Sparse Gaussian Elimination Modulo p: An Update

Abstract

This paper considers elimination algorithms for sparse matrices over finite fields. We mostly focus on computing the rank, because it raises the same challenges as solving linear systems, while being slightly simpler. We developed a new sparse elimination algorithm inspired by the Gilbert-Peierls sparse LU factorization, which is well-known in the numerical computation community. We benchmarked it against the usual right-looking sparse gaussian elimination and the Wiedemann algorithm using the Sparse Integer Matrix Collection of Jean-Guillaume Dumas. We obtain large speedups (1000 $$\times $$ and more) on many cases. In particular, we are able to compute the rank of several large sparse matrices in seconds or minutes, compared to days with previous methods.