Section: New Results

Elliptic curve cryptology

Participants : Jean-Marc Couveignes, Andreas Enge, Damien Robert.

Couveignes and Lercier study in [26] the problem of parameterisations by radicals of low genus algebraic curves. They prove that for $q$ a prime power that is large enough and prime to 6, a fixed positive proportion of all genus 2 curves over the field with $q$ elements can be parameterised by 3-radicals. This results in the existence of a deterministic encoding into these curves when $q$ is congruent to 2 modulo 3. Deterministic encodings into curves are useful in numerous situations, for instance in discrete logarithm cryptography. The parameterisation found by Couveignes and Lercier is in some sense the first generic one for genus 2 curves.

A software for this method is in preparation.

The survey [21] , published in the Handbook of Finite Fields, presents the state of the art of the use of elliptic curves in cryptography.