Welcome to the resource topic for
**2023/1958**

**Title:**

Revisiting Pairing-friendly Curves with Embedding Degrees 10 and 14

**Authors:**
Yu Dai, Debiao He, Cong Peng, Zhijian Yang, Chang-an Zhao

**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 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 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 \mathbb{G}_1 and \mathbb{G}_2, as well as subgroup membership testing for \mathbb{G}_2 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 \mathbb{G}_1 and \mathbb{G}_2, group exponentiations and subgroup membership testings. Our results demonstrate that BW14-351 is a strong candidate for building pairing-based cryptographic protocols.

**ePrint:**
https://eprint.iacr.org/2023/1958

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