An Elementary Method For Fast Modular Exponentiation With Factored Modulus

Abstract

We present a fast algorithm for modular exponentiation when the factorization of the modulus is known. Let $a,n,m$ be positive integers and suppose $m$ factors canonically as $\prod_{i=1}^k p_i^{e_i}$. Choose integer parameters $t_i\in [1, e_i]$ for $1\le i\le k$. Then we can compute the modular exponentiation $a^n\pmod{m}$ in $O(\max(e_i/t_i)+\sum_{i=1}^k t_i\log p_i)$ steps (i.e., modular operations). We go on to analyze this algorithm mathematically and programmatically, showing significant asymptotic improvement in specific cases. Specifically, for an infinite family of $m$ we achieve a complexity of $O(\sqrt{\log m})$ steps, much faster than the Repeated Squaring Algorithm, which has complexity $O(\log m)$. Additionally, we extend our algorithm to matrices and hence general linear recurrences. The complexity is similar; with the same setup we can exponentiate matrices in $GL_d(\mathbb{Z}/m\mathbb{Z})$ in less than $O(\max(e_i/t_i)+d^2\sum_{i=1}^k t_i\log p_i)$ steps. This improves Fiduccia's algorithm and the results of Bostan and Mori in the case of $\mathbb{Z}/m\mathbb{Z}$. We prove analogous results for $\mathbb{Z}/p^k\mathbb{Z}$ ring extensions.