Fast, Parallel Algorithm for Multiplying Polynomials with Integer Coefficients

This paper aims to develop and analyze an effective parallel algorithm for multiplying polynomials and power series with integer coefficients. Such operations are of fundamental importance when generating parameters for public key cryptosystems, whereas their effective implementation translates directly into the speed of such algorithms in practical applications. The algorithm has been designed specifically to accelerate the process of generating modular polynomials, but due to its good numerical properties it may surely be used to solve other computational problems as well. The basic idea behind this new method was to adapt it to parallel computing. Nowadays, it is a very important property, as it allows to fully exploit the computing power offered by modern processors. The combination of the Chinese Remainder Theorem and the Fast Fourier Transform made it possible to develop a highly effective multiplication method. Under certain conditions, it is asymptotically faster than the algorithm based on Fast Fourier Transform when applied to multiply both: numbers and polynomials. Undoubtedly, this result is the major theoretical conclusion of this paper.