Section: New Results

Sparse linear algebra modulo $p$

Participants : Hamza Jeljeli, Emmanuel Thomé [contact] .

The resolution of linear algebra problems with subexponential methods, which is the topic of the ANR-CATREL project (to begin in 2013) calls for the resolution of large sparse linear systems defined over finite fields. In preparation for this, H. Jeljeli has developed software for performing sparse matrix times vector multiplication on NVIDIA GPUS [16] . This code provides a very significant speedup over the use of CPUs for this task, and achieves this speedup by a clever use of a “residue number system” representation of the finite field elements.

As a complement, a recent re-implementation of Thomé's algorithm for the (matrix) Berlekamp-Massey step in the block Wiedemann algorithm has been done. This program can of course be special-cased to the simple non-matrix case. The GPU code above and this special case, together, form the needed software to have a sparse linear system solver over finite fields using Wiedemann's algorithm. This has been put to use, and led to the completion of a discrete logarithm record in ${𝔽}_{2^{619}}$, the linear system part taking only 17 hours in total on one GPU (plus 1 hour on one CPU for the Berlekamp-Massey step).