An efficient parallel block-reduction algorithm

In this paper, we present a new parallel block-reduction algorithm for reducing lattice bases which allows the use of an arbitrarily chosen block-size between two and n where n denotes the dimension of the lattice. Thus, we are building a hierarchy of parallel lattice basis reduction algorithms between the known parallel all-swap algorithm which is a parallelization for block-size two and the reduction algorithm for block-size n which corresponds to the known sequential lattice basis reduction algorithm. We show that even though the parallel all-swap algorithm as well as the parallel block-reduction algorithm have the same asymptotic complexity in respect to arithmetic operations in theory, in practice neither block-size two nor block-size n are a priori the best choices. The optimal block-size in respect to minimizing the reduction time rather depends strongly on the used parallel system and the corresponding communication costs.