Improving the Gaudry-Schost algorithm for multidimensional discrete logarithms

The discrete logarithm problem arises from various areas, including counting the number of points of certain curves and diverse cryptographic schemes. The Gaudry-Schost algorithm and its variants are state-of-the-art low-memory methods solving the multi-dimensional discrete logarithm problem through finding collisions between pseudorandom tame walks and wild walks. In this work, we explore the impact on the choice of tame and wild sets of the Gaudry-Schost algorithm, and give two variants with improved average case time complexity for the multidimensional case under certain heuristic assumptions. We explain why the second method is asymptotically optimal.