Woensdag 8 december 2010
||David Gruenewald (Radboud Universiteit)
||Hyperelliptic Curves, Cryptography and Factorization Algorithms
||11:00 - 12:00
Vanaf kwart voor 11 is er koffie en thee voor de zaal.
We give a gentle introduction to the arithmetic on Jacobians of hyperelliptic
curves of genus g and describe how these objects can be used in cryptography.
As a special case, this includes elliptic curve cryptography (g=1)
which is ubiquitous in modern day cryptosystems.
Then we look at Lenstra's elliptic curve factorization method (ECM),
inspired by Pollard's p-1 algorithm.
One can replace elliptic curves (g=1) with Jacobians of hyperelliptic curves
of higher genus to get an analogous algorithm (HECM), but in general
this will not be competitive with elliptic curves as the arithmetic
is slower for higher genus.
We report on a recent implementation of HECM in genus 2 due to R. Cosset
which is competitive with ECM and we describe some of the objects
(Kummer surfaces) and ideas used to bridge the gap.