Elliptic curves: number theory and cryptography

Abstract

INTRODUCTION THE BASIC THEORY Weierstrass Equations The Group Law Projective Space and the Point at Infinity Proof of Associativity Other Equations for Elliptic Curves Other Coordinate Systems The j-Invariant Elliptic Curves in Characteristic 2 Endomorphisms Singular Curves Elliptic Curves mod n TORSION POINTS Torsion Points Division Polynomials The Weil Pairing The Tate-Lichtenbaum Pairing Elliptic Curves over Finite Fields Examples The Frobenius Endomorphism Determining the Group Order A Family of Curves Schoof's Algorithm Supersingular Curves The Discrete Logarithm Problem The Index Calculus General Attacks on Discrete Logs Attacks with Pairings Anomalous Curves Other Attacks Elliptic Curve Cryptography The Basic Setup Diffie-Hellman Key Exchange Massey-Omura Encryption ElGamal Public Key Encryption ElGamal Digital Signatures The Digital Signature Algorithm ECIES A Public Key Scheme Based on Factoring A Cryptosystem Based on the Weil Pairing Other Applications Factoring Using Elliptic Curves Primality Testing Elliptic Curves over Q The Torsion Subgroup: The Lutz-Nagell Theorem Descent and the Weak Mordell-Weil Theorem Heights and the Mordell-Weil Theorem Examples The Height Pairing Fermat's Infinite Descent 2-Selmer Groups Shafarevich-Tate Groups A Nontrivial Shafarevich-Tate Group Galois Cohomology Elliptic Curves over C Doubly Periodic Functions Tori Are Elliptic Curves Elliptic Curves over C Computing Periods Division Polynomials The Torsion Subgroup: Doud's Method Complex Multiplication Elliptic Curves over C Elliptic Curves over Finite Fields Integrality of j-Invariants Numerical Examples Kronecker's Jugendtraum DIVISORS Definitions and Examples The Weil Pairing The Tate-Lichtenbaum Pairing Computation of the Pairings Genus One Curves and Elliptic Curves Equivalence of the Definitions of the Pairings Nondegeneracy of the Tate-Lichtenbaum Pairing ISOGENIES The Complex Theory The Algebraic Theory Velu's Formulas Point Counting Complements Hyperelliptic Curves Basic Definitions Divisors Cantor's Algorithm The Discrete Logarithm Problem Zeta Functions Elliptic Curves over Finite Fields Elliptic Curves over Q Fermat's Last Theorem Overview Galois Representations Sketch of Ribet's Proof Sketch of Wiles's Proof APPENDIX A: NUMBER THEORY APPENDIX B: GROUPS APPENDIX C: FIELDS APPENDIX D: COMPUTER packages REFERENCES INDEX Exercises appear at the end of each chapter.