Fast evaluation of iterated multiplication of very large polynomials: an application to Chinese remainder theory

Abstract

We consider the problem of exactly computing the number of integers in a Chinese Remainder Representation (CRR) whose pseudorank does not equal the rank. We call this number the census. The rank is key in developing CRR-intrinsic methods for comparing integers in CRR, a problem known to be notoriously difficult. Pseudorank can be computed in highly restrictive computation models. We have developed and implemented a fast, efficient algorithm for computing the census based on using a variant of the FFT to compute iterated products of polynomials of very large degree, and with arbitrary size integer coefficients. Experimental census results are tabulated. This census information makes possible a new approach to exploring the fine structure of CRR.