Paper 2025/2230 Original: eprint.iacr.org/2025/2230 ↗

Efficient Algorithms for $\mathbb{G}_2$ Subgroup Membership testing on Pairing-friendly Curves

Abstract

Subgroup membership testing serves as a crucial countermeasure against small subgroup attacks, thereby ensuring the security of pairing-based cryptographic protocols. Despite its vital importance, the expensive computational requirements for membership testing on specific pairing-friendly curves pose a non-negligible challenge. In this paper, we revisit the $\mathbb{G}_2$ membership testing algorithms on KSS16 curves and propose a novel approach specifically designed for the families constructed by the KSS method (Kachisa-Schaefer-Scott method). Moreover, we generalize several previous methods for $\mathbb{G}_2$ membership testing, rendering them applicable to more generic pairing-friendly curves. Specifically, we implement an efficient $\mathbb{G}_2$ membership testing on three well-known curves KSS16-329, KSS16-330, and KSS16-766 for verification. The experimental results illustrate that our new method achieves improvements of $24.0\%$, $33.3\%$, and $29.2\%$ in terms of clock cycles compared to the state-of-the-art, respectively.