Nonces appear in responses
and are encrypted during certain portions of the exchange to secure their use.
Three different authentication methods can be used with IKE key determination:
■ Digital signatures: The exchange is authenticated by signing a mutually ob- tainable hash; each party encrypts the hash with its private key. The hash is
generated over important parameters, such as user IDs and nonces.
■ Public-key encryption: The exchange is authenticated by encrypting param- eters such as IDs and nonces with the sender’s private key.
■ Symmetric-key encryption: A key derived by some out-of-band mechanism can be used to authenticate the exchange by symmetric encryption of ex-
change parameters.
IKEV2 EXCHANGES The IKEv2 protocol involves the exchange of messages in pairs. The first two pairs of exchanges are referred to as the initial exchanges (Figure 20.11a). In the first exchange, the two peers exchange information concern-
ing cryptographic algorithms and other security parameters they are willing to use
along with nonces and Diffie–Hellman (DH) values. The result of this exchange is to
set up a special SA called the IKE SA (see Figure 20.2). This SA defines parameters
for a secure channel between the peers over which subsequent message exchanges
take place. Thus, all subsequent IKE message exchanges are protected by encryp-
tion and message authentication. In the second exchange, the two parties authenti-
cate one another and set up a first IPsec SA to be placed in the SADB and used for
688 CHAPTER 20 / IP SECURITY
Figure 20.11 IKEv2 Exchanges
HDR, SAi1, KEi, Ni
ResponderInitiator
(a) Initial exchanges
HDR, SAr1, KEr, Nr, [CERTREQ]
HDR, SK {IDi, [CERT,] [CERTREQ,] [IDr,] AUTH, SAi2, TSi, TSr}
HDR, SK {IDr, [CERT,] AUTH, SAr2, TSi, TSr}
HDR, SK {[N], SA, Ni, [KEi], [TSi, TSr]}
(b) CREATE_CHILD_SA exchange
HDR, SK {SA, Nr, [KEr], [TSi, TSr]}
HDR, SK {[N,] [D,] [CP,] ...}
(c) Informational exchange
HDR, SK {[N,] [D,] [CP], ...}
HDR = IKE header SAx1 = offered and chosen algorithms, DH group KEx = Diffie–Hellman public key Nx= nonces CERTREQ = Certificate request IDx = identity CERT = certificate
SK {...} = MAC and encrypt AUTH = Authentication SAx2 = algorithms, parameters for IPsec SA TSx = traffic selectors for IPsec SA N = Notify D = Delete CP = Configuration
protecting ordinary (i.e. non-IKE) communications between the peers. Thus, four
messages are needed to establish the first SA for general use.
The CREATE_CHILD_SA exchange can be used to establish further SAs for protecting traffic. The informational exchange is used to exchange management information, IKEv2 error messages, and other notifications.
Header and Payload Formats
IKE defines procedures and packet formats to establish, negotiate, modify, and de-
lete security associations. As part of SA establishment, IKE defines payloads for
exchanging key generation and authentication data. These payload formats provide
a consistent framework independent of the specific key exchange protocol, encryp-
tion algorithm, and authentication mechanism.
IKE HEADER FORMAT An IKE message consists of an IKE header followed by one or more payloads. All of this is carried in a transport protocol. The specification dic-
tates that implementations must support the use of UDP for the transport protocol.
20.5 / INTERNET KEY EXCHANGE 689
Figure 20.12a shows the header format for an IKE message. It consists of the
following fields.
■ Initiator SPI (64 bits): A value chosen by the initiator to identify a unique IKE security association (SA).
■ Responder SPI (64 bits): A value chosen by the responder to identify a unique IKE SA.
■ Next Payload (8 bits): Indicates the type of the first payload in the message; payloads are discussed in the next subsection.
■ Major Version (4 bits): Indicates major version of IKE in use.
■ Minor Version (4 bits): Indicates minor version in use.
■ Exchange Type (8 bits): Indicates the type of exchange; these are discussed later in this section.
■ Flags (8 bits): Indicates specific options set for this IKE exchange. Three bits are defined so far. The initiator bit indicates whether this packet is sent by
the SA initiator. The version bit indicates whether the transmitter is capable
of using a higher major version number than the one currently indicated. The
response bit indicates whether this is a response to a message containing the
same message ID.
■ Message ID (32 bits): Used to control retransmission of lost packets and matching of requests and responses.
■ Length (32 bits): Length of total message (header plus all payloads) in octets.
IKE PAYLOAD TYPES All IKE payloads begin with the same generic payload header shown in Figure 20.12b. The Next Payload field has a value of 0 if this is the last
Figure 20.12 IKE Formats
MjVer MnVer Exchange Type FlagsNext Payload
Message ID
Length
(a) IKE header
(b) Generic Payload header
Initiator’s Security Parameter Index (SPI)
Responder’s Security Parameter Index (SPI)
0Bit: 8 16 24 31
RESERVED Payload LengthNext Payload C
0Bit: 8 16 31
690 CHAPTER 20 / IP SECURITY
Type Parameters
Security Association Proposals
Key Exchange DH Group #, Key Exchange Data
Identification ID Type, ID Data
Certificate Cert Encoding, Certificate Data
Certificate Request Cert Encoding, Certification Authority
Authentication Auth Method, Authentication Data
Nonce Nonce Data
Notify Protocol-ID, SPI Size, Notify Message Type, SPI, Notification Data
Delete Protocol-ID, SPI Size, # of SPIs, SPI (one or more)
Vendor ID Vendor ID
Traffic Selector Number of TSs, Traffic Selectors
Encrypted IV, Encrypted IKE payloads, Padding, Pad Length, ICV
Configuration CFG Type, Configuration Attributes
Extensible Authentication
Protocol
EAP Message
Table 20.3 IKE Payload Types
payload in the message; otherwise its value is the type of the next payload. The
Payload Length field indicates the length in octets of this payload, including the
generic payload header.
The critical bit is 0 if the sender wants the recipient to skip this payload if it
does not understand the payload type code in the Next Payload field of the previous
payload. It is set to 1 if the sender wants the recipient to reject this entire message if
it does not understand the payload type.
Table 20.3 summarizes the payload types defined for IKE and lists the fields,
or parameters, that are part of each payload. The SA payload is used to begin the establishment of an SA. The payload has a complex, hierarchical structure. The
payload may contain multiple proposals. Each proposal may contain multiple pro-
tocols. Each protocol may contain multiple transforms. And each transform may
contain multiple attributes. These elements are formatted as substructures within
the payload as follows.
■ Proposal: This substructure includes a proposal number, a protocol ID (AH, ESP, or IKE), an indicator of the number of transforms, and then a transform
substructure. If more than one protocol is to be included in a proposal, then
there is a subsequent proposal substructure with the same proposal number.
■ Transform: Different protocols support different transform types. The trans- forms are used primarily to define cryptographic algorithms to be used with a
particular protocol.
■ Attribute: Each transform may include attributes that modify or complete the specification of the transform. An example is key length.
20.5 / INTERNET KEY EXCHANGE 691
The Key Exchange payload can be used for a variety of key exchange tech- niques, including Oakley, Diffie–Hellman, and the RSA-based key exchange used
by PGP. The Key Exchange data field contains the data required to generate a ses-
sion key and is dependent on the key exchange algorithm used.
The Identification payload is used to determine the identity of communicating peers and may be used for determining authenticity of information. Typically the
ID Data field will contain an IPv4 or IPv6 address.
The Certificate payload transfers a public-key certificate. The Certificate Encoding field indicates the type of certificate or certificate-related information,
which may include the following:
■ PKCS #7 wrapped X.509 certificate
■ PGP certificate
■ DNS signed key
■ X.509 certificate—signature
■ X.509 certificate—key exchange
■ Kerberos tokens
■ Certificate Revocation List (CRL)
■ Authority Revocation List (ARL)
■ SPKI certificate
At any point in an IKE exchange, the sender may include a Certificate Request payload to request the certificate of the other communicating entity. The payload
may list more than one certificate type that is acceptable and more than one certifi-
cate authority that is acceptable.
The Authentication payload contains data used for message authentication purposes. The authentication method types so far defined are RSA digital signa-
ture, shared-key message integrity code, and DSS digital signature.
The Nonce payload contains random data used to guarantee liveness during an exchange and to protect against replay attacks.
The Notify payload contains either error or status information associated with this SA or this SA negotiation. The following table lists the IKE notify messages.
Error Messages Status Messages
Unsupported Critical Initial Contact
Payload Set Window Size
Invalid IKE SPI Additional TS Possible
Invalid Major Version IPCOMP Supported
Invalid Syntax NAT Detection Source IP
Invalid Payload Type NAT Detection Destination IP
Invalid Message ID Cookie
Invalid SPI Use Transport Mode
692 CHAPTER 20 / IP SECURITY
Error Messages Status Messages
No Proposal Chosen HTTP Cert Lookup Supported
Invalid KE Payload Rekey SA
Authentication Failed ESP TFC Padding Not Supported
Single Pair Required Non First Fragments Also
No Additional SAS
Internal Address Failure
Failed CP Required
TS Unacceptable
Invalid Selectors
The Delete payload indicates one or more SAs that the sender has deleted from its database and that therefore are no longer valid.
The Vendor ID payload contains a vendor-defined constant. The constant is used by vendors to identify and recognize remote instances of their implementa-
tions. This mechanism allows a vendor to experiment with new features while main-
taining backward compatibility.
The Traffic Selector payload allows peers to identify packet flows for process- ing by IPsec services.
The Encrypted payload contains other payloads in encrypted form. The en- crypted payload format is similar to that of ESP. It may include an IV if the encryp-
tion algorithm requires it and an ICV if authentication is selected.
The Configuration payload is used to exchange configuration information be- tween IKE peers.
The Extensible Authentication Protocol (EAP) payload allows IKE SAs to be authenticated using EAP, which was discussed in Chapter 16.
20.6 CRYPTOGRAPHIC SUITES
The IPsecv3 and IKEv3 protocols rely on a variety of types of cryptographic algo-
rithms. As we have seen in this book, there are many cryptographic algorithms of
each type, each with a variety of parameters, such as key size. To promote interop-
erability, two RFCs define recommended suites of cryptographic algorithms and
parameters for various applications.
RFC 4308 defines two cryptographic suites for establishing virtual private net-
works. Suite VPN-A matches the commonly used corporate VPN security used in
older IKEv1 implementations at the time of the issuance of IKEv2 in 2005. Suite
VPN-B provides stronger security and is recommended for new VPNs that imple-
ment IPsecv3 and IKEv2.
Table 20.4a lists the algorithms and parameters for the two suites. There are
several points to note about these two suites. Note that for symmetric cryptography,
20.6 / CRYPTOGRAPHIC SUITES 693
VPN-A relies on 3DES and HMAC, while VPN-B relies exclusively on AES. Three
types of secret-key algorithms are used:
■ Encryption: For encryption, the cipher block chaining (CBC) mode is used.
■ Message authentication: For message authentication, VPN-A relies on HMAC with SHA-1 with the output truncated to 96 bits. VPN-B relies on a variant of
CMAC with the output truncated to 96 bits.
■ Pseudorandom function: IKEv2 generates pseudorandom bits by repeated use of the MAC used for message authentication.
RFC 6379 defines four optional cryptographic suites that are compatible with
the United States National Security Agency’s Suite B specifications. In 2005, the
NSA issued Suite B, which defined the algorithms and strengths needed to pro-
tect both sensitive but unclassified (SBU) and classified information for use in
its Cryptographic Modernization program [LATT09]. The four suites defined in
RFC 6379 provide choices for ESP and IKE. The four suites are differentiated by
the choice of cryptographic algorithm strengths and a choice of whether ESP is to
provide both confidentiality and integrity or integrity only. All of the suites offer
greater protection than the two VPN suites defined in RFC 4308.
VPN-A VPN-B
ESP encryption 3DES-CBC AES-CBC (128-bit key)
ESP integrity HMAC-SHA1-96 AES-XCBC-MAC-96
IKE encryption 3DES-CBC AES-CBC (128-bit key)
IKE PRF HMAC-SHA1 AES-XCBC-PRF-128
IKE Integrity HMAC-SHA1-96 AES-XCBC-MAC-96
IKE DH group 1024-bit MODP 2048-bit MODP
(a) Virtual private networks (RFC 4308)
GCM-128 GCM-256 GMAC-128 GMAC-256
ESP encryption/
Integrity
AES-GCM
(128-bit key)
AES-GCM
(256-bit key)
Null Null
ESP integrity Null Null AES-GMAC
(128-bit key)
AES-GMAC
(256-bit key)
IKE encryption AES-CBC
(128-bit key)
AES-CBC
(256-bit key)
AES-CBC
(128-bit key)
AES-CBC
(256-bit key)
IKE PRF HMAC-SHA-256 HMAC-SHA-384 HMAC-SHA-256 HMAC-SHA-384
IKE Integrity HMAC-SHA-
256-128
HMAC-SHA-
384-192
HMAC-SHA-
256-128
HMAC-SHA-
384-192
IKE DH group 256-bit random
ECP
384-bit random ECP 256-bit random
ECP
384-bit random
ECP
(b) NSA Suite B (RFC 6379)
Table 20.4 Cryptographic Suites for IPsec
694 CHAPTER 20 / IP SECURITY
Key Terms
Table 20.4b lists the algorithms and parameters for the two suites. As with
RFC 4308, three categories of secret key algorithms are listed:
■ Encryption: For ESP, authenticated encryption is provided using the GCM mode with either 128-bit or 256-bit AES keys. For IKE encryption, CBC is
used, as it was for the VPN suites.
■ Message authentication: For ESP, if only authentication is required, then GMAC is used. As discussed in Chapter 12, GMAC is simply the authenti-
cation portion of GMC. For IKE, message authentication is provided using
HMAC with one of the SHA-3 hash functions.
■ Pseudorandom function: As with the VPN suites, IKEv2 in these suites gen- erates pseudorandom bits by repeated use of the MAC used for message
authentication.
For the Diffie–Hellman algorithm, the use of elliptic curve groups modulo
a prime is specified. For authentication, elliptic curve digital signatures are listed.
The original IKEv2 documents used RSA-based digital signatures. Equivalent or
greater strength can be achieved using ECC with fewer key bits.
20.7 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS
anti-replay service
Authentication Header (AH)
Encapsulating Security
Payload (ESP)
Internet Key Exchange
(IKE)
Internet Security Association
and Key Management
Protocol (ISAKMP)
IP Security (IPsec)
IPv4
IPv6
Oakley key determination
protocol
replay attack
security association (SA)
transport mode
tunnel mode
Review Questions
20.1 List and briefly describe some benefits of IPsec. 20.2 List and briefly define different categories of IPsec documents. 20.3 What parameters identify an SA and what parameters characterize the nature of a
particular SA?
20.4 What is the difference between transport mode and tunnel mode? 20.5 What are the types of secret key algorithm used in IPsec? 20.6 Why does ESP include a padding field? 20.7 What are the basic approaches to bundling SAs? 20.8 What are the roles of the Oakley key determination protocol and ISAKMP in IPsec?
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20.7 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 695
Problems
20.1 Describe and explain each of the entries in Table 20.2. 20.2 Draw a figure similar to Figure 20.8 for AH. 20.3 List the major security services provided by AH and ESP, respectively. 20.4 In discussing AH processing, it was mentioned that not all of the fields in an IP header
are included in MAC calculation. a. For each of the fields in the IPv4 header, indicate whether the field is immutable,
mutable but predictable, or mutable (zeroed prior to ICV calculation). b. Do the same for the IPv6 header. c. Do the same for the IPv6 extension headers. In each case, justify your decision for each field.
20.5 Suppose that the current replay window spans from 120 to 530. a. If the next incoming authenticated packet has sequence number 340, what will the
receiver do with the packet, and what will be the parameters of the window after that?
b. If instead the next incoming authenticated packet has sequence number 598, what will the receiver do with the packet, and what will be the parameters of the win- dow after that?
c. If instead the next incoming authenticated packet has sequence number 110, what will the receiver do with the packet, and what will be the parameters of the win- dow after that?
20.6 When tunnel mode is used, a new outer IP header is constructed. For both IPv4 and IPv6, indicate the relationship of each outer IP header field and each extension header in the outer packet to the corresponding field or extension header of the inner IP packet. That is, indicate which outer values are derived from inner values and which are constructed independently of the inner values.
20.7 End-to-end authentication and encryption are desired between two hosts. Draw figures similar to Figure 20.8 that show each of the following. a. Transport adjacency with encryption applied before authentication. b. A transport SA bundled inside a tunnel SA with encryption applied before
authentication. c. A transport SA bundled inside a tunnel SA with authentication applied before
encryption.
20.8 The IPsec architecture document states that when two transport mode SAs are bundled to allow both AH and ESP protocols on the same end-to-end flow, only one ordering of security protocols seems appropriate: performing the ESP protocol before performing the AH protocol. Why is this approach recommended rather than authentication before encryption?
20.9 For the IKE key exchange, indicate which parameters in each message go in which ISAKMP payload types.
20.10 Where does IPsec reside in a protocol stack?
696696
APPENDIX A
PROJECTS FOR TEACHING CRYPTOGRAPHY
A.1 Sage Computer Algebra Projects
A.2 Hacking Project
A.3 Block Cipher Projects
A.4 Laboratory Exercises
A.5 Research Projects
A.6 Programming Projects
A.7 Practical Security Assessments
A.8 Firewall Projects
A.9 Case Studies
A.10 Writing Assignments
A.11 Reading/Report Assignments
A.12 Discussion Topics
A.1 / SAGE COMPUTER ALGEBRA PROJECTS 697
Analysis and observation, theory and experience must never disdain or exclude each other; on the contrary, they support each other.
—On War, Carl Von Clausewitz
Many instructors believe that research or implementation projects are crucial to the
clear understanding of cryptography and network security. Without projects, it may
be difficult for students to grasp some of the basic concepts and interactions among
components. Projects reinforce the concepts introduced in the book, give the stu-
dent a greater appreciation of how a cryptographic algorithm or protocol works,
and can motivate students and give them confidence that they are capable of not
only understanding but implementing the details of a security capability.
In this text, I have tried to present the concepts of cryptography and network
security as clearly as possible and have provided numerous homework problems to
reinforce those concepts. However, many instructors will wish to supplement this
material with projects. This appendix provides some guidance in that regard and
describes support material available in the Instructor’s Resource Center (IRC) for this book, accessible to instructors from Pearson Education. The support material
covers 12 types of projects and other student exercises:
■ Sage computer algebra projects
■ Hacking project
■ Block cipher projects
■ Laboratory exercises
■ Research projects
■ Programming projects
■ Practical security assessments
■ Firewall projects
■ Case studies
■ Writing assignments
■ Reading/report assignments
■ Discussion topics
A.1 SAGE COMPUTER ALGEBRA PROJECTS
One of the most important new features for this edition is the use of Sage for cryp-
tographic examples and homework assignments. Sage is an open-source, multiplat-
form, freeware package that implements a very powerful, flexible, and easily learned
mathematics and computer algebra system. A computer algebra system (CAS) is
software that can perform symbolic as well as numerical calculations. CASs have
been used for teaching since their inception some decades ago, and there is now
a considerable literature on their use. A CAS is a natural tool for extending the
learning experience in a cryptography course.
698 APPENDIX A / PROJECTS FOR TEACHING CRYPTOGRAPHY
Unlike competing systems such as Mathematica, Maple, and MATLAB, there
are no licensing agreements or fees involved with Sage. Thus, Sage can be made
available on computers and networks at school, and students can individually down-
load the software to their own personal computers for use at home. Another advan-
tage of using Sage is that students learn a powerful, flexible tool that can be used for
virtually any mathematical application, not just cryptography. The Sage Web site
(http://www.sagemath.org) provides considerable documentation on how to install,
set up, and use Sage on a variety of computers and how to use it online via the Web.
The use of Sage can make a significant difference to the teaching of the
mathematics of cryptographic algorithms. Appendix B provides a large number of
examples of the use of Sage covering many cryptographic concepts. Appendix C lists
exercises in each of these topic areas to enable the student to gain hands-on experi-
ence with cryptographic algorithms. Copies of both appendices are available online
so that students do not have to key in lines of code that are printed in the appendices.
The IRC contains solutions to all of the exercises in Appendix C.
Dan Shumow of Microsoft and the University of Washington developed all of
the examples and assignments in Appendices B and C.
A.2 HACKING PROJECT
The aim of this project is to hack into a corporation’s network through a series of
steps. The corporation is named Extreme In Security Corporation. As the name
indicates, the corporation has some security holes in it, and a clever hacker is able
to access critical information by hacking into its network. The IRC includes what is
needed to set up the Web site. The student’s goal is to capture the secret informa-
tion about the price on the quote the corporation is placing next week to obtain
a contract for a governmental project.
The student should start at the Web site and find his or her way into the
network. At each step, if the student succeeds, there are indications as to how to
proceed on to the next step as well as the grade until that point.
The project can be attempted in three ways:
1. Without seeking any sort of help
2. Using some provided hints
3. Using exact directions
The IRC includes the files needed for this project:
1. Web Security project
2. Web Hacking exercises (XSS and Script-attacks) covering client-side and server-side vulnerability exploitations, respectively
3. Documentation for installation and use for the above
4. A PowerPoint file describing Web hacking. This file is crucial to understand- ing how to use the exercises since it clearly explains the operation using
screen shots.
A.5 / RESEARCH PROJECTS 699
This project was designed and implemented by Professor Sreekanth Malladi
of Dakota State University.
A.3 BLOCK CIPHER PROJECTS
This is a lab that explores the operation of the AES encryption algorithm by tracing
its execution, computing one round by hand, and then exploring the various block
cipher modes of use. The lab also covers DES. In both cases, an online Java applet
is used (or can be downloaded) to execute AES or DES.
