Elliptic curves: number theory and cryptography
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boca Raton [u.a.]
Chapman & Hall/CRC
2008
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Discrete mathematics and its applications
50 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVIII, 513 S. graph. Darst. |
| ISBN: | 1420071467 9781420071467 |
Internformat
MARC
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| 020 | |a 9781420071467 |9 978-1-4200-7146-7 | ||
| 035 | |a (OCoLC)192045762 | ||
| 035 | |a (DE-599)BSZ27862586X | ||
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| 100 | 1 | |a Washington, Lawrence C. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Elliptic curves |b number theory and cryptography |c Lawrence C. Washington |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Boca Raton [u.a.] |b Chapman & Hall/CRC |c 2008 | |
| 300 | |a XVIII, 513 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Discrete mathematics and its applications |v 50 | |
| 650 | 4 | |a Cryptography | |
| 650 | 4 | |a Curves, Elliptic | |
| 650 | 4 | |a Number theory | |
| 650 | 0 | 7 | |a Elliptische Kurve |0 (DE-588)4014487-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Zahlentheorie |0 (DE-588)4067277-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Kryptologie |0 (DE-588)4033329-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Elliptische Kurve |0 (DE-588)4014487-2 |D s |
| 689 | 0 | 1 | |a Zahlentheorie |0 (DE-588)4067277-3 |D s |
| 689 | 0 | 2 | |a Kryptologie |0 (DE-588)4033329-2 |D s |
| 689 | 0 | |5 DE-604 | |
| 830 | 0 | |a Discrete mathematics and its applications |v 50 |w (DE-604)BV023551867 |9 50 | |
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Datensatz im Suchindex
| _version_ | 1805066807625646080 |
|---|---|
| adam_text |
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
η
. 64
Exercises
. 71
3
Torsion Points
77
3.1
Torsion Points
. 77
3.2
Division Polynomials
. 80
3.3
The Weil Pairing
. 86
3.4
The Tate-Lichtenbaum 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
XVJ
4.3.2
Legendre
Symbols.
Ю4
4.3.3 Orders
of
Points.
Ю6
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
ρ
and
λ
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-Riick 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
xvu
8.7
2-Selmer
Groups; Shafarevich-Tate Groups .
236
8.8
A Noritrivial Shafarevich-Tate Group
. 239
8.9
Galois Cohomology
. 244
Exercises
. 253
9
Elliptic Curves over
С
257
9.1
Doubly Periodic Functions
. 257
9.2
Tori are Elliptic Curves
. 267
9.3
Elliptic Curves over
С
. 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
С
. 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
Vélu's
Formulas
. 392
12.4
Point Counting
. 396
12.5
Complements
. 401
Exercises
. 402
XVIII
13
Hyperelliptłc
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
Ribeťs
Proof
. 454
15.4
Sketch of Wiles's Proof
. 461
A Number Theory
471
В
Groups
477
С
Fields
481
D
Computer Packages
489
D.I
Pari
. 489
D.
2
Magma
. 492
D.3 SAGE
. 494
References
501
Index
509 |
| adam_txt |
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
η
. 64
Exercises
. 71
3
Torsion Points
77
3.1
Torsion Points
. 77
3.2
Division Polynomials
. 80
3.3
The Weil Pairing
. 86
3.4
The Tate-Lichtenbaum 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
XVJ
4.3.2
Legendre
Symbols.
Ю4
4.3.3 Orders
of
Points.
Ю6
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
ρ
and
λ
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-Riick 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
xvu
8.7
2-Selmer
Groups; Shafarevich-Tate Groups .
236
8.8
A Noritrivial Shafarevich-Tate Group
. 239
8.9
Galois Cohomology
. 244
Exercises
. 253
9
Elliptic Curves over
С
257
9.1
Doubly Periodic Functions
. 257
9.2
Tori are Elliptic Curves
. 267
9.3
Elliptic Curves over
С
. 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
С
. 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
Vélu's
Formulas
. 392
12.4
Point Counting
. 396
12.5
Complements
. 401
Exercises
. 402
XVIII
13
Hyperelliptłc
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
Ribeťs
Proof
. 454
15.4
Sketch of Wiles's Proof
. 461
A Number Theory
471
В
Groups
477
С
Fields
481
D
Computer Packages
489
D.I
Pari
. 489
D.
2
Magma
. 492
D.3 SAGE
. 494
References
501
Index
509 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Washington, Lawrence C. |
| author_facet | Washington, Lawrence C. |
| author_role | aut |
| author_sort | Washington, Lawrence C. |
| author_variant | l c w lc lcw |
| building | Verbundindex |
| bvnumber | BV023057625 |
| callnumber-first | Q - Science |
| callnumber-label | QA567 |
| callnumber-raw | QA567.2.E44 |
| callnumber-search | QA567.2.E44 |
| callnumber-sort | QA 3567.2 E44 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 180 ST 276 |
| classification_tum | MAT 145f DAT 465f |
| ctrlnum | (OCoLC)192045762 (DE-599)BSZ27862586X |
| dewey-full | 516.3/52 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/52 |
| dewey-search | 516.3/52 |
| dewey-sort | 3516.3 252 |
| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
| discipline_str_mv | Informatik Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV023057625 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:27:25Z |
| indexdate | 2024-07-20T03:24:13Z |
| institution | BVB |
| isbn | 1420071467 9781420071467 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016260894 |
| oclc_num | 192045762 |
| open_access_boolean | |
| owner | DE-703 DE-91G DE-BY-TUM DE-20 DE-19 DE-BY-UBM DE-706 |
| owner_facet | DE-703 DE-91G DE-BY-TUM DE-20 DE-19 DE-BY-UBM DE-706 |
| physical | XVIII, 513 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Chapman & Hall/CRC |
| record_format | marc |
| series | Discrete mathematics and its applications |
| 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 XVIII, 513 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Discrete mathematics and its applications 50 Cryptography Curves, Elliptic Number theory 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 Discrete mathematics and its applications 50 (DE-604)BV023551867 50 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016260894&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Washington, Lawrence C. Elliptic curves number theory and cryptography Discrete mathematics and its applications Cryptography Curves, Elliptic Number theory 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_exact_search_txtP | 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 | Cryptography Curves, Elliptic Number theory Elliptische Kurve (DE-588)4014487-2 gnd Zahlentheorie (DE-588)4067277-3 gnd Kryptologie (DE-588)4033329-2 gnd |
| topic_facet | Cryptography Curves, Elliptic Number theory Elliptische Kurve Zahlentheorie Kryptologie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016260894&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV023551867 |
| work_keys_str_mv | AT washingtonlawrencec ellipticcurvesnumbertheoryandcryptography |