Q-adic Transform revisited

TL;DR

This paper introduces an improved algorithm for fast modular polynomial multiplication using $q$-adic and $X$-adic representations, optimizing conversions and reducing divisions for efficient finite field arithmetic.

Contribution

It presents a new version of conversions between $X$-adic and $q$-adic representations with fewer divisions and more tabulations, enhancing modular polynomial multiplication.

Findings

Faster modular polynomial multiplication algorithm

Reduced number of divisions in conversions

Effective arithmetic in small finite field extensions

Abstract

We present an algorithm to perform a simultaneous modular reduction of several residues. This algorithm is applied fast modular polynomial multiplication. The idea is to convert the $X$-adic representation of modular polynomials, with $X$ an indeterminate, to a $q$-adic representation where $q$ is an integer larger than the field characteristic. With some control on the different involved sizes it is then possible to perform some of the $q$-adic arithmetic directly with machine integers or floating points. Depending also on the number of performed numerical operations one can then convert back to the $q$-adic or $X$-adic representation and eventually mod out high residues. In this note we present a new version of both conversions: more tabulations and a way to reduce the number of divisions involved in the process are presented. The polynomial multiplication is then applied to arithmetic in small finite field extensions.