Verifying Correctness of the Number Theoretic Transform and Fast Number Theoretic Transform in F⋆

Abstract

As engineers continue to develop more sophisticated algorithms to optimize cryptographic algorithms, their often simple mathematical specifications become convoluted in the algorithms, from which a class of correctness bugs arise. Because cryptographic algorithms often secure sensitive information, their correctness, and in turn their security is a top priority. The Number Theoretic Transform (NTT) is an algorithm that enables efficient polynomial multiplication and has recently gained importance in post-quantum cryptography. This thesis presents a proof of correctness of the NTT in F⋆ , a proof-oriented programming language that extracts to OCaml, and shows that we can use the NTT to perform polynomial multiplications. We provide an implementation of the Cooley-Tukey fast NTT algorithm and a proof that it matches the original NTT specification. This thesis also presents a representation of polynomials in the F⋆ subset Low*, which extracts to performant C code.