Lecture Notes on Cryptography
Shafi Goldwasser1 Mihir Bellare2
August 2001
1
MIT Laboratory of Computer Science, 545 Technology Square, Cambridge, MA 02139, USA. E-
mail: ; Web page: http://theory.lcs.mit.edu/ shafi
2
Department of Computer Science and Engineering, Mail Code 0114, University of California
at San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA. E-mail: ; Web
page: http://www-cse.ucsd.edu/users/mihir
, Foreword
This is a set of lecture notes on cryptography compiled for 6.87s, a one week long course on cryptography
taught at MIT by Shafi Goldwasser and Mihir Bellare in the summers of 1996–2001. The notes were
formed by merging notes written for Shafi Goldwasser’s Cryptography and Cryptanalysis course at MIT with
notes written for Mihir Bellare’s Cryptography and network security course at UCSD. In addition, Rosario
Gennaro (as Teaching Assistant for the course in 1996) contributed Section 9.6, Section 11.4, Section 11.5,
and Appendix D to the notes, and also compiled, from various sources, some of the problems in Appendix E.
Cryptography is of course a vast subject. The thread followed by these notes is to develop and explain the
notion of provable security and its usage for the design of secure protocols.
Much of the material in Chapters 2, 3 and 7 is a result of scribe notes, originally taken by MIT graduate
students who attended Professor Goldwasser’s Cryptography and Cryptanalysis course over the years, and
later edited by Frank D’Ippolito who was a teaching assistant for the course in 1991. Frank also contributed
much of the advanced number theoretic material in the Appendix. Some of the material in Chapter 3 is
from the chapter on Cryptography, by R. Rivest, in the Handbook of Theoretical Computer Science.
Chapters 4, 5, 6, 8 and 10, and Sections 9.5 and 7.4.6, were written by Professor Bellare for his Cryptography
and network security course at UCSD.
All rights reserved.
Shafi Goldwasser and Mihir Bellare Cambridge, Massachusetts, August 2001.
2
, Table of Contents
1 Introduction to Modern Cryptography 11
1.1 Encryption: Historical Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Modern Encryption: A Computational Complexity Based Theory . . . . . . . . . . . . . . . . 12
1.3 A Short List of Candidate One Way Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Security Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 The Model of Adversary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Road map to Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 One-way and trapdoor functions 17
2.1 One-Way Functions: Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 One-Way Functions: Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 (Strong) One Way Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Weak One-Way Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.3 Non-Uniform One-Way Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.4 Collections Of One Way Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.5 Trapdoor Functions and Collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 In Search of Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 The Discrete Logarithm Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.2 The RSA function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.3 Connection Between The Factorization Problem And Inverting RSA . . . . . . . . . . 30
2.3.4 The Squaring Trapdoor Function Candidate by Rabin . . . . . . . . . . . . . . . . . . 30
2.3.5 A Squaring Permutation as Hard to Invert as Factoring . . . . . . . . . . . . . . . . . 34
2.4 Hard-core Predicate of a One Way Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.1 Hard Core Predicates for General One-Way Functions . . . . . . . . . . . . . . . . . . 35
2.4.2 Bit Security Of The Discrete Logarithm Function . . . . . . . . . . . . . . . . . . . . . 36
2.4.3 Bit Security of RSA and SQUARING functions . . . . . . . . . . . . . . . . . . . . . . 38
2.5 One-Way and Trapdoor Predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5.1 Examples of Sets of Trapdoor Predicates . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 Pseudo-random bit generators 41
3.0.2 Generating Truly Random bit Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 41
3
, 4 Goldwasser and Bellare
3.0.3 Generating Pseudo-Random Bit or Number Sequences . . . . . . . . . . . . . . . . . . 42
3.0.4 Provably Secure Pseudo-Random Generators: Brief overview . . . . . . . . . . . . . . 43
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 The Existence Of A Pseudo-Random Generator . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Next Bit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Examples of Pseudo-Random Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.1 Blum/Blum/Shub Pseudo-Random Generator . . . . . . . . . . . . . . . . . . . . . . . 49
4 Block ciphers and modes of operation 51
4.1 What is a block cipher? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Data Encryption Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.1 A brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.3 Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Advanced Encryption Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4 Some Modes of operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4.1 Electronic codebook mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4.2 Cipher-block chaining mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4.3 Counter mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 Key recovery attacks on block ciphers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.6 Limitations of key-recovery based security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.7 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5 Pseudo-random functions 58
5.1 Function families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 Random functions and permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3 Pseudorandom functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.4 Pseudorandom permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.4.1 PRP under CPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.4.2 PRP under CCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4.3 Relations between the notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.5 Sequences of families of PRFs and PRPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.6 Usage of PRFs and PRPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.6.1 The shared random function model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.6.2 Modeling block ciphers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.7 Example Attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.8 Security against key-recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.9 The birthday attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.10 PRFs versus PRPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.11 Constructions of PRF families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.11.1 Extending the domain size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.12 Some applications of PRFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.12.1 Cryptographically Strong Hashing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.12.2 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.12.3 Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.12.4 Identify Friend or Foe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.12.5 Private-Key Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Shafi Goldwasser1 Mihir Bellare2
August 2001
1
MIT Laboratory of Computer Science, 545 Technology Square, Cambridge, MA 02139, USA. E-
mail: ; Web page: http://theory.lcs.mit.edu/ shafi
2
Department of Computer Science and Engineering, Mail Code 0114, University of California
at San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA. E-mail: ; Web
page: http://www-cse.ucsd.edu/users/mihir
, Foreword
This is a set of lecture notes on cryptography compiled for 6.87s, a one week long course on cryptography
taught at MIT by Shafi Goldwasser and Mihir Bellare in the summers of 1996–2001. The notes were
formed by merging notes written for Shafi Goldwasser’s Cryptography and Cryptanalysis course at MIT with
notes written for Mihir Bellare’s Cryptography and network security course at UCSD. In addition, Rosario
Gennaro (as Teaching Assistant for the course in 1996) contributed Section 9.6, Section 11.4, Section 11.5,
and Appendix D to the notes, and also compiled, from various sources, some of the problems in Appendix E.
