- Published 2013 in Selected Areas in Cryptography

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, this algorithm achieves heuristic complexity LQ(1/4 + o(1)). For technical reasons, unless n is already a composite with factors of the right size, this is done by embedding FQ in a small extension FQe with e ≤ 2dlogp ne.

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@inproceedings{Joux2013ANI,
title={A New Index Calculus Algorithm with Complexity \$\$L(1/4+o(1))\$\$ in Small Characteristic},
author={Antoine Joux},
booktitle={Selected Areas in Cryptography},
year={2013}
}