Washington Elliptic Curves

Number Theory and Cryptography, Second Edition
2. Auflage 2012
ISBN: 978-1-4200-7147-4
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 536 Seiten

Reihe: Discrete Mathematics and Its Applications

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.

Zielgruppe

Computer scientists, electrical engineers, number theorists, and cryptographers; students of mathematics, computer science, and electrical engineering.

Autoren/Hrsg.

Washington, Lawrence C.

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Zahlentheorie
• Mathematik | Informatik Mathematik Geometrie

Inhaltsverzeichnis

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.