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

*Razvan Barbulescu and Pierrick Gaudry and Antoine Joux and Emmanuel Thomé*

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

**Category / Keywords: **public-key cryptography / cryptanalysis, number theory, discrete logarithm problem, finite fields

**Date: **received 18 Jun 2013, last revised 25 Nov 2013

**Contact author: **Emmanuel Thome at gmail com

**Available format(s): **PDF | BibTeX Citation

**Note: **significantly improved version, further details given.

**Version: **20131125:121045 (All versions of this report)

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