Algorithms and arithmetic operators for computing the $\eta_T$ pairing in characteristic three

Jean-Luc Beuchat 1 Nicolas Brisebarre 2 Jérémie Detrey 3 Eiji Okamoto 4 Masaaki Shirase 5 Tsuyoshi Takagi 5
1 LCIS - Laboratory of Cryptography and Information Security
2 LIP - Laboratoire de l'Informatique du Parallélisme
3 Cosec, B-IT - Computer Security group [Bonn]
4 LCIS - Laboratory of Cryptography and Information Security
5 Information Security Takagi Lab
Abstract : Since their introduction in constructive cryptographic applications, pairings over (hyper)elliptic curves are at the heart of an ever increasing number of protocols. Software implementations being rather slow, the study of hardware architectures became an active research area. In this paper, we discuss several algorithms to compute the $\eta_T$ pairing in characteristic three and suggest further improvements. These algorithms involve addition, multiplication, cubing, inversion, and sometimes cube root extraction over $\mathbb{F}_{3^m}$. We propose a hardware accelerator based on a unified arithmetic operator able to perform the operations required by a given algorithm. We describe the implementation of a compact coprocessor for the field $\mathbb{F}_{3^{97}}$ given by $\mathbb{F}_3[x]/(x^{97}+x^{12}+2)$, which compares favorably with other solutions described in the open literature.
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IEEE Transactions on Computers, Institute of Electrical and Electronics Engineers, 2008, Special Section on Special-Purpose Hardware for Cryptography and Cryptanalysis, 57 (11), pp.1454-1468. 〈10.1109/TC.2008.103〉
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Jean-Luc Beuchat, Nicolas Brisebarre, Jérémie Detrey, Eiji Okamoto, Masaaki Shirase, et al.. Algorithms and arithmetic operators for computing the $\eta_T$ pairing in characteristic three. IEEE Transactions on Computers, Institute of Electrical and Electronics Engineers, 2008, Special Section on Special-Purpose Hardware for Cryptography and Cryptanalysis, 57 (11), pp.1454-1468. 〈10.1109/TC.2008.103〉. 〈inria-00423993〉

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