For both AES and DES, the lab is divided into three separate parts:
■ Block cipher internals: This part involves encrypting plaintext and analyzing the intermediate results after each round. There is an online calculator for both
AES and DES that provides the intermediate results and the final ciphertext.
■ Block cipher round: This part involves calculating one round by hand and comparing the results to those produced by the calculator.
■ Block cipher modes of use: Enables the student to compare the operation of CBC and CFB modes.
The IRC contains the .html and .jar files needed to set up these labs on your
own Web site. Alternatively, the material is available from the Student Resources
section of this book’s Web site. Click on the rotating globe.
Lawrie Brown of the Australian Defence Force Academy developed these
projects.
A.4 LABORATORY EXERCISES
Professor Sanjay Rao and Ruben Torres of Purdue University have prepared a set
of laboratory exercises that are included in the IRC. These are implementation
projects designed to be programmed on Linux but could be adapted for any
Unix environment. These laboratory exercises provide realistic experience in
implementing security functions and applications.
A.5 RESEARCH PROJECTS
An effective way of reinforcing basic concepts from the course and for teaching
students research skills is to assign a research project. Such a project could involve
a literature search as well as an Internet search of vendor products, research lab
activities, and standardization efforts. Projects could be assigned to teams or, for
smaller projects, to individuals. In any case, it is best to require some sort of project
proposal early in the term, giving the instructor time to evaluate the proposal for
appropriate topic and appropriate level of effort. Student handouts for research
projects should include
700 APPENDIX A / PROJECTS FOR TEACHING CRYPTOGRAPHY
■ A format for the proposal
■ A format for the final report
■ A schedule with intermediate and final deadlines
■ A list of possible project topics
The students can select one of the topics listed in the IRC or devise their own
comparable project. The IRC includes a suggested format for the proposal and final
report as well as a list of 15 possible research topics.
A.6 PROGRAMMING PROJECTS
The programming project is a useful pedagogical tool. There are several attractive
features of stand-alone programming projects that are not part of an existing
security facility:
1. The instructor can choose from a wide variety of cryptography and network security concepts to assign projects.
2. The projects are platform and language independent. Students can program the projects on any available computer and in any appropriate language.
3. The instructor need not download, install, and configure any particular infra- structure for stand-alone projects.
There is also flexibility in the size of projects. Larger projects give students
more a sense of achievement, but students with less ability or fewer organizational
skills can be left behind. Larger projects usually elicit more overall effort from
the best students. Smaller projects can have a higher concepts-to-code ratio and,
because more of them can be assigned, the opportunity exists to address a variety
of different areas.
Again, as with research projects, the students should first submit a proposal.
The student handout should include the same elements listed in the preceding
section. The IRC includes a set of 12 possible programming projects.
The following individuals have supplied the research and programming proj-
ects suggested in the IRC: Henning Schulzrinne of Columbia University; Cetin Kaya
Koc of Oregon State University; and David M. Balenson of Trusted Information
Systems and George Washington University.
A.7 PRACTICAL SECURITY ASSESSMENTS
Examining the current infrastructure and practices of an existing organization is
one of the best ways of developing skills in assessing its security posture. The IRC
contains a list of such activities. Students, working either individually or in small
groups, select a suitable small- to medium-sized organization. They then interview
some key personnel in that organization in order to conduct a suitable selection
of security risk assessment and review tasks as it relates to the organization’s IT
A.10 / WRITING ASSIGNMENTS 701
infrastructure and practices. As a result, they can then recommend suitable changes,
which can improve the organization’s IT security. These activities help students
develop an appreciation of current security practices and the skills needed to review
these and recommend changes.
Lawrie Brown of the Australian Defence Force Academy developed these
projects.
A.8 FIREWALL PROJECTS
The implementation of network firewalls can be a difficult concept for students
to grasp initially. The IRC includes a Network Firewall Visualization tool to con-
vey and teach network security and firewall configuration. This tool is intended to
teach and reinforce key concepts including the use and purpose of a perimeter fire-
wall, the use of separated subnets, the purposes behind packet filtering, and the
shortcomings of a simple packet filter firewall.
The IRC includes a .jar file that is fully portable, and a series of exercises.
The tool and exercises were developed at U.S. Air Force Academy.
A.9 CASE STUDIES
Teaching with case studies engages students in active learning. The IRC includes
case studies in the following areas:
■ Disaster recovery
■ Firewalls
■ Incidence response
■ Physical security
■ Risk
■ Security policy
■ Virtualization
Each case study includes learning objectives, case description, and a series
of case discussion questions. Each case study is based on real-world situations and
includes papers or reports describing the case.
The case studies were developed at North Carolina A&T State University.
A.10 WRITING ASSIGNMENTS
Writing assignments can have a powerful multiplier effect in the learning process
in a technical discipline such as cryptography and network security. Adherents of
the Writing Across the Curriculum (WAC) movement (http://wac.colostate.edu/)
report substantial benefits of writing assignments in facilitating learning. Writing
assignments lead to more detailed and complete thinking about a particular topic. In
702 APPENDIX A / PROJECTS FOR TEACHING CRYPTOGRAPHY
addition, writing assignments help to overcome the tendency of students to pursue
a subject with a minimum of personal engagement, just learning facts and problem-
solving techniques without obtaining a deep understanding of the subject matter.
The IRC contains a number of suggested writing assignments, organized
by chapter. Instructors may ultimately find that this is an important part of their
approach to teaching the material. I would greatly appreciate any feedback on this
area and any suggestions for additional writing assignments.
A.11 READING/REPORT ASSIGNMENTS
Another excellent way to reinforce concepts from the course and to give students
research experience is to assign papers from the literature to be read and analyzed.
The student is then asked to write a brief report on the assigned paper. The IRC
includes a suggested list of papers, one or two per chapter, to be assigned. The
IRC provides a PDF copy of each of the papers. The IRC also includes a suggested
assignment wording.
A.12 DISCUSSION TOPICS
One way to provide a collaborative experience is discussion topics, a number of
which are included in the IRC. Each topic relates to material in the book. The
instructor can set it up so that students can discuss a topic either in a class setting,
an online chat room, or a message board. Again, I would greatly appreciate any
feedback on this area and any suggestions for additional discussion topics.
703
APPENDIX B
SAGE EXAMPLES By Dan Shumow
University of Washington
B.1 Linear Algebra and Matrix Functionality
B.2 Chapter 2: Number Theory
B.3 Chapter 3: Classical Encryption
B.4 Chapter 4: Block Ciphers and the Data Encryption Standard
B.5 Chapter 5: Basic Concepts in Number Theory and Finite Fields
B.6 Chapter 6: Advanced Encryption Standard
B.7 Chapter 8: Pseudorandom Number Generation and Stream Ciphers
B.8 Chapter 9: Public-Key Cryptography and RSA
B.9 Chapter 10: Other Public-Key Cryptosystems
B.10 Chapter 11: Cryptographic Hash Functions
B.11 Chapter 13: Digital Signatures
Hiva-Network.Com
704 APPENDIX B / SAGE EXAMPLES
This appendix contains a number of examples that illustrate cryptographic concepts,
organized by the chapter in which those concepts were discussed. All the examples
are in Sage.1 See Appendix C for how to get started using Sage and for a brief intro-
duction to Sage syntax and operations. We begin with a brief introduction to some
basic Sage matrix and linear algebra operations.
You should be able to follow the examples in this section as written. However,
if you have difficulty interpreting the Sage code, please refer to Section C.2
in Appendix C.
B.1 LINEAR ALGEBRA AND MATRIX FUNCTIONALITY
Sage includes linear algebra and matrix functionality. The following shows some of
the basic functionality applicable to cryptography.
In Sage you specify a matrix as a list of lists of numbers, passed to the matrix function. For example, passing a list of lists of integers as follows:
sage: M = matrix([[1, 3],[7,9]]); M
[1 3]
[7 9]
Alternately, passing a list of lists of rationals as follows:
sage: M = matrix([[1/2, 2/3, 3/4],[5, 7, 8]]); M
[1/2 2/3 3/4]
[ 5 7 8]
You can specify that the input should be reduced by a modulus, using the
IntegerModRing (functionality to be described later)
Sage: R = IntegerModRing(100)
sage: M = matrix(R, [[1],[102],[1003]]); M
[1]
[2]
[3]
Or that the input should be considered in a finite field (also to be described
later).
sage: F = GF(2);
sage: M = matrix(F, [[1, 2, 0, 3]]); M
[1 0 0 1]
1All of the Sage code in this appendix is available at this book’s Companion Web site in .sage files, so that you can load and execute the programs if you wish. See Preface for access information.
B.2 / NUMBER THEORY 705
Sage also supports multiplication, addition, and inversion of matrices as
follows:
sage: M1 = matrix([[1, 2],[3,4]]);
sage: M2 = matrix([[1,−1],[1, 1]]);
sage: M1*M2
[3 1]
[7 1]
sage: M1 + M2
[2 1]
[4 5]
sage: M2^−1
[ 1/2 1/2]
[−1/2 1/2]
B.2 CHAPTER 2: NUMBER THEORY
Example 1: Chinese Remainder Theorem.
def chinese_remainder_theorem(moduli, residues): r""" Function that implements the chinese remainder theorem.
INPUT:
moduli − list or positive integers.
residues − list of remainders such that remainder at position j results when divided by the corresponding modulus at position j in moduli.
OUTPUT:
x − integer such that division by moduli[j] gives remainder residue[j].
"""
if (len(moduli) != len(residues)):
raise ValueError, "expected len(moduli) == len(residues)"
M = prod(moduli);
x = 0;
706 APPENDIX B / SAGE EXAMPLES
for j in xrange(len(moduli)): Mj = moduli[j] Mpr = M/Mj
(Mj_Mpr_gcd, Mpr_inv, Mj_inv) = xgcd(Mpr, Mj)
Mpr_inv = Mpr_inv
if (Mj_Mpr_gcd != 1):
raise ValueError, "Expected all moduli are coprime."
x += residues[j]*Mpr*Mpr_inv;
return x;
Example 2: Miller–Rabin Primality Test.
r""" EXAMPLES:
sage: MILLER_RABIN_TEST(101) False
sage: MILLER_RABIN_TEST(592701729979) True
"""
def MILLER_RABIN_TEST(n): r"""
This function implements the Miller-Rabin Test. It either returns "inconclusive" or "composite."
INPUT:
n − positive integer to probabilistically determine the primality of.
OUTPUT:
If the function returns False, then the test was inconclusive.
If the function returns True, then the test was conclusive and n is composite.
"""
R = IntegerModRing(n); # object for integers mod n # (1) Find integers k, q w/ k > 0 and q odd so that (n−1) == 2^k * q q = n−1 k = 0
B.2 / NUMBER THEORY 707
while (1 == (q % 2)): k += 1 q = q.quo_rem(2)[0] # q/2 but with result of type Integer
# (2) select random a in 1 < a < n−1
a = randint(1,n−1)
a = R(a) # makes it so modular exponentiation is done fast
# if a^q mod n == 1 then return inconclusive if (1 == a^q):
return False
# (3) for j = 0 to k−1 do: if a^(2^j * q) mod n = n−1 return inconclusive
e = q
for j in xrange(k): if (n−1) == (a^e):
return False e = 2*e
# (4) if you've made it here return composite. return True
Example 3: Modular Exponentiation (Square and Multiply).
def ModExp(x,e,N): r""" Calculates x^e mod N using square and multiply.
INPUT:
x − an integer. e − a nonnegative integer. N − a positive integer modulus.
OUTPUT:
y − x^e mod N
"""
e_bits = e.bits() e_bitlen = len(e_bits)
y = 1
for j in xrange(e_bitlen):
y = y^2 % N
708 APPENDIX B / SAGE EXAMPLES
if (1 == e_bits[e_bitlen−1−j]): y = x*y % N
return y
Example 4: Using built-in Sage functionality for CRT. Sage has built in functions to perform the Chinese Remainder Theorem.
There are several functions that produce a wide array of CRT functionality.
The simplest function performs the CRT with two modulii. Specifically CRT
(or the lowercase crt) when called as:
crt(a,b,m,n)
will return a number that is simultaneously congruent to a mod m and b mod n. All parameters are assumed to be integers and the parameters m, n must be relatively prime. Some examples of this function are:
sage: CRT(8, 16, 17, 49) −3120
sage: CRT(1,2,5,7) 16
sage: CRT(50,64,101,127) −62166
If you want to perform the CRT with a list of residues and moduli, Sage
includes the function CRT_list.
CRT_list(v, modulii)
requires that v and modulii be lists of integers of the same length. Furthermore, the elements of modulii must be relatively prime. Then the output is an integer
that reduces to v[i] mod modulii[i] (for i in range(len(v))). For example, the last
call to CRT would have been
sage: CRT_list([50,64],[101,127]) 1969
Note that this answer is different. However, you can check that both answers
satisfy the requirements of the CRT. Here are examples with longer lists:
sage: CRT_list([8, 20, 13], [49, 101, 127]) 608343
sage: CRT_list([10,11,12,13,14],[29,31,37,41,43]) 36657170
The function CRT_basis can be used to precompute the values associated to
the given set of modulii. If modulii is a list of relatively prime modulii, then
CRT_basis(modulii) returns a list a. This list a is such that if x is a list of residues of the modulii, then the output of the CRT can be found by summing:
a[0]*x[0] + a[1]*x[1] + ... + a[len(a)−1]*x[len(a)−1]
B.2 / NUMBER THEORY 709
In the case of the modulii used in the last call to CRT_list this function returns
as follows:
sage: CRT_basis([29,31,37,41,43]) [32354576, 20808689, 23774055, 17163708, 23184311]
The last CRT function that Sage provides is CRT_vectors. This function
performs CRT_list on several different lists (with the same set of modulii) and
returns a list of the simultaneous answers. It is efficient in that it uses CRT_
basis and does not recompute those values for each list. For example:
sage: CRT_vectors([[1,10],[2,11],[3,12],[4,13],[5,14]], [29,31,37,41,43]) [36657161, 36657170]
Example 5: Using built-in Sage functionality for Modular Exponentiation. Sage can perform modular exponentiation using fast algorithms (like
square and multiply) and without allowing the intermediate computations
to become huge. This is done through IntegerModRing objects. Specifically,
creating an IntegerModRing object indicates that arithmetic should be done
with a modulus. Then you cast your integers in this ring to indicate that all
arithmetic should be done with the modulus. Then for elements of this ring,
exponentiation is done efficiently. For example:
sage: R = IntegerModRing(101) sage: x = R(10) sage: x^99 91
sage: R = IntegerModRing(1024) sage: x = R(111) sage: x^345 751
sage: x = R(100) sage: x^200 0
sage: N = 127*101 sage: R = IntegerModRing(N) sage: x = R(54) sage: x^95 9177
Creating an IntegerModRing is similar to creating a FiniteField with GF(...)
except that the modulus can be a general composite.
Example 6: Using built-in Sage functionality for Euler’s totient. Sage has the Euler totient functionality built in. The function is called
euler_phi because of the convention of using the Greek letter phi to represent
710 APPENDIX B / SAGE EXAMPLES
this function. The operation of this function is simple. Just call euler_phi on an
integer and it computes the totient function. This function factors the input,
and hence requires exponential time.
sage: euler_phi(101) 100
sage: euler_phi(1024) 512
sage: euler_phi(333) 216
sage: euler_phi(125) 100
sage: euler_phi(423) 276
B.3 CHAPTER 3: CLASSICAL ENCRYPTION
The following functions are useful for classical cipher examples and exercises:
en_alphabet = "abcdefghijklmnopqrstuvwxyz"
# # This function returns true if and only if the character c is an # alphabetic character # def is_alphabetic_char(c):
return (c.lower() in en_alphabet)
# # This function converts a single character into its numeric value # def char_to_num(c):
return en_alphabet.index(c.lower())
# # This function returns the character corresponding to x mod 26 # in the English alphabet # def num_to_char(x):
return en_alphabet[x % 26]
Example 1: Implement Sage encryption/decryption functions that take a key (as an integer in 0, 1, 2, . . . , 25), and a string. The function should only operate
B.3 / CLASSICAL ENCRYPTION 711
on the characters ‘a’, ‘b’, . . . ‘z’ (both upper and lower case), and it should
leave any other characters unchanged.
Solution:
def CaesarEncrypt(k, plaintext):
ciphertext = ""
for j in xrange(len(plaintext)):
p = plaintext[j]
if is_alphabetic_char(p):
x = (k + char_to_num(p)) % 26 c = num_to_char(x)
else:
c = p
ciphertext += c
return ciphertext
def CaesarDecrypt(k, ciphertext):
plaintext = ""
for j in xrange(len(ciphertext)):
c = ciphertext[j]
if is_alphabetic_char(c):
x = (char_to_num(c) − k) % 26 p = num_to_char(x)
else:
p = c
plaintext += p
return plaintext
Example 2: Implement a function that performs a brute force attack on a ciphertext, it should print a list of the keys and associated decryptions. It
should also take an optional parameter that takes a substring and only prints
out potential plaintexts that contain that decryption.
Solution:
def BruteForceAttack(ciphertext, keyword=None):
for k in xrange(26):
plaintext = CaesarDecrypt(k, ciphertext)
if (None==keyword) or (keyword in plaintext): print "key", k, "decryption", plaintext
return
712 APPENDIX B / SAGE EXAMPLES
Example 3: Show the output of your encrypt function (Example 1) on the following (key, plaintext) pairs:
■ k = 16 plaintext = “Get me a vanilla ice cream, make it a double.” ■ k = 15 plaintext = “I don’t much care for Leonard Cohen.” ■ k = 16 plaintext = “I like root beer floats.”
Solution: sage: k = 6; plaintext = 'Get me a vanilla ice cream, make it a double.' sage: CaesarEncrypt(k, plaintext) 'mkz sk g bgtorrg oik ixkgs, sgqk oz g juahrk.'
sage: k = 15; plaintext = "I don't much care for Leonard Cohen." sage: CaesarEncrypt(k, plaintext) "x sdc'i bjrw rpgt udg atdcpgs rdwtc."
sage: k = 16; plaintext = "I like root beer floats." sage: CaesarEncrypt(k, plaintext) 'y byau heej ruuh vbeqji.'
Example 4: Show the output of your decrypt function (Example 1) on the following (key, ciphertext) pairs:
■ k = 12 ciphertext = ‘nduzs ftq buzq oazqe.’ ■ k = 3 ciphertext = “fdhvdu qhhgv wr orvh zhljkw.” ■ k = 20 ciphertext = “ufgihxm uly numnys.”
Solution: sage: k = 12; ciphertext = "nduzs ftq buzq oazqe." sage: CaesarDecrypt(k, ciphertext) 'bring the pine cones.'
sage: k = 3; ciphertext = "fdhvdu qhhgv wr orvh zhljkw." sage: CaesarDecrypt(k, ciphertext) 'caesar needs to lose weight.'
sage: k = 20; ciphertext = "ufgihxm uly numnys." sage: CaesarDecrypt(k, ciphertext) 'almonds are tastey.'
Example 5: Show the output of your attack function (Example 4) on the following ciphertexts, if an optional keyword is specified, pass that to your
attack function:
■ ciphertext = ‘gryy guru gob tab gb nzoebfr puncry.’ keyword = ‘chapel’ ■ ciphertext = ‘wziv kyv jyfk nyve kyv tpdsrcj tirjy.’ keyword = ‘cymbal’ ■ ciphertext = ‘baeeq klwosjl osk s esf ozg cfwo lgg emuz.’ no keyword
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B.4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD 713
Solution:
sage: ciphertext = 'gryy gurz gb tb gb nzoebfr puncry.' sage: BruteForceAttack(ciphertext, 'chapel') key 13 decryption tell them to go to ambrose chapel.
sage: ciphertext = 'wziv kyv jyfk nyve kyv tpdsrcj tirjy.' sage: BruteForceAttack(ciphertext, 'cymbal') key 17 decryption fire the shot when the cymbals crash.
sage: ciphertext = 'baeeq klwosjl osk s esf ozg cfwo lgg emuz.' sage: BruteForceAttack(ciphertext) key 0 decryption baeeq klwosjl osk s esf ozg cfwo lgg emuz. key 1 decryption azddp jkvnrik nrj r dre nyf bevn kff dlty. key 2 decryption zycco ijumqhj mqi q cqd mxe adum jee cksx. key 3 decryption yxbbn hitlpgi lph p bpc lwd zctl idd bjrw. key 4 decryption xwaam ghskofh kog o aob kvc ybsk hcc aiqv. key 5 decryption wvzzl fgrjneg jnf n zna jub xarj gbb zhpu. key 6 decryption vuyyk efqimdf ime m ymz ita wzqi faa ygot. key 7 decryption utxxj dephlce hld l xly hsz vyph ezz xfns. key 8 decryption tswwi cdogkbd gkc k wkx gry uxog dyy wemr. key 9 decryption srvvh bcnfjac fjb j vjw fqx twnf cxx vdlq. key 10 decryption rquug abmeizb eia i uiv epw svme bww uckp. key 11 decryption qpttf zaldhya dhz h thu dov ruld avv tbjo. key 12 decryption posse yzkcgxz cgy g sgt cnu qtkc zuu sain. key 13 decryption onrrd xyjbfwy bfx f rfs bmt psjb ytt rzhm. key 14 decryption nmqqc wxiaevx aew e qer als oria xss qygl. key 15 decryption mlppb vwhzduw zdv d pdq zkr nqhz wrr pxfk. key 16 decryption lkooa uvgyctv ycu c ocp yjq mpgy vqq owej. key 17 decryption kjnnz tufxbsu xbt b nbo xip lofx upp nvdi. key 18 decryption jimmy stewart was a man who knew too much. key 19 decryption ihllx rsdvzqs vzr z lzm vgn jmdv snn ltbg. key 20 decryption hgkkw qrcuypr uyq y kyl ufm ilcu rmm ksaf. key 21 decryption gfjjv pqbtxoq txp x jxk tel hkbt qll jrze. key 22 decryption feiiu opaswnp swo w iwj sdk gjas pkk iqyd. key 23 decryption edhht nozrvmo rvn v hvi rcj fizr ojj hpxc. key 24 decryption dcggs mnyquln qum u guh qbi ehyq nii gowb. key 25 decryption cbffr lmxptkm ptl t ftg pah dgxp mhh fnva.
