Introduction to Number Theoretic Transform

Abstract: The Number Theoretic Transform (NTT) can be regarded as a variant of the Discrete Fourier Transform. NTT has been quite a powerful mathematical tool in developing Post-Quantum Cryptography and Homomorphic Encryption. The Fourier Transform essentially decomposes a signal into its frequencies. They are traditionally sine or cosine waves. NTT works more over groups or finite fields rather than on a continuous signal and polynomials work as the analog of sine waves in case of NTT. Fast Fourier Trnasform (FFT) style NTT or fast NTT has been proven to be useful in lattice-based cryptography due to its ability to reduce the complexity of polynomial multiplication from quadratic to quasilinear. We have introduced the concepts of cyclic, negacyclic convolutions along with NTT and its inverse and their fast versions.