Skew-polynomial-sparse matrix multiplication

Based on the observation that Q(p-1)x(p-1) is isomorphic to a quotient skew polynomial ring, we propose a new deterministic algorithm for (p - 1) x (p - 1) matrix multiplication over Q, where p is a prime number. The algorithm has complexity O(T & omega;-2p2), where T & LE; p - 1 is a parameter determined by the skew-polynomial-sparsity of input matrices and & omega; 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 & SIM;(t & omega;-2p2 + p2 log 1 & nu;), where t & LE; p -1 is the skew-polynomial-sparsity of the product and & nu; 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.& COPY; 2023 Elsevier Ltd. All rights reserved.