B.4 CHAPTER 4: BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD
Example 1: This example implements simplified DES, which is described in Appendix G.
# # The Expansions/Permutations are stored as lists of bit positions #
714 APPENDIX B / SAGE EXAMPLES
P10_data = [3, 5, 2, 7, 4, 10, 1, 9, 8, 6];
P8_data = [6, 3, 7, 4, 8, 5, 10, 9];
LS1_data = [2, 3, 4, 5, 1];
LS2_data = [3, 4, 5, 1, 2];
IP_data = [2, 6, 3, 1, 4, 8, 5, 7];
IPinv_data = [4, 1, 3, 5, 7, 2, 8, 6];
EP_data = [4, 1, 2, 3, 2, 3, 4, 1];
P4_data = [2, 4, 3, 1];
SW_data = [5, 6, 7, 8, 1, 2, 3, 4];
# # SDES lookup tables #
S0_data = [[1, 0, 3, 2],
[3, 2, 1, 0],
[0, 2, 1, 3],
[3, 1, 3, 2]];
S1_data = [[0, 1, 2, 3],
[2, 0, 1, 3],
[3, 0, 1, 0],
[2, 1, 0, 3]];
def ApplyPermutation(X, permutation): r""" This function takes a permutation list (list of bit positions.) And outputs a bit list with the bits taken from X. """
# permute the list X l = len(permutation); return [X[permutation[j]−1] for j in xrange(l)];
def ApplySBox(X, SBox):
r""" This function Applies the SDES SBox (by table look up """
r = 2*X[0] + X[3]; c = 2*X[1] + X[2]; o = SBox[r][c]; return [o & 2, o & 1];
B.4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD 715
# # Each of these functions uses ApplyPermutation # and a permutation list to perform an SDES # Expansion/Permutation #
def P10(X): return ApplyPermutation(X, P10_data);
def P8(X): return ApplyPermutation(X, P8_data);
def IP(X): return ApplyPermutation(X, IP_data);
def IPinv(X): return ApplyPermutation(X, IPinv_data);
def EP(X): return ApplyPermutation(X, EP_data);
def P4(X): return ApplyPermutation(X, P4_data);
def SW(X): return ApplyPermutation(X, SW_data);
def LS1(X): return ApplyPermutation(X, LS1_data);
def LS2(X): return ApplyPermutation(X, LS2_data);
# # These two functions perform the SBox substitutions #
def S0(X): return ApplySBox(X, S0_data);
def S1(X): return ApplySBox(X, S1_data);
def concatenate(left, right): r""" Joins to bit lists together. """ ret = [left[j] for j in xrange(len(left))]; ret.extend(right); return ret;
def LeftHalfBits(block): r""" Returns the left half bits from block. """
716 APPENDIX B / SAGE EXAMPLES
l = len(block); return [block[j] for j in xrange(l/2)];
def RightHalfBits(block): r""" Returns the right half bits from block. """ l = len(block); return [block[j] for j in xrange(l/2, l)];
def XorBlock(block1, block2): r""" Xors two blocks together. """ l = len(block1); if (l != len(block2)):
raise ValueError, "XorBlock arguments must be same length"
return [(block1[j]+block2[j]) % 2 for j in xrange(l)];
def SDESKeySchedule(K): r""" Expands an SDES Key (bit list) into the two round keys. """ temp_K = P10(K);
left_temp_K = LeftHalfBits(temp_K); right_temp_K = RightHalfBits(temp_K);
K1left = LS1(left_temp_K); K1right = LS1(right_temp_K);
K1temp = concatenate(K1left, K1right); K1 = P8(K1temp);
K2left = LS2(K1left); K2right = LS2(K1right);
K2temp = concatenate(K2left, K2right);
K2 = P8(K2temp);
return (K1, K2);
def f_K(block, K): r""" Performs the f_K function supplied block and K. """ left_block = LeftHalfBits(block); right_block = RightHalfBits(block);
B.5 / BASIC CONCEPTS IN NUMBER THEORY AND FINITE FIELDS 717
temp_block1 = EP(right_block);
temp_block2 = XorBlock(temp_block1, K);
left_temp_block2 = LeftHalfBits(temp_block2); right_temp_block2 = RightHalfBits(temp_block2);
S0_out = S0(left_temp_block2); S1_out = S1(right_temp_block2);
temp_block3 = concatenate(S0_out, S1_out);
temp_block4 = P4(temp_block3)
temp_block5 = XorBlock(temp_block4, left_block);
output_block = concatenate(temp_block5, right_ block)
return output_block;
def SDESEncrypt(plaintext_block, K): r""" Performs a single SDES plaintext block encryption. (Given plaintext and key as bit lists.) """
(K1, K2) = SDESKeySchedule(K);
temp_block1 = IP(plaintext_block);
temp_block2 = f_K(temp_block1, K1);
temp_block3 = SW(temp_block2);
temp_block4 = f_K(temp_block3, K2);
output_block = IPinv(temp_block4);
return output_block;
B.5 CHAPTER 5: BASIC CONCEPTS IN NUMBER THEORY AND FINITE FIELDS
Example 1: The Euclidean algorithm for the greatest common divisor.
def EUCLID(a,b): r""" The Euclidean algorithm for finding the gcd of a and b. This algorithm assumes that a > b => 0
INPUT: a − positive integer b − nonnegative integer less than a
718 APPENDIX B / SAGE EXAMPLES
OUTPUT: g − greatest common divisor of a and b """ if (b < 0) or ( a <= b):
raise ValueError, "Expected 0 < a < b"
(A, B) = (a,b);
while (True):
if (0 == B): return A;
R = A % B; A = B; B = R;
Example 2: The extended Euclidean algorithm for the greatest common divisor.
def EXTENDED_EUCLID(m,b): r""" The extended Euclidean algorithm to find gcd(m,b). The input is expected to be such that 0 <= b < m.
INPUT:
m − positive integer
b − nonnegative integer less than m
OUTPUT:
(g, b_inv) − g is the gcd of m and b, b_inv is the multiplicative inverse of b mod m.
"""
if (m < b) or (b < 0):
raise ValueError, "Expected input (0 < b < m)"
(A1,A2,A3) = (1,0,m); (B1,B2,B3) = (0,1,b);
while (True):
if (0 == B3): return (A3, None)
if (1 == B3): return (B3, B2)
Q = floor(A3/B3)
(T1,T2,T3) = (A1−Q*B1, A2−Q*B2, A3−Q*B3)
B.5 / BASIC CONCEPTS IN NUMBER THEORY AND FINITE FIELDS 719
(A1, A2, A3) = (B1, B2, B3) (B1, B2, B3) = (T1, T2, T3)
Example 3: Euclidean algorithm to find gcd of polynomials (with coefficients in a field).
def POLYNOMIAL_EUCLID(A, B): r""" Euclidian algorithm for polynomial GCD: Given two polynomials over the same base field, Assuming degree(A) => degree(B) => 0.
INPUT:
A − polynomial over a field.
B − polynomial over the same field as A, and 0 <= degree(B) <= degree(A).
OUTPUT:
G − greatest common divisor of A and B.
""" degA = A.degree(); degB = B.degree();
if ((degB < 0) or (degA < degB)): raise ValueError, "Expected 0 <= degree(B) <= degree(A)"
while(True):
if (0 == B): return A;
R = A % B;
A = B; B = R;
Example 4: Extended Euclidean algorithm for the gcd of two polynomials (with coefficients in the same field).
def POLYNOMIAL_EXTENDED_EUCLID(m, b): r""" Extended Euclidian algorithm for polynomial GCD: Given two polynomials over the same base field, Assuming degree(m) => degree(b) => 0
INPUT:
m − polynomial over a field.
720 APPENDIX B / SAGE EXAMPLES
b − polynomial over the same field as A, and 0 <= degree(B) <= degree(M).
OUTPUT:
(g,b_inv) − the pair where:
g − greatest common divisor of m and b.
m_inv − is None if G is not of degree 0, and otherwise it is the polynomial such that b(X)*b_inv(X) = 1 mod m(X)
""" degm = m.degree(); degb = b.degree();
if(degb < 0) or (degm < degb): raise ValueError, "expected 0 <= degree(b) <= degree(m)"
(A1, A2, A3) = (1, 0, m); (B1, B2, B3) = (0, 1, b);
while (True):
if (0 == B3): return (A3, None);
if (0 == B3.degree()): return (B3/B3, B2/B3);
Q = A3.quo_rem(B3)[0];
(T1, T2, T3) = (A1 − Q*B1, A2 − Q*B2, A3 − Q*B3); (A1, A2, A3) = (B1, B2, B3); (B1, B2, B3) = (T1, T2, T3);
Example 5: Sage has built in functionality for the gcd function. The regular greatest common divisor function can simply be called as:
sage: gcd(15,100) 5
sage: gcd(90,65311) 1
You can also call it as a method on Integer objects:
sage: x = 10456890 sage: x.gcd(100) 10
The extended Euclidean algorithm for the greatest common divisor is
also built into Sage. Calling xgcd(a,b) returns a tuple, the first element
B.5 / BASIC CONCEPTS IN NUMBER THEORY AND FINITE FIELDS 721
is the gcd, the second and third elements are coefficients u, v such that
gcd(a,b) = u* a + v* b. This can be called as:
sage: xgcd(17,31) (1, 11, −6) sage: xgcd(10, 115) (5, −11, 1)
This can also be called as a method on Integer objects
sage: x = 300 sage: x.xgcd(36) (12, 1, −8)
Example 6: Sage includes robust support for working with finite fields and performing finite field arithmetic. To initialize a finite field with prime order,
use the GF command passing the order as the parameter.
sage: F = GF(2) sage: F Finite Field of size 2
sage: F = GF(37) sage: F Finite Field of size 37
sage: p = 95131 sage: K = GF(p) sage: K Finite Field of size 95131
To initialize a field with a prime power order use the GF command with
the following syntax (to keep track of the primitive element of the extension
field).
sage: F.<a> = GF(128) sage: F Finite Field in a of size 2^7
To do arithmetic in finite fields use the following syntax:
sage: K = GF(37) sage: a = K(3) sage: b = K(18) sage: a − b 22 sage: a + b 21 sage: a * b 17 sage: a/b 31
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722 APPENDIX B / SAGE EXAMPLES
sage: a^−1 25 sage: 1/a 25
To do arithmetic in a finite field with a prime power order, specify
elements using the primitive element:
sage: F.<a> = GF(128) sage: b = a^2 + 1 sage: c = a^5 + a^3 + 1 sage: b − c a^5 + a^3 + a^2 sage: b + c a^5 + a^3 + a^2 sage: b*c a^3 + a^2 + a sage: b/c a^5 + a^3 + a^2 + a sage: b^−1 a^5 + a^3 + a sage: 1/b a^5 + a^3 + a
Example 7: With Sage you can create rings of polynomials over finite fields and do arithmetic with them. To create polynomial rings over finite fields do the following:
sage: R.<x> = GF(2)[] sage: R
Univariate Polynomial Ring in x over Finite Field of size 2 (using NTL) sage: R.<x> = GF(101)[]
sage: R sage: R.<x> = F[] sage: R Univariate Polynomial Ring in x over Finite Field in a of size 2^7
After initializing a polynomial ring, you can then just perform arithmetic
as you would expect:
sage: R.<x> = GF(2)[] sage: f = x^3 + x + 1 sage: g = x^5 + x sage: f + g x^5 + x^3 + 1 sage: f*g x^8 + x^6 + x^5 + x^4 + x^2 + x
B.5 / BASIC CONCEPTS IN NUMBER THEORY AND FINITE FIELDS 723
Division is accomplished by the quo_rem function:
sage: g.quo_rem(f) (x^2 + 1, x^2 + 1)
You can also compute the greatest common divisor:
sage: f.gcd(g) 1
sage: g.gcd(g^2) x^5 + x
sage: R.<x> = GF(17)[] sage: f = 3*x^3 + 2*x^2 + x sage: g = x^2 + 5 sage: f − g 3*x^3 + x^2 + x + 12 sage: f * g 3*x^5 + 2*x^4 + 16*x^3 + 10*x^2 + 5*x sage: f.quo_rem(g) (3*x + 2, 3*x + 7)
And computing gcds in this polynomial ring we see:
sage: f.gcd(g) 1
sage: f.gcd(x^2 + x) x
When creating a Sage finite field with a prime power order, Sage finds an
irreducible polynomial for you. For example:
sage: F.<a> = GF(32) a^5 + a^2 + 1
However, there are many irreducible polynomials over GF(2) of degree 5, such
as x^5 + x^3 + 1. Suppose that you want to create your own extension of the binary field with degree 5, and an irreducible polynomial of your choice. Then
you can do so as follows:
sage: R.<x> = GF(2)[] sage: F = GF(2).extension(x^5 + x^3 + 1, 'a') sage: a = F.gen()
You need to do this last step to inject the primitive element into the
interpreter’s name space. This is done automatically when using the GF
function to create an extension field, but not when you use the member
function extension on a field object.
724 APPENDIX B / SAGE EXAMPLES
B.6 CHAPTER 6: ADVANCED ENCRYPTION STANDARD
Example 1: Simplified AES.
# # These structures are the underlying # Galois Field and corresponding Vector Space # of the field used in the SAES algorithm # These structures allow us to easily compute with these fields.
# F = GF(2); L.<a> = GF(2^4); V = L.vector_space(); VF8 = VectorSpace(F, 8);
# # The MixColumns and its Inverse matrices are stored # as 2x2 matrices with elements in GF(2^4) (as are state matrices.) # The MixColumns operation (and its inverse) are performed by # matrix multiplication. #
MixColumns_matrix = Matrix(L, [[1,a^2],[a^2,1]]);
InverseMixColumns_matrix = MixColumns_matrix.inverse();
SBox_matrix = Matrix(L, [ [ 1 + a^3, a^2, a + a^3, 1 + a + a^3], [ 1 + a^2 + a^3, 1, a^3, 1 + a^2], [ a + a^2, 0, a, 1 + a], [ a^2 + a^3, a + a^2 + a^3, 1 + a + a^2 + a^3, 1 + a + a^2] ]);
InverseSBox_matrix = Matrix(L, [ [ a + a^3, 1 + a^2, 1 + a^3, 1 + a + a^3], [ 1, 1 + a + a^2, a^3, 1 + a + a^2 + a^3], [ a + a^2, 0, a, 1 + a], [ a^2 + a^3, a^2, 1 + a^2 + a^3, a + a^2 + a^3] ]);
RCON = [ VF8([F(0), F(0), F(0), F(0), F(0), F(0), F(0), F(1)]), VF8([F(0), F(0), F(0), F(0), F(1), F(1), F(0), F(0)]) ];
B.6 / ADVANCED ENCRYPTION STANDARD 725
def SAES_ToStateMatrix(block): r""" Converts a bit list into an SAES state matrix. """
B = block;
# form the plaintext block into a matrix of GF(2^n) elements S00 = L(V([B[0], B[1], B[2], B[3]])); S01 = L(V([B[4], B[5], B[6], B[7]])); S10 = L(V([B[8], B[9], B[10], B[11]])); S11 = L(V([B[12], B[13], B[14], B[15]]));
state_matrix = Matrix(L, [[S00,S01],[S10,S11]]);
return state_matrix;
def SAES_FromStateMatrix(State Matrix): r""" Converts an SAES State Matrix to a bit list. """
output = [];
# convert State Matrix back into bit list for r in xrange(2):
for c in xrange(2): v = V(State Matrix[r,c]); for j in xrange(4):
output.append(Integer(v[j]));
return output;
def SAES_AddRoundKey(state_matrix, K): r""" Adds a round key to an SAES state matrix. """
K_matrix = SAES_ToStateMatrix(K);
next_state_matrix = K_matrix + state_matrix;
return next_state_matrix;
def SAES_MixColumns(state_matrix): r""" Performs the Mix Columns operation. """
next_state_matrix = MixColumns_matrix*state_matrix;
return next_state_matrix;
726 APPENDIX B / SAGE EXAMPLES
def SAES_InverseMixColumns(state_matrix): r""" Performs the Inverse Mix Columns operation. """
next_state_matrix = InverseMixColumns_matrix* state_matrix; return next_state_matrix;
def SAES_ShiftRow(state_matrix): r""" Performs the Shift Row operation. """
M = state_matrix; next_state_matrix = Matrix(L, [
[M[0,0], M[0,1]], [M[1,1], M[1,0]] ]);
return next_state_matrix;
def SAES_SBox(nibble): r""" Performs the SAES SBox look up in the SBox matrix (lookup table.) """
v = nibble._vector_(); c = Integer(v[0]) + 2*Integer(v[1]); r = Integer(v[2]) + 2*Integer(v[3]); return SBox_matrix[r,c];
def SAES_NibbleSubstitution(state_matrix): r""" Performs the SAES SBox on each element of an SAES state matrix. """
M = state_matrix; next_state_matrix = Matrix(L,
[ [ SAES_SBox(M[0,0]), SAES_SBox(M[0,1])], [ SAES_SBox(M[1,0]), SAES_SBox(M[1,1])] ]);
return next_state_matrix;
def SAES_InvSBox(nibble): r""" Performs the SAES Inverse SBox look up in the SBox matrix (lookup table.) """
v = nibble._vector_(); c = Integer(v[0]) + 2*Integer(v[1]);
B.6 / ADVANCED ENCRYPTION STANDARD 727
r = Integer(v[2]) + 2*Integer(v[3]); return InverseSBox_matrix[r,c];
def SAES_InvNibbleSub(state_matrix): r""" Performs the SAES Inverse SBox on each element of an SAES state matrix. """
M = state_matrix; next_state_matrix = Matrix(L, [ [ SAES_InvSBox(M[0,0]), SAES_InvSBox(M[0,1])], [ SAES_InvSBox(M[1,0]), SAES_InvSBox(M[1,1])] ]);
return next_state_matrix;
def RotNib(w): r""" Splits an 8 bit list into two elements of GF(2^4) """ N_0 = L(V([w[j] for j in xrange(4)])); N_1 = L(V([w[j] for j in xrange(4,8)])); return (N_1, N_0);
def SAES_g(w, i): r""" Performs the SAES g function on the 8 bit list w. """ (N0, N1) = RotNib(w); N0 = V(SAES_SBox(N0)); N1 = V(SAES_SBox(N1)); temp1 = VF8( [ N0[0], N0[1], N0[2], N0[3], N1[0], N1[1], N1[2], N1[3] ] ); output = temp1 + RCON[i]; return output;
def SAES_KeyExpansion(K): r""" Expands an SAES key into two round keys. """ w0 = VF8([K[j] for j in xrange(8)]); w1 = VF8([K[j] for j in xrange(8,16)]);
w2 = w0 + SAES_g(w1, 0); w3 = w1 + w2;
w4 = w2 + SAES_g(w3, 1); w5 = w3 + w4;
K0 = [w0[j] for j in xrange(8)]; K0.extend([w1[j] for j in xrange(8)]);
K1 = [w2[j] for j in xrange(8)]; K1.extend([w3[j] for j in xrange(8)]);
728 APPENDIX B / SAGE EXAMPLES
K2 = [w4[j] for j in xrange(8)]; K2.extend([w4[j] for j in xrange(8)]);
return (K0, K1, K2);
# # Encrypts one plaintext block with key K #
def SAES_Encrypt(plaintext, K): r""" Performs a SAES encryption on a single plaintext block. (Both block and key passed as bit lists.) """
# get the key schedule (K0, K1, K2) = SAES_KeyExpansion(K);
state_matrix0 = SAES_ToStateMatrix(plaintext);
state_matrix1 = SAES_AddRoundKey(state_matrix0, K0);
state_matrix2 = SAES_NibbleSubstitution (state_matrix1);
state_matrix3 = SAES_ShiftRow(state_matrix2);
state_matrix4 = SAES_MixColumns(state_matrix3);
state_matrix5 = SAES_AddRoundKey(state_matrix4, K1);
state_matrix6 = SAES_NibbleSubstitution (state_matrix5);
state_matrix7 = SAES_ShiftRow(state_matrix6);
state_matrix8 = SAES_AddRoundKey(state_matrix7, K2);
output = SAES_FromStateMatrix(state_matrix8); return output;
# # Decrypts one ciphertext block with key K #
def SAES_Decrypt(ciphertext, K): r""" Performs a single SAES decryption operation on a − ciphertext block. (Both block and key passed as bit lists.) """
# perform key expansion (K0, K1, K2) = SAES_KeyExpansion(K);
B.7 / PSEUDORANDOM NUMBER GENERATION AND STREAM CIPHERS 729
# form the ciphertext block into a matrix of GF(2^n) elements
state_matrix0 = SAES_ToStateMatrix(ciphertext);
state_matrix1 = SAES_AddRoundKey(state_matrix0, K2);
state_matrix2 = SAES_ShiftRow(state_matrix1);
state_matrix3 = SAES_InvNibbleSub(state_matrix2);
state_matrix4 = SAES_AddRoundKey(state_matrix3, K1);
state_matrix5 = SAES_InverseMixColumns (state_matrix4);
state_matrix6 = SAES_ShiftRow(state_matrix5);
state_matrix7 = SAES_InvNibbleSub(state_matrix6);
state_matrix8 = SAES_AddRoundKey(state_matrix7, K0);
output = SAES_FromStateMatrix(state_matrix8);
return output;
B.7 CHAPTER 8: PSEUDORANDOM NUMBER GENERATION AND STREAM CIPHERS
Example 1: Blum Blum Shub RNG.
def BlumBlumShub_Initialize(bitlen, seed): r""" Initializes a Blum-Blum-Shub RNG State.
