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

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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 construc