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.

翻译：本文对商环中的多项式，如 $\mathbb{Z}_{m}[x]/\left<x^{n}-a\right>$，给出了有效的基于快速傅里叶变换的乘法算法，并对这些乘法算法适用的必要和/或充分条件进行了详细的研究。这篇论文允许我们统一不同的方法来解决在这些商环中高效计算两个多项式乘积的问题。