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This paper proposes the computation of the Tate pairing, Ate pairing and its variations on the special Jacobi quartic elliptic curve Y2=dX4+Z4. We improve the doubling and addition steps in Miller's algorithm to compute the Tate pairing. We use the birational equivalence between Jacobi quartic curves and Weierstrass curves, together with a specific point representation to obtain the best result to date among curves with quartic twists. For the doubling and addition steps in Miller's algorithm for the computation of the Tate pairing, we obtain a theoretical gain up to 27% and 39%, depending on the embedding degree and the extension field arithmetic, with respect to Weierstrass curves and previous results on Jacobi quartic curves. Furthermore and for the first time, we compute and implement Ate, twisted Ate and optimal pairings on the Jacobi quartic curves. Our results are up to 27% more efficient compared to the case of Weierstrass curves with quartic twists.

quartic form Throughout this subsection, K denotes a fixed field of odd characteristic and L an algebraic extension of K. Let d; a 2 K. Assume that d is a square in L unless stated otherwise. Definition 2.8. An extended Jacobi quartic curve defined over K is the curve EQ;d;a W y 2 D dx4 C 2ax2 C 1: This curve is non-singular if and only if d.a2 d/ ¤ 0. The j -invariant is given by 64.a2C 3d/3=.d.a2 d/2/ 2 K. The projective closure of EQ is given by the equation EQ;d;a W Y 2Z2 D dX4 C 2aX2Z2 CZ4: A point .X WY WZ/withZ ¤ 0 onEQ corresponds to the affine point .X=Z; Y

.-H. Wong Gary Carter Ed Dawson Twisted Edwards curves revisited Pieprzyk J. Advances in Cryptology – ASIACRYPT 2008, volume 5350 of Lecture Notes in Computer Science 326 343 Springer 2008 10.1007/978-3-540-89255-7_20 [31] Hüseyin Hışıl, Kenneth K.-H. Wong, Gary Carter, and Ed Dawson. Jacobi quartic curves revisited. In C. Boyd and J. M. G. Nieto, editors, Information Security and Privacy (ACISP 2009) , volume 5594 of Lecture Notes in Computer Science , pages 452–468. Springer, 2009. 10.1007/978-3-642-02620-1_31. Hüseyin Hışıl Kenneth K.-H. Wong Gary Carter Ed Dawson