Automorphism groups of hyperelliptic function fields [Elektronische Ressource] / Norbert Göb

Norbert Göb

Sujets

• Informatik
• Mathematics

Informations

• Publié par: technische_universitat_kaiserslautern
• Publié le: 01 janvier 2004
• Nombre de lectures: 20
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Norbert Gob
Automorphism Groups
of Hyperelliptic Function Fields
Vom Fachbereich Mathematik
der Technischen Universit at Kaiserslautern
zur Verleihung des akademischen Grades
Doktor der Naturwissenschaften
(Doctor rerum naturalium, Dr. rer. nat.)
genehmigte Dissertation
Gutachter:
Privatdozent Dr. Andreas Guthmann
Prof. Dr. Wolfram Decker
Datum der Disputation:
28. Januar 2004
D 386For my son HendrikContents
Notation 3
Introduction 5
Chapter I. Hyperelliptic Function Fields 9
1. Elementary Notations 9
2. Algebraic Function Fields 9
3. The Riemann-Roch Theorem 12
4. Sub elds of Algebraic Function Fields 15
5. Automorphisms of Algebraic Function Fields 19
6. Hyperelliptic Function Fields 21
7. Weierstra Points 25
8. Cantor’s Representation of Divisor Classes 29
Chapter II. Cryptographic Aspects 33
1. Hyperelliptic Crypto Systems 33
2. Attacks on HECC 36
3. Known Algorithms for Computing the Order of Jacobians 37
3.1. Zeta Functions 37
3.2. CM-Method 38
3.3. Weil Descent 39
3.4. AGM Method (Gaudry-Harley) 40
4. A Theorem by Madan and its Practical Consequences 41
Chapter III. De ning Equations 49
1. A Simple Criterion 49
2. A Theorem by Lockhart 51
3. Basis Transformations 52
3.1. Relations Between the Variable Symbols 53
3.2.on Between the Square Roots 54
3.3. Putting Both Relations Together 57
4. Isomorphisms 60
Chapter IV. Isomorphisms and Normal Forms 63
1. Checking for Dening Equations 63
1.1. Solving Over the Given Constant Field 64
1.2. Constructing Necessary Constant Field Extensions 75
1.3. Checking for Solutions Over the Algebraic Closure 76
2. Explicit Determination of Isomorphisms 77
3. Normal Forms 79
Chapter V. Computing the Automorphism Group 85
1. Elementary Facts on Automorphism Groups 86
2. Automorphism Groups and Associated Normal Forms 88
3. Subgroup Checking 94
12 CONTENTS
3.1. Checking for Cyclic Subgroups Whose Order is Prime to char(k) 94
3.2. Checking for Elementary Abelian char(k)-Groups 95
3.3. Checking for Semidirect Product Groups 96
4. Computing the Automorphism Group 97
5. Stoll’s Algorithm 98
Chapter VI. Computational Aspects 101
1. A Secure Jacobian 101
2. The Number of Non-Trivial Automorphism Groups 102
3. Fixed Fields and Their Class Numbers 103
3.1. Fixed Fields of C , where (n,char(k))=1 and =0 104n
3.2. Fixed Fields of C , (n,char(k))=1 and =1 106n
m3.3. Fixed Fields of C 109p
4. Eciency Considerations and Comparison to Stoll’s Algorithm 111
Conclusion 115
Appendix A. Function Fields with Nontrivial Automorphism Groups 117
1. All Imaginary Quadratic Hyperelliptic Fields of Genus 2 over F 1173
2. Allary Quadratic Hyptic Fields of Genus 3 over F 1183
3. All Imaginary Quadratic Hyperelliptic Fields of Genus 2 over F 1215
4. Random Hyperelliptic Fields of Genus 3 130
5. Random Hyperelliptic of Genus 4 131
Appendix B. Jacobian Orders of Sub elds 133
1. Fixed Fields of C , where (n,char(k))=1 and =0 133n
2. Fixed of C , (n,char(k))=1 and =1 137n
Bibliography 141
Index 145
Lebenslauf des Autors 147
Author’s Curriculum Vitae 149Notation
In this work, the following notations are used—some of them without further de-
nition.
(k,+) additive group of the eld k
A , A (A) adele spaces of FF F
k algebraic closure of k
1A alternating group of order n!n 2
Aut(F/k) eld automorphisms of F xing k
Aut(G) automorphisms of G
|M| cardinality of the set M
char(k) characteristic of the