Conceptual Review on Number Theoretic Transform and Comprehensive Review on Its Implementations

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
$O(n \log {n})$
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, which relies on the hardness of certain mathematical problems to ensure security. Its importance in these fields continues to grow as quantum computing technology advances and traditional encryption methods become vulnerable. In this report, we discuss the mathematical concepts of polynomial multiplications using NTT and provide a comprehensive review of the latest implementation and state-of-the-art of NTT in both PQC and HE schemes.