Many Rational Points: Coding Theory and Algebraic Geometry
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Dordrecht
Springer Netherlands
2003
|
| Schriftenreihe: | Mathematics and Its Applications
564 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | 2 Triangle Groups: An Introduction 279 3 Elementary Shimura Curves 281 4 Examples of Shimura Curves 282 5 Congruence Zeta Functions 283 6 Diophantine Properties of Shimura Curves 284 7 Klein Quartic 285 8 Supersingular Points 289 Towers of Elkies 9 289 7. CRYPTOGRAPHY AND APPLICATIONS 291 1 Introduction 291 Discrete Logarithm Problem 2 291 Curves for Public-Key Cryptosystems 3 295 Hyperelliptic Curve Cryptosystems 4 297 CM-Method 5 299 6 Cryptographic Exponent 300 7 Constructive Descent 302 8 Gaudry and Harley Algorithm 306 9 Picard Jacobians 307 Drinfeld Module Based Public Key Cryptosystems 10 308 11 Drinfeld Modules and One Way Functions 308 12 Shimura's Map 309 13 Modular Jacobians of Genus 2 Curves 310 Modular Jacobian Surfaces 14 312 15 Modular Curves of Genus Two 313 16 Hecke Operators 314 8. REFERENCES 317 345 Index Xll Preface The history of counting points on curves over finite fields is very ex tensive, starting with the work of Gauss in 1801 and continuing with the work of Artin, Schmidt, Hasse and Weil in their study of curves and the related zeta functions Zx(t), where m Zx(t) = exp (2: N t ) m m 2': 1 m with N = #X(F qm). If X is a curve of genus g, Weil's conjectures m state that L(t) Zx(t) = (1 - t)(l - qt) where L(t) = rr~!l (1 - O' |
| Beschreibung: | 1 Online-Ressource (XXI, 346 p) |
| ISBN: | 9789401702515 9789048164967 |
| DOI: | 10.1007/978-94-017-0251-5 |
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Datensatz im Suchindex
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|---|---|
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| author | Hurt, Norman E. |
| author_facet | Hurt, Norman E. |
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| author_sort | Hurt, Norman E. |
| author_variant | n e h ne neh |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
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| dewey-sort | 3516.35 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-94-017-0251-5 |
| format | Electronic eBook |
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| indexdate | 2024-07-10T01:21:15Z |
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| isbn | 9789401702515 9789048164967 |
| language | English |
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| spelling | Hurt, Norman E. Verfasser aut Many Rational Points Coding Theory and Algebraic Geometry by Norman E. Hurt Dordrecht Springer Netherlands 2003 1 Online-Ressource (XXI, 346 p) txt rdacontent c rdamedia cr rdacarrier Mathematics and Its Applications 564 2 Triangle Groups: An Introduction 279 3 Elementary Shimura Curves 281 4 Examples of Shimura Curves 282 5 Congruence Zeta Functions 283 6 Diophantine Properties of Shimura Curves 284 7 Klein Quartic 285 8 Supersingular Points 289 Towers of Elkies 9 289 7. CRYPTOGRAPHY AND APPLICATIONS 291 1 Introduction 291 Discrete Logarithm Problem 2 291 Curves for Public-Key Cryptosystems 3 295 Hyperelliptic Curve Cryptosystems 4 297 CM-Method 5 299 6 Cryptographic Exponent 300 7 Constructive Descent 302 8 Gaudry and Harley Algorithm 306 9 Picard Jacobians 307 Drinfeld Module Based Public Key Cryptosystems 10 308 11 Drinfeld Modules and One Way Functions 308 12 Shimura's Map 309 13 Modular Jacobians of Genus 2 Curves 310 Modular Jacobian Surfaces 14 312 15 Modular Curves of Genus Two 313 16 Hecke Operators 314 8. REFERENCES 317 345 Index Xll Preface The history of counting points on curves over finite fields is very ex tensive, starting with the work of Gauss in 1801 and continuing with the work of Artin, Schmidt, Hasse and Weil in their study of curves and the related zeta functions Zx(t), where m Zx(t) = exp (2: N t ) m m 2': 1 m with N = #X(F qm). If X is a curve of genus g, Weil's conjectures m state that L(t) Zx(t) = (1 - t)(l - qt) where L(t) = rr~!l (1 - O' Mathematics Computational complexity Geometry, algebraic Number theory Computer engineering Algebraic Geometry Discrete Mathematics in Computer Science Number Theory Electrical Engineering Mathematik Codierungstheorie (DE-588)4139405-7 gnd rswk-swf Rationaler Punkt (DE-588)4177004-3 gnd rswk-swf Galois-Feld (DE-588)4155896-0 gnd rswk-swf Algebraische Kurve (DE-588)4001165-3 gnd rswk-swf Algebraische Kurve (DE-588)4001165-3 s 1\p DE-604 Codierungstheorie (DE-588)4139405-7 s 2\p DE-604 Galois-Feld (DE-588)4155896-0 s 3\p DE-604 Rationaler Punkt (DE-588)4177004-3 s 4\p DE-604 https://doi.org/10.1007/978-94-017-0251-5 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Hurt, Norman E. Many Rational Points Coding Theory and Algebraic Geometry Mathematics Computational complexity Geometry, algebraic Number theory Computer engineering Algebraic Geometry Discrete Mathematics in Computer Science Number Theory Electrical Engineering Mathematik Codierungstheorie (DE-588)4139405-7 gnd Rationaler Punkt (DE-588)4177004-3 gnd Galois-Feld (DE-588)4155896-0 gnd Algebraische Kurve (DE-588)4001165-3 gnd |
| subject_GND | (DE-588)4139405-7 (DE-588)4177004-3 (DE-588)4155896-0 (DE-588)4001165-3 |
| title | Many Rational Points Coding Theory and Algebraic Geometry |
| title_auth | Many Rational Points Coding Theory and Algebraic Geometry |
| title_exact_search | Many Rational Points Coding Theory and Algebraic Geometry |
| title_full | Many Rational Points Coding Theory and Algebraic Geometry by Norman E. Hurt |
| title_fullStr | Many Rational Points Coding Theory and Algebraic Geometry by Norman E. Hurt |
| title_full_unstemmed | Many Rational Points Coding Theory and Algebraic Geometry by Norman E. Hurt |
| title_short | Many Rational Points |
| title_sort | many rational points coding theory and algebraic geometry |
| title_sub | Coding Theory and Algebraic Geometry |
| topic | Mathematics Computational complexity Geometry, algebraic Number theory Computer engineering Algebraic Geometry Discrete Mathematics in Computer Science Number Theory Electrical Engineering Mathematik Codierungstheorie (DE-588)4139405-7 gnd Rationaler Punkt (DE-588)4177004-3 gnd Galois-Feld (DE-588)4155896-0 gnd Algebraische Kurve (DE-588)4001165-3 gnd |
| topic_facet | Mathematics Computational complexity Geometry, algebraic Number theory Computer engineering Algebraic Geometry Discrete Mathematics in Computer Science Number Theory Electrical Engineering Mathematik Codierungstheorie Rationaler Punkt Galois-Feld Algebraische Kurve |
| url | https://doi.org/10.1007/978-94-017-0251-5 |
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