A BBS-RNG State is a list with two elements: [N, X] N is a 2*bitlen modulus (product of two primes) X is the current state of the PRNG.
INPUT: bitlen − the bit length of each of the prime factors of n
seed − a large random integer to start out the prng
OUTPUT:
state − a BBS-RNG internal state
"""
# note that this is not the most cryptographically secure
730 APPENDIX B / SAGE EXAMPLES
# way to generate primes, because we do not know how the # internal sage random_prime function works.
p = 3; while (p < 2^(bitlen−1)) or (3 != (p % 4)):
p = random_prime(2^bitlen);
q = 3; while (q < 2^(bitlen−1)) or (3 != (q % 4)):
q = random_prime(2^bitlen);
N = p*q;
X = (seed^2 % N)
state = [N, X]
return state;
def BlumBlumShub_Generate(num_bits, state): r""" Blum−Blum−Shum random number generation function.
INPUT:
num_bits − the number of bits (iterations) to generate with this RNG.
state − an internal state of the BBS−RNG (a list [N, X].)
OUTPUT:
random_bits − a num_bits length list of random bits.
"""
random_bits = [];
N = state[0] X = state[1]
for j in xrange(num_bits):
X = X^2 % N random_bits.append(X % 2)
# update the internal state state[1] = X;
return random_bits;
Example 2: Linear Congruential RNG.
def LinearCongruential_Initialize(a, c, m, X0): r"""
Hiva-Network.Com
B.8 / PUBLIC-KEY CRYPTOGRAPHY AND RSA 731
This functional initializes a linear congruential RNG state.
This state is a list of four integers: [a, c, m, X]
a,c,m are the parameters of the linear congruential instantiation X is the current state of the PRNG.
INPUT:
a − The coefficient c − The offset m − The modulus X0 − The initial state
OUTPUT:
state − The initial internal state of the RNG
"""
return [a,c,m,X0]
def LinearCongruential_Generate(state): r"""
Generates a single linear congruential RNG output and updates the state.
INPUT:
state − an internal RNG state.
OUTPUT:
X − a single output of the linear congruential RNG.
""" a = state[0] c = state[1] m = state[2] X = state[3] X_next = (a*X + c) % m state[3] = X_next return X_next
B.8 CHAPTER 9: PUBLIC-KEY CRYPTOGRAPHY AND RSA
Example 1: Using Sage we can simulate an RSA encryption and decryption.
sage: # randomly select some prime numbers sage: p = random_prime(1000); p 191
732 APPENDIX B / SAGE EXAMPLES
sage: q = random_prime(1000); q 601 sage: # compute the modulus sage: N = p*q sage: R = IntegerModRing(N) sage: phi_N = (p−1)*(q−1) sage: # we can choose the encrypt key to be anything sage: # relatively prime to phi_N sage: e = 17 sage: gcd(d, phi_N) 1 sage: # the decrypt key is the multiplicative inverse sage: # of d mod phi_N sage: d = xgcd(d, phi_N)[1] % phi_N sage: d 60353 sage: # Now we will encrypt/decrypt some random 7 digit numbers
sage: P = randint(1,127); P 97 sage: # encrypt sage: C = R(P)^e; C 46685 sage: # decrypt sage: R(C)^d 97
sage: P = randint(1,127); P 46 sage: # encrypt sage: C = R(P)^e; C 75843 sage: # decrypt sage: R(C)^d 46
sage: P = randint(1,127); P 3 sage: # encrypt sage: C = R(P)^e; C 288 sage: # decrypt sage: R(C)^d 3
Also, Sage can just as easily do much larger numbers:
sage: p = random_prime(1000000000); p
B.8 / PUBLIC-KEY CRYPTOGRAPHY AND RSA 733
114750751 sage: q = random_prime(1000000000); q 8916569 sage: N = p*q sage: R = IntegerModRing(N) sage: phi_N = (p−1)*(q−1) sage: e = 2^16 + 1 sage: d = xgcd(e, phi_N)[1] % phi_N sage: d 237150735093473
sage: P = randint(1,1000000); P 955802 sage: C = R(P)^e sage: R(C)^d 955802
Example 2: In Sage, we can also see an example of RSA signing/verifying.
sage: p = random_prime(10000); p 1601 sage: q = random_prime(10000); q 4073 sage: N = p*q sage: R = IntegerModRing(N) sage: phi_N = (p−1)*(q−1) sage: e = 47 sage: gcd(e, phi_N) 1 sage: d = xgcd(e,phi_N)[1] % phi_N sage: # Now by exponentiating with the private key sage: # we are effectively signing the data sage: # a few examples of this
sage: to_sign = randint(2,2^10); to_sign 650 sage: # the signature is checked by exponentiating sage: # and checking vs the to_sign value sage: signed = R(to_sign)^d; signed 2910116 sage: to_sign == signed^e True sage: to_sign = randint(2,2^10); to_sign 362 sage: signed = R(to_sign)^d; signed 546132 sage: to_sign == signed^e True
734 APPENDIX B / SAGE EXAMPLES
sage: # we can also see what happens if we try to verify a bad signature
sage: to_sign = randint(2,2^10); to_sign 605 sage: signed = R(to_sign)^d; signed 1967793 sage: bad_signature = signed − randint(2,100) sage: to_sign == bad_signature^e False
B.9 CHAPTER 10: OTHER PUBLIC-KEY CRYPTOSYSTEMS
Example 1: Here is an example of Alice and Bob performing a Diffie–Hellman Key Exchange done in Sage:
sage: # Alice and Bob agree on the domain parameters: sage: p = 619 sage: F = GF(p) sage: g = F(2) sage: # Alice picks a random value x in 1 . . . 618 sage: x = randint(1,618); x 571 sage: # Alice computes X = g^x and sends this to Bob sage: X = g^571; X 591 sage: # Bob picks a random value y in 1 . . . 618 sage: y = randint(1,618);y 356 sage: # Bob computes Y = g^y and sends this to Alice sage: Y = g^y; Y 199 sage: # Alice computes Y^x sage: Y^x 563 sage: # Bob computes X^y sage: X^y 563 sage: # Alice and Bob now share a secret value
Example 2: In reality to prevent what is known as small subgroup attacks, the prime p is chosen so that p - 2q + 1 where p is a prime as well.
sage: q = 761 sage: p = 2*q + 1 sage: is_prime(q) True
B.9 / OTHER PUBLIC-KEY CRYPTOSYSTEMS 735
sage: is_prime(p) True sage: F = GF(p) sage: g = F(3) sage: g^q 1 sage: # note that g^q = 1 implies g is of order q sage: # Alice picks a random value x in 2 . . . q−1 sage: x = randint(2,q−1); x 312 sage: # Alice computes X = g^x and sends it to Bob sage: X = g^x; X 26 sage: # Bob computes a random value y in 2 . . . q−1 sage: y = randint(2,q−1); y 24 sage: # Bob computes Y = g^y and sends it to Alice sage: Y = g^y; Y 1304 sage: # Alice computes Y^x sage: Y^x 541 sage: # Bob computes X^y sage: X^y 541 sage: # Alice and Bob now share the secret value 541
Example 3: Sage has a significant amount of support for elliptic curves. This functionality can be very useful when learning, because it allows you to easily
calculate things and get the big picture. Doing the examples by hand may
cause you to get mired in the details. First you instantiate an elliptic curve,
by specifying the field that it is over, and the coefficients of the defining
Weierstrass equation. For this purpose, we write the Weierstrass equation as
y2 + a1xy + a3y = x3 + a2x2 + a4x + a6
Then the Sage function EllipticCurve(R, [a1, a2, a3, a4, a6]) creates the elliptic
curve over the ring R.
sage: E = EllipticCurve(GF(17), [1,2,3,4,5]) sage: E Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Finite Field of size 17
sage: E = EllipticCurve(GF(29), [0,0,0,1,1]) sage: E Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 29
736 APPENDIX B / SAGE EXAMPLES
sage: E = EllipticCurve(GF(127), [0,0,0,2,17]) sage: E Elliptic Curve defined by y^2 = x^3 + 2*x + 17 over Finite Field of size 127
sage: F.<theta> = GF(2^10) sage: E = EllipticCurve(F, [1,0,0,1,0]) sage: E Elliptic Curve defined by y^2 + x*y = x^3 + x over Finite Field in theta of size 2^10
Example 4: Koblitz curves. A Koblitz curve is an elliptic curve over a binary field defined by an equation of the form
y2 + xy = x3 + ax2 + 1
where a = 0 or 1. FIPS 186-3 recommends a number of Koblitz curves for use with the Digital Signature Standard (DSS). Here we give an example of a curve
of similar form to the Koblitz curves:
sage: F.<theta> = GF(2^17) sage: E = EllipticCurve(F,[1,0,0,theta,1]) sage: E Elliptic Curve defined by y^2 + y = x^3 + theta* x^2 = 1 over Finite Field in theta of size 2^17
Example 5: Sage can even easily instantiate curves of cryptographic sizes, like K163, which is one of the FIPS 186-3 curves.
sage: F.<theta> = GF(2^163) sage: E = EllipticCurve(F, [1,0,0,1,1]) sage: E Elliptic Curve defined by y^2 + x*y = x^3 + x^2 + 1 over Finite Field in theta of size 2^163
However, you should be careful that when instantiating a curve of cryptographic
sizes, some of the functions on the curve object will not work because they
require exponential time to run. While you can compute some things with
these objects, it is best to leave your experimentation to the smaller sized
curves.
You can calculate some values of the curve, such as the number of points:
sage: E = EllipticCurve(GF(107), [0,0,0,1,0]) sage: E.order() 108
You can also determine the generators of a curve:
sage: E = EllipticCurve(GF(101), [0,0,0,1,0]) sage: E.gens() ((7 : 42 : 1), (36 : 38 : 1))
B.9 / OTHER PUBLIC-KEY CRYPTOSYSTEMS 737
Note that this output is printed (x : y : z). This is a minor technical consideration
because Sage stores points in what is known as “projective coordinates.” The
precise meaning is not important, because for non-infinite points the value z will
always be 1 and the first two values in a coordinate will be the x and y coordinates,
exactly as you would expect. This representation is useful because it allows the
point at infinity to be specified as a point with the z coordinate equal to 0:
sage: E(0) (0 : 1 : 0)
This shows how you can recognize a point at infinity as well as specify it. If you
want to get the x and y coordinates out of a point on the curve, you can do so
as follows:
sage: P = E.random_point(); P (62 : 38 : 1) sage: (x,y) = P.xy(); (x,y) (62, 38)
You can specify a point on the curve by casting an ordered pair to the curve as:
sage: P = E((62,−38)); P (62 : 63 : 1)
Now that you can find the generators on a curve and specify points you can
experiment with these points and do arithmetic as well. Continuing to use E
as the curve instantiated in the previous example, we can set G1 and G2 to the
generators:
sage: (G1, G2) = E.gens() sage: P = E.random_point(); P (49 : 29 : 1)
You can compute the sum of two points as in the following examples:
sage: G1 + G2 + P (69 : 96 : 1) sage: G1 + P (40 : 62 : 1) sage: P + P + G2 (84 : 25 : 1)
You can compute the inverse of a point using the unary minus ( - ) operator:
sage: −P (49 : 72 : 1) sage: −G1 (7 : 59 : 1)
You can also compute repeated point addition (adding a point to itself many
times) with the * operator:
sage: 13*G1 (72 : 23 : 1)
738 APPENDIX B / SAGE EXAMPLES
sage: 2*G2 (9 : 58 : 1) sage: 88*P (87 : 75 : 1)
And for curves over small finite fields you can also compute the order (discrete
log of the point at infinity with respect to that point).
sage: G1.order() 10
sage: G2.order() 10
sage: P.order() 10
Example 6: Using the Sage elliptic curve functionality to perform a simulated elliptic curve Diffie–Hellman (ECDH) key exchange.
sage: # calculate domain parameters sage: F = GF(127) sage: E = EllipticCurve(F, [0, 0, 0, 3, 4]) sage: G = E.gen(0); G (94 : 6 : 1) sage: q = E.order(); q 122
sage: # Alice computes a secret value x in 2 . . . q−1 sage: x = randint(2,q−1); x 33
sage: # Alice computes a public value X = x*G sage: X = x*G; X (55 : 89 : 1)
sage: # Bob computes a secret value y in 2 . . . q−1 sage: y = randint(2,q−1); y 55
sage: # Bob computes a public value Y = y*G sage: Y = y*G; Y (84 : 39 : 1)
sage: # Alice computes the shared value sage: x*Y (91 : 105 : 1)
sage: # Bob computes the shared value sage: y*X (91 : 105 : 1)
B.10 / CRYPTOGRAPHIC HASH FUNCTIONS 739
However, in practice most curves that are used have a prime order:
sage: # Calculate the domain parameters sage: F = GF(101) sage: E = EllipticCurve(F, [0, 0, 0, 25, 7]) sage: G = E((97,34)) sage: q = E.order() sage: # Alice computes a secret values x in 2 . . . q−1 sage: x = randint(2,q−1) sage: # Alice computes a public value X = x*G sage: X = x*G sage: # Bob computes a secret value y in 2 . . . q−1 sage: y = randint(2,q−1) sage: # Bob computes a public value Y = y*G sage: Y = y*G sage: # Alice computes the shared secret value sage: x*Y (23 : 15 : 1) sage: # Bob computes the shared secret value sage: y*X (23 : 15 : 1)
B.10 CHAPTER 11: CRYPTOGRAPHIC HASH FUNCTIONS
Example 1: The following is an example of the MASH hash function in Sage. MASH is a function based on the use of modular arithmetic. It involves use
of an RSA-like modulus M, whose bit length affects the security. M should be
difficult to factor, and for M of unknown factorization, the security is based in
part on the difficulty of extracting modular roots. M also determines the block
size for processing messages. In essence, MASH is defined as:
Hi = ((xi ⊕ Hi - 1) 2 OR Hi - 1)(mod M)
where
A = 0xFF00 c 00 Hi - 1 = the largest prime less than M xi = the ith digit of the base M expansion of input n. That is, we express n as a number of base M. Thus:
n = x0 + x1M + x2M2 + c
The following is an example of the MASH hash function in Sage.
# # This function generates a mash modulus # takes a bit length, and returns a Mash # modulus l or l−1 bits long (if n is odd)
Hiva-Network.Com
740 APPENDIX B / SAGE EXAMPLES
# returns p, q, and the product N # def generate_mash_modulus(l):
m = l.quo_rem(2)[0]
p = 1 while (p < 2^(m−1)): p = random_prime(2^m)
q = 1 while (q < 2^(m−1)): q = random_prime(2^m)
N = p*q return (N, p, q)
# # Mash Hash # the value n is the data to be hashed. # the value N is the modulus # Returns the hash value. # def MASH(n, N):
H = previous_prime(N)
q = n
while (0 != q): (q, a) = q.quo_rem(N) H = ((H+a)^2 + H) % N
return H
The output of these functions running;
sage: data = ZZ(randint(1,2^1000)) sage: (N, p, q) = generate_mash_modulus(20) sage: MASH(data, N) 220874 sage: (N, p, q) = generate_mash_modulus(50) sage: MASH(data, N) 455794413217080 sage: (N, p, q) = generate_mash_modulus(100) sage: MASH(data, N) 268864504538508517754648285037 sage: data = ZZ(randint(1,2^1000)) sage: MASH(data, N) 236862581074736881919296071248 sage: data = ZZ(randint(1,2^1000)) sage: MASH(data, N) 395463068716770866931052945515
B.11 / DIGITAL SIGNATURES 741
B.11 CHAPTER 13: DIGITAL SIGNATURES
Example 1: Using Sage, we can perform a DSA sign and verify:
sage: # First we generate the domain parameters sage: # Generate a 16 bit prime q sage: q = 1; sage: while (q < 2^15): q = random_prime(2^16) . . . .: sage: q 42697 sage: # Generate a 64 bit p, such that q divides (p−1) sage: p = 1 sage: while (not is_prime(p)): . . . .: p = (2^48 + randint(1,2^46)*2)*q + 1 . . . .: sage: p 12797003281321319017 sage: # Generate h and g sage: h = randint(2,p−2) sage: h 5751574539220326847 sage: F = GF(p) sage: g = F(h)^((p−1)/q) sage: g 9670562682258945855
sage: # Generate a user public / private key sage: # private key sage: x = randint(2,q−1) sage: x 20499 sage: # public key sage: y = F(g)^x sage: y 7955052828197610751 sage: # Sign and verify a random value sage: H = randint(2,p−1) sage: # Signing sage: # random blinding value sage: k = randint(2,q−1) sage: r = F(g)^k % q sage: r = F(g)^k sage: r = r.lift() % q sage: r 6805 sage: kinv = xgcd(k,q)[1] % q
742 APPENDIX B / SAGE EXAMPLES
sage: s = kinv*(H + x*r) % q sage: s 26026
sage: # Verifying sage: w = xgcd(s,q)[1]; w 12250 sage: u1 = H*w % q; u1 6694 sage: u2 = r*w % q; u2 16706 sage: v = F(g)^u1 * F(y)^u2 sage: v = v.lift() % q sage: v 6805 sage: v == r True
sage: # Sign and verify another random value sage: H = randint(2,p−1) sage: k = randint(2,q−1) sage: r = F(g)^k sage: r = r.lift() % q sage: r 3284 sage: kinv = xgcd(k,q)[1] % q sage: s = kinv*(H + x*r) % q sage: s 2330
sage: # Verifying sage: w = xgcd(s,q)[1]; w 4343 sage: u1 = H*w % q; u1 32191 sage: u2 = r*w % q; u2 1614 sage: v = F(g)^u1 * F(y)^u2 sage: v = v.lift() % q sage: v 3284 sage: v == r True
Example 2: The following functions implement DSA domain parameter generation, key generation, and DSA Signing:
# # Generates a 16 bit q and 64 bit p, both prime # such that q divides p−1 #
B.11 / DIGITAL SIGNATURES 743
def DSA_generate_domain_parameters():
g = 1
while (1 == g):
# first find a q q = 1 while (q < 2^15): q = random_prime(2^16) # next find a p p = 1 while (not is_prime(p)):
p = (2^47 + randint(1,2^45)*2)*q + 1
F = GF(p)
h = randint(2,p−1)
g = (F(h)^((p−1)/q)).lift()
return (p, q, g)
# # Generates a users private and public key # given domain parameters p, q, and g # def DSA_generate_keypair(p, q, g):
x = randint(2,q−1)
F = GF(p)
y = F(g)^x y = y.lift()
return (x,y)
# # Given domain parameters p, q and g # as well as a secret key x # and a hash value H # this performs the DSA signing algorithm # def DSA_sign(p, q, g, x, H):
k = randint(2,q−1)
F = GF(p)
r = F(g)^k
r = r.lift() % q
kinv = xgcd(k,q)[1] % q
s = kinv*(H + x*r) % q
return (r, s)
744744
REFERENCES
ABBREVIATIONS
ACM Association for Computing Machinery
IBM International Business Machines Corporation
IEEE Institute of Electrical and Electronics Engineers
NIST National Institute of Standards and Technology
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BELL94 Bellare, M., and Rogaway, P. “Optimal Asymmetric Encryption—How to Encrypt with RSA.” Proceedings, Eurocrypt ’94, 1994.
BELL96a Bellare, M.; Canetti, R.; and Krawczyk, H. “Keying Hash Functions for Message Authentication.” Proceedings, CRYPTO ’96, August 1996; published by Springer- Verlag. An expanded version is available at http://www-cse.ucsd.edu/users/mihir.
BELL96b Bellare, M.; Canetti, R.; and Krawczyk, H. “The HMAC Construction.” CryptoBytes, Spring 1996.
BELL96c Bellare, M., and Rogaway, P. “The Exact Security of Digital Signatures – How to Sign with RSA and Rabin.” Advances in Cryptology – Eurocrypt ’96, 1996.
BELL97 Bellare, M., and Rogaway, P. “Collision-Resistant Hashing: Towards Making UOWHF’s Practical.” Proceedings, CRYPTO ’97, 1997; published by Springer-Verlag.
BELL98 Bellare, M., and Rogaway, P. “PSS: Provably Secure Encoding Method for Digital Signatures.” Submission to IEEE P1363, August 1998. Available from http://grouper .ieee.org/groups/1363.
BELL00 Bellare, M.; Kilian, J.; and Rogaway, P. “The Security of the Cipher Block Chaining Message Authentication Code.” Journal of Computer and System Sciences, December 2000.
BELL09 Bellare, M., et al. “Format Preserving Encryption”. Proceedings of SAC 2009 (Selected Areas in Cryptography), November 2009. Available at Cryptology ePrint Archive http:// eprint.iacr.org/2009/.
BELL10a Bellare, M.; Rogaway, P.; and Spies, T. The FFX Mode of Operation for Format- Preserving Encryption, Draft 1.1. NIST, http://csrc.nist.gov/groups/ST/toolkit/BCM/documents/ proposedmodes/ffx/ffx-spec.pdf, February, 2010.