Cryptography is of course a vast subject. The thread followed by these notes is to develop and explain the
notion of provable security and its usage for the design of secure protocols.
Much of the material in Chapters 2, 3 and 7 is a result of scribe notes, originally taken by MIT graduate
students who attended Professor Goldwasser’s Cryptography and Cryptanalysis course over the years, and
later edited by Frank D’Ippolito who was a teaching assistant for the course in 1991. Frank also contributed
much of the advanced number theoretic material in the Appendix. Some of the material in Chapter 3 is
from the chapter on Cryptography, by R. Rivest, in the Handbook of Theoretical Computer Science.
Chapters 4, 5, 6, 8 and 10, and Sections 9.5 and 7.4.6, were written by Professor Bellare for his Cryptography
and network security course at UCSD.
All rights reserved.
Shafi Goldwasser and Mihir Bellare Cambridge, Massachusetts, August 2001.
2
, Table of Contents
1 Introduction to Modern Cryptography 11
1.1 Encryption: Historical Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Modern Encryption: A Computational Complexity Based Theory . . . . . . . . . . . . . . . . 12
1.3 A Short List of Candidate One Way Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Security Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 The Model of Adversary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Road map to Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 One-way and trapdoor functions 17
2.1 One-Way Functions: Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 One-Way Functions: Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 (Strong) One Way Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Weak One-Way Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.3 Non-Uniform One-Way Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.4 Collections Of One Way Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.5 Trapdoor Functions and Collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 In Search of Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 The Discrete Logarithm Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.2 The RSA function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.3 Connection Between The Factorization Problem And Inverting RSA . . . . . . . . . . 30
2.3.4 The Squaring Trapdoor Function Candidate by Rabin . . . . . . . . . . . . . . . . . . 30
2.3.5 A Squaring Permutation as Hard to Invert as Factoring . . . . . . . . . . . . . . . . . 34
2.4 Hard-core Predicate of a One Way Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.1 Hard Core Predicates for General One-Way Functions . . . . . . . . . . . . . . . . . . 35
2.4.2 Bit Security Of The Discrete Logarithm Function . . . . . . . . . . . . . . . . . . . . . 36
2.4.3 Bit Security of RSA and SQUARING functions . . . . . . . . . . . . . . . . . . . . . . 38
2.5 One-Way and Trapdoor Predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5.1 Examples of Sets of Trapdoor Predicates . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 Pseudo-random bit generators 41
3.0.2 Generating Truly Random bit Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 41
3
, 4 Goldwasser and Bellare
3.0.3 Generating Pseudo-Random Bit or Number Sequences . . . . . . . . . . . . . . . . . . 42
3.0.4 Provably Secure Pseudo-Random Generators: Brief overview . . . . . . . . . . . . . . 43
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 The Existence Of A Pseudo-Random Generator . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Next Bit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Examples of Pseudo-Random Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.1 Blum/Blum/Shub Pseudo-Random Generator . . . . . . . . . . . . . . . . . . . . . . . 49
4 Block ciphers and modes of operation 51
4.1 What is a block cipher? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Data Encryption Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.1 A brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.3 Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Advanced Encryption Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4 Some Modes of operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4.1 Electronic codebook mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4.2 Cipher-block chaining mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4.3 Counter mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 Key recovery attacks on block ciphers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.6 Limitations of key-recovery based security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.7 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5 Pseudo-random functions 58
5.1 Function families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 Random functions and permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3 Pseudorandom functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.4 Pseudorandom permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.4.1 PRP under CPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.4.2 PRP under CCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4.3 Relations between the notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.5 Sequences of families of PRFs and PRPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.6 Usage of PRFs and PRPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.6.1 The shared random function model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.6.2 Modeling block ciphers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.7 Example Attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.8 Security against key-recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.9 The birthday attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.10 PRFs versus PRPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.11 Constructions of PRF families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.11.1 Extending the domain size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.12 Some applications of PRFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.12.1 Cryptographically Strong Hashing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.12.2 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.12.3 Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.12.4 Identify Friend or Foe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.12.5 Private-Key Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
