Revisiting Fast Fourier multiplication algorithms on quotient rings

TL;DR

This paper formalizes and unifies efficient Fast Fourier Transform-based algorithms for polynomial multiplication in quotient rings with composite moduli, analyzing conditions for their applicability.

Contribution

It provides a formal framework and unification of various FFT-based algorithms for polynomial multiplication in quotient rings with composite moduli.

Findings

Conditions for applicability of FFT algorithms in quotient rings analyzed

Unified framework for different FFT approaches developed

Enhanced understanding of polynomial multiplication in composite modulus rings

Abstract

This work formalizes efficient Fast Fourier-based multiplication algorithms for polynomials in quotient rings such as $\mathbb{Z}_{m}[x]/\left<x^{n}-a\right>$, with $n$ a power of 2 and $m$ a non necessarily prime integer. We also present a meticulous study on the necessary and/or sufficient conditions required for the applicability of these multiplication algorithms. This paper allows us to unify the different approaches to the problem of efficiently computing the product of two polynomials in these quotient rings.