Elliptic Curves: Number Theory and

Like its bestselling predecessor, Elliptic Curves: Number Theory and Cryptography, Second Editiondevelops 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 systemsTaking 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	p. 1
Exercises	p. 8
The Basic Theory	p. 9
Weierstrass Equations	p. 9
The Group Law	p. 12
Projective Space and the Point at Infinity	p. 18
Proof of Associativity	p. 20
The Theorems of Pappus and Pascal	p. 33
Other Equations for Elliptic Curves	p. 35
Legendre Equation	p. 35
Cubic Equations	p. 36
Quartic Equations	p. 37
Intersection of Two Quadratic Surfaces	p. 39
Other Coordinate Systems	p. 42
Projective Coordinates	p. 42
Jacobian Coordinates	p. 43
Edwards Coordinates	p. 44
The j-invariant	p. 45
Elliptic Curves in Characteristic 2	p. 47
Endomorphisms	p. 50
Singular Curves	p. 59
Elliptic Curves mod n	p. 64
Exercises	p. 71
Torsion Points	p. 77
Torsion Points	p. 77
Division Polynomials	p. 80
The Weil Pairing	p. 86
The Tate-Lichtenbaum Pairing	p. 90
Exercises	p. 92
Elliptic Curves over Finite Fields	p. 95
Examples	p. 95
The Frobenius Endomorphism	p. 98
Determining the Group Order	p. 102
Subfield Curves	p. 102
Legendre Symbols	p. 104
Orders of Points	p. 106
Baby Step, Giant Step	p. 112
A Family of Curves	p. 115
Schoof's Algorithm	p. 123
Supersingular Curves	p. 130
Exercises	p. 139
The Discrete Logarithm Problem	p. 143
The Index Calculus	p. 144
General Attacks on Discrete Logs	p. 146
Baby Step, Giant Step	p. 146
Pollard's [rho] and [lambda] Methods	p. 147
The Pohlig-Hellman Method	p. 151
Attacks with Pairings	p. 154
The MOV Attack	p. 154
The Frey-Ruck Attack	p. 157
Anomalous Curves	p. 159
Other Attacks	p. 165
Exercises	p. 166
Elliptic Curve Cryptography	p. 169
The Basic Setup	p. 169
Diffie-Hellman Key Exchange	p. 170
Massey-Omura Encryption	p. 173
ElGamal Public Key Encryption	p. 174
ElGamal Digital Signatures	p. 175
The Digital Signature Algorithm	p. 179
ECIES	p. 180
A Public Key Scheme Based on Factoring	p. 181
A Cryptosystem Based on the Weil Pairing	p. 184
Exercises	p. 187
Other Applications	p. 189
Factoring Using Elliptic Curves	p. 189
Primality Testing	p. 194
Exercises	p. 197
Elliptic Curves over Q	p. 199
The Torsion Subgroup. The Lutz-Nagell Theorem	p. 199
Descent and the Weak Mordell-Weil Theorem	p. 208
Heights and the Mordell-Weil Theorem	p. 215
Examples	p. 223
The Height Pairing	p. 230
Fermat's Infinite Descent	p. 231
2-Selmer Groups; Shafarevich-Tate Groups	p. 236
A Nontrivial Shafarevich-Tate Group	p. 239
Galois Cohomology	p. 244
Exercises	p. 253
Elliptic Curves over C	p. 257
Doubly Periodic Functions	p. 257
Tori are Elliptic Curves	p. 267
Elliptic Curves over C	p. 272
Computing Periods	p. 286
The Arithmetic-Geometric Mean	p. 288
Division Polynomials	p. 294
The Torsion Subgroup: Doud's Method	p. 302
Exercises	p. 307
Complex Multiplication	p. 311
Elliptic Curves over C	p. 311
Elliptic Curves over Finite Fields	p. 318
Integrality of j-invariants	p. 322
Numerical Examples	p. 330
Kronecker's Jugendtraum	p. 336
Exercises	p. 337
Divisors	p. 339
Definitions and Examples	p. 339
The Weil Pairing	p. 349
The Tate-Lichtenbaum Pairing	p. 354
Computation of the Pairings	p. 358
Genus One Curves and Elliptic Curves	p. 364
Equivalence of the Definitions of the Pairings	p. 370
The Weil Pairing	p. 371
The Tate-Lichtenbaum Pairing	p. 374
Nondegeneracy of the Tate-Lichtenbaum Pairing	p. 375
Exercises	p. 379
Isogenies	p. 381
The Complex Theory	p. 381
The Algebraic Theory	p. 386
Velu's Formulas	p. 392
Point Counting	p. 396
Complements	p. 401
Exercises	p. 402
Hyperelliptic Curves	p. 407
Basic Definitions	p. 407
Divisors	p. 409
Cantor's Algorithm	p. 417
The Discrete Logarithm Problem	p. 420
Exercises	p. 426
Zeta Functions	p. 429
Elliptic Curves over Finite Fields	p. 429
Elliptic Curves over Q	p. 433
Exercises	p. 442
Fermat's Last Theorem	p. 445
Overview	p. 445
Galois Representations	p. 448
Sketch of Ribet's Proof	p. 454
Sketch of Wiles's Proof	p. 461
Number Theory	p. 471
Groups	p. 477
Fields	p. 481
Computer Packages	p. 489
Pari	p. 489
Magma	p. 492
SAGE	p. 494
References	p. 501
Index	p. 509
Table of Contents provided by Ingram. All Rights Reserved.

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Elliptic Curves: Number Theory and Cryptography, Second Edition

Summary

Table of Contents

Supplemental Materials