Blocked Lanczos-style Algorithms over Small Finite Fields

Solutions to very large sparse systems of linear equations over small finite fields are essential in the best known algorithms for factoring integers and computing discrete logarithms. Due to the sizes of such systems, elimination-based approaches are impractical, so attention has shifted to iterative or black-box methods. This thesis examines one brand of such algorithms: those based on the Lanczos iteration. Blocked versions of the Lanczos procedure enhance reliability over the scalar algorithm and facilitate parallelization. These methods are randomized and thus require a probabilistic analysis of reliability. In this thesis, Montgomery’s algorithm [Mon95], which currently the most commonly used method, is examined in detail and it is shown that it is provably unreliable on certain input matrices. Furthermore, it is difficult to identify a class of matrices over which Montgomery’s algorithm can be proved reliable. A new block Lanczos approach is then introduced. It is based on lookahead, which is a technique designed to improve the reliability of the scalar Lanczos algorithm over small finite fields. The reliability of this method over arbitrary fields is then established. There is also discussion of the expected memory requirements of the lookahead-based approach. Empirical data show that the lookahead-based algorithm is competitive with Montgomery’s algorithm when their relative reliability is taken into account. Furthermore, the memory requirements of the lookahead-based approach appear to be similar to what is predicted by the conjecture given here.