Skew-polynomial-sparse matrix multiplication

Abstract

Based on the observation that Q(p−1)×(p−1) is isomorphic to a quotient skew polynomial ring, we propose a new deterministic algorithm for (p−1)×(p−1) matrix multiplication over Q, where p is a prime number. The algorithm has complexity O(Tω−2p2), where T≤p−1 is a parameter determined by the skew-polynomial-sparsity of input matrices and ω is the asymptotic exponent of matrix multiplication. Here a matrix is skew-polynomial-sparse if its corresponding skew polynomial is sparse. Moreover, by introducing randomness, we also propose a probabilistic algorithm with complexity O∼(tω−2p2+p2log⁡1ν), where t≤p−1 is the skew-polynomial-sparsity of the product and ν is the probability parameter. The main feature of the algorithms is the acceleration for matrix multiplication if the input matrices or their products are skew-polynomial-sparse.