A New Index Calculus Algorithm with Complexity \(L(1/4+o(1))\) in Small Characteristic

Conference paper

DOI: 10.1007/978-3-662-43414-7_18

Part of the Lecture Notes in Computer Science book series (LNCS, volume 8282)
Cite this paper as:
Joux A. (2014) A New Index Calculus Algorithm with Complexity \(L(1/4+o(1))\) in Small Characteristic. In: Lange T., Lauter K., Lisoněk P. (eds) Selected Areas in Cryptography -- SAC 2013. SAC 2013. Lecture Notes in Computer Science, vol 8282. Springer, Berlin, Heidelberg


In this paper, we describe a new algorithm for discrete logarithms in small characteristic. This algorithm is based on index calculus and includes two new contributions. The first is a new method for generating multiplicative relations among elements of a small smoothness basis. The second is a new descent strategy that allows us to express the logarithm of an arbitrary finite field element in terms of the logarithm of elements from the smoothness basis. For a small characteristic finite field of size \(Q=p^n\), this algorithm achieves heuristic complexity \(L_Q(1/4+o(1)).\) For technical reasons, unless \(n\) is already a composite with factors of the right size, this is done by embedding \({\mathbb F}_{Q}\) in a small extension \({\mathbb F}_{Q^e}\) with \(e\le 2\lceil \log _p n \rceil \).

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Laboratoire PRISMCryptoExperts and Université de Versailles Saint-Quentin-en-YvelinesVersailles CedexFrance

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