This tutorial aims to establish connections between polynomial modular multiplication over a ring to circular convolution and discrete Fourier transform (DFT). The main goal is to extend the well-known theory of DFT in signal processing (SP) to other applications involving polynomials in a ring such as homomorphic encryption (HE). HE allows any third party to operate on the encrypted data without decrypting it in advance. Since most HE schemes are constructed from the ring-learning with errors (R-LWE) problem, efficient polynomial modular multiplication implementation becomes critical. Any improvement in the execution of these building blocks would have significant consequences for the global performance of HE. This lecture note describes three approaches to implementing long polynomial modular multiplication using the number theoretic transform (NTT): zero-padded convolution, without zero-padding, also referred to as negative wrapped convolution (NWC), and low-complexity NWC (LC-NWC).

翻译：本教程旨在建立环上多项式模乘法与循环卷积及离散傅里叶变换（DFT）之间的联系。主要目标是拓展信号处理（SP）中成熟的DFT理论，使其适用于同态加密（HE）等涉及环上多项式的应用场景。HE允许任何第三方在不预先解密的情况下对加密数据进行操作。由于大多数HE方案基于环学习误差（R-LWE）问题构建，高效实现多项式模乘法变得至关重要。这些基本模块运算效率的任何提升都将对HE的整体性能产生重大影响。本讲义阐述了使用数论变换（NTT）实现长多项式模乘法的三种方法：零填充卷积法、无零填充法（亦称负包裹卷积（NWC）），以及低复杂度NWC（LC-NWC）。