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Chapter 1．Cryptography 1

1．Early History 1

2．The Idea of Public Key Cryptography 2

3．The RSA Cryptosystem 5

4．Diffie-Hellman and the Digital Signature Algorithm 8

5．Secret Sharing,Coin Flipping,and Time Spent on Homework 10

6．Passwords,Signatures,and Ciphers 12

7．Practical Cryptosystems and Useful Impractical Ones 13

Exercises 17

Chapter 2．Complexity of Computations 18

1．The Big-O Notation 18

Exercises 21

2．Length of Numbers 22

Exercises 23

3．Time Estimates 24

Exercises 31

4．P,NP,and NP-Completeness 34

Exercises 41

5．Promise Problems 44

6．Randomized Algorithms and Complexity Classes 45

Exercises 48

7．Some Other Complexity Classes 48

Exercises 52

Chapter 3．Algebra 53

1．Fields 53

Exercises 55

2．Finite Fields 55

Exercises 61

3．The Euclidean Algorithm for Polynomials 63

Exercises 64

4．Polynomial Rings 65

Exercises 70

5．Gr？bner Bases 70

Exercises 78

Chapter 4．Hidden Monomial Cryptosystems 80

1．The Imai-Matsumoto System 80

Exercises 86

2．Patarin's Little Dragon 87

Exercises 95

3．Systems That Might Be More Secure 96

Exercises 102

Chapter 5．Combinatorial-Algebraic Cryptosystems 103

1．History 103

2．Irrelevance of Brassard's Theorem 104

Exercises 105

3．Concrete Combinatorial-Algebraic Systems 105

Exercises 109

4．The Basic Computational Algebra Problem 111

Exercises 112

5．Cryptographic Version of Ideal Membership 112

6．Linear Algebra Attacks 113

7．Designing a Secure System 114

Chapter 6．Elliptic and Hyperelliptic Cryptosystems 117

1．Elliptic Curves 117

Exercises 129

2．Elliptic Curve Cryptosystems 131

Exercises 136

3．Elliptic Curve Analogues of Classical Number Theory Problems 137

Exercises 139

4．Cultural Background:Conjectures on Elliptic Curves and Surprising Relations with Other Problems 139

5．Hyperelliptic Curves 144

Exercises 148

6．Hyperelliptic Cryptosystems 148

Exercises 154

Appendix．An Elementary Introduction to Hyperelliptic Curves&by Alfred J.Menezes,Yi-Hong Wu,and Robert J.Zuccherato 155

1．Basic Definitions and Properties 156

2．Polynomial and Rational Functions 159

3．Zeros and Poles 161

4．Divisors 167

5．Representing Semi-Reduced Divisors 169

6．Reduced Divisors 171

7．Adding Reduced Divisors 172

Exercises 178

Answers to Exercises 179

Bibliography 193

Subject Index 201