Preface Introduction Chapter 1. Three Motivating Problems §1.1.Fermat's Last Theorem §1.2.The Congruent Number Problem §1.3.Cryptography Chapter 2. Back to the Beginning §2.1.The Unit Circle: Real vs. Rational Points §2.2.Parametrizing the Rational Points on the Unit Circle §2.3.Finding all Pythagorean Triples §2.4.Looking for Underlying Structure: Geometry vs. Algebra §2.5.More about Points on Curves §2.6.Gathering Some Insight about Plane Curves §2.7.Additional Exercises Chapter 3. Some Elementary Number Theory §3.1.The Integers §3.2.Some Basic Properties of the Integers §3.3.Euclid's Algorithm §3.4.A First Pass at Modular Arithmetic §3.5.Elementary Cryptography:Caesar Cipher §3.6.Affine Ciphers and Linear Congruences §3.7.Systems of Congruences Chapter 4. A Second View of Modular Arithmetic: 7§n and Un §4.1.Groups and Rings §4.2.Fractions and the Notion of an Equivalence Relation §4.3.Modular Arithmetic §4.4.A Few More Comments on the Euler Totient Function §4.5.An Application to Factoring Chapter 5. Public-Key Cryptography and RSA §5.1.A Brief Overview of Cryptographic Systems §5.2 . RSA §5.3.Hash Functions §5.4.Breaking Cryptosystems and Practical RSA Security Considerations Chapter 6. A Little More Algebra §6.1.Towards a Classification of Groups §6.2.Cayley Tables §6.3.A Couple of Non-abelian Groups §6.4.Cyclic Groups and Direct Products §6.5.Fundamental Theorem of Finite Abelian Groups §6.6.Primitive Roots §6.7.Diffie-Hellman Key Exchange §6.8.E1Gamal Encryption Chapter 7. Curves in Affine and Projective Space §7.1.Affine and Projective Space §7.2.Curves in the Affine and Projective Plane §7.3.Rational Points on Curves §7.4.The Group Law for Points on an Elliptic Curve §7.5.A Formula for the Group Law on an Elliptic Curve §7.6.The Number of Points on an Elliptic Curve Chapter 8. Applications of Elliptic Curves
§8.1.Elliptic Curves and Factoring §8.2.Elliptic Curves and Cryptography §8.3.Remarks on a Post-Quantum Cryptographic World Appendix A. Deeper Results and Concluding Thoughts §A.I. The Congruent Number Problem and Tunnell's Solution §A.2.A Digression on Functions of a Complex Variable §A.3.Return to the Birch and Swinnerton-Dyer Conjecture §A.4.Elliptic Curves over C Appendix B. Answers to Selected Exercises §B.1.Chapter 2 §B.2.Chapter 3 §B.3.Chapter 4 §B.4.Chapter 5 §B.5.Chapter 6 §B.6.Chapter 7 Bibliography Index
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