Speeding up the number theoretic transform for faster ideal lattice-based cryptography

Abstract

The Number Theoretic Transform (NTT) provides efficient algorithms for cyclic and nega-cyclic convolutions, which have many applications in computer arithmetic, e.g., for multiplying large integers and large degree polynomials. It is commonly used in cryptographic schemes that are based on the hardness of the Ring LearningWith Errors (R-LWE) problem to efficiently implement modular polynomial multiplication. We present a new modular reduction technique that is tailored for the special moduli required by the NTT. Based on this reduction, we speed up the NTT and propose faster, multi-purpose algorithms. We present two implementations of these algorithms: a portable C implementation and a high-speed implementation using assembly with AVX2 instructions. To demonstrate the improved efficiency in an application example, we benchmarked the algorithms in the context of the R-LWE key exchange protocol that has recently been proposed by Alkim, Ducas, P¨oppelmann and Schwabe. In this case, our C and assembly implementations compute the full key exchange 1.44 and 1.21 times faster, respectively. These results are achieved with full protection against timing attacks.