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

2010; Springer Science+Business Media; Linguagem: Inglês

10.1007/978-3-642-17455-1_2

1611-3349

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

Algebraic Geometry and Number Theory

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 2.33 million of clock cycles on a single core of an Intel Core i7 2.8GHz processor, which implies that the pairing computation takes 0.832msec. 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 = 262 − 254 + 244. 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.