Faster $\mathbb{F}_p$ -Arithmetic for Cryptographic Pairings on Barreto-Naehrig Curves

Abstract

This paper describes a new method to speed up $\mathbb{F}_p$ -arithmetic for Barreto-Naehrig (BN) curves. We explore the characteristics of the modulus defined by BN curves and choose curve parameters such that $\mathbb{F}_p$ multiplication becomes more efficient. The proposed algorithm uses Montgomery reduction in a polynomial ring combined with a coefficient reduction phase using a pseudo-Mersenne number. With this algorithm, the performance of pairings on BN curves can be significantly improved, resulting in a factor 5.4 speed-up compared with the state-of-the-art hardware implementations. Using this algorithm, we implemented a pairing processor in hardware, which runs at 204 MHz and finishes one ate and R-ate pairing computation over a 256-bit BN curve in 4.22 ms and 2.91 ms, respectively.