Fine Tuning the Function Field Sieve Algorithm for the Medium Prime Case

Abstract

This paper builds on the variant of the function field sieve (FFS) algorithm for the medium prime case introduced by Joux and Lercier in 2006. We make several contributions. The first contribution uses a divisibility and smoothness technique and goes on to develop a sieving method based on the technique. This leads to significant practical efficiency improvements in the descent phase and also provides improvement to Joux's pinpointing technique. The second contribution is a detailed analysis of the degree of freedom and the use of a walk technique in the descent phase of the algorithm. Such analysis shows that it is possible to compute discrete logarithms over certain fields, which are excluded by the earlier analyses performed by Joux and Lercier (2006) and Joux (2013). In concrete terms, we present computations of discrete logs for fields with 16 and 19-bit prime characteristic. We also provide concrete analysis of the effectiveness of the FFS algorithm for certain fields of characteristic ranging from 16 to 32-bit primes. The final contribution is to perform a complete asymptotic analysis of the FFS algorithm for fields F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</sub> with p = L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</sub> (1/3, c). This closes gaps and corrects errors in the analysis earlier performed by Joux-Lercier and Joux and also provides new insights into the asymptotic behavior of the algorithm.