Faster Number Theoretic Transform on Graphics Processors for Ring Learning with Errors Based Cryptography

Abstract

The Number Theoretic Transform (NTT) has been revived recently by the advent of the Ring-Learning with Errors (Ring-LWE) Homomorphic Encryption (HE) schemes. In these schemes, the NTT is used to calculate the product of high degree polynomials with multi-precision coefficients in quasilinear time. This is known as the most time-consuming operation in Ring–based HE schemes. Therefore; accelerating NTT is key to realize efficient implementations. As such, in its current version, a fast NTT implementation is included in cuHE, which is a publicly available HE library in Compute Unified Device Architecture (CUDA). We analyzed cuHE NTT kernels and found out that they suffer from two performance pitfalls: shared memory conflicts and thread divergence. We show that by using a set of CUDA tailored-made optimizations, we can improve on the speed of cuHE NTT computation by 20%-50% for different problem sizes.