International Association
for Cryptologic Research

Many lattice based cryptosystems are based on the Ring learning with errors (Ring-LWE) problem. The most critical and computationally intensive operation of these Ring-LWE based cryptosystems is polynomial multiplication over rings. In this paper, we exploit the number theoretic transform (NTT) to build a family of scalable polynomial multiplier architectures, which provide designers with a trade-off choice of speed vs. area. Our polynomial multipliers are capable to calculate the product of two $n$-degree polynomials in about $(1.5n\lg n + 1.5n)/b$ clock cycles, where $b$ is the number of the butterfly operators. In addition, we exploit the cancellation lemma to reduce the required ROM storage. The experimental results on a Spartan-6 FPGA show that the proposed polynomial multiplier architectures achieve a speedup of 3 times on average and consume less Block RAMs and slices when compared with the compact design. Compared with the state of the art of high-speed design, the proposed hardware architectures save up to 46.64\% clock cycles and improve the utilization rate of the main data processing units by 42.27\%. Meanwhile, our designs can save up to 29.41\% block RAMs.