Number Theory/Cryptography Seminar

Abstract:

The Hasse Weil bound restricts the number of points of a curve which are defined over a finite field; if the number of points meets this bound, the curve is called maximal.  Giulietti and Korchmaros introduced a curve C_3 which is maximal over F_{q^6} and determined its automorphism group.  Garcia, Guneri, and Stichtenoth generalized this construction to a family of curves C_n, indexed by an odd integer n >= 3, such that C_n is maximal over F_{q^{2n}}.  In this talk, I will first explain why this family of maximal curves is interesting and then describe recent work with Guralnick and Malmskog in which we determine the automorphism group

I will explain how to use classical complex analytic methods in order to arrive at a deterministic polynomial-time reduction of the problem of factoring to the computation of some multiple series.

We define quadratic number fields and discuss algorithmic aspects of computing their ideal class groups, regulators and fundamental units. We also present number theoretic and cryptographic applications of these algorithms, such as Pell equation solving or computing discrete logarithms.

In this talk, I will focus on encryption schemes. I will first remind everyone of their security definitions, and then give an overview of some of the main constructions based on general assumptions.

The talk is intended for general audience of basic familiarity with algebraic number theory.