  We describe a provably quasi-polynomial algorithm to compute discrete
logarithms in the multiplicative groups of finite fields of small
characteristic, that is finite fields whose characteristic is logarithmic in
the order. We partially follow the heuristically quasi-polynomial algorithm
presented by Barbulescu, Gaudry, Joux and Thome'. The main difference is to use
a presentation of the finite field based on elliptic curves: the abundance of
elliptic curves ensures the existence of such a presentation.
