  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 small characteristic finite field
discrete logarithm algorithms. 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
don't 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 haven't performed any record
computation with this new method but our experiments with the field F 3 1345
indicate that switching to elliptic representations might be possible with
performances comparable to the current best practical methods.
