Going by the abstract alone, this seems to me to be theoretically devastating, and should in my view kill all DLP cryptosystems over fields with small characteristic. "Kill" meaning nobody should want to use them anymore. --- http://link.springer.com/chapter/10.1007%2F978-3-642-55220-5_1 Advances in Cryptology – EUROCRYPT 2014 Lecture Notes in Computer Science Volume 8441, 2014, pp 1-16 A Heuristic Quasi-Polynomial Algorithm for Discrete Logarithm in Finite Fields of Small Characteristic Razvan Barbulescu, Pierrick Gaudry, Antoine Joux, Emmanuel Thome ABSTRACT The difficulty of computing discrete logarithms 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 factorization problem. In 2013, Joux designed a new algorithm with a complexity of L(1/4 + epsilon) in small characteristic. In the same spirit, we propose in this article another heuristic algorithm that provides a quasi-polynomial complexity when q is of size at most comparable with k. By quasi-polynomial, we mean a runtime of N^O(logN) where N is the bit-size of the input... Complexity results in this article rely on heuristics which have been checked experimentally.