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Pairing-Friendly Twisted Hessian Curves

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Progress in Cryptology – INDOCRYPT 2018 (INDOCRYPT 2018)

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Abstract

This paper presents efficient formulas to compute Miller doubling and Miller addition utilizing degree-3 twists on curves with j-invariant 0 written in Hessian form. We give the formulas for both odd and even embedding degrees and for pairings on both \(\mathbb {G}_1 \times \mathbb {G}_2\) and \(\mathbb {G}_{2} \times \mathbb {G}_{1}\). We propose the use of embedding degrees 15 and 21 for 128-bit and 192-bit security respectively in light of the NFS attacks and their variants. We give a comprehensive comparison with other curve models; our formulas give the fastest known pairing computation for embedding degrees 15, 21, and 24.

Chitchanok Chuengsatiansup acknowledges the support of Bpifrance in the context of the national projet RISQ (P141580). Chloe Martindale was supported by the Commission of the European Communities through the Horizon 2020 program under CHIST-ERA USEIT (NWO project 651.002.004).

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Chuengsatiansup, C., Martindale, C. (2018). Pairing-Friendly Twisted Hessian Curves. In: Chakraborty, D., Iwata, T. (eds) Progress in Cryptology – INDOCRYPT 2018. INDOCRYPT 2018. Lecture Notes in Computer Science(), vol 11356. Springer, Cham. https://doi.org/10.1007/978-3-030-05378-9_13

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  • DOI: https://doi.org/10.1007/978-3-030-05378-9_13

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