Paper 2024/585 Original: eprint.iacr.org/2024/585 ↗
A Complete Beginner Guide to the Number Theoretic Transform (NTT)
Abstract
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.
Metadata
- Available format(s)
PDF- Category
- Foundations
- Publication info
- Preprint.
- History
- 2024-04-29: last updated
- 2024-04-16: published
- Short URL
- https://ia.cr/2024/585
- License
- https://creativecommons.org/licenses/by/4.0/
BibTeX
@misc{cryptoeprint:2024/585,
author = {Ardianto Satriawan and Rella Mareta and Hanho Lee},
title = {A Complete Beginner Guide to the Number Theoretic Transform (NTT)},
howpublished = {Cryptology {ePrint} Archive, Paper 2024/585},
year = {2024},
url = {https://eprint.iacr.org/2024/585}
}