Paper 2010/354

High-Speed Software Implementation of the Optimal Ate Pairing over Barreto-Naehrig Curves

Jean-Luc Beuchat, Jorge Enrique González Díaz, Shigeo Mitsunari, Eiji Okamoto, Francisco Rodríguez-Henríquez, and Tadanori Teruya


This paper describes the design of a fast software library for the computation of the optimal ate pairing on a Barreto--Naehrig elliptic curve. Our library is able to compute the optimal ate pairing over a $254$-bit prime field $\mathbb{F}_{p}$, in just $2.63$ million of clock cycles on a single core of an Intel Core i7 $2.8$GHz processor, which implies that the pairing computation takes $0.942$msec. We are able to achieve this performance by a careful implementation of the base field arithmetic through the usage of the customary Montgomery multiplier for prime fields. The prime field is constructed via the Barreto--Naehrig polynomial parametrization of the prime $p$ given as, $p = 36t^4 +36t^3 +24t^2 +6t+1$, with $t = 2^{62} - 2^{54} + 2^{44}$. This selection of $t$ allows us to obtain important savings for both the Miller loop as well as the final exponentiation steps of the optimal ate pairing.

Note: fix typo

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francisco @ cs cinvestav mx
2010-09-13: last of 7 revisions
2010-06-18: received
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      author = {Jean-Luc Beuchat and Jorge Enrique González Díaz and Shigeo Mitsunari and Eiji Okamoto and Francisco Rodríguez-Henríquez and Tadanori Teruya},
      title = {High-Speed Software Implementation of the Optimal Ate Pairing over Barreto-Naehrig Curves},
      howpublished = {Cryptology ePrint Archive, Paper 2010/354},
      year = {2010},
      note = {\url{}},
      url = {}
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