Parallelization of the Wiedemann Large Sparse System Solver over Large Prime Fields For the partial fulfilment of the degree of Master of Technology

The discrete logarithm problem over finite fields serves as the source of security for several cryptographic primitives. The fastest known algorithms for solving the discrete logarithm problem require solutions of large sparse linear systems over large prime fields, and employ iterative solvers for this purpose. The published results on this topic are mainly focused on systems over binary fields, that is, systems coming from integer-factoring algorithms. Solving systems over large prime fields has not yet received much research attention. In this thesis, our main goal is to efficiently implement the Wiedemann method to solve large sparse linear systems of equations over large prime fields. The second phase of the Wiedemann method (computation of the minimal polynomial of a linear sequence) offers several choices including the Berlekamp-Massey and the Levinson-Durbin algorithms. Assessing the relative performance of the above two variants of the second phase is another important goal of this work. We first detail our optimized sequential implementation of the Wiedemann method. Subsequently, we deal with shared-memory parallel implementations of the Wiedemann method using a small number of cores. We have been able to achieve a speedup of about four using eight cores. Our experiments also suggest that the Levinson-Durbin algorithm in the second stage is more suitable to parallelization than the Berlekamp-Massey algorithm.