Elliptic Curves: Number Theory and Cryptography(Discrete Mathematics and Its Applications)

Elliptic curves have played an increasingly important role in number theory and related fields over the last several decades, most notably in areas such as cryptography, factorization, and the proof of Fermat's Last Theorem. However, most books on the subject assume a rather high level of mathematical sophistication, and few are truly accessible to senior undergraduate or beginning graduate students. Assuming only a modest background in elementary number theory, groups, and fields, Elliptic Curves: Number Theory and Cryptography introduces both the cryptographic and number theoretic sides of elliptic curves, interweaving the theory of elliptic curves with their applications. The author introduces elliptic curves over finite fields early in the treatment, leading readers directly to the intriguing cryptographic applications, but the book is structured so that readers can explore the number theoretic aspects independently if desired. By side-stepping algebraic geometry in favor an approach based on basic formulas, this book clearly demonstrates how elliptic curves are used and opens the doors to higher-level studies. Elliptic Curves offers a solid introduction to the mathematics and applications of elliptic curves that well prepares its readers to tackle more advanced problems in cryptography and number theory.

Table of Contents:
INTRODUCTION Exercises THE BASIC THEORY Weierstrass Equations The Group Law Projective Space and the Point at Infinity Proof of Associativity Other Equations for Elliptic Curves The j-Invariant Elliptic Curves in Characteristic Endomorphisms Singular Curves Elliptic Curves mod n Exercises TORSION POINTS Torsion Points Division Polynomials The Weil Pairing Exercises ELLIPTIC CURVES OVER FINITE FIELDS Examples The Frobenius Endomorphism Determining the Group Order A Family of Curves Schoof's Algorithm Supersingular Curves Exercises THE DISCRETE LOGARITHM PROBLEM The Index Calculus General Attacks on Discrete Logs The MOV Attack Anomalous Curves The Tate-Lichtenbaum Pairing Other Attacks Exercises ELLIPTIC CURVE CRYPTOGRAPHY The Basic Setup Diffie-Hellman Key Exchange Massey-Omura Encryption ElGamal Public Key Encryption ElGamal Digital Signatures The Digital Signature Algorithm A Public Key Scheme Based on Factoring A Cryptosystem Based on the Weil Pairing Exercises OTHER APPLICATIONS Factoring Using Elliptic Curves Primality Testing Exercises 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 Exercises ELLIPTIC CURVES OVER C Doubly Periodic Functions Tori are Elliptic Curves Elliptic Curves over C Computing Periods Division Polynomials Exercises COMPLEX MULTIPLICATION Elliptic Curves over C Elliptic Curves over Finite Fields Integrality of j-Invariants A Numerical Example Kronecker's Jugendtraum Exercises DIVISORS Definitions and Examples The Weil Pairing The Tate-Lichtenbaum Pairing Computation of the Pairings Genus One Curves and Elliptic Curves Exercises ZETA FUNCTIONS Elliptic Curves over Finite Fields Elliptic Curves over Q Exercises FERMAT'S LAST THEOREM Overview Galois Representations Sketch of Ribet's Proof Sketch of Wiles' s Proof APPENDICES Number Theory Groups Fields REFERENCES INDEX