Elliptic Curves Number Theory and Cryptography, Second Edition

Like its bestselling predecessor, this book develops the theory of elliptic curves to provide a basis for both number theoretic and cryptographic applications. This edition now includes alternative coordinate systems and related computational issues, a more elementary treatment of the Tate???Lichtenbaum pairing, additional elliptic curve cryptosystems, and Doud??'s analytic method for computing torsion on elliptic curves over Q. It also contains new chapters on isogenies and hyperelliptic curves as well as discusses how to compute elliptic curves in some popular computer algebra systems. Basic exercises appear at the end of each chapter, with selected answers in an appendix.

Like its bestselling predecessor, Elliptic Curves: Number Theory and Cryptography, Second Edition develops the theory of elliptic curves to provide a basis for both number theoretic and cryptographic applications. With additional exercises, this edition offers more comprehensive coverage of the fundamental theory, techniques, and applications of elliptic curves. New to the Second Edition Chapters on isogenies and hyperelliptic curves A discussion of alternative coordinate systems, such as projective, Jacobian, and Edwards coordinates, along with related computational issues A more complete treatment of the Weil and Tate–Lichtenbaum pairings Doud’s analytic method for computing torsion on elliptic curves over Q An explanation of how to perform calculations with elliptic curves in several popular computer algebra systems Taking a basic approach to elliptic curves, this accessible book prepares readers to tackle more advanced problems in the field. It introduces elliptic curves over finite fields early in the text, before moving on to interesting applications, such as cryptography, factoring, and primality testing. The book also discusses the use of elliptic curves in Fermat’s Last Theorem. Relevant abstract algebra material on group theory and fields can be found in the appendices.

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 Vélu’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.