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Neeraj Yadav

Cryptography pdf Lecture Notes

Neeraj Yadav | 25-Feb-2016 |
Introduction , Notions of Security , Pseudorandom Generators , Encryption Using Pseudorandom Functions , Encryption Using Pseudorandom Permutations , Authentication , CCA-Secure Encryption , Collision-Resistant Hash Functions , One-Way Functions and Hardcore Predicates , PRGs from One-Way Permutations , Pseudorandom Functions from PRGs , Pseudorandom Permutations from PRFs , Public-key Encryption , CPA-secure Public-Key Encryption , Signature Schemes , Signature Schemes in the Random Oracle Model , Zero Knowledge Proofs , Proofs of Knowledge and Commitment Schemes , Introduction , Notions of Security , Pseudorandom Generators , Encryption Using Pseudorandom Functions , Encryption Using Pseudorandom Permutations , Authentication , CCA-Secure Encryption , Collision-Resistant Hash Functions , One-Way Functions and Hardcore Predicates , PRGs from One-Way Permutations , Pseudorandom Functions from PRGs , Pseudorandom Permutations from PRFs , Public-key Encryption , CPA-secure Public-Key Encryption , Signature Schemes , Signature Schemes in the Random Oracle Model , Zero Knowledge Proofs , Proofs of Knowledge and Commitment Schemes ,

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1 Introduction 1
1.1 Alice, Bob, Eve, and the others . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 The Pre-history of Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Perfect Security and One-Time Pad . . . . . . . . . . . . . . . . . . . . . . 5
2 Notions of Security 7
2.1 Semantic Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Security for Multiple Encryptions: Plain Version . . . . . . . . . . . . . . . 12
2.3 Security against Chosen Plaintext Attack . . . . . . . . . . . . . . . . . . . 13
3 Pseudorandom Generators 15
3.1 Pseudorandom Generators And One-Time Encryption . . . . . . . . . . . . 15
3.2 Description of RC4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Encryption Using Pseudorandom Functions 21
4.1 Pseudorandom Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Encryption Using Pseudorandom Functions . . . . . . . . . . . . . . . . . . 22
4.3 The Randomized Counter Mode . . . . . . . . . . . . . . . . . . . . . . . . 24
5 Encryption Using Pseudorandom Permutations 27
5.1 Pseudorandom Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.1.1 Some Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.1.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2 The AES Pseudorandom Permutation . . . . . . . . . . . . . . . . . . . . . 28
5.3 Encryption Using Pseudorandom Permutations . . . . . . . . . . . . . . . . 29
5.3.1 ECB Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.3.2 CBC Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6 Authentication 31
6.1 Message Authentication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.2 Construction for Short Messages . . . . . . . . . . . . . . . . . . . . . . . . 32
6.3 Construction for Messages of Arbitrary Length . . . . . . . . . . . . . . . . 33
7 CCA-Secure Encryption 37
7.1 CBC-MAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
7.2 Combining MAC and Encryption . . . . . . . . . . . . . . . . . . . . . . . . 38
8 Collision-Resistant Hash Functions 43
8.1 Combining Encryption and Authentication . . . . . . . . . . . . . . . . . . 43
8.1.1 Encrypt-Then-Authenticate . . . . . . . . . . . . . . . . . . . . . . . 43
8.1.2 Encrypt-And-Authenticate . . . . . . . . . . . . . . . . . . . . . . . 44
8.1.3 Authenticate-Then-Encrypt . . . . . . . . . . . . . . . . . . . . . . . 44
8.2 Cryptographic Hash Functions . . . . . . . . . . . . . . . . . . . . . . . . . 45
8.2.1 Denition and Birthday Attack . . . . . . . . . . . . . . . . . . . . . 45
8.2.2 The Merkle-Damgard Transform . . . . . . . . . . . . . . . . . . . . 47
8.3 Hash Functions and Authentication . . . . . . . . . . . . . . . . . . . . . . . 49
9 One-Way Functions and Hardcore Predicates 51
9.1 One-way Functions and One-way Permutations . . . . . . . . . . . . . . . . 52
9.2 A Preview of What is Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . 53
9.3 Hard-Core Predicate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
9.4 The Goldreich-Levin Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 54
9.5 The Goldreich-Levin Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 59
9.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
10 PRGs from One-Way Permutations 63
10.1 Pseudorandom Generators from One-Way Permutations . . . . . . . . . . . 63
11 Pseudorandom Functions from PRGs 69
11.1 Pseudorandom generators evaluated on independent seeds . . . . . . . . . . 69
11.2 Construction of Pseudorandom Functions . . . . . . . . . . . . . . . . . . . 70
11.2.1 Considering a tree of small depth . . . . . . . . . . . . . . . . . . . . 71
11.2.2 Proving the security of the GGM construction . . . . . . . . . . . . 72
12 Pseudorandom Permutations from PRFs 75
12.1 Pseudorandom Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . 75
12.2 Feistel Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
12.3 The Luby-Racko Construction . . . . . . . . . . . . . . . . . . . . . . . . . 77
12.4 Analysis of the Luby-Racko Construction . . . . . . . . . . . . . . . . . . . 78
13 Public-key Encryption 85
13.1 Public-Key Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
13.2 Public Key Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
13.3 Denitions of Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
13.4 The Decision Die-Hellman Assumption . . . . . . . . . . . . . . . . . . . . 88
13.5 Decision Die Hellman and Quadratic Residues . . . . . . . . . . . . . . . 89
13.6 El Gamal Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
14 CPA-secure Public-Key Encryption 93
14.1 Hybrid Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
14.2 RSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
14.3 Trapdoor Permutations and Encryption . . . . . . . . . . . . . . . . . . . . 96
15 Signature Schemes 99
15.1 Signature Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
15.2 One-Time Signatures and Key Refreshing . . . . . . . . . . . . . . . . . . . 101
15.3 From One-Time Signatures to Fully Secure Signatures . . . . . . . . . . . . 104
16 Signature Schemes in the Random Oracle Model 109
16.1 The Hash-and-Sign Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
16.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
17 CCA Security with a Random Oracle 113
17.1 Hybrid Encryption with a Random Oracle . . . . . . . . . . . . . . . . . . . 113
17.2 Security Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
18 Zero Knowledge Proofs 119
18.1 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
18.2 The Graph Non-Isomorphism Protocol . . . . . . . . . . . . . . . . . . . . . 120
18.3 The Graph Isomorphism Protocol . . . . . . . . . . . . . . . . . . . . . . . . 122
18.4 A Simulator for the Graph Isomorphism Protocol . . . . . . . . . . . . . . . 125
19 Zero Knowledge Proofs of Quadratic Residuosity 129
19.1 The Quadratic Residuosity Problem . . . . . . . . . . . . . . . . . . . . . . 129
19.2 The Quadratic Residuosity Protocol . . . . . . . . . . . . . . . . . . . . . . 131
20 Proofs of Knowledge and Commitment Schemes 133
20.1 Proofs of Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
20.2 Uses of Zero Knowledge proofs . . . . . . . . . . . . . . . . . . . . . . . . . 134
20.3 Commitment Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
21 Zero Knowledge Proofs of 3-Colorability 139
21.1 A Protocol for 3-Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
21.2 Simulability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
21.3 Computational Zero Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . 142
21.4 Proving that the Simulation is Indistinguishable . . . . . . . . . . . . . . . 142

 

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