Authors: Yu Dai, Debiao He, Cong Peng, Zhijian Yang, Changan Zhao

 

Title: Revisiting Pairing-Friendly Curves with Embedding Degrees 10 and 14

 

Conference:  The 30th Asian Cryptography Conference

 

Abstract: Since 2015, there has been a significant decrease in the asymptotic complexity of computing discrete logarithms in finite fields. As a result, the key sizes of many mainstream pairing-friendly curves have to be updated to maintain the desired security level. In PKC'20, Guillevic conducted a comprehensive assessment of the security of a series of pairing-friendly curves with embedding degrees ranging from $9$ to $17$.  In this paper, we focus on five pairing-friendly curves with embedding degrees 10 and 14 at the 128-bit security level, with BW14-351 emerging as the most competitive candidate. First, we extend the optimized formula for the optimal pairing on BW13-310, a 128-bit secure curve with a prime $p$ in 310 bits and embedding degree $13$, to our target curves. This generalization allows us to compute the optimal pairing in approximately $\log r/(2\varphi(k))$ Miller iterations, where $r$ and $k$ are the order of pairing groups and the embedding degree respectively. Second, we develop optimized algorithms for cofactor multiplication for $\G_1$ and $\G_2$, as well as subgroup membership testing for $\G_2$ on these curves. Finally, we provide detailed performance comparisons between BW14-351 and other popular curves on a 64-bit platform in terms of pairing computation, hashing to $\G_1$ and $\G_2$, group exponentiations, and subgroup membership testings. Our results demonstrate that BW14-351 is a strong candidate for building pairing-based cryptographic protocols.