Elliptic curves : number theory and cryptography / Lawrence C. Washington.

Format:

Book

Author/Creator:

Washington, Lawrence C.

Contributor:

Alumni and Friends Memorial Book Fund.

Series:

CRC Press series on discrete mathematics and its applications

Discrete mathematics and its applications

Language:

English

Subjects (All):

Curves, Elliptic.

Number theory.

Cryptography.

Physical Description:

xi, 428 pages : illustrations ; 24 cm.

Place of Publication:

Boca Raton : Chapman & Hall/CRC, [2003]

Summary:

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 Curoes: 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 of 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.

Contents:

2 The Basic Theory 9

2.1 Weierstrass Equations 9

2.2 The Group Law 12

2.3 Projective Space and the Point at Infinity 18

2.4 Proof of Associativity 20

2.4.1 The Theorems of Pappus and Pascal 32

2.5 Other Equations for Elliptic Curves 35

2.5.1 Legendre Equation 35

2.5.2 Cubic Equations 35

2.5.3 Quartic Equations 36

2.5.4 Intersection of Two Quadratic Surfaces 39

2.6 The j-invariant 41

2.7 Elliptic Curves in Characteristic 2 44

2.8 Endomorphisms 46

2.9 Singular Curves 55

2.10 Elliptic Curves mod n 59

3 Torsion Points 73

3.2 Division Polynomials 76

3.3 The Weil Pairing 82

4 Elliptic Curves over Finite Fields 89

4.2 The Frobenius Endomorphism 92

4.3 Determining the Group Order 96

4.3.1 Subfield Curves 96

4.3.2 Legendre Symbols 98

4.3.3 Orders of Points 100

4.3.4 Baby Step, Giant Step 103

4.4 A Family of Curves 105

4.5 Schoof's Algorithm 113

4.6 Supersingular Curves 120

5 The Discrete Logarithm Problem 133

5.1 The Index Calculus 134

5.2 General Attacks on Discrete Logs 136

5.2.1 Baby Step, Giant Step 136

5.2.2 Pollard's [rho] and [lambda] Methods 137

5.2.3 The Pohlig-Hellman Method 141

5.3 The MOV Attack 144

5.4 Anomalous Curves 147

5.5 The Tate-Lichtenbaum Pairing 153

5.6 Other Attacks 156

6 Elliptic Curve Cryptography 159

6.2 Diffie-Hellman Key Exchange 160

6.3 Massey-Omura Encryption 163

6.4 ElGamal Public Key Encryption 164

6.5 ElGamal Digital Signatures 165

6.6 The Digital Signature Algorithm 168

6.7 A Public Key Scheme Based on Factoring 169

6.8 A Cryptosystem Based on the Weil Pairing 173

7 Other Applications 179

7.1 Factoring Using Elliptic Curves 179

7.2 Primality Testing 184

8 Elliptic Curves over Q 189

8.1 The Torsion Subgroup. The Lutz-Nagell Theorem 189

8.2 Descent and the Weak Mordell-Weil Theorem 198

8.3 Heights and the Mordell-Weil Theorem 206

8.5 The Height Pairing 221

8.6 Fermat's Infinite Descent 222

8.7 2-Selmer Groups; Shafarevich-Tate Groups 227

8.8 A Nontrivial Shafarevich-Tate Group 229

8.9 Galois Cohomology 234

9 Elliptic Curves over C 247

9.1 Doubly Periodic Functions 247

9.2 Tori are Elliptic Curves 257

9.3 Elliptic Curves over C 262

9.4 Computing Periods 275

9.4.1 The Arithmetic-Geometric Mean 277

9.5 Division Polynomials 283

10 Complex Multiplication 295

10.1 Elliptic Curves over C 295

10.2 Elliptic Curves over Finite Fields 302

10.3 Integrality of j-invariants 306

10.4 Numerical Examples 314

10.5 Kronecker's Jugendtraum 320

11 Divisors 323

11.2 The Weil Pairing 333

11.3 The Tate-Lichtenbaum Pairing 338

11.4 Computation of the Pairings 341

11.5 Genus One Curves and Elliptic Curves 346

12 Zeta Functions 355

12.1 Elliptic Curves over Finite Fields 355

12.2 Elliptic Curves over Q 359

13 Fermat's Last Theorem 371

13.2 Galois Representations 374

13.3 Sketch of Ribet's Proof 380

13.4 Sketch of Wiles's Proof 387

A Number Theory 397

B Groups 403

C Fields 407.

Notes:

Includes bibliographical references (pages 415-423) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

1584883650

OCLC:

51667781