A block Lanczos algorithm for finding dependencies over GF(2)

Abstract

Some integer factorization algorithms require several vectors in the null space of a sparse m x n matrix over the field GF(2). We modify the Lanczos algorithm to produce a sequence of orthogonal subspaces of GF(2)n, each having dimension almost N, where N is the computer word size, by applying the given matrix and its transpose to N binary vectors at once. The resulting algorithm takes about n/(N - 0.76) iterations. It was applied to matrices larger than 106 × 106 during the factorizations of 105-digit and 119-digit numbers via the general number field sieve.