0 Introduction 1

0.1 The Meaning of the Word Cryptography 2

0.2 Symmetric Key Cryptography 2

0.3 Public Key (Asymmetric) Cryptography 5

0.4 Key Establishment 8

0.5 Cryptography — more than just Hiding Secrets 9

0.6 Standards 10

0.7 Attacks 11

1 Congruence Equations 13

1.1 Congruence Arithmetic 13

1.1.1 Computer Examples 17

1.1.2 Problems 18

1.2 The Euclidean Algorithm — Finding Inverses 19

1.2.1 Computer Examples 26

1.2.2 Problems 27

1.3 Discrete Logarithms and Diffie-Hellman Key Exchange 27

1.3.1 Computer Examples 34

1.3.2 Problems 35

1.4 Attacking the Discrete Logarithm 37

1.4.1 Computer Examples 45

1.4.2 Problems 46

2 The ElGamal Scheme 49

2.1 Primitive Roots 49

2.1.1 Computer Examples 55

2.1.2 Problems 55

2.2 The ElGamal Scheme 56

2.2.1 Computer Examples 58

2.2.2 Problems 60

2.3 Security of the ElGamal Scheme 62

2.3.1 Computer Examples 64

2.3.2 Problems 64

3 The RSA Scheme 67

3.1 Euler's Theorem 67

3.1.1 Computer Examples 71

3.1.2 Problems 71

3.2 The RSA Algorithm 71

3.2.1 Computer Examples 76

3.2.2 Problems 78

3.3 RSA Security 79

3.3.1 Computer Examples 85

3.3.2 Problems 87

3.4 Implementing RSA 88

3.4.1 Computer Examples 90

3.4.2 Problems 91

4 Elliptic Curve Cryptography 93

4.1 Elliptic Curves and Elliptic Curve Groups 93

4.1.1 Computer Examples 102

4.1.2 Problems 103

4.2 Elliptic Curve Cryptography 104

4.2.1 Computer Examples 109

4.2.2 Problems 112

4.3 The Elliptic Curve Factoring Scheme 113

4.3.1 Computer Examples 115

4.3.2 Problems 116

5 Digital Signatures 117

5.1 Hash Functions 117

5.1.1 Computer Examples 129

5.1.2 Problems 130

5.2 Digital Signature Schemes 132

5.2.1 Computer Examples 139

5.2.2 Problems 141

5.3 Attacks on Digital Signatures 142

5.3.1 Computer Examples 150

5.3.2 Problems 152

6 Primality Testing 155

6.1 Fermat's Approach and Wilson's Theorem 156

6.1.1 Computer Examples 157

6.1.2 Problems 158

6.2 The Miller-Selfridge-Rabin Primality Test 158

6.2.1 Computer Examples 164

6.2.2 Problems 164

6.3 True Primality Tests  165

6.3.1 Computer Examples 168

6.4 Mersenne Primes and the Lucas-Lehmer Test 169

6.4.1 Computer Examples 171

6.4.2 Problems 171

6.5 Primes is in P 171

6.5.1 Computer Examples 175

6.5.2 Problems 176

7 Factoring Methods 179

7.1 Fermat Again 180

7.1.1 Computer Examples 183

7.1.2 Problems 185

7.2 The Quadratic Sieve 185

7.2.1 Computer Examples 187

7.2.2 Problems 189

7.3 Pollard's p - 1 and rho Methods 189

7.3.1 Computer Examples 193

7.3.2 Problems194

7.4 Continued Fractions and Factoring194

7.4.1 Computer Examples 200

7.4.2 Problems 203

Appendix: Solutions to Problems 207

References 231

Index 236

Notation 239

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