Modular exponentiation via the explicit Chinese remainder theorem

Abstract

Fix pairwise coprime positive integers p 1 , p 2 , … , p s p_1,p_2,\dots ,p_s . We propose representing integers u u modulo m m , where m m is any positive integer up to roughly p 1 p 2 ⋯ p s \sqrt {p_1p_2\cdots p_s} , as vectors ( u mod p 1 , u mod p 2 , … , u mod p s ) (u\bmod p_1,u\bmod p_2,\dots ,u\bmod p_s) . We use this representation to obtain a new result on the parallel complexity of modular exponentiation: there is an algorithm for the Common CRCW PRAM that, given positive integers x x , e e , and m m in binary, of total bit length n n , computes x e mod m x^e\bmod m in time O ( n / lg ⁡ lg ⁡ n ) O(n/{\lg \lg n}) using n O ( 1 ) n^{O(1)} processors. For comparison, a parallelization of the standard binary algorithm takes superlinear time; Adleman and Kompella gave an O ( ( lg ⁡ n ) 3 ) O((\lg n)^3) expected time algorithm using exp ⁡ ( O ( n lg ⁡ n ) ) \exp ( O(\sqrt {n\lg n})) processors; von zur Gathen gave an NC algorithm for the highly special case that m m is polynomially smooth.