International Association
for Cryptologic Research

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 $L(1/4+o(1))$.