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CREDITS
754754
A Abelian groups, 144–145, 322–323
associative, 322 closure, 322 commutative, 322 elliptic curve, 323 identity element, 322 inverse element, 322
Absorbing phase, 367, 368 Access control, 30–32, 528, 584, 591–592,
596, 601 Access point (AP), 583, 584–585, 590, 592 Access requestor (AR), 520 Accidental association, 583 Accountability, 23 Active attacks, 27–29
denial of service, 29 masquerade, 28 modification of messages, 28 replay, 28
Ad hoc networks, 583 Adaptive chosen message attack, 422 Adaptive Proportion Test, 276 Addition, 145, 146, 164
algebraic description of, 325 geometric description of, 323–325
Additive inverse, 56 AddRoundKey, 174, 177, 180, 198–200
forward add round key transformation, 189
inputs for single AES round, 190 inverse add round key transformation,
189 Administrative management domain
(ADMD), 615, 651–652 Advanced Encryption Standard (AES),
119, 129, 142, 160, 172, 724–729 AddRoundKey and InvMixColumns,
198–199 avalanche effect, 194–197 byte-level operations, 180 data structures, 176 detailed structure, 177–179
AddRoundKey, 177 MixColumns, 177 ShiftRows, 177 substitute bytes, 177
encryption and decryption, 178 process, 175 round, 179
equivalent inverse cipher, 197–199 example, 193–197 vs. FPE, 231 general structure, 174–177
State, 174 implementation, 197–201
8-bit processor, 199–200 32-bit processor, 200–201
inputs for, 190 InvShiftRows and InvSubByte, 198 key expansion, 190–193 parameters, 177 row and column operations, 186 State array, 174 transformation functions (See
Transformation functions, AES) AH. See Authentication Header (AH) Alert protocol, 554–555 Algorithm
decryption, 288 design, 257–258
asymmetric ciphers, 258 cryptographic algorithms, 257–258
hash functions, 258 message authentication codes, 258 purpose-built algorithms, 257 symmetric block ciphers, 258
encryption, 286 negotiation, 571
ANSI X9.17 PRNG, 263–264 input, 263 keys, 263 output, 264
Anti-replay service, 675–676 receiver, 676 replay attack, 675 sender, 676
AP. See Access point (AP) AR. See Access requestor (AR) Arbitrary reversible substitution cipher,
122 Associative group, 143, 322 Associative laws, 56 Associativity of multiplication, 145 Asymmetric card authentication key, 512 Asymmetric cipher, 258, 334–336 Asymmetric encryption, 20
keys, 288 PKI, 285 public key certificate, 285 public key cryptographic algorithm,
285 terminology related to, 285
Attack surfaces, 37–38 Attack trees, 38–40 Authenticated encryption (AE), 402–408
CMAC, 402–405 GCM, 405–408
authentication and encryption functions, 406
message authentication code, 407 Authentication
data origin, 30 ESP, 681 exchange, 33 IEEE 802.11i wireless LAN security,
596, 601–603 IKE key determination, 687 payload, 691 peer entity, 29, 30 protocols, 20 public-key cryptography, 290, 291 server, 525 S/MIME, 628–630
Authentication Header (AH), 666, 669 Authentication service exchange, 496 Authenticator, 290, 383, 525, 527, 530 Authenticity, 23 Authenticity-related threats, 625 Authority key identifier, 466 Autokey system, 104 Automated key management, 684 Availability, 22, 24 Availability service, 32 Availability-related threats, 625 Avalanche effect, 194–197
DES, 131–133
B Backward unpredictability, 256 Barrier security, 589 Base64 transfer encoding, 623 Basic service set (BSS), 590, 592 BIC. See Bit independence criterion
(BIC) Big-endian format, 358 Bijection, 71 Binary curve, 325
Binary operator, 53 Binary operator mod, 83 BIO, 513 BIO-A, 513 Birthday attack, 355 Birthday paradox, 351 Bit independence criterion (BIC), 136 56-Bit keys, 134 Bit length, 238 8-Bit processor, 199–200 32-Bit processor, 200–201 Bitrate, 365, 366 Blinding, 306 Block cipher, 120–121, 713–717
advantage of, 268 CBC mode, 216–218 CFB mode, 218–220
CTR mode, 218, 222–224 encryption, 218 OFB mode, 218, 220–222 s-bit, 219 segments, 218
CTR mode, 218, 222–224 advantages of, 223–224 hardware efficiency, 223 preprocessing, 224 provable security, 224 random access, 224 simplicity, 224 software efficiency, 224
design principles, 135–137 BIC, 136 design of function F, 136–137 key schedule algorithm, 137 number of rounds, 136 SAC, 136
double DES, 208–210 ECB, 213–216
diffusion, 216 error propagation, 215 error recovery, 215 modes of operation, 213 overhead, 215 security, 216
FPE, 231–245 AES vs., 231 difficulties in designing an,
232–233 Feistel structure for, 233–238 motivation, 231–232 NIST methods for, 238–245 notation in, 236 parameters in, 236
internals, 699 MAC on, 399–401
CMAC, 400–401 DAA, 399–400
modes of operation, 214 modes of use, 699 multiple encryption, 208–213 OFB mode, 218, 220–222 PRNGs, 261–267
ANSI X9.17 PRNG, 263–264 mechanisms, 262 NIST CTR_DRBG, 264–267
processes, 89 projects, 699 round, 699 symmetric, 258 triple DES
known-plaintext attack on, 212 with three keys, 213 with two keys, 210–213
TRNG, 274 tweakable, 225–226
INDEX
INDEX 755
XTS-AES mode, 224–231 ciphertext-stealing technique, 229 definition, 230 feedback characteristic of modes of
operation, 225 operation on sector, 229–231 operation on single block, 227–229 storage encryption requirements,
226–227 tweakable block ciphers, 225–226
Block size, 126 Blum Blum Shub (BBS) generator,
260–261, 729–730 Bring-your-own-device (BYOD) policy,
587 Broad network access, 530–531 Brute-force approach, 253 Brute-force attacks, 89, 91, 255, 302, 350–353
birthday paradox, 351 collision resistant attacks, 351–353 cryptanalysis, 353–354 MAC, 393–394 preimage and second preimage
attacks, 351 BSS. See Basic service set (BSS) Business continuity and disaster
recovery, 543 BYOD policy. See Bring-your-own-
device (BYOD) policy
C Caesar cipher, 92–94, 102–103 Canonical form, 625 Capacity, 367 Card authentication key, 513 Cardholder unique identifier (CHUID),
511, 512 CBC mode. See Cipher block chaining
(CBC) mode CBC-MAC/CMAC, 278 CCA. See Chosen ciphertext attack
(CCA) CCMP. See Counter Mode-CBC MAC
Protocol (CCMP) Certificate Association Data, 644 Certificate payload, 691 Certificate policies, 466 Certificate Request payload, 691 Certification authority (CA)
forward certificates, 463 reverse certificates, 463
CFB mode. See Cipher feedback (CFB) mode
Change Cipher Spec protocol, 553 Character marking, 110 Character strings, 235–237 Chi step function, 372, 375–376 Chinese remainder theorem (CRT),
71–73, 300, 705–706 bijection, 71 first assertion, 71–72 second assertion, 72
Chosen ciphertext attack (CCA), 90, 302, 307–308
Chosen text attack, 90 Chosen-plaintext approach, 211 Chosen-plaintext attack, 90 CHUID. See Cardholder unique
identifier (CHUID) CIA triad, 22
accountability, 23 authenticity, 23 availability, 22, 24 confidentiality, 22 high level, 23 integrity, 22, 24 low level, 23 moderate level, 23
Cipher, 86 block (See Block cipher)
design principles, 135–137 design of function F, 136–137 key schedule algorithm, 137 number of rounds, 136 SAC, 136
Cipher block chaining (CBC) mode, 216–218, 347
hash function based on, 354–355 Cipher feedback (CFB) mode, 218–220
CTR mode, 218, 222–224 encryption, 218 OFB mode, 218, 220–222 s-bit, 219 segments, 218
Cipher spec, 551 Cipher-Based Message Authentication
Code (CMAC), 400–405 Ciphertext, 86, 87, 287
plaintext transforming to, 89 Ciphertext only attack, 90 Ciphertext-stealing technique, 229 Claimant, 476 Classical encryption, 86–111, 710–713 Client write key, 551 Client write MAC secret, 551 Client/server authentication exchange, 498 Closure, 143
under multiplication, 145 Closure group, 322 Cloud auditor, 534, 535 Cloud broker, 534, 535
service aggregation, 535 service arbitrage, 535 service intermediation, 535
Cloud carrier, 534, 535 Cloud computing, 529–535
characteristics of, 530–532 broad network access, 530–531 measured service, 531 on-demand self-service, 531 rapid elasticity, 531 resource pooling, 531–532
context, 533 deployment models
community cloud, 532 hybrid cloud, 532 private cloud, 532 public cloud, 532
elements, 530–533 reference architecture, 534–535
cloud auditor, 534, 535 cloud broker, 534, 535 cloud carrier, 534, 535 cloud consumer, 534 cloud provider, 534
service models IaaS, 532 PaaS, 532 SaaS, 532
Cloud consumer, 534 Cloud provider, 534 Cloud security
addressing, 544 risks and countermeasures, 535–537
abuse and nefarious use, 536 account or service hijacking, 537 data loss or leakage, 537 insecure interfaces and APIs, 536 malicious insiders, 536 shared technology issues, 536–537 unknown risk profile, 537
as service, 541–544 CMAC. See Cipher-Based Message
Authentication Code (CMAC) Coefficient set, 151 Collision, 348 Collision resistant, 349
attacks, 351–353 brute-force attacks, 351–353
Communications channel (CC), 40
Community cloud, 532 Commutative, 144 Commutative group, 322 Commutative laws, 56 Commutative ring, 145 Commutativity of multiplication, 145 Complete mediation, 35 Composite number, 69 Composition, 370 Comprehensive email security, 625–627 Compression
function, 354 method, 551 S/MIME, 631–632
Computation resistance, MAC, 393 Computational aspects, 297–302 Computationally secure encryption
scheme, 91 Computer algebra system (CAS), 697 Computer security
availability, 22, 24 challenges, 25–26 confidentiality
data, 21 privacy, 21
definition of, 21 integrity, 22, 24
Conditioning algorithms, 273 Confidentiality, 21, 22, 24, 551
public-key cryptosystem, 289 S/MIME, 629–630
Confidentiality-related threats, 625 Configuration payload, 692 Confusion, 124–125 Congruences
properties of, 53 relation, 53 relation mod, 83–84
Congruent modulo n, 35 Connection confidentiality, 30 Connection integrity
recovery and, 30 selective-field, 30
Connection protocol, 574–578 channel mechanism, 574–575
close a channel, 575 data transfer, 575 open a new channel, 575
channel types, 575–576 direct-tcpip, 576 forwarded-tcpip, 576 session, 575 x11, 575
port forwarding, 576–578 Connectionless confidentiality, 30 Connectionless integrity, 30
selective-field, 30 Consistency, 255 Constant exponentiation time, 306 Constant polynomial, 151 Content types, 620–622
application type, 622 message type, 622 message/external-body subtype, 622 message/partial subtype, 622 message/rfc822 subtype, 622 multipart type, 621 multipart/alternative subtype, 621–622 multipart/digest subtype, 622 multipart/mixed subtype, 621 multipart/parallel subtype, 621 text type, 621
Content-Description header fields, 620 Content-ID header fields, 620 Content-Transfer-Encoding header
fields, 620 Content-Type header fields, 619 Conventional encryption, 86, 89, 289
attacking, 89 secure use of, 87
756 INDEX
Cookie exchange, 686 Counter (CTR) mode, 218, 222–224
advantages of, 223–224 hardware efficiency, 223 preprocessing, 224 provable security, 224 random access, 224 simplicity, 224 software efficiency, 224
Counter Mode-CBC MAC Protocol (CCMP), 608
CREATE_CHILD_SA exchange, 688 Credential, 476 Credential service provider (CSP), 476 CRT. See Chinese remainder theorem
(CRT) Cryptanalysis, 86, 353–354
and brute-force attack, 89–91 computationally secure, 91 types of attacks on encrypted
messages, 90 unconditionally secure, 91
compression function, 354 computational effort for, 333 MAC, 394 public-key, 294 RSA algorithm, 303 structure of secure hash code, 353
Cryptographic algorithms, 632–633 MUST, 632 and protocols, 20
asymmetric encryption, 20 authentication protocols, 20 data integrity algorithms, 20 symmetric encryption, 20
SHOULD, 632–633 Cryptographic checksum, 388 Cryptographic hash functions, 340–376,
414, 739–741 applications of, 341–346 collision resistant, 349 digital signatures, 344–345 intrusion detection, 345 message authentication, 341–344 one-way password file, 345 preimage resistant, 349 PRF, 346 PRNG, 346 properties, relationship, 350 pseudorandomness, 350 requirements and security, 348–354
brute-force attacks, 350–353 collision, 348 cryptanalysis, 353–354 preimage, 348 second preimage resistant, 349
resistance properties, 350 virus detection, 345
Cryptographic suites, 692–694 encryption, 693, 694 message authentication, 693, 694 PRF, 693, 694
Cryptographic system, 86 Cryptographically secure pseudorandom
bit generator (CSPRBG), 260 Cryptography, 86, 89
and network security block cipher projects, 699 case studies, 701 firewall projects, 701 hacking project, 698–699 laboratory exercises, 699 practical security assessments,
700–701 programming projects, 700 reading/report assignments, 702 research projects, 699–700 sage computer algebra projects,
697–698 writing assignments, 701–702
number of keys used, 89 conventional encryption, 89 secret-key, 89 single-key, 89 symmetric key, 89
plaintext, 89 block cipher, 89 stream cipher, 89
transforming plaintext to ciphertext, 89 product systems, 89
Cryptology, 86 CSPRBG. See Cryptographically secure
pseudorandom bit generator (CSPRBG)
CTR mode. See Counter (CTR) mode CTR_DRBG, 262, 264–267 Cubic equation, 323, 325, 328–329 Cyclic group, 145
D DANE. See DNS-based authentication
of named entities (DANE) Data Authentication Algorithm (DAA),
399–400 Data confidentiality, 21, 30, 31
CCMP, 608 TKIP, 607
Data encryption algorithm (DEA), 129 Data Encryption Standard (DES), 110,
127, 129–131, 284, 713–717 avalanche effect, 131–133 DAA, 399–400 decryption, 131 double, 208–210
meet-in-the-middle attack, 210 reduction to single stage, 209–210
encryption, 130–131 example, 131–133 permuted input, 130 preoutput, 131 strength of, 134–135
nature of DES algorithm, 134–135 timing attacks, 135 use of 56-Bit keys, 134
subkey, 131 triple
known-plaintext attack on, 212 with three keys, 213 with two keys, 210–213
Data integrity, 20, 22, 30–32 Data loss prevention (DLP), 542–543 Data origin authentication, 30 Data protection in the cloud, 537–541
attributes, 540 entities
client, 539 data owner, 539 server, 540 user, 539
multi-instance model, 539 multi-tenant model, 539 primary key, 540 relation, 540 tuples, 540
Database, 639–640 distributed, 640 SAD (See Security association
database (SAD)) DEA. See Data encryption algorithm
(DEA) Deciphering, 86 Decryption, 86, 292
algorithm, 87, 288 DES, 131 elliptic curve, 331–333 Feistel cipher, 126, 127–129 FPE, 233–235 signature verification, 436 tables for substitution, 122
Defense in depth, 37
Delete payload, 692 Denial of service (DoS), 29, 584 DES. See Data Encryption Standard
(DES) Deskewing algorithms, 273 Determinant, 99 Deterministic primality algorithm, 70 Device security, 587–589 DH. See Diffie-Hellman (DH) DHCP. See Dynamic Host Configuration
Protocol (DHCP) Diffie-Hellman (DH)
key exchange, 314–315 algorithm, 315–316, 685–686 analog, 331 discrete logarithm, 315 example, 734, 738–739 key exchange protocols, 317 man-in-the-middle attack, 317–318
values, 687 Diffusion, 124–125 Digital random number generator
(DRNG), 276–279 hardware architecture, 277–278
CBC-MAC/CMAC, 278 Intel DRNG logical structure, 279 Intel processor chip, 277
logical structure, 278–279 Digital Signature Algorithm (DSA), 420,
426–430 approach, 426–428 signing and verifying, 429
Digital signature key, 512 Digital signatures, 32, 286, 290, 292, 687,
741–744 attacks and forgeries
adaptive chosen message attack, 422 directed chosen message attack, 422 existential forgery, 423 generic chosen message attack, 422 key-only attack, 422 known message attack, 422 selective forgery, 423 total break, 423 universal forgery, 423
cryptographic hash functions, 344–345 definition, 420 direct, 423–424 ECDSA, 430–433 Elgamal signature scheme, 424–425 essential elements, 421 NIST digital signature algorithm,
426–430 properties, 421–422 requirements, 423 Schnorr signature scheme, 425–426 simplified examples, 345
Digrams, 96–98 Direct digital signature, 423–424 Directed chosen message attack, 422 Discrete logarithms, 73–78, 315
calculation of, 77–78 for modular arithmetic, 75–77 powers of integer, 73–75
Disk drives, 271–272 Distributed database, 640 Distribution system (DS), 590, 592, 594 Distributive laws, 56, 145 Divides, 47, 154 Divisibility, 47–48 Division algorithm, 48–49 Divisor, 47, 154 DNS Security Extensions (DNSSEC),
625, 639–643 operation, 642 resource records for, 642–643
DNS-based authentication of named entities (DANE), 625, 643–645
S/MIME, 645 SMTP, 645
INDEX 757
TLSA record, 643–644 Certificate Association Data, 644 Matching Type field, 644 Selector field, 644
DNSSEC. See DNS Security Extensions (DNSSEC)
Domain Name System (DNS), 615 database, 639–640 distributed database, 640 domain name space, 639 elements, 639 name resolution, 641 name servers, 639 operation, 640–641 resolvers, 639 variable-depth hierarchy for names,
639 Domain-Based Message Authentication,
Reporting and Conformance (DMARC), 626–627, 654–658
functional flow, 657 identifier alignment, 654 on receiver side, 655–657 reports, 658 on sender side, 655 tag and value descriptions, 656
DomainKeys Identified Mail (DKIM), 626, 648–654
deployment example, 651 email threats, 649–650
capabilities, 649–650 characteristics, 649 location, 650
functional flow, 651–654 strategy, 650–651
DoS. See Denial of service (DoS) Double encryption, 496 Dynamic biometrics, 476 Dynamic Host Configuration Protocol
(DHCP), 523
E EAP. See Extensible Authentication
Protocol (EAP) EAP authenticator, 525, 530 EAP over LAN (EAPOL)
-EAP packet, 529 packets, 528–529
body, 529 body length, 529 protocol version, 529 type, 529
-Start packet, 529 EAP peer, 525, 530 EAP-GPSK (EAP Generalized Pre-
Shared Key), 524 EAP-IKEv2, 524 EAPOL (EAP over LAN), 528 EAP-TLS (EAP Transport Layer
Security), 524 EAP-TTLS (EAP Tunneled TLS), 524 Ease of analysis, 127 Economy of mechanism, 34 EEPROM. See Electrically erasable
programmable ROM (EEPROM) Electrically erasable programmable
ROM (EEPROM), 509 Electronic codebook (ECB), 213–216
characteristic of, 214 diffusion, 216 error propagation, 215 error recovery, 215 modes of operation, 213 overhead, 215 security, 216
Electronic facial image, 512 Electronic mail security, 613–658
DANE, 625, 643–645 Secure/Multipurpose Internet Mail
Extension, 645
Simple Mail Transfer Protocol, 645 TLSA record, 643–644
DKIM, 626, 648–654 email threats, 649–650 functional flow, 651–654 strategy, 650–651
DMARC, 654–658 functional flow, 657 identifier alignment, 654 on receiver side, 655–657 reports, 658 on sender side, 655 tag and value descriptions, 656
DNSSEC, 625, 639–643 operation, 642 resource records for, 642–643
email format, 617–625 MIME, 618–625 RFC 5322, 618
email threats and comprehensive email security, 625–627
Internet mail architecture, 613–617 email components, 614–615 email protocols, 615–617
PGP, 638–639 S/MIME, 627–638
certificate processing, 637 cryptographic algorithms, 632–633 enhanced security services, 637–638 message content types, 632 messages, 633–637 operational description, 628–632
SPF, 626, 645–648 mechanisms, 647 modifiers, 647 operation, 648 on receiver side, 647–648 on sender side, 647
Elgamal cryptographic system, 318–321 Elgamal digital signature scheme,
424–425 Elliptic curve, 323 Elliptic curve arithmetic, 321–330
abelian groups, 322–323 associative, 322 closure, 322 commutative, 322 elliptic curve, 323 identity element, 322 inverse element, 322
over GF(2m), 328–330 finite field, 328 points on, 328
over real numbers, 323–325 algebraic description of addition,
325 example of, 324 geometric description of addition,
323–325 Weierstrass equation, 323
over Zp, 325–328 binary curve, 325 points on, 326 prime curve, 325
Elliptic curve cryptography (ECC), 321–322, 325
computational effort for cryptanalysis, 333
Diffie-Hellman key exchange analog, 331
encryption/decryption, 331–333 order, 331 PRNG on, 336 security of, 333–334
Elliptic Curve Digital Signature Algorithm (ECDSA)
generation and authentication, 431–433
global domain parameters, 431 key generation, 431
process involved in, 430 signing and verifying, 432
Email compatibility, 630–631 components, 614–615
ADMD, 615 DNS, 615 MDA, 615 MHS, 614 MS, 615 MSA, 615 MTA, 615 MUA, 614–615
format, 617–625 MIME, 618–625 RFC 5322, 618
protocols, 615–617 IMAP, 617 POP3, 617 SMTP, 615–617
security, 543 threats, 625–627, 649–650
capabilities, 649–650 characteristics, 649 location, 650
Encapsulating Security Payload (ESP), 666, 673–680
anti-replay service, 675–676 receiver, 676 replay attack, 675 sender, 676
encryption and authentication algorithms, 675, 678
format, 674–675 information, 669 padding, 675 protocol operation, 680 transport and tunnel modes, 676–681
Encapsulation, 36 Enciphering, 86 Encipherment, 32 Encoded message (EM) verification,
436–438 Encrypted messages, types of attacks
on, 90 chosen ciphertext, 90 chosen plaintext, 90 chosen text, 90 ciphertext only, 90 known plaintext, 90
Encrypted payload, 692 Encryption, 86, 292, 543
algorithm, 86, 286 asymmetric, 20 CFB mode, 218 classical, 710–713 conventional, 289 cryptographic suites, 693, 694 and decryption tables for substitution,
122 DES, 130–131 elliptic curve, 331–333 Feistel cipher, 126 FPE, 233–235 message (See Message encryption) public-key, 288, 289 scheme
computationally secure, 91 unconditionally secure, 91
storage requirements, 226–227 symmetric, 20 wireless security measures, 584
End-to-end encryption, 442 Enhanced nondeterministic random
number generator (ENRNG), 278 Enhanced security services, 637–638
secure mailing lists, 638 security labels, 638 signed receipts, 637 signing certificates, 638
Hiva-Network.Com
758 INDEX
ENRNG. See Enhanced nondeterministic random number generator (ENRNG)
Entropy rate, 273 Entropy source, 254
NIST CTR_DRBG, 265 TRNG, 271–272
disk drives, 271–272 sound/video input, 271
Equivalent inverse cipher, 197–199 Error control
external, 386 internal, 386
