Modern Cryptography and Elliptic Curves: A Beginner’s Guide

Description

Contents......Page 3 Preface......Page 6 3 Motivating Problems......Page 12 Fermat’s Last Theorem......Page 14 The Congruent Number Problem......Page 16 Cryptography......Page 17 Back to the Beginning......Page 19 The Unit Circle: Real vs. Rational Points......Page 20 Parametrizing the Rational Points on the Unit Circle......Page 22 Finding all Pythagorean Triples......Page 26 Looking for Underlying Structure: Geometry vs. Algebra......Page 37 More about Points on Curves......Page 44 Gathering Some Insight about Plane Curves......Page 48 Additional Exercises......Page 53 Elementary Number Theory......Page 55 The Integers......Page 56 Some Basic Properties of the Integers......Page 57 Euclid’s Algorithm......Page 62 A First Pass at Modular Arithmetic......Page 66 Elementary Cryptography: Caesar Cipher......Page 73 Affine Ciphers and Linear Congruences......Page 76 Systems of Congruences......Page 80 Groups and Rings......Page 82 Fractions and the Notion of an Equivalence Relation......Page 86 Modular Arithmetic......Page 88 A Few More Comments on the Euler Totient Function......Page 102 An Application to Factoring......Page 104 Public-Key Cryptography & RSA......Page 109 A Brief Overview of Cryptographic Systems......Page 110 RSA......Page 115 Hash Functions......Page 122 Breaking Cryptosystems and Practical RSA Security Considerations......Page 131 More Algebra......Page 135 Cayley Tables......Page 136 A Couple of Non-abelian Groups......Page 139 Cyclic Groups and Direct Products......Page 142 Fundamental Theorem of Finite Abelian Groups......Page 146 Primitive Roots......Page 149 Diffie–Hellman Key Exchange......Page 151 ElGamal Encryption......Page 152 Affine and Projective Space......Page 154 Curves in the Affine and Projective Plane......Page 160 Rational Points on Curves......Page 163 The Group Law for Points on an Elliptic Curve......Page 166 A Formula for the Group Law on an Elliptic Curve......Page 186 The Number of Points on an Elliptic Curve......Page 192 Applications of Elliptic Curves......Page 195 Elliptic Curves and Factoring......Page 196 Elliptic Curves and Cryptography......Page 202 Remarks on a Post-Quantum Cryptographic World......Page 204 Congruent Number Problem & Tunnell’s Solution......Page 208 Digression on Functions of Complex Variable......Page 214 Return to Birch & Swinnerton-Dyer Conjecture......Page 216 Elliptic Curves over C......Page 217 B.1. Chapter 2......Page 223 B.2. Chapter 3......Page 235 B.3. Chapter 4......Page 237 B.4. Chapter 5......Page 240 B.5. Chapter 6......Page 242 B.6. Chapter 7......Page 245 Biblio......Page 248 Index......Page 251