Modular Matrix Multiplication on GPU for Polynomial System Solving

Abstract

The bottleneck of the SPARSE-FGLM algorithm for Gröbner bases change of order is an iterative matrix - tall and skinny matrix product over a finite prime field. Our contribution is twofold. First, we port existing CPU-only algorithms for matrix products over prime fields to GPU architectures, and carry out a performance analysis of our implementation that shows that we can nearly achieve the maximum theoretical throughput of the hardware. Second, existing CPU-only algorithms could not handle primes with more than 26 bits, other than the GMP-based implementation in FLINT; we overcome this limitation by proposing an efficient multiword matrix product algorithm that can deal with primes with at most 35 bits; we benchmarked it on GPU.