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

Razvan Barbulescu 1 Pierrick Gaudry 1 Antoine Joux 2, 3 Emmanuel Thomé 1
1 CARAMEL - Cryptology, Arithmetic: Hardware and Software
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
2 Chaire de cryptologie
UPMC - Université Pierre et Marie Curie - Paris 6
Abstract : 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 $L(1/4+o(1))$.
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Razvan Barbulescu, Pierrick Gaudry, Antoine Joux, Emmanuel Thomé. A heuristic quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic. Eurocrypt 2014, May 2014, Copenhagen, Denmark. pp.1-16, ⟨10.1007/978-3-642-55220-5_1⟩. ⟨hal-00835446v2⟩

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