Solving linear equations over GF(2): block Lanczos algorithm

Abstract

We propose a method of solving large sparse systems of linear equations over GF(2), the field with two elements. We use the Lanczos algorithm, modified in two ways. A lookahead Lanczos algorithm is needed for the problem of dividing by an inner product whose value happens to be 0. A block version of the algorithm allows us to perform 32 matrix-vector operations for the cost of one. The resulting algorithm is competitive with structured Gaussian elimination in terms of time, and has much lower space requirements. It may be useful in the last stage of integer factorization. © 1993.