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：本文形式化了商环（如 $\mathbb{Z}_{m}[x]/\left<x^{n}-a\right>$）中基于快速傅里叶变换的高效多项式乘法算法，其中 $n$ 为 2 的幂次，$m$ 不一定为素数。我们同时对这些乘法算法适用性所需的必要和/或充分条件进行了细致研究。本文使我们能够统一处理此类商环中两个多项式乘积高效计算问题的不同方法。