Error propagation, 215 Error recovery, 215 ESMTP. See Extended SMTP (ESMTP) ESS. See Extended service set (ESS) Euclidean algorithm, 49–52
example, 52, 717–719 extended, 59–61, 719–720 greatest common divisor, 49–50 for polynomials, 156, 163 relatively prime, 49 revisited, 58–59
Euler totient functionality built in, 709–710 Euler’s theorem, 66–67 Euler’s totient function, 65–66 Event detection, 32 Existential forgery, 423 Extended Euclidean algorithm, 719–720 Extended service set (ESS), 590, 593 Extended SMTP (ESMTP), 615 Extensible Authentication Protocol
(EAP), 523–527 authentication methods
EAP-GPSK, 524 EAP-IKEv2, 524 EAP-TLS, 524 EAP-TTLS, 524
exchanges, 524–527 fields, 525–526 -Key packet, 529 layered context, 523 -Logoff packet, 529 messages
code, 525 data, 526 identifier, 525 length, 525
methods, 524 pass-through mode, 525 payload, 692
Extensible Markup Language (XML), 506 External error control, 386
F Factor, 154 Factoring problem, 302–305 Fail-safe defaults, 34–35 Family Educational Rights and Privacy
Act (FERPA), 24 Fast software encryption/decryption, 127 Fault-based attack, 306 FCS. See Frame check sequence (FCS) Federated identity management, 502
identity federation, 504–508 identity management, 503–504
Feedback characteristic of modes of operation, 225
Feistel cipher, 123–129 confusion, 124–125 decryption, 126, 127–129 design features, 126–127 diffusion, 124–125 encryption, 126 example, 129 parameters, 126–127 structure, 121–123, 125–127
Feistel structure for FPE, 233–238 bit length, 238
character strings, 235–237 encryption and decryption, 233–235 function FK, 237–238 message length, 238 radix, 238
Fermat’s theorem, 64–65 FERPA. See Family Educational Rights
and Privacy Act (FERPA) Fields, 142, 146–147, 172
multiplicative inverse, 146 types of, 148
Fingerprint templates, 512 Finite fields, 142, 328, 717–723
abelian group, 144–145 arithmetic, 172–174
irreducible, 173 cyclic group, 145 fields, 146–147 of form GF(2n), 157–168
computational considerations, 163–165 generator, 166–168 modular polynomial arithmetic,
159–161 motivation, 157–159 multiplicative inverse, 161–163
of form GF(p), 147–150 multiplicative inverse, 149–150 order p, 147–149
groups, 143–144 polynomial arithmetic
with coefficients in Zp, 152–155 greatest common divisor, 156–157 ordinary, 151–152
rings, 145–146 Finite group, 144 FIPS PUB 199, 23 Firewall, 523, 589
projects, 701 First assertion, 71–72 Format-preserving encryption (FPE),
231–245 AES vs., 213 applications, 231–232 difficulties in designing an, 232–233 Feistel structure for, 233–238
bit length, 238 character strings, 235–237 encryption and decryption, 233–235 function FK, 237–238 message length, 238 radix, 238
motivation, 231–232 NIST methods for, 238–245
FF1 algorithm, 239–242 FF2 algorithm, 242–244 FF3 algorithm, 244–245
notation in, 236 parameters in, 236
Forward add round key transformation (AddRoundKey), 189
Forward mix column transformation (MixColumns), 186
Forward shift row transformation (ShiftRows), 185
Forward substitute byte transformation (SubBytes), 180
Forward unpredictability, 256 4-way handshake, 606 FPE. See Format-preserving encryption
(FPE) Frame check sequence (FCS), 386, 592 Frequency test, 256
G Galois/counter mode (GCM), 405–408
authentication and encryption functions, 406
message authentication code, 407 Generalized number field sieve (GNFS),
303
Generate function, 266 Generator, 145, 166–168 Generic chosen message attack, 422 GMK. See Group master key (GMK) GNFS. See Generalized number field
sieve (GNFS) Greatest common divisor, 49–50
finding, 50–52, 156–157 Group master key (GMK), 605 Group temporal key (GTK), 605 Groups, 143–144
associative, 143 closure, 143 commutative, 144 cyclic, 145 distribution, 607 finite, 144 generate, 145 generator, 145 identity element, 144 infinite, 144 inverse element, 144 keys, 605–607 order of, 144 permutation, 144
H Hacking project, 698–699 Handshake protocol
action, 557 CipherSpec
Cipher algorithm, 558 Cipher type, 558 hash size, 558 is exportable, 558 IV size, 558 key material, 558 MAC algorithm, 558
CipherSuite parameter anonymous Diffie-Hellman, 558 ephemeral Diffie-Hellman, 558 fixed Diffie-Hellman, 558 Fortezza, 558 RSA, 558
client authentication and key exchange, 559–560
certificate message, 560 ephemeral or anonymous Diffie-
Hellman, 559 fixed Diffie-Hellman, 560 Fortezza, 560 RSA, 560
finished message, 561 security capabilities, 556–558
cipher suite, 557 compression method, 557 random, 556 session ID, 556 version, 556
server authentication and key exchange, 559–560
anonymous Diffie-Hellman, 559 ephemeral Diffie-Hellman, 559 Fortezza, 559 RSA key exchange, 559
Hardware fault-based attack, 302 Hash code, 353
digital signature, 345 message authentication, 343–344 secure, general structure of, 353
Hash functions, 340, 384 attack against, 342 based on cipher block chaining,
354–355 birthday attack, 355 meet-in-the-middle-attack, 355
cryptographic, 340–376, 739–741 applications of, 341–346 brute-force attacks, 350–353 collision, 348
INDEX 759
collision resistant, 349 cryptanalysis, 353–354 digital signatures, 344–345 intrusion detection, 345 message authentication, 341–344 one-way password file, 345 preimage, 348 preimage resistant, 349 PRF, 346 PRNG, 346 properties, relationship, 350 pseudorandomness, 350 requirements and security, 348–354 resistance properties, 350 second preimage resistant, 349 virus detection, 345
keyed, 344 and message authentication codes,
258, 394–398 PRNG on, 413–414 resistance properties, 350 strong, 349 TRNG, 273–274 two simple, 346–348
Hash value, 349, 351, 356 Header fields, 619–620
Content-Description, 620 Content-ID, 620 Content-Transfer-Encoding, 620 Content-Type, 619 MIME-Version, 619
Health testing, 274–276 on conditioning function, 276 on noise source, 274–276
Hill cipher, 99–102 algorithm, 100–102 concepts from linear algebra, 99–100 determinant, 99
HMAC, 394–398 algorithm, 395–398 design objectives, 395 efficient implementation of, 397 security of, 398 structure, 396
HTTPS (HTTP over SSL), 566–567 connection closure, 567 connection initiation, 566–567
Human attack surface, 38 Hybrid cloud, 532
I IaaS. See Infrastructure as a service
(IaaS) Ideal block cipher, 121–123 Identification payload, 691 Identification string exchange, 570 Identities, 56 Identity and access management (IAM),
542 Identity element, 56, 144, 322 Identity federation, 504–508
examples, 507–508 scenarios, 507 standards, 506–507
SAML, 506 SOAP, 506 WS-Security, 506 XML, 506
Identity management system administrators, 504 attribute service, 503 authorization, 503 data consumers, 504 identity provider, 503 identity services, 503 key services, 503 management, 503 point of contact, 503 principal, 503 provisioning, 503
SSO protocol services, 503 trust services, 503
Identity provider, 503 Identity theft (MAC spoofing), 583 IEEE 802.11 wireless LAN, 589–595
association-related services, 594–595 association, 595 BSS transition, 595 disassociation, 595 ESS transition, 595 no transition, 595 reassociation, 595
MPDU format, 592 network components and architectural
model, 592–593 ESS, 593
protocol architecture, 590–592 logical link control, 592 media access control, 591–592 physical layer, 590
protocol stack, 591 services, 593–595
association-related services, 594–595 distribution of messages within a
DS, 594 terminology, 590 Wi-Fi alliance, 590
IEEE 802.11i wireless LAN security, 595–609
authentication phase, 601–603 access control approach, 601 EAP exchange, 602–603 MPDU exchange, 602
discovery phase, 599–601 MPDU exchange, 600–601 security capabilities, 600
elements of, 597 key management phase, 603–607
group key distribution, 607 group keys, 605–606 pairwise key distribution, 606–607 pairwise keys, 605
phases of operation, 596–599 authentication, 598 connection termination, 599 discovery, 598 key generation and distribution, 598 protected data transfer, 598
protected data transfer phase, 607–608
CCMP, 608 TKIP, 607–608
pseudorandom function, 608–609 services, 596
access control, 596 authentication, 596 privacy with message integrity, 596
IEEE 802.1X Port-Based NAC, 527–529 access control, 528 EAPOL, 528 terminology, 527
IKE. See Internet Key Exchange (IKE) IKEv2 Exchanges, 687–688 IMAP. See Internet Mail Access Protocol
(IMAP) Independent BSS (IBSS), 592 Indeterminate, 151 Index, 76 Infinite field, 147 Infinite group, 144 Information access threats, 42 Informational exchange, 688 Infrastructure as a service (IaaS), 532 Initialization value (IV), 675 Initialization vectors, 551 Injection of commands, 40 Inputs
ANSI X9.17 PRNG, 263 for single AES round, 190 sound/video, 271
Instructor’s Resource Center (IRC), 697 Integral domain, 146 Integration, 594 Integrity, 22, 24
data, 22 system, 22
Integrity-related threats, 625 Intel digital random number generator,
276–279 hardware architecture, 277–278 logical structure, 278–279
Internal error control, 386 International Organization for
Standardization (ISO), 44 Internet Architecture Board (IAB),
662 Internet banking server (IBS), 40 Internet Key Exchange (IKE), 666
header and payload formats, 688–692 key determination
authentication, 687 cookie exchange, 686 features, 686–687 IKEv2 Exchanges, 687–688 nonces, 687 protocol, 684–688
payload types, 689–692 requirements, 686
Internet Mail Access Protocol (IMAP), 617
Internet mail architecture, 613–617 email components, 614–615
ADMD, 615 DNS, 615 MDA, 615 MHS, 614 MS, 615 MSA, 615 MTA, 615 MUA, 614–615
email protocols, 615–617 IMAP, 617 POP3, 617 SMTP, 615–617
Internet security, 20 Internet Security Association and
Key Management Protocol (ISAKMP), 684
Internet Service Provider (ISP), 663 Internet Society (ISOC), 43 Intruder, 42–43 Intrusion detection, 345 Intrusion management, 543 Inverse add round key transformation,
189 Inverse element, 55, 144, 322 Inverse mix column transformation
(InvMixColumns), 187 Inverse shift row transformation
(InvShiftRows), 185 Inverse substitute byte transformation
(InvSubBytes), 184 Invisible ink, 110 InvMixColumns, 198–199 InvShiftRows, 198 InvSubByte, 198 Iota step function, 376 IP security (IPsec), 662–694
applications, 663–664 architecture, 669 authentication plus confidentiality,
681–682 benefits of, 664–665 destination address, 668 documents, 665–666
AH, 666 architecture, 665 cryptographic algorithms, 666 ESP, 666 IKE, 666
760 INDEX
IP security (IPsec) (Continued) ESP, 673–680
anti-replay service, 675–676 encryption and authentication
algorithms, 675, 678 format, 674–675 information, 669 padding, 675 protocol operation, 680 transport and tunnel modes,
676–680 IKE, 666
header and payload formats, 688–692
key determination protocol, 684–688
routing applications, 665 SA, 668
combinations of, 682–684 IP destination address, 668 Security Protocol Identifier, 668 SPI, 668
SAD, 668, 669–670 AH information, 669 Anti-Replay Window, 669 ESP information, 669 IPsec Protocol Mode, 670 Lifetime of this Security
Association, 669 Path MTU, 670 Sequence Counter Overflow, 669 Sequence Number Counter, 669 SPI, 669
services, 666 SPD, 668, 670–671
local and remote ports, 671 local IP address, 670 name, 671 next layer protocol, 670 remote IP address, 670
traffic processing, 671–673 inbound packets, 672–673 outbound packets, 671–672
transport and tunnel modes, 666–668 VPN scenario, 664
IPv4, 663 IPv6, 663, 667 Iris images, 512 Irreducible polynomial, 154, 173 Irreversible mapping, 121 ISAKMP. See Internet Security
Association and Key Management Protocol (ISAKMP)
IS-Box, 181, 182 Isolation, 36 ISP. See Internet Service Provider (ISP) Iteration function, 365 ITU Telecommunication Standardization
Sector (ITU-T), 43
K Keccak, 365, 367, 371, 373 KEK. See Key encryption key (KEK) Kerberos, 482–500
environmental shortcomings authentication forwarding, 495 encryption system dependence, 495 internet protocol dependence, 495 interrealm authentication, 495 message byte ordering, 495 ticket lifetime, 495
exchanges, 491 motivation, 483–484
reliable, 484 scalable, 484 secure, 484 transparent, 484
overview of, 489 principal, 493
technical deficiencies double encryption, 496 password attacks, 496 PCBC encryption, 496 session keys, 496
Version 4, 484–494 authentication dialogue, 488–490 authentication service exchange, 491 client/server authentication
exchange, 492 message exchanges, 488 protocol, 491 secure authentication dialogue,
486–488 simple authentication dialogue,
484–485 ticket-granting service exchange,
492 Version 5, 495–500
authentication dialogue, 496–498 authentication service exchange, 496 message exchanges, 497 nonce, 497 options, 497 realm, 497 ticket flags, 498–500 ticket-granting service exchange,
497 times, 497
Kerberos realm, 490, 493–494 Key
ANSI X9.17 PRNG, 263 asymmetric encryption, 288 determination protocol, 684–688 expansion algorithm, 190–192 generation, 301–302 length, 265 schedule algorithm, 137 size, 127 3DES, 210–213 unwrapping, 410–413
Key distribution center (KDC), 444–446, 478–481
Key distribution, symmetric using asymmetric encryption, 451–454
hybrid scheme, 454 secret key distribution, 453 simple secret key distribution,
451–453 using symmetric encryption, 442–451
controlling key usage, 449–451 decentralized key control, 448–449 hierarchical key control, 446 key distribution scenario, 444–446 session key lifetime, 446–447 transparent key control scheme,
447–448 Key distribution technique, 442 Key encryption key (KEK), 408 Key exchange, 292, 571
Diffie-Hellman, 314–315 algorithm, 315–316, 685–686 analog, 331 discrete logarithm, 315 example, 734, 738–739 key exchange protocols, 317 man-in-the-middle attack, 317–318
payload, 691 Key management and distribution
hierarchy, 444 public keys distribution, 454–459 public-key infrastructure, 467–469 symmetric key distribution
using asymmetric encryption, 451–454
using symmetric encryption, 442–451 X.509 certificates, 459–467
Key management key, 512 Key usage, 466 Key Wrap (KW) mode, 408–409
Key wrapping algorithm, 409–410 KEK, 408 operation for 256-bit key, 411 and unwrapping, 410–413
Keyed hash function, 344 Key-only attack, 422 Keystream, 267, 268 Known message attack, 422 Known-plaintext, 90
attack on triple Data Encryption Standard, 211, 212
Koblitz curve, 736 KW mode. See Key Wrap (KW) mode
L Lanes, 369, 370 Layering, 37 Least astonishment, 37 Least common mechanism, 36 Least privilege, 35–36 Linear algebra and matrix functionality,
704–705 Linear congruential generators, 258–259,
730–731 Local forwarding, 578 Lucifer cipher, 233
M MAC protocol data unit (MPDU), 590, 591
CRC, 592 destination MAC address, 591 exchange, 600–601
AS, 602 association, 600–601 EAP exchange, 602 network and security capability
discovery, 600 open system authentication, 600 secure key delivery, 602
format, 592 MAC Control, 591 MAC header, 591 MAC trailer, 592 MSDU, 591 source MAC address, 591
MAC service data unit (MSDU), 590, 591, 594
Mail Delivery Agent (MDA), 615 Mail Submission Agent (MSA), 615 Malicious association, 583 Management information base (MIB)
content, 666 Man-in-the-middle attacks, 317–318, 451,
452, 583 Manual key management, 684 Mapping
definition, 123 irreversible, 121 nonsingular, 121 policy, 466 reversible, 121
Mask generation function (MGF), 307, 433–434
Masquerade, 28, 383 Master key, 444, 448 Master secret, 551, 560
Diffie-Hellman, 560 RSA, 560
Master session key (MSK), 605 Matching Type field, 644 Mathematical attacks, 302 Maurer’s universal statistical test, 256 MD4, 356 MD5, 353, 365 MDA. See Mail Delivery Agent (MDA) Measured service, 531 Media access control (MAC), 591 Media gateway, 521 Meet-in-the-middle attack, 210, 355
INDEX 761
Message authentication, 341–344 attack against hash function, 342 cryptographic suites, 693, 694 functions, 383–390
hash function, 384 MAC, 388–390 message encryption, 384–388
hash code, 343–344 keyed hash function, 344 message digest, 341 requirements, 382–383
content modification, 383 destination repudiation, 383 disclosure, 382 masquerade, 383 sequence modification, 383 source repudiation, 383 timing modification, 383 traffic analysis, 383
simplified examples, 343 Message authentication code (MAC),
344, 382, 552 authenticated encryption, 402–408
CMAC, 402–405 GCM, 405–408
basic uses of, 389 on block ciphers, 399–401
CMAC, 400–401 DAA, 399–400
HMAC, 394–398 algorithm, 395–398 design objectives, 395 efficient implementation of, 397 security of, 398 structure, 396
key wrapping, 408–413 algorithm, 409–410 KEK, 408 operation for 256-bit key, 411 and unwrapping, 410–413
PRNG using, 415 requirements for, 391–393 security of, 393–394
brute-force attacks, 393–394 computation resistance, 393 cryptanalysis, 394
Message digest, 341 generation using SHA-512, 357
Message encryption, 384–388 basic uses of, 384 public-key encryption, 387–388 symmetric encryption, 384–387
external error control, 386 internal error control, 386
TCP segment, 387 Message Handling Service (MHS), 614 Message integrity, 551
CCMP, 608 TKIP, 607
Message integrity code (MIC), 607 Message length, 238 Message Store (MS), 615 Message Transfer Agent (MTA), 615 Message type, 622 Message User Agents (MUA), 614–615 Message/external-body subtype, 622 Message/partial subtype, 622 Message/rfc822 subtype, 622 MIC. See Message integrity code (MIC) Micali-Schnorr PRNG, 334 Michael, 607, 608 Miller-Rabin algorithm, 68–70
details of, 69 properties of prime numbers, 68–69
first property, 68 second property, 68
repeated use of, 70 Miller–Rabin primality test, 706–707 MIME. See Multipurpose Internet Mail
Extension (MIME)
MIME-Version header fields, 619 MixColumns, 174, 177, 200
transformation, 186–189, 205–206 Mobile device security, 585–589
cloud-based applications, 585 de-perimeterization, 586 external business requirements, 586 growing use of new devices, 585 strategy, 587–589
barrier security, 589 device security, 587–589 elements, 586 traffic security, 589
threats, 586–587 interaction with other systems, 587 lack of physical security controls,
586 location services, 587 by unknown parties, 587 untrusted content, 587 untrusted mobile devices, 586 untrusted networks, 587
Modification of messages, 28 Modular arithmetic, 53–61, 149, 157, 161
congruences, 53 congruent modulo n, 35 Euclidean algorithm
extended, 59–61 revisited, 58–59
exponentiation in, 298–299 modulus, 53 operations, 54–55 properties of, 55–58, 298
reducing k modulo n, 56 set of residues/residue classes, 56
Modular exponentiation, 707–708 built-in Sage functionality for, 709
Modular polynomial arithmetic, 159–161 Modularity, 36–37 Modulus, 53, 83 Monic polynomial, 151 Monoalphabetic ciphers, 94–97
digrams, 96 permutation, 94 relative frequency, 95, 96 substitution cipher, 94
Multi-instance model, 539 Multipart type, 621 Multipart/alternative subtype, 621–622 Multipart/digest subtype, 622 Multipart/mixed subtype, 621 Multipart/parallel subtype, 621 Multiple encryption, 208–213
double DES, 208–210 meet-in-the-middle attack, 210 reduction to single stage, 209–210
triple DES known-plaintext attack on, 212 with three keys, 213 with two keys, 210–213
Multiplication, 145–146, 164–165, 206 Multiplicative identity, 146 Multiplicative inverse, 146, 149–150,
161–163 Multipurpose Internet Mail Extension
(MIME), 618–625 canonical form, 625 content types, 620–622
application type, 622 message type, 622 message/external-body subtype, 622 message/partial subtype, 622 message/rfc822 subtype, 622 multipart type, 621 multipart/alternative subtype,
621–622 multipart/digest subtype, 622 multipart/mixed subtype, 621 multipart/parallel subtype, 621 text type, 621
header fields, 619–620 Content-Description, 620 Content-ID, 620 Content-Transfer-Encoding, 620 Content-Type, 619 MIME-Version, 619
message structure, 624 multipart example, 623 native form, 623–625 specification, 619 transfer encodings, 622–623
base64 transfer encoding, 623 quoted-printable, 623
use of, 619 Multirate padding, 366 Multi-tenant model, 539 Mutual authentication
asymmetric encryption, 500–501 remote user-authentication principles,
477–478 symmetric encryption, 478–481
N NAC. See Network access control (NAC) NAS. See Network access server (NAS) National Institute of Standards and
Technology (NIST), 43, 129, 172 digital signature algorithm, 426–430 for electronic user authentication,
475–476 FPE, 238–245
FF1 algorithm, 239–242 FF2 algorithm, 242–244 FF3 algorithm, 244–245
Native form, 623–625 Network access control (NAC), 520–523
context, 521 elements of
AR, 520 NAS, 521 policy server, 520
enforcement methods, 522–523 DHCP management, 523 firewall, 523 IEEE 802.1X, 522 VLANs, 522
Network access server (NAS), 521 Network attack surface, 37 Network injection attack, 584 Network security, 20, 543–544. See also
Cryptography access model, 42 basic tasks, 42 cryptography and
block cipher projects, 699 case studies, 701 firewall projects, 701 hacking project, 698–699 laboratory exercises, 699 practical security assessments,
700–701 programming projects, 700 reading/report assignments, 702 research projects, 699–700 sage computer algebra projects,
697–698 writing assignments, 701–702
model for, 41–43 secret information, 41 security-related transformation, 41 threats
information access, 42 service, 42
Next-bit test, 260 NIST. See National Institute of Standards
and Technology (NIST) NIST CTR_DRBG, 264–267
entropy source, 265 functions, 266 generate, 266
762 INDEX
NIST CTR_DRBG (Continued) initialize, 265 key length, 265 output block length, 265 parameters, 265 reseed interval, 265 seed length, 265 update, 266–267
No zero divisors, 146 Nonce, 217, 445, 478, 687
