Paper 2023/1958

Revisiting Pairing-friendly Curves with Embedding Degrees 10 and 14

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 $$ to $$. In this paper, we focus on pairing-friendly curves with embedding degrees of 10 and 14. First, we extend the optimized formula of the optimal pairing on BW13-310, a 128-bit secure curve with a prime $$ in 310 bits and embedding degree $$, to our target curves. This generalization allows us to compute the optimal pairing in approximately $$ Miller iterations, where $$ and $$ are the order of pairing groups and the embedding degree respectively. Second, we develop optimized algorithms for cofactor multiplication for $$ and $$, as well as subgroup membership testing for $$ on these curves. Based on these theoretical results a new 128-bit secure curve emerges: BW14-351. 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 $$ and $$, group exponentiations and subgroup membership testings. Our results demonstrate that BW14-351 is a strong candidate for building pairing-based cryptographic protocols.