Multivariate polynomial products over modular rings using residue arithmetic

Abstract

A computational system called the polynomial residue number system (PRNS) has previously been proposed and analyzed. It solves the problem of multiplying two univariate polynomials modulo (x/sup N//spl plusmn/1) over the modular ring Z/sub p/. In the present paper, extensions of PRNS for computing the product of two multivariate polynomials modulo a polynomial are developed. Such a number system is termed as multivariate polynomial residue number system (MPRNS). MPRNS is essentially an isomorphic representation between the ring Z/sub p/[x]//spl Pi//sub i=1//sup L/ (x/sub i/(N/sub i/)/spl plusmn/1) of L-variate polynomials in the indeterminate vector x=(x/sub 1/, x/sub 2/, ..., x/sub L/) and the ring Z/sub p/(N/sub 1/N/sub 2/...N/sub L/). Issues related to existence of isomorphic mappings, their properties and multiplicative complexity of the resulting algorithm have been addressed. The applications of the MPRNS scheme are also presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>