Cryptology ePrint Archive: Report 2009/155

Faster Computation of the Tate Pairing

Christophe Arene and Tanja Lange and Michael Naehrig and Christophe Ritzenthaler

Abstract: This paper proposes new explicit formulas for the doubling and addition steps in Miller's algorithm to compute the Tate pairing on elliptic curves in Weierstrass and in Edwards form. For Edwards curves the formulas come from a new way of seeing the arithmetic. We state the first geometric interpretation of the group law on Edwards curves by presenting the functions which arise in addition and doubling. The Tate pairing on Edwards curves can be computed by using these functions in Miller's algorithm.

Computing the sum of two points or the double of a point and the coefficients of the corresponding functions is faster with our formulas than with all previ ously proposed formulas for pairings on Edwards curves. They are even competitive with all published formulas for pairing computation on Weierstrass curves. We also improve the formulas for Tate pairing computation on Weierstrass curve s in Jacobian coordinates. Finally, we present several examples of pairing-friendly Edwards curves.

Category / Keywords: public-key cryptography / Pairing, Miller function, explicit formulas, Edwards curves

Date: received 3 Apr 2009, last revised 22 May 2010

Contact author: tanja at hyperelliptic org

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Version: 20100523:014032 (All versions of this report)

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