  In the present work, we present a new discrete logarithm algorithm, in the
same vein as in recent works by Joux, using an asymptotically more efficient
descent approach. The main result gives a quasi-polynomial heuristic complexity
for the discrete logarithm problem in finite field of small characteristic. By
quasi-polynomial, we mean a complexity of type $n^{O(\log n)}$ where $n$ is the
bit-size of the cardinality of the finite field. Such a complexity is smaller
than any $L(\varepsilon)$ for $\epsilon>0$. It remains super-polynomial in the
size of the input, but offers a major asymptotic improvement compared to
$L(1/4+o(1))$.
