Black box linear algebra: extending wiedemann's analysis of a sparse matrix preconditioner for computations over small fields

Abstract

Wiedemann's paper, introducing his algorithm for sparse and structured matrix computations over arbitrary fields, also presented a pair of matrix preconditioners for computations over small fields. The analysis of the second of these is extended here in order to provide more explicit statements of the expected number of nonzero entries in the matrices obtained as well as bounds on the probability that the matrices being considered have maximal rank. It is hoped that this will make Wiedemann's second preconditioner of more practical use. This is part of ongoing work to establish that this matrix preconditioner can be used to bound the number of nontrivial nilpotent blocks in the Jordan normal form of a preconditioned matrix, in such a way that one can also sample uniformly from the null space of the originally given matrix. If successful this will result in a black box algorithm for the type of matrix computation required when using the number field sieve for integer factorization that is provably reliable (unlike some heuristics, presently in use) and --- by a small factor --- asymptotically more efficient than alternative provably reliable techniques that make use of other matrix preconditioners or require computations over field extensions.