Black Box Linear Algebra: Extending Wiedemann's Analysis of a Sparse Matrix Preconditioner for Computations over Small Fields

TL;DR

This paper extends Wiedemann's analysis of a matrix preconditioner for small fields, providing explicit bounds and probabilistic guarantees, aiming to improve reliability and efficiency in sparse matrix computations relevant to integer factorization.

Contribution

It offers a more explicit analysis of a matrix preconditioner over small fields, including bounds on nonzero entries and rank probability, advancing the theoretical foundation for reliable black box algorithms.

Findings

Explicit bounds on nonzero entries in preconditioned matrices

Probabilistic bounds on maximal rank achievement

Potential for more efficient integer factorization algorithms

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 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 such matrices have maximal rank. This is part of ongoing work to establish that this matrix preconditioner can also 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 and - by a small factor - asymptotically more efficient than alternative techniques that make use of other matrix preconditioners or require computations over field extensions.