Elliptic curves: number theory and cryptography
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
Chapman & Hall/CRC
2008
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| Edition: | 2. ed. |
| Series: | Discrete mathematics and its applications
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| ISBN: | 1420071467 9781420071467 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Titel: Elliptic curves
Autor: Washington, Lawrence C.
Jahr: 2008
Contents
1 Introduction 1
Exercises 8
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 33
2.5 Other Equations for Elliptic Curves 35
2.5.1 Legendre Equation 35
2.5.2 Cubic Equations 36
2.5.3 Quartic Equations 37
2.5.4 Intersection of Two Quadratic Surfaces 39
2.6 Other Coordinate Systems 42
2.6.1 Projective Coordinates 42
2.6.2 Jacobian Coordinates 43
2.6.3 Edwards Coordinates 44
2.7 The j-invariant 45
2.8 Elliptic Curves in Characteristic 2 47
2.9 Endomorphisms 50
2.10 Singular Curves 59
2.11 Elliptic Curves mod n 64
Exercises 71
3 Torsion Points 77
3.1 Torsion Points 77
3.2 Division Polynomials 80
3.3 The Weil Pairing 80
3.4 The Tate-Liehtenbaum Pairing 90
Exercises 92
4 Elliptic Curves over Finite Fields 95
4.1 Examples 95
4.2 The Frobenius Endomorphism 98
4.3 Determining the Group Order 102
4.3.1 Subfield Curves 102
xv
xvi
4.3.2 Legendre Symbols 104
4.3.3 Orders of Points 106
4.3.4 Baby Step. Giant Step 112
4.4 A Family of Curves 115
4.5 Schoofs Algorithm 123
4.6 Supersingular Curves 130
Exercises 139
5 The Discrete Logarithm Problem 143
5.1 The Index Calculus 144
5.2 General Attacks on Discrete Logs 146
5.2.1 Baby Step. Giant Step 146
5.2.2 Pollard s p and A Methods 147
5.2.3 The Pohlig-Hellman Method 151
5.3 Attacks with Pairings 154
5.3.1 The MOV Attack 154
5.3.2 The Frey-Ruck Attack 157
5.4 Anomalous Curves 159
5.5 Other Attacks 165
Exercises 166
6 Elliptic Curve Cryptography 169
6.1 The Basic Setup 169
6.2 Diffie-Hellman Key Exchange 170
6.3 Massey-Omura Encryption 173
6.4 ElGamal Public Key Encryption 174
6.5 ElGamal Digital Signatures 175
6.6 The Digital Signature Algorithm 179
6.7 ECIES 180
6.8 A Public Key Scheme Based on Factoring 181
6.9 A Cryptosystem Based on the Weil Pairing 184
Exercises 187
7 Other Applications 189
7.1 Factoring Using Elliptic Curves 189
7.2 Primality Testing 194
Exercises 197
8 Elliptic Curves over Q 199
8.1 The Torsion Subgroup. The Lutz-Nagell Theorem 199
8.2 Descent and the Weak Mordell-Weil Theorem 208
8.3 Heights and the Mordell-Weil Theorem 215
8.4 Examples 223
8.5 The Height Pairing 230
8.6 Fermat s Infinite Descent 231
xvii
8.7 2-Selmer Groups; Shafarevich-Tate Groups 236
8.8 A Nontrivial Shafarevich-Tate Group 239
8.9 Galois Cohomology 244
Exercises 253
9 Elliptic Curves over C 257
9.1 Doubly Periodic Functions 257
9.2 Tori are Elliptic Curves 267
9.3 Elliptic Curves over C 272
9.4 Computing Periods 286
9.4.1 The Arithmetic-Geometric Mean 288
9.5 Division Polynomials 294
9.6 The Torsion Subgroup: Doud s Method 302
Exercises 307
10 Complex Multiplication 311
10.1 Elliptic Curves over C 311
10.2 Elliptic Curves over Finite Fields 318
10.3 Integrality of j-invariants 322
10.4 Numerical Examples 330
10.5 Kronecker s Jugendtraum 336
Exercises 337
11 Divisors 339
11.1 Definitions and Examples 339
11.2 The Weil Pairing 349
11.3 The Tate-Lichtenbaum Pairing 354
11.4 Computation of the Pairings 358
11.5 Genus One Curves and Elliptic Curves 364
11.6 Equivalence of the Definitions of the Pairings 370
11.6.1 The Weil Pairing 371
11.6.2 The Tate-Lichtenbaum Pairing 374
11.7 Nondegeneracy of the Tate-Lichtenbaum Pairing 375
Exercises 379
12 Isogenies 381
12.1 The Complex Theory 381
12.2 The Algebraic Theory 386
12.3 Velu s Formulas 392
12.4 Point Counting 396
12.5 Complements 401
Exercises 402
xviii
13 Hyperelliptic Curves 407
13.1 Basic Definitions 407
13.2 Divisors 409
13.3 Cantor s Algorithm 417
13.4 The Discrete Logarithm Problem 420
Exercises 426
14 Zeta Functions 429
14.1 Elliptic Curves over Finite Fields 429
14.2 Elliptic Curves over Q 433
Exercises 442
15 Fermat s Last Theorem 445
15.1 Overview 445
15.2 Galois Representations 448
15.3 Sketch of Ribet s Proof 454
15.4 Sketch of Wiles s Proof 461
A Number Theory 471
B Groups 477
C Fields 481
D Computer Packages 489
D.I Pari 489
D.2 Magma 492
D.3 SAGE 494
References 501
Index 509
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| author | Washington, Lawrence C. |
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| discipline | Informatik Mathematik |
| edition | 2. ed. |
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| language | English |
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| series2 | Discrete mathematics and its applications |
| spelling | Washington, Lawrence C. Verfasser aut Elliptic curves number theory and cryptography Lawrence C. Washington 2. ed. Boca Raton [u.a.] Chapman & Hall/CRC 2008 txt rdacontent n rdamedia nc rdacarrier Discrete mathematics and its applications Elliptische Kurve (DE-588)4014487-2 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Kryptologie (DE-588)4033329-2 gnd rswk-swf Elliptische Kurve (DE-588)4014487-2 s Zahlentheorie (DE-588)4067277-3 s Kryptologie (DE-588)4033329-2 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020134914&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Washington, Lawrence C. Elliptic curves number theory and cryptography Elliptische Kurve (DE-588)4014487-2 gnd Zahlentheorie (DE-588)4067277-3 gnd Kryptologie (DE-588)4033329-2 gnd |
| subject_GND | (DE-588)4014487-2 (DE-588)4067277-3 (DE-588)4033329-2 |
| title | Elliptic curves number theory and cryptography |
| title_auth | Elliptic curves number theory and cryptography |
| title_exact_search | Elliptic curves number theory and cryptography |
| title_full | Elliptic curves number theory and cryptography Lawrence C. Washington |
| title_fullStr | Elliptic curves number theory and cryptography Lawrence C. Washington |
| title_full_unstemmed | Elliptic curves number theory and cryptography Lawrence C. Washington |
| title_short | Elliptic curves |
| title_sort | elliptic curves number theory and cryptography |
| title_sub | number theory and cryptography |
| topic | Elliptische Kurve (DE-588)4014487-2 gnd Zahlentheorie (DE-588)4067277-3 gnd Kryptologie (DE-588)4033329-2 gnd |
| topic_facet | Elliptische Kurve Zahlentheorie Kryptologie |
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