Modular exponentiation via the explicit Chinese remainder theorem

Abstract

Fix pairwise coprime positive integers p 1 ,P 2 ,..., p a . We propose representing integers u modulo m, where m is any positive integer up to roughly p 1 p 2 ... P s , as vectors (u mod p 1 , u mod p 2 ,.., u mod 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, e, and m in binary, of total bit length n, computes x e mod m in time O(n/lglgn) using 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 ) expected time algorithm using exp(O(n lg n)) processors; von zur Gathen gave an NC algorithm for the highly special case that m is polynomially smooth.