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

Norbert GobAutomorphism Groupsof Hyperelliptic Function FieldsVom Fachbereich Mathematikder Technischen Universit at Kaiserslauternzur Verleihung des akademischen GradesDoktor der Naturwissenschaften(Doctor rerum naturalium, Dr. rer. nat.)genehmigte DissertationGutachter:Privatdozent Dr. Andreas GuthmannProf. Dr. Wolfram DeckerDatum der Disputation:28. Januar 2004D 386For my son HendrikContentsNotation 3Introduction 5Chapter I. Hyperelliptic Function Fields 91. Elementary Notations 92. Algebraic Function Fields 93. The Riemann-Roch Theorem 124. Sub elds of Algebraic Function Fields 155. Automorphisms of Algebraic Function Fields 196. Hyperelliptic Function Fields 217. Weierstra Points 258. Cantor’s Representation of Divisor Classes 29Chapter II. Cryptographic Aspects 331. Hyperelliptic Crypto Systems 332. Attacks on HECC 363. Known Algorithms for Computing the Order of Jacobians 373.1. Zeta Functions 373.2. CM-Method 383.3. Weil Descent 393.4. AGM Method (Gaudry-Harley) 404. A Theorem by Madan and its Practical Consequences 41Chapter III. De ning Equations 491. A Simple Criterion 492. A Theorem by Lockhart 513. Basis Transformations 523.1. Relations Between the Variable Symbols 533.2.on Between the Square Roots 543.3. Putting Both Relations Together 574. Isomorphisms 60Chapter IV. Isomorphisms and Normal Forms 631. Checking for Dening Equations 631.1. Solving Over the Given Constant Field 641.2.