ALGORITHMIC ASPECTS OF ELLIPTIC BASES IN FINITE FIELD DISCRETE LOGARITHM ALGORITHMS

Abstract

Elliptic bases, introduced by Couveignes and Lercier in 2009, give an elegant way of representing finite field extensions. A natural question which seems to have been considered independently by several groups is to use this representation as a starting point for discrete logarithm algorithms in small characteristic finite fields. This idea has been recently proposed by two groups working on it, in order to achieve provable quasi-polynomial time for discrete logarithms in small characteristic finite fields. In this paper, we do not try to achieve a provable algorithm but, instead, investigate the practicality of heuristic algorithms based on elliptic bases. Our key idea is to use a different model of the elliptic curve used for the elliptic basis that allows for a relatively simple adaptation of the techniques used with former Frobenius representation algorithms. We have not performed any record computation with this new method but our experiments with the field F31345 indicate that switching to elliptic repre-sentations might be possible with performances comparable to the current best practical methods.