payload, 691 Non-deterministic random bit
generators (NRBGs) model, 275 Nonrepudiation, 30, 31 Nonsingular mapping, 121 Nontraditional networks, 583 Notarization, 33 Notify payload, 691 NRBGs model. See Non-deterministic
random bit generators (NRBGs) model
Number of rounds, 127, 136 Number theory, 705–710
Chinese remainder theorem, 71–73, 705–706
discrete logarithms, 73–78 calculation of, 77–78 for modular arithmetic, 75–77 powers of integer, 73–75
divisibility, 47–48 division algorithm, 48–49 Euclidean algorithm, 49–52 Euler totient functionality built in,
709–710 Euler’s theorem, 66–67 Euler’s totient function, 65–66 Fermat’s theorem, 64–65 and finite fields, 717–723 Miller–Rabin primality test, 706–707 modular arithmetic, 53–61
Euclidean algorithm revisited, 58–59 extended Euclidean algorithm,
59–61 modular arithmetic operations,
54–55 modulus, 53 properties of, 55–58 properties of congruences, 53
modular exponentiation, 707–709 primality, testing for, 68–70
algorithm, 70 distribution of primes, 70 Miller-Rabin algorithm, 68–70
prime numbers, 61–64
O OAEP. See Optimal asymmetric
encryption padding (OAEP) Oakley Key Determination Protocol,
684 OFB mode. See Output feedback (OFB)
mode On-demand self-service, 531 One-time pad, 105–106 One-way authentication
asymmetric encryption, 501–502 remote user-authentication principles,
478 symmetric encryption, 482
One-way function, 293 One-way password file, 345 Open design, 35 Open Shortest Path First (OSPF), 665 Optimal asymmetric encryption padding
(OAEP), 307–308 Order, 70, 74 Order of group, 144 Ordinary polynomial arithmetic, 151–152 OSI security architecture
ITU-T3 Recommendation X.800, 26
security attack, 26 security mechanism, 26 security service, 26 threats and attacks, 27
OSPF. See Open Shortest Path First (OSPF) Output, 264 Output block length, 265 Output feedback (OFB) mode, 218,
220–222
P PaaS. See Platform as a service (PaaS) Pairwise master key (PMK), 605 Pairwise transient key (PTK), 605 Parameters
SHA, 356 SHA-3, 369
Passive attack, 27 release of message contents, 27 traffic analysis, 27
Password attacks, 496 Path MTU, 670 Peer certificate, 550 Peer entity authentication, 29, 30 Perfect secrecy, 106 Permutation, 94, 125, 131, 144 Permuted input, 130 Personal identification number (PIN),
476, 511 Personal identity verification (PIV)
authentication, 512–514 authentication key, 512 card application administration key,
512 card issuance and management
subsystem, 509 credentials and keys, 511–512 documentation, 510–511 FIPS 201 PIV system model, 510 front-end subsystem, 509 system model, 509–510
PGP. See Pretty Good Privacy (PGP) Pi step function, 374–375 PIN. See Personal identification number
(PIN) Pin punctures, 110 PIV. See Personal identity verification
(PIV) PKI. See Public key infrastructure (PKI) Plaintext, 86, 286
transforming to ciphertext, 89 Plaintext–ciphertext pair, 210 Platform as a service (PaaS), 532 Playfair cipher, 97–99
monarchy, 97 plaintext, 98 relative frequency of letters, 98
PMK. See Pairwise master key (PMK) Point at infinity, 323 Policy mappings, 466 Policy server, 520, 521 Pollard rho method, 333 Polyalphabetic ciphers, 102–105
autokey system, 104 one-time pad, 105–106 polyalphabetic substitution cipher, 102 substitution cipher, 102 Vernam cipher, 104–105 Vigenère cipher, 102–104
Polynomial arithmetic
coefficient set, 151 with coefficients in Zp, 152–155 constant polynomial, 151 Euclidean algorithm for, 156, 163 examples of, 153, 155 greatest common divisor, 156–157 indeterminate, 151 modular, 159–161 monic polynomial, 151
ordinary, 151–152 treatment of, 151
with coefficients in GF(28), 203–206 ring, 152 root of, 166
Port, 576–578 Post Office Protocol (POP3), 617 Practical security assessments, 700–701 Preimage, 348
attacks, 351 brute-force attacks, 351
Preimage resistant, 349 Preoutput, 131 Pre-shared key (PSK), 605 Pretty Good Privacy (PGP), 638–639 PRF. See Pseudorandom function (PRF) Primality, testing for, 68–70
algorithm, 70 distribution of primes, 70 Miller–Rabin algorithm, 68–70, 706–707
details of, 69 repeated use of, 70 two properties of prime numbers,
68–69 Prime curve, 325 Prime number, 61–64, 149, 158 Prime polynomial, 154 Primitive root, 74, 314 Privacy, 21 Private cloud, 532 Private keys, 287, 288
certificate, 285 RSA algorithm, 300–301
Private-key usage period, 466 PRNG. See Pseudorandom number
generator (PRNG) Product cipher, 123, 124 Product systems, 89 Programming projects, 700 Propagating cipher block chaining
(PCBC) encryption, 496 Pseudorandom function (PRF), 253–255,
334, 346, 413 cryptographic suites, 693, 694
Pseudorandom number generator (PRNG), 253–255, 346, 729–731
Blum Blum Shub generator, 260–261, 729–730
on elliptic curve cryptography, 336 on hash function, 413–414 linear congruential generators,
258–259, 730–731 MAC function, 415 next-bit test, 260 principles of, 252–258
algorithm design, 257–258 requirements, 255–257 TRNGs, PRNGs, and PRFs, 253–255 use of random numbers, 252–253
randomness, 255–256 consistency, 255 frequency test, 256 Maurer’s universal statistical test, 256 runs test, 256 scalability, 255 uniformity, 255
requirements, 255–257 on RSA, 334–335 seed requirements, 256–257 unpredictability
backward, 256 forward, 256
using block cipher, 261–267 ANSI X9.17 PRNG, 263–264 mechanisms, 262 NIST CTR_DRBG, 264–267
Pseudorandom numbers, 253 Pseudorandomness, 350 PSK. See Pre-shared key (PSK) Psychological acceptability, 36
INDEX 763
PTK. See Pairwise transient key (PTK) Public cloud, 532 Public key infrastructure (PKI), 285, 513 Public keys, 287, 288, 454–459
(asymmetric) cryptographic algorithm, 285
authority, 456–457 certificates, 457–459 cryptanalysis, 294 public announcement of, 454–455 publicly available directory, 455–456 RSA algorithm, 299–300 uncontrolled distribution, 455
Public-key cryptography, 284, 731–739 applications for, 291–292
decryption, 292 digital signature, 292 encryption, 292 key exchange, 292
authentication, 290, 291 characteristics, 286 ciphertext, 287 confidentiality, 289 conventional encryption, 289 decryption algorithm, 288 digital signature, 290 encryption algorithm, 286 misconception, 284 plaintext, 286 principles of, 285–294 public and private keys, 287, 288 public-key cryptanalysis, 294 public-key encryption, 286–288 requirements for, 292–294
one-way function, 293 trap-door one-way function, 293
secrecy, 291 secret key, 288
Public-key encryption, 286–289, 387–388, 687
Public-key infrastructure X.509 (PKIX) elements
certification authority, 467 CRL issuer, 467 end entity, 467 registration authority, 467 repository, 467
management functions certification, 468 cross certification, 469 initialization, 468 key pair recovery, 468–469 key pair update, 469 registration, 468 revocation request, 469
management protocols, 469 Publicly available directory, 455–456 Purpose-built algorithms, 257 Puzzle for Inspector Morse, 111
Q Quoted-printable transfer encodings, 623
R Radix, 236, 238 Rail fence cipher, 107 Random delay, 306 Random numbers
generator, 218, 254 randomness, 252–253
independence, 252 uniform distribution, 252
unpredictability, 253 use of, 252–253
Randomization approach, 253 Randomness, 252–253
characteristics, 255 consistency, 255 frequency test, 256 independence, 252
Maurer’s universal statistical test, 256 runs test, 256 scalability, 255 tests, 256 uniform distribution, 252 uniformity, 255
Rapid elasticity, 531 RAS. See Remote access server (RAS) RC4, 268–271
initialization of S, 269 stream generation, 270 strength of, 271
Read-only memory (ROM), 509 Realm, 493 Registration authority (RA), 475 Relatively prime, 49, 57, 148 Release of message contents, 27 Relying party (RP), 476 Relying subsystem, 510 Remote access server (RAS), 521 Remote forwarding, 578 Remote user-authentication
principles challenge/response, 478 identification step, 474 mutual authentication, 477–478 NIST model, 475–476 one-way authentication, 478 something the individual does
(dynamic biometrics), 476 something the individual is (static
biometrics), 476 something the individual knows, 476 something the individual possesses,
476 verification step, 474
using asymmetric encryption mutual authentication, 500–501 one-way authentication, 501–502
using symmetric encryption mutual authentication, 478–481 one-way authentication, 482
Repetition Count Test, 275–276 Replay, 28 Replay attack, 477, 675 Research projects, 699–700 Reseed interval, 265–267 Residue, 49, 83, 148, 161 Residue classes, 56, 84 Resource pooling, 531–532 Resource records (RRs), 639, 640 Reversible mapping, 121 RFC 4686, 649–650
capabilities, 649–650 characteristics, 649 location, 650
RFC 5322, 618 Rho step function, 373–374 Rijndael, 172, 174, 185, 192, 199 Rings, 145–146
associativity of multiplication, 145 closure under multiplication, 145 commutativity of multiplication, 145 distributive laws, 145 integral domain, 146 multiplicative identity, 146 no zero divisors, 146
Rivest-Shamir-Adleman (RSA) algorithm, 286, 294–308, 731–734
computational aspects, 297–302 exponentiation in modular
arithmetic, 298–299 key generation, 301–302 private key, 300–301 public key, 299–300
description of, 295–297 example of, 297 PRNG on, 334–335 processing of multiple blocks, 298 security of, 302–308
brute force attacks, 302 CCA, 302, 307–308 factoring problem, 302–305 fault-based attack, 306 hardware fault-based attack, 302 mathematical attacks, 302 MIPS-years needed to factor, 304 OAEP, 307–308 timing attacks, 302, 305–306
Robust Security Network (RSN), 596 Rotor machines, 108–110
DES, 110 multiple cylinders, 109 single-cylinder system, 109 with wiring represented by numbered
contacts, 109 Round, 131, 132
constants in SHA-3, 376 function, 125, 127, 360–364
Routing control, 33 RRs. See Resource records (RRs) RSA algorithm. See Rivest-Shamir-
Adleman (RSA) algorithm RSA-PSS digital signature algorithm
mask generation function, 433–434 signature verification, 436–438 signing operation, 434–436
RSN. See Robust Security Network (RSN)
Runs test, 256
S SaaS. See Software as a service (SaaS) Sage computer algebra projects, 697–698 SAML. See Security Assertion Markup
Language (SAML) S-Box, 180, 181, 185 Scalability, 255 Schnorr digital signature scheme, 425–426 SecaaS. See Security as a service (SecaaS) Second assertion, 72 Second preimage
attacks, 351 brute-force attacks, 351
Second preimage resistant, 349 Secret information, 41 Secret key, 86, 288 Secret-key encryption, 89 Secure Hash Algorithm (SHA), 355–364 Secure shell (SSH)
connection protocol, 574–578 transport layer security, 567–578 User Authentication Protocol, 573–574
Secure/Multipurpose Internet Mail Extension (S/MIME), 625, 627–638
certificate processing, 637 cryptographic algorithms, 632–633
MUST, 632 SHOULD, 632–633
DANE, 645 enhanced security services, 637–638
secure mailing lists, 638 security labels, 638 signed receipts, 637 signing certificates, 638
messages, 633–637 certificates-only message, 637 clear signing, 636 compressedData, 632 envelopedData, 632, 634–635 registration request, 636–637 signedData, 632, 635–636
operational description, 628–632 authentication, 628–630 compression, 631–632 confidentiality, 629–630 email compatibility, 630–631
simplified functional flow, 631 user agent role, 637
764 INDEX
Security as a service (SecaaS), 541 Security Assertion Markup Language
(SAML), 506 Security assessments, 543 Security association database (SAD),
668, 669–670 AH information, 669 Anti-Replay Window, 669 ESP information, 669 IPsec Protocol Mode, 670 Lifetime of this Security Association,
669 Path MTU, 670 Sequence Counter Overflow, 669 Sequence Number Counter, 669 SPI, 669
Security associations (SA), 668 authentication plus confidentiality,
681–682 ESP with authentication option, 681 transport adjacency, 682 transport-tunnel bundle, 682
combinations of, 682–684 IP destination address, 668 lifetime of, 669 payload
attribute, 690 proposal, 690 transform, 690
Security Protocol Identifier, 668 SPI, 668
Security attacks, 26–29 active attacks, 27–29
denial of service, 29 masquerade, 28 modification of messages, 28 replay, 28
passive attack, 27 release of message contents, 27 traffic analysis, 27
Security audit trail, 32 Security design principles
complete mediation, 35 economy of mechanism, 34 encapsulation, 36 fail-safe defaults, 34–35 isolation, 36 layering, 37 least astonishment, 37 least common mechanism, 36 least privilege, 35–36 modularity, 36–37 open design, 35 psychological acceptability, 36 separation of privilege, 35
“Security in the Internet Architecture” (RFC 1636), 662
Security information and event management (SIEM), 543
Security label, 32 Security mechanisms
cryptographic hash functions, 348–350 collision resistant, 349 preimage resistant, 349 pseudorandomness, 350 second preimage resistant, 349
ECB, 216 MAC, 393–394
based on hash functions, 398 brute-force attacks, 393–394 computation resistance, 393 cryptanalysis, 394
pervasive event detection, 32 security audit trail, 32 security label, 32 security recovery, 32 trusted functionality, 32
RSA algorithm, 302–308 brute force attacks, 302
CCA, 302, 307–308 factoring problem, 302–305 fault-based attack, 306 hardware fault-based attack, 302 mathematical attacks, 302 MIPS-years needed to factor, 304 OAEP, 307–308 timing attacks, 302, 305–306
services, 29–33 access control, 31 availability service, 32–33 data confidentiality, 30, 31 data integrity, 30, 31 nonrepudiation, 30, 31
specific access control, 32 authentication exchange, 33 data integrity, 32 digital signature, 32 encipherment, 32 notarization, 33 routing control, 33 traffic padding, 33
Security Parameter Index (SPI), 668, 669
Security policy database (SPD), 668, 670–671
local and remote ports, 671 local IP address, 670 name, 671 next layer protocol, 670 remote IP address, 670
Security policy violation, 40 Security Protocol Identifier, 668 Security recovery, 32 Security services (X.800), 29–30 Security-related transformation, 41 Seed, 254, 265
input to PRNG, 257 requirements, 256–257
Seed length, 265 Selective forgery, 423 Selective-field confidentiality, 30 Selector field, 644 Selectors, 670 Sender Policy Framework (SPF), 626,
645–648 mechanisms, 647 modifiers, 647 operation, 648 on receiver side, 647–648 on sender side, 647
Separation of privilege, 35 Sequence Counter Overflow, 669 Sequence Number Counter, 669 Sequence numbers, 498, 551 Server and client random, 551 Server write key, 551 Server write MAC secret, 551 Service aggregation, 535 Service arbitrage, 535 Service intermediation, 535 Service threats, 42 Session identifier, 550 Session key, 444, 496 Session security module (SSM),
447–448 Set of residues, 56 SHA. See Secure Hash Algorithm
(SHA) SHA-0, 356 SHA-1, 356 SHA-2, 356 SHA-3, 365–376
iteration function f, 369–376 Chi step function, 375–376 composition, 370 constants in SHA-3, 376 Iota step function, 376 Pi step function, 374–375
Rho step function, 373–374 structure of, 370–371 theta step function, 371–373
parameters, 369 sponge construction, 365–369
absorbing phase, 367, 368 bitrate, 365 capacity, 367 iteration function, 365 multirate padding, 366 simple padding, 366 sponge function input and output,
366 squeezing phase, 368
state matrix, 370 step functions in, 371
SHA-224, 356 SHA-256, 356 SHA-384, 356 SHA-512, 356
constants, 359 logic, 356–359
big-endian format, 358 step 1 append padding bits, 357 step 2 append length, 357 step 3 initialize hash buffer, 357–358 step 4 process message in 1024-bit
(128-byte) blocks, 358–359 step 5 output, 359
message digest generation using, 357 round function, 360–364
ShiftRows, 174, 177, 200 AES row and column operations, 186 forward shift row transformation, 185 inverse shift row transformation, 185
SIEM. See Security information and event management (SIEM)
Signal-hiding techniques, 584 Signature verification
decryption, 436 EM verification, 436–438
Signing operation forming the signature, 436 message encoding, 434–436
Simple Mail Transfer Protocol (SMTP), 615–617, 645
Simple Network Management Protocol Version 3 (SNMPv3), 390
Simple Object Access Protocol (SOAP), 506
Simple padding, 366 Simplified AES (S-AES), 724–729 Single sign-on (SSO), 503 Single-cylinder system, 109 Single-key encryption, 86, 89 Skew, 273 S/MIME. See Secure/Multipurpose
Internet Mail Extension (S/ MIME)
SMTP. See Simple Mail Transfer Protocol (SMTP)
Software as a service (SaaS), 532 Software attack surface, 38 Sound/video input, 271 SPD. See Security policy database (SPD) Special number field sieve (SNFS), 303,
304 Sponge construction, 365–369
absorbing phase, 367, 368 bitrate, 365 capacity, 367 iteration function, 365 multirate padding, 366 simple padding, 366 sponge function input and output, 366 squeezing phase, 368
Sponge function input and output, 366 Squeezing phase, 368 Standards, 43–44 STARTTLS, 617, 625
INDEX 765
State, 174 State array, 174 State matrix, 370 Static biometrics, 476 Steganography, 110–111
advantage, 111 character marking, 110 drawbacks, 111 invisible ink, 110 pin punctures, 110 typewriter correction ribbon, 110
Stream ciphers, 120–121, 267–268, 729–731
advantage of, 268 design considerations for, 268 processes, 89 RC4, 268–271
initialization of S, 269 stream generation, 270 strength of, 271
Stream generation, 270 Strict avalanche criterion (SAC), 136 SubBytes, 174, 180, 200 Subject key identifier, 466 Subkey, 127, 131, 498 Subscriber, 475–476 Substitute bytes, 177, 180–185
AES byte-level operations, 180 constuction of S-Box and IS-Box, 182 forward substitute byte
transformation, 180 inverse substitute byte transformation,
184 Substitution techniques, 92–106, 122,
123, 125 Caesar cipher, 92–94 Hill cipher, 99–102 monoalphabetic ciphers, 94–97 one-time pad, 105–106 playfair cipher, 97–99 polyalphabetic ciphers, 102–105
Substitution-permutation network (SPN), 125
Supplicants, 520, 525 Suppress-replay attacks, 480 Symmetric block ciphers, 258 Symmetric card authentication key, 512 Symmetric cipher model, 86–91
ciphertext, 87 cryptanalysis and brute-force attack,
89–91 attacks on encrypted messages, 90 computationally secure encryption
scheme, 91 cryptanalysis, 89 unconditionally secure encryption
scheme, 91 cryptography
keys used, 89 plaintext, processed, 89 plaintext to ciphertext, 89
decryption algorithm, 87 encryption algorithm, 86 model of symmetric cryptosystem, 88 plaintext, 86 secret key, 86 secure use of conventional encryption,
87 simplified model of symmetric
encryption, 87 Symmetric cryptosystem, 88 Symmetric encryption, 20, 86–91,
384–387 external error control, 386 internal error control, 386 remote user-authentication using
mutual authentication, 478–481 one-way authentication, 482
Symmetric key encryption, 89, 687 System integrity, 22
T Tag, 391, 656 TDEA. See Triple Data Encryption
Algorithm (TDEA) Temporal Key Integrity Protocol
(TKIP), 607–608 Text type, 621 TFC. See Traffic flow confidentiality
(TFC) Theta step function, 371–373 Threats
mobile device security, 586–587 interaction with other systems, 587 lack of physical security controls,
586 location services, 587 by unknown parties, 587 untrusted content, 587 untrusted mobile devices, 586 untrusted networks, 587
wireless network security, 583–584 Ticket, 485 Ticket-granting server (TGS), 486 Ticket-granting service exchange, 497 Time complexity, 291, 293 Timestamp, 424, 477 Timing attacks, 302, 305–306
blinding, 306 constant exponentiation time, 306 DES, 135 random delay, 306
TKIP. See Temporal Key Integrity Protocol (TKIP)
TKIP sequence counter (TSC), 608 Total break, 423 Traditional block cipher structure,
119–129 arbitrary reversible substitution
cipher, 122 block cipher, 120–121 confusion, 124–125 diffusion, 124–125 encryption and decryption tables for
substitution, 122 Feistel cipher, 123–129
block size, 126 ease of analysis, 127 fast software encryption/decryption,
127 key size, 127 number of rounds, 127 round function, 127 structure, 121–123, 125–127 subkey generation algorithm, 127
ideal block cipher, 121–123 motivation for Feistel cipher structure,
121–123 permutation, 123, 125 reversible or nonsingular, 121 round function, 125 SPN, 125 stream cipher, 120–121 substitution, 123, 125
Traffic analysis, 27 Traffic flow confidentiality (TFC), 675 Traffic padding, 33 Traffic processing, IP security (IPsec),
671–673 inbound packets, 672–673 outbound packets, 671–672
Traffic security, 589 Traffic Selector payload, 692 Traffic-flow confidentiality, 30 Transfer encodings, 622–623
base64 transfer encoding, 623 quoted-printable, 623
Transformation functions, AES, 179–190 AddRoundKey transformation
forward add round key transformation, 189
inputs for single AES round, 190 inverse add round key
transformation, 189 MixColumns transformation, 186–189
forward mix column transformation, 186
inverse mix column transformation, 187
ShiftRows transformation AES row and column operations, 186 forward shift row transformation, 185 inverse shift row transformation, 185
substitute bytes transformation, 180–185 AES byte-level operations, 180 constuction of S-Box and IS-Box, 182 forward substitute byte
transformation, 180 inverse substitute byte
transformation, 184 Transport layer protocol
host keys, 568–569 key generation, 572–573 packet exchange, 569–572
algorithm negotiation, 571 key exchange, 571 message authentication code, 570 packet length, 569 padding length, 569 payload, 569 random padding, 570
Transport Layer Security (TLS), 546–578
alert protocol, 554–555 architecture, 549–551
cipher spec, 551 compression method, 551 connection, 550 is resumable, 551 master secret, 551 peer certificate, 550 session, 550 session identifier, 550
