A new index calculus algorithm with complexity $L(1/4+o(1))$ in very small characteristic

Antoine Joux

Abstract

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 $\GF{Q}$ in a small extension $\GF{Q^e}$ with $e\leq 2\lceil \log_p n \rceil$.

Available format(s)
Category
Foundations
Publication info
Published elsewhere. Will appear in the proceedings of SAC 2013
Keywords
Number TheoryDiscrete Logarithms
Contact author(s)
antoine joux @ m4x org
History
2013-07-24: revised
See all versions
Short URL
https://ia.cr/2013/095

CC BY

BibTeX

@misc{cryptoeprint:2013/095,
author = {Antoine Joux},
title = {A new index calculus algorithm with complexity $L(1/4+o(1))$ in very small characteristic},
howpublished = {Cryptology ePrint Archive, Paper 2013/095},
year = {2013},
note = {\url{https://eprint.iacr.org/2013/095}},
url = {https://eprint.iacr.org/2013/095}
}

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