When NTT Meets Karatsuba: Preprocess-then-NTT Technique Revisited

Abstract

The Number Theoretic Transform (NTT) technique is widely used to implement cryptographic schemes based on the Ring Learning With Errors problem(RLWE), since it provides efficient algorithm for multiplication of polynomials over the finite field. However, the module in NTT must be big enough such that the finite field has some special root of unity, which makes the corresponding schemes a bit less efficient. At Inscrypt 2018, Zhou et al. proposed a technique called Preprocess-then-NTT to relax the constraint for the modulus while keeping NTT work, at the cost of time complexity. In this paper, we improve the Preprocess-then-NTT technique by mixing it with Karatsuba multiplication such that the time complexity is better than the original NTT algorithm asymptotically. The claim was also verified in our experiments. As a result, our new algorithm not only relaxes the constraint for modulus but also improves efficiency in practice. In addition, we also present some advantages and applications of our new algorithm.