A Complete Beginner Guide to the Number Theoretic Transform (NTT)
The Number Theoretic Transform (NTT) is a powerful mathematical tool that has become increasingly important in developing Post Quantum Cryptography (PQC) and Homomorphic Encryption (HE). Its ability to efficiently calculate polynomial multiplication using the convolution theorem with a quasi-linear complexity $O(n \log{n})$ instead of $O(n^2)$ when implemented with Fast Fourier Transform-style algorithms has made it a key component in modern cryptography. FFT-style NTT algorithm or fast-NTT is particularly useful in lattice-based cryptography. In this short note, we briefly introduce the basic concepts of linear, cyclic, and negacyclic convolutions via traditional schoolbook algorithms, traditional NTT, its inverse (INTT), and FFT-like versions of NTT/INTT. We then provide consistent toy examples through different concepts and algorithms to understand the basics of NTT concepts.
Note: This is a partial re-publication of the concepts part of the paper: "Conceptual Review on Number Theoretic Transform and Comprehensive Review on Its Implementations", DOI: 10.1109/ACCESS.2023.3294446 The original paper consists of the NTT concepts and a comparison between its hardware implementation, while this paper only contains the concept part.