attacks categories, 564–565 TLSv1.3, 565–566
Change Cipher Spec protocol, 553 connection state
client write key, 551 client write MAC secret, 551 initialization vectors, 551 sequence numbers, 551 server and client random, 551 server write key, 551 server write MAC secret, 551
cryptographic computations, 561–563 generation, 562–563 heartbeat protocol, 563–564 master secret creation, 561
handshake protocol, 556–561 HTTPS, 566–567
connection closure, 567 connection initiation, 566–567
message authentication code, 570 padding, 569 pseudorandom function, 562–563 record protocol, 551–553
compressed length (16 bits), 553 compression, 551–552 confidentiality, 551 content type (8 bits), 553 fragmentation, 551 MAC, 553 major version (8 bits), 553 message integrity, 551–553 minor version (8 bits), 553
secure shell, 567–578 connection protocol, 574–579 transport layer protocol, 568–572 user authentication protocol,
573–574
766 INDEX
Transport Layer Security (TLS) (Continued)
session state Cipher spec, 551 compression method, 551 is resumable, 551 master secret, 551 peer certificate, 550 session identifier, 550
web security considerations, 547–549 Transport modes, 666–668, 676–681 Transposition cipher, 107–108 Trap-door one-way function, 293 Triple Data Encryption Algorithm
(TDEA), 124 Triple Data Encryption Standard
(3DES) known-plaintext attack on, 212 with three keys, 213 with two keys, 210–213
True random number generator (TRNG), 253–255, 271–279
block cipher, 274 entropy sources, 271–272
disk drives, 271–272 sound/video input, 271
hash function, 273–274 health testing, 274–276
on conditioning function, 276 on noise source, 274–276
Intel digital random number generator, 276–279
hardware architecture, 277–278 logical structure, 278–279
PRNGs vs., 254 Trusted functionality, 32 TSC. See TKIP sequence counter (TSC) Tunnel modes, 574, 666–668, 676–681 Tweakable block ciphers, 225–226 Two simple hash function, 346–348 Typewriter correction ribbon, 110
U Unconditionally secure encryption
scheme, 91 Uniform distribution, 252 Uniformity, 255 Universal forgery, 423 Unpredictability, 253
backward, 256 forward, 256
Update function, 266–267 U.S. National Security Agency (NSA),
336 Use of known authenticated session, 40 User authentication
federated identity management identity federation, 504–508 identity management, 503–504
Kerberos, 482–500 motivation, 483–484 Version 4, 484–494 Version 5, 495–500
personal identity verification authentication, 512–514 credentials and keys, 511–512 documentation, 510–511 system model, 509–510
remote user-authentication principles identification step, 474 mutual authentication, 477–478
NIST model, 475–476 one-way authentication, 478 something the individual does
(dynamic biometrics), 476 something the individual is (static
biometrics), 476 something the individual knows, 476 something the individual possesses,
476 verification step, 474
using asymmetric encryption mutual authentication, 500–501 one-way authentication, 501–502
using symmetric encryption mutual authentication, 478–481 one-way authentication, 482
User Authentication Protocol authentication methods, 574 message exchange, 573–574 message types and formats, 573
User credential compromise, 40 User credential guessing, 40 User terminal and user (UT/U), 40
V Vendor ID payload, 692 Verifier, 476 Vernam cipher, 104–105 Vigenère cipher, 102–104 Virtual local area networks (VLANs),
522 Virtual private networks (VPNs), 589,
677 Virus detection, 345 Viruses, 43
W Weak collision resistant, 349 Web security, 543
considerations, 547–549 threats, 548 traffic security approaches, 548–549
Weierstrass equation, 323 WEP. See Wired Equivalent Privacy
(WEP) Wi-Fi, 582 Wi-Fi Protected Access (WPA), 590, 596 Wired Equivalent Privacy (WEP), 596,
607 Wireless network security, 582–609
components, 583 IEEE 802.11 wireless LAN, 589–595
association-related services, 594–595 MPDU format, 592 network components and
architectural model, 592–593 protocol architecture, 590–592 services, 593–595 terminology, 590 Wi-Fi alliance, 590
IEEE 802.11i wireless LAN security, 595–609
authentication phase, 601–603 discovery phase, 599–601 elements of, 597 key management phase, 603–607 phases of operation, 596–599 protected data transfer phase,
607–608 pseudorandom function, 608–609 services, 596
measures, 584–585 securing wireless access points, 584 securing wireless networks, 585 securing wireless transmissions, 584
mobile device security, 585–589 strategy, 587–589 threats, 586–587
threats, 583–584 wireless security, 582–585
Worms, 43 WPA. See Wi-Fi Protected Access
(WPA) Writing Across the Curriculum (WAC)
movement, 701 WS-Security, 506
X X.509 certificates, 459–467
certificate subject and issuer attributes issuer alternative name, 466 subject alternative name, 466 subject directory attributes, 466
certification authority forward certificates, 463 reverse certificates, 463
certification path constraints basic constraints, 467 name constraints, 467 policy constraints, 467
formats extensions, 461 issuer name, 461 issuer unique identifier, 461 period of validity, 461 serial number, 460 signature, 461 signature algorithm identifier, 460 subject name, 461 subject unique identifier, 461 subject’s public-key information,
461 version, 460
hierarchy, 464 key and policy information, 465–466
authority key identifier, 466 certificate policies, 466 key usage, 466 policy mappings, 466 private-key usage period, 466 subject key identifier, 466
revocation of, 464–465 user’s, 462–464 Version 3, 465
XML. See Extensible Markup Language (XML)
XTS-AES mode, 224–231 ciphertext-stealing technique, 229 definition, 230 feedback characteristic of modes of
operation, 225 operation on sector, 229–231 operation on single block, 227–229 storage encryption requirements,
226–227 tweakable block ciphers, 225–226
Z Zero point, 323 ZIP, 94
Hiva-Network.Com
- Cover
- Contents
- Notation
- Preface��������������
- About the Author�����������������������
- Part One: Background���������������������������
- Chapter 1 Computer and Network Security Concepts�������������������������������������������������������
- 1.1 Computer Security Concepts�������������������������������������
- 1.2 The OSI Security Architecture����������������������������������������
- 1.3 Security Attacks���������������������������
- 1.4 Security Services����������������������������
- 1.5 Security Mechanisms������������������������������
- 1.6 Fundamental Security Design Principles�������������������������������������������������
- 1.7 Attack Surfaces and Attack Trees�������������������������������������������
- 1.8 A Model for Network Security���������������������������������������
- 1.9 Standards��������������������
- 1.10 Key Terms, Review Questions, and Problems�����������������������������������������������������
- Chapter 2 Introduction to Number Theory����������������������������������������������
- 2.1 Divisibility and the Division Algorithm��������������������������������������������������
- 2.2 The Euclidean Algorithm����������������������������������
- 2.3 Modular Arithmetic�����������������������������
- 2.4 Prime Numbers������������������������
- 2.5 Fermat's and Euler's Theorems
- 2.6 Testing for Primality��������������������������������
- 2.7 The Chinese Remainder Theorem����������������������������������������
- 2.8 Discrete Logarithms������������������������������
- 2.9 Key Terms, Review Questions, and Problems����������������������������������������������������
- Appendix 2A The Meaning of Mod�������������������������������������
- Part Two: Symmetric Ciphers����������������������������������
- Chapter 3 Classical Encryption Techniques������������������������������������������������
- 3.1 Symmetric Cipher Model���������������������������������
- 3.2 Substitution Techniques����������������������������������
- 3.3 Transposition Techniques�����������������������������������
- 3.4 Rotor Machines�������������������������
- 3.5 Steganography������������������������
- 3.6 Key Terms, Review Questions, and Problems����������������������������������������������������
- Chapter 4 Block Ciphers and the Data Encryption Standard���������������������������������������������������������������
- 4.1 Traditional Block Cipher Structure���������������������������������������������
- 4.2 The Data Encryption Standard���������������������������������������
- 4.3 A DES Example������������������������
- 4.4 The Strength of DES������������������������������
- 4.5 Block Cipher Design Principles�����������������������������������������
- 4.6 Key Terms, Review Questions, and Problems����������������������������������������������������
- Chapter 5 Finite Fields������������������������������
- 5.1 Groups�����������������
- 5.2 Rings����������������
- 5.3 Fields�����������������
- 5.4 Finite Fields of the Form GF(p)������������������������������������������
- 5.5 Polynomial Arithmetic��������������������������������
- 5.6 Finite Fields of the Form GF(2n)�������������������������������������������
- 5.7 Key Terms, Review Questions, and Problems����������������������������������������������������
- Chapter 6 Advanced Encryption Standard���������������������������������������������
- 6.1 Finite Field Arithmetic����������������������������������
- 6.2 AES Structure������������������������
- 6.3 AES Transformation Functions���������������������������������������
- 6.4 AES Key Expansion����������������������������
- 6.5 An AES Example�������������������������
- 6.6 AES Implementation�����������������������������
- 6.7 Key Terms, Review Questions, and Problems����������������������������������������������������
- Appendix 6A Polynomials with Coefficients in GF(28)
- Chapter 7 Block Cipher Operation���������������������������������������
- 7.1 Multiple Encryption and Triple DES���������������������������������������������
- 7.2 Electronic Codebook������������������������������
- 7.3 Cipher Block Chaining Mode�������������������������������������
- 7.4 Cipher Feedback Mode�������������������������������
- 7.5 Output Feedback Mode�������������������������������
- 7.6 Counter Mode�����������������������
- 7.7 XTS-AES Mode for Block-Oriented Storage Devices����������������������������������������������������������
- 7.8 Format-Preserving Encryption���������������������������������������
- 7.9 Key Terms, Review Questions, and Problems����������������������������������������������������
- Chapter 8 Random Bit Generation and Stream Ciphers���������������������������������������������������������
- 8.1 Principles of Pseudorandom Number Generation�������������������������������������������������������
- 8.2 Pseudorandom Number Generators�����������������������������������������
- 8.3 Pseudorandom Number Generation Using a Block Cipher��������������������������������������������������������������
- 8.4 Stream Ciphers�������������������������
- 8.5 RC4��������������
- 8.6 True Random Number Generators����������������������������������������
- 8.7 Key Terms, Review Questions, and Problems����������������������������������������������������
- Part Three: Asymmetric Ciphers 283�����������������������������������������
- Chapter 9 Public-Key Cryptography and RSA������������������������������������������������
- 9.1 Principles of Public-Key Cryptosystems�������������������������������������������������
- 9.2 The RSA Algorithm����������������������������
- 9.3 Key Terms, Review Questions, and Problems����������������������������������������������������
- Chapter 10 Other Public-Key Cryptosystems������������������������������������������������
- 10.1 Diffie-Hellman Key Exchange���������������������������������������
- 10.2 Elgamal Cryptographic System����������������������������������������
- 10.3 Elliptic Curve Arithmetic�������������������������������������
- 10.4 Elliptic Curve Cryptography���������������������������������������
- 10.5 Pseudorandom Number Generation Based on an Asymmetric Cipher������������������������������������������������������������������������
- 10.6 Key Terms, Review Questions, and Problems�����������������������������������������������������
- Part Four: Cryptographic Data Integrity Algorithms���������������������������������������������������������
- Chapter 11 Cryptographic Hash Functions����������������������������������������������
- 11.1 Applications of Cryptographic Hash Functions��������������������������������������������������������
- 11.2 Two Simple Hash Functions�������������������������������������
- 11.3 Requirements and Security�������������������������������������
- 11.4 Hash Functions Based on Cipher Block Chaining���������������������������������������������������������
- 11.5 Secure Hash Algorithm (SHA)���������������������������������������
- 11.6 SHA-3�����������������
- 11.7 Key Terms, Review Questions, and Problems�����������������������������������������������������
- Chapter 12 Message Authentication Codes����������������������������������������������
- 12.1 Message Authentication Requirements�����������������������������������������������
- 12.2 Message Authentication Functions��������������������������������������������
- 12.3 Requirements for Message Authentication Codes���������������������������������������������������������
- 12.4 Security of MACs����������������������������
- 12.5 MACs Based on Hash Functions: HMAC����������������������������������������������
- 12.6 MACs Based on Block Ciphers: DAA and CMAC�����������������������������������������������������
- 12.7 Authenticated Encryption: CCM and GCM�������������������������������������������������
- 12.8 Key Wrapping������������������������
- 12.9 Pseudorandom Number Generation Using Hash Functions and MACs������������������������������������������������������������������������
- 12.10 Key Terms, Review Questions, and Problems������������������������������������������������������
- Chapter 13 Digital Signatures������������������������������������
- 13.1 Digital Signatures������������������������������
- 13.2 Elgamal Digital Signature Scheme��������������������������������������������
- 13.3 Schnorr Digital Signature Scheme��������������������������������������������
- 13.4 NIST Digital Signature Algorithm��������������������������������������������
- 13.5 Elliptic Curve Digital Signature Algorithm������������������������������������������������������
- 13.6 RSA-PSS Digital Signature Algorithm�����������������������������������������������
- 13.7 Key Terms, Review Questions, and Problems�����������������������������������������������������
- Part Five: Mutual Trust������������������������������
- Chapter 14 Key Management and Distribution�������������������������������������������������
- 14.1 Symmetric Key Distribution Using Symmetric Encryption�����������������������������������������������������������������
- 14.2 Symmetric Key Distribution Using Asymmetric Encryption������������������������������������������������������������������
- 14.3 Distribution of Public Keys���������������������������������������
- 14.4 X.509 Certificates������������������������������
- 14.5 Public-Key Infrastructure�������������������������������������
- 14.6 Key Terms, Review Questions, and Problems�����������������������������������������������������
- Chapter 15 User Authentication�������������������������������������
- 15.1 Remote User-Authentication Principles�������������������������������������������������
- 15.2 Remote User-Authentication Using Symmetric Encryption�����������������������������������������������������������������
- 15.3 Kerberos��������������������
- 15.4 Remote User-Authentication Using Asymmetric Encryption������������������������������������������������������������������
- 15.5 Federated Identity Management�����������������������������������������
- 15.6 Personal Identity Verification������������������������������������������
- 15.7 Key Terms, Review Questions, and Problems�����������������������������������������������������
- Part Six: Network And Internet Security����������������������������������������������
- Chapter 16 Network Access Control and Cloud Security�����������������������������������������������������������
- 16.1 Network Access Control����������������������������������
- 16.2 Extensible Authentication Protocol����������������������������������������������
- 16.3 IEEE 802.1X Port-Based Network Access Control���������������������������������������������������������
- 16.4 Cloud Computing���������������������������
- 16.5 Cloud Security Risks and Countermeasures����������������������������������������������������
- 16.6 Data Protection in the Cloud����������������������������������������
- 16.7 Cloud Security as a Service���������������������������������������
- 16.8 Addressing Cloud Computing Security Concerns��������������������������������������������������������
- 16.9 Key Terms, Review Questions, and Problems�����������������������������������������������������
- Chapter 17 Transport-Level Security������������������������������������������
- 17.1 Web Security Considerations���������������������������������������
- 17.2 Transport Layer Security������������������������������������
- 17.3 HTTPS�����������������
- 17.4 Secure Shell (SSH)������������������������������
- 17.5 Key Terms, Review Questions, and Problems�����������������������������������������������������
- Chapter 18 Wireless Network Security�������������������������������������������
- 18.1 Wireless Security�����������������������������
- 18.2 Mobile Device Security����������������������������������
- 18.3 IEEE 802.11 Wireless LAN Overview���������������������������������������������
- 18.4 IEEE 802.11i Wireless LAN Security����������������������������������������������
- 18.5 Key Terms, Review Questions, and Problems�����������������������������������������������������
- Chapter 19 Electronic Mail Security������������������������������������������
- 19.1 Internet Mail Architecture��������������������������������������
- 19.2 Email Formats�������������������������
- 19.3 Email Threats and Comprehensive Email Security����������������������������������������������������������
- 19.4 S/MIME������������������
- 19.5 Pretty Good Privacy�������������������������������
- 19.6 DNSSEC������������������
- 19.7 DNS-Based Authentication of Named Entities������������������������������������������������������
- 19.8 Sender Policy Framework�����������������������������������
- 19.9 DomainKeys Identified Mail��������������������������������������
- 19.10 Domain-Based Message Authentication, Reporting, and Conformance����������������������������������������������������������������������������
- 19.11 Key Terms, Review Questions, and Problems������������������������������������������������������
- Chapter 20 IP Security�����������������������������
- 20.1 IP Security Overview��������������������������������
- 20.2 IP Security Policy������������������������������
- 20.3 Encapsulating Security Payload������������������������������������������
- 20.4 Combining Security Associations�������������������������������������������
- 20.5 Internet Key Exchange���������������������������������
- 20.6 Cryptographic Suites��������������������������������
- 20.7 Key Terms, Review Questions, and Problems�����������������������������������������������������
- APPENDICES 696���������������������
- Appendix A Projects for Teaching Cryptography and Network Security�������������������������������������������������������������������������
- A.1 Sage Computer Algebra Projects�����������������������������������������
- A.2 Hacking Project��������������������������
- A.3 Block Cipher Projects��������������������������������
- A.4 Laboratory Exercises�������������������������������
- A.5 Research Projects����������������������������
- A.6 Programming Projects�������������������������������
- A.7 Practical Security Assessments�����������������������������������������
- A.8 Firewall Projects����������������������������
- A.9 Case Studies�����������������������
- A.10 Writing Assignments�������������������������������
- A.11 Reading/Report Assignments��������������������������������������
- A.12 Discussion Topics�����������������������������
- Appendix B Sage Examples�������������������������������
- B.1 Linear Algebra and Matrix Functionality��������������������������������������������������
- B.2 Chapter 2: Number Theory�����������������������������������
- B.3 Chapter 3: Classical Encryption������������������������������������������
- B.4 Chapter 4: Block Ciphers and the Data Encryption Standard��������������������������������������������������������������������
- B.5 Chapter 5: Basic Concepts in Number Theory and Finite Fields�����������������������������������������������������������������������
- B.6 Chapter 6: Advanced Encryption Standard��������������������������������������������������
- B.7 Chapter 8: Pseudorandom Number Generation and Stream Ciphers�����������������������������������������������������������������������
- B.8 Chapter 9: Public-Key Cryptography and RSA�����������������������������������������������������
- B.9 Chapter 10: Other Public-Key Cryptosystems�����������������������������������������������������
- B.10 Chapter 11: Cryptographic Hash Functions����������������������������������������������������
- B.11 Chapter 13: Digital Signatures������������������������������������������
- References�����������������
- Credits��������������
- Index������������
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