Skew-polynomial-sparse matrix multiplication

Based on the observation that $Q^{(p-1)\times (p-1)}$ is isomorphic to a quotient skew polynomial ring, we propose a new deterministic algorithm for $(p-1)\times (p-1)$ matrix multiplication over $Q$, where p is a prime number. The algorithm has complexity $O\left(T^{\omega -2}{p}^2\right)$, where $T\le p-1$ 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 $O^{\sim }(t^{\omega -2}{p}^2+p^2\log \frac{1}{\nu })$, where $t\le p-1$ 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.