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Title:
Efficient Pairings Final Exponentiation Using Cyclotomic Cubing for Odd Embedding Degrees Curves
Authors: Walid Haddaji, Loubna Ghammam, Nadia El Mrabet, Leila Ben Abdelghani
Abstract:In pairings-based cryptographic applications, final exponentiation with a large fixed exponent ensures distinct outputs for the Tate pairing and its derivatives. Despite notable advancements in optimizing elliptic curves with even embedding degrees, improvements for those with odd embedding degrees, particularly those divisible by (3), remain underexplored. This paper introduces three methods for applying cyclotomic cubing in final exponentiation and enhancing computational efficiency. The first allows for the execution of one cyclotomic cubing based on the final exponentiation structure. The second leverages some existing seeds structure to enable the use of cyclotomic cubing and extends this strategy to generate new seeds. The third allows generating sparse ternary representation seeds to apply cyclotomic cubing as an alternative to squaring. These optimizations improve performance by up to 19.3\% when computing the final exponentiation for the optimal Ate pairing on BLS15 and BLS27, the target elliptic curves of this study.
ePrint: https://eprint.iacr.org/2025/958
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