High-Speed Polynomial Multiplication Architecture for Ring-LWE and SHE Cryptosystems

Abstract

Polynomial multiplication is the basic and most computationally intensive operation in ring-learning with errors (ring-LWE) encryption and somewhat homomorphic encryption (SHE) cryptosystems. In this paper, the fast Fourier transform (FFT) with a linearithmic complexity of $O(n\log n)$ , is exploited in the design of a high-speed polynomial multiplier. A constant geometry FFT datapath is used in the computation to simplify the control of the architecture. The contribution of this work is three-fold. First, parameter sets which support both an efficient modular reduction design and the security requirements for ring-LWE encryption and SHE are provided. Second, a versatile pipelined architecture accompanied with an improved dataflow are proposed to obtain a high-speed polynomial multiplier. Third, the proposed architecture supports polynomial multiplications for different lengths $n$ and moduli $p$ . The experimental results on a Spartan-6 FPGA show that the proposed design results in a speedup of 3.5 times on average when compared with the state of the art. It performs a polynomial multiplication in the ring-LWE scheme $(n=256,p=1049089)$ and the SHE scheme $(n=1024,p=536903681)$ in only 6.3 $\mu{\rm s}$ and 33.1 $\mu{\rm s}$ , respectively.