April 14, 2024

Fall 2006 Thematic Program in Cryptography

Coxeter Lecture Series
September 25-27, 2006 at 3:30 p.m.
Fields Institute

Professor Gerhard Frey
Institut für Experimentelle Mathematik, Universität Duisburg-Essen

Duality Theorems in Arithmetic Geometry and Applications

September 25
Bilinear Structures in Theory and Practice

Duality theorems are at the heart of class field theory both for number fields and geometric objects like curves and abelian varieties. They relate abelian Galois extensions with invariants of the base object. In particular, class groups of rings of integers and group schemes attached to Jacobians of curves are involved in this game. Since these groups are the most popular for producing crypto primitives based on discrete logarithms (which use a priori only the Z-linear structure) they carry unavoidably a bilinear structure. In the first lecture we want to sketch the mathematical background and destructive and constructive consequences.

September 26
From Curves to Brauer Groups

In the second lecture we describe the Lichtenbaum-Tate pairing and explain why Brauer groups of local fields are closely related to torsion points of (generalized) Jacobian varieties over finite fields. This includes theoretical as well as computational aspects

September 27
Computing in Brauer Groups of Number Fields

Finally we use local classifield theory to describe the Brauer group of local fields by cyclic algebras classified by their invariant. We explain how this is related to the classical discrete logarithm in finite fields. By the celebrated Hasse-Brauer-Noether-Sequence we can globalize and find an Index-Calculus-Algorithm to determine local invariants. We shall apply this both to the discrete logarithm in finite fields and to the computation of the Euler totient function.

Thematic Program Home page

The Fields Institute Coxeter Lecture Series (CLS) brings a leading mathematician to the Institute to give a series of three lectures in the field of the current thematic program. The first talk is an overview for a general mathematical audience, postdoctoral fellows and graduate students. The other two talks are chosen, in collaboration with the organizers of the thematic program, to target specialists in the field.


Index of Fields Distinguished and Coxeter Lectures.