Razvan Basbulescu |

Date de l'exposé : 3 octobre 2014

### A heuristic quasi-polynomial algorithm for discrete logarithm

in finite fields of small characteristic The difficulty of discrete logarithm computations in fields GF(q^k) depends on the relative sizes of k and q. Until recently all the cases had a sub-exponential complexity of type L(1/3), similar to the complexity of factoring. If n is the bit-size of q^k, then L(1/3) can be approximated by 2^(n^(1/3)). In 2013, Joux designed a new algorithm for constant characteristic of complexity L(1/4+o(1)), approximatively 2^(n^(1/4)). Inspired by Joux' algorithm, we propose a heuristic algorithm that provides a quasi-polynomial complexity when q is of size O(poly(k)). By quasi-polynomial, we mean a runtime of n^O(log n).Hence, small characteristic pairings have an asymptotic complexity which is inapropiate for cryptography. In addition, in practice we expect the algorithm to be much faster in the case GF(q^2k), when q and k are roughly equal. The small characteristic pairings which were previously evaluated to 128 bits of security correspond to this case, and were reevaluated to a much lower security. It allows to conclude that small characteristic pairings must be avoided in cryptography.