Long Polynomial Modular Multiplication using Low-Complexity Number Theoretic Transform

TL;DR

This paper explores efficient methods for polynomial modular multiplication over rings using the number theoretic transform, aiming to improve homomorphic encryption performance by extending DFT theory to polynomial rings.

Contribution

It introduces three approaches—zero-padded convolution, negative wrapped convolution, and low-complexity NWC—for implementing long polynomial modular multiplication with reduced complexity.

Findings

Proposes three novel methods for polynomial multiplication.

Analyzes the complexity and efficiency of each approach.

Highlights potential improvements for homomorphic encryption implementations.

Abstract

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).