A reliable block Lanczos algorithm over small finite fields

Abstract

Blocked versions of the Lanczos procedure have been successfully applied to sample nullspace elements of very large sparse matrices over small finite fields. The heuristic currently in use, namely, Montgomery's method [10], is unreliable for certain input matrices. This paper introduces a new biconditional block Lanczos approach based on lookahead, a technique designed to improve the reliability of the scalar Lanczos algorithm. Empirical data show that the performance of the lookahead-based algorithm is competitive with that of Montgomery's heuristic when their relative reliability is taken into account. The reliability of this new algorithm for arbitrary matrices over small finite fields is then established. In the process, some results on the ranks of certain submatrices of a randomly determined block Hankel matrix are established. These results may be applicable in other contexts, such as Coppersmith's block Wiedemann algorithm [3].