On the Function Field Sieve and the Impact of Higher Splitting Probabilities: Application to Discrete Logarithms in $\mathbb{F}_{2^{1971}}$ and $\mathbb{F}

G\"olo\u{g}lu, FarukGranger, RobertMcGuire, GaryZumbrägel, Jens2016-01-192016-01-192016-01-19201310.1007/978-3-642-40084-1_7https://infoscience.epfl.ch/handle/20.500.14299/122368In this paper we propose a binary field variant of the Joux-Lercier medium-sized Function Field Sieve, which results not only in complexities as low as $L_{q^n}(1/3,(4/9)1/3)$ for computing arbitrary logarithms, but also in an heuristic polynomial time algorithm for finding the discrete logarithms of degree one and two elements when the field has a subfield of an appropriate size. To illustrate the efficiency of the method, we have successfully solved the DLP in the finite fields with $2^{1971}$ and $2^{3164}$ elements, setting a record for binary fields.Discrete logarithm problemFunction field sieveOn the Function Field Sieve and the Impact of Higher Splitting Probabilities: Application to Discrete Logarithms in $\mathbb{F}_{2^{1971}}$ and $\mathbb{F}_{2^{3164}}$text::conference output::conference proceedings::conference paper