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

Abstract

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