Elliptic Curves: Number Theory and Cryptography

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CRC Press, May 28, 2003 - Mathematics - 440 pages
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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.
 

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Contents

Introduction
v
Exercises
8
The Basic Theory
9
22 The Group Law
12
23 Projective Space and the Point at Infinity
18
24 Proof of Associativity
20
241 The Theorems of Pappus and Pascal
32
25 Other Equations for Elliptic Curves
35
Exercises
175
Other Applications
179
72 Primality Testing
184
Exercises
187
Elliptic Curves over Q
189
82 Descent and the Weak MordellWeil Theorem
198
83 Heights and the MordellWeil Theorem
206
84 Examples
214

253 Quartic Equations
36
254 Intersection of Two Quadratic Surfaces
39
26 The jinvariant
41
27 Elliptic Curves in Characteristic 2
44
28 Endomorphisms
46
29 Singular Curves
55
210 Elliptic Curves mod n
59
Exercises
67
Torsion Points
73
32 Division Polynomials
76
33 The Weil Pairing
82
Exercises
86
Elliptic Curves over Finite Fields
89
42 The Frobenius Endomorphism
92
43 Determining the Group Order
96
432 Legendre Symbols
98
433 Orders of Points
100
434 Baby Step Giant Step
103
44 A Family of Curves
105
45 Schoofs Algorithm
113
46 Supersingular Curves
120
Exercises
130
The Discrete Logarithm Problem
133
51 The Index Calculus
134
52 General Attacks on Discrete Logs
136
522 Pollards p and A Methods
137
523 The PohligHellman Method
141
53 The MOV Attack
144
54 Anomalous Curves
147
55 The TateLichtenbaum Pairing
153
56 Other Attacks
156
Elliptic Curve Cryptography
159
62 DiffieHellman Key Exchange
160
63 MasseyOmura Encryption
163
64 ElGamal Public Key Encryption
164
65 ElGamal Digital Signatures
165
66 The Digital Signature Algorithm
168
67 A Public Key Scheme Based on Factoring
169
68 A Cryptosystem Based on the Weil Pairing
173
85 The Height Pairing
221
86 Fermats Infinite Descent
222
87 2Selmer Groups ShafarevichTate Groups
227
88 A Nontrivial ShafarevichTate Group
229
89 Galois Cohomology
234
Exercises
244
Elliptic Curves over C
247
92 Tori are Elliptic Curves
257
93 Elliptic Curves over C
262
94 Computing Periods
275
941 The ArithmeticGeometric Mean
277
95 Division Polynomials
283
Exercises
291
Complex Multiplication
295
102 Elliptic Curves over Finite Fields
302
103 Integrality of jinvariants
306
104 Numerical Examples
314
105 Kroneckers Jugendtraum
320
Exercises
321
Divisors
323
112 The Weil Pairing
333
113 The TateLichtenbaum Pairing
338
114 Computation of the Pairings
341
115 Genus One Curves and Elliptic Curves
346
Exercises
353
Zeta Functions
355
122 Elliptic Curves over Q
359
Exercises
368
Fermats Last Theorem
371
132 Galois Representations
374
133 Sketch of Ribets Proof
380
134 Sketch of Wiless Proof
387
Number Theory
397
Groups
403
Fields
407
References
415
Index
425
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