International Association
for Cryptologic Research

The block Lanczos algorithm proposed by Peter Montgomery is an efficient means to tackle the sparse linear algebra problem which arises in the context of the number field sieve factoring algorithm and its predecessors. We present here a modified version of the algorithm, which incorporates several improvements: we discuss how to efficiently handle homogeneous systems and how to reduce the number of vectors stored in the course of the computation. We also provide heuristic justification for the success probability of our modified algorithm.

While the overall complexity and expected number of steps of the block Lanczos is not changed by the modifications presented in this article, we expect these to be useful for implementations of the block Lanczos algorithm where the storage of auxiliary vectors sometimes has a non-negligible cost.