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

A quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic