Faster Ate pairing computation on Selmer's model of elliptic curves

Emmanuel Fouotsa 1  and Abdoul Aziz Ciss 2
  • 1 Laboratoire de Mathématiques Nicolas Oresme(LMNO), Université de Caen, Basse Normandie B.P. 5186, 14032 Caen Cedex, France; and Department of Mathematics, Higher Teacher Training College, University of Bamenda, P.O. Box 39, Bambili, Cameroon
  • 2 Laboratoire de Traitement de l'Information et Systèmes Intelligents, Ecole Polytechnique de Thies, Senegal


This paper revisits the computation of pairings on a model of elliptic curve called Selmer curves. We extend the work of Zhang, Wang, Wang and Ye to the computation of other variants of the Tate pairing on this curve. Especially, we show that the Selmer model of an elliptic curve presents faster formulas for the computation of the Ate and optimal Ate pairings with respect to Weierstrass elliptic curves. We show how to parallelise the computation of these pairings and we obtained very fast results. We also present an example of optimal pairing on a pairing-friendly Selmer curve of embedding degree k = 12.

  • 1

    D. F. Aranha, L. Fuentes-Castañeda, E. Knapp, A. Menezes and F. Rodríguez-Henríquez, Implementing pairings at the 192-bit security level, Pairing-Based Cryptography – Pairing 2012 (Cologne 2012), Lecture Notes in Comput. Sci. 7708, Springer, Berlin (2013), 177–195.

  • 2

    P. S. L. M. Barreto, S. D. Galbraith, C. O'Eigeartaigh and M. Scott, Efficient pairing computation on supersingular abelian varieties, Des. Codes Cryptogr. 42 (2007), 3, 239–271.

  • 3

    D. Boneh and M. K. Franklin, Identity-based encryption from the weil pairing, Advances in Cryptology – CRYPTO 2001 (Santa Barbara 2001), Lecture Notes in Comput. Sci. 2139, Springer, Berlin (2001), 213–229.

  • 4

    W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), 3–4, 235–265.

  • 5

    C. Costello, T. Lange and M. Naehrig, Faster pairing computations on curves with high-degree twists, Public Key Cryptography – PKC 2010 (Paris 2010), Lecture Notes in Comput. Sci. 6056, Springer, Berlin (2010), 224–242.

  • 6

    S. Duquesne and G. Frey, Background on pairings, Handbook of Elliptic and Hyperelliptic Curve Cryptography, CRC Press, Boca Raton (2005), 115–124.

  • 7

    R. Dutta, R. Barua and P. Sarkar, Pairing-based cryptographic protocols: A survey, IACR Cryptol. ePrint Arch. 2004 (2004), Paper No. 64.

  • 8

    R. R. Farashahi and M. Joye, Efficient arithmetic on Hessian curves, Public Key Cryptography – PKC 2010 (Paris 2010), Lecture Notes in Comput. Sci. 6056, Springer, Berlin (2010), 243–260.

  • 9

    D. Freeman, M. Scott and E. Teske, A taxonomy of pairing-friendly elliptic curves, J. Cryptology 23 (2010), 2, 224–280.

  • 10

    G. Frey, M. Müller and H. Rück, The tate pairing and the discrete logarithm applied to elliptic curve cryptosystems, IEEE Trans. Inform. Theory 45 (1999), 5, 1717–1719.

  • 11

    S. Galbraith, Pairings, Advances in Elliptic Curve Cryptography, London Math. Soc. Lecture Note Ser. 317, Cambridge University Press, Cambridge (2005), 183–213.

  • 12

    F. Hess, Pairing lattices, Pairing-Based Cryptography – Pairing 2008 (Egham 2008), Lecture Notes in Comput. Sci. 5209, Springer, Berlin (2008), 18–38.

  • 13

    F. Hess, N. P. Smart and F. Vercauteren, The eta pairing revisited, IEEE Trans. Inform. Theory 52 (2006), 10, 4595–4602.

  • 14

    A. Joux, A one round protocol for tripartite Diffie–Hellman, Algorithmic Number Theory – ANTS-IV (Leiden 2000), Lecture Notes in Comput. Sci. 1838, Springer, Berlin (2008), 385–393.

  • 15

    V. S. Miller, The Weil pairing, and its efficient calculation, J. Cryptology 17 (2004), 4, 235–261.

  • 16

    F. Vercauteren, Optimal pairings, IEEE Trans. Inform. Theory 56 (2010), 1, 455–461.

  • 17

    L. Zhang, K. Wang, H. Wang and D. Ye, Another elliptic curve model for faster pairing computation, Information Security Practice and Experience – ISPEC 2011 (Guangzhou 2011), Lecture Notes in Comput. Sci. 6672, Springer, Berlin (2011), 432–446.

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Groups – Complexity – Cryptology is a journal for speedy publication of articles in the areas of combinatorial and computational group theory, computer algebra, complexity theory, and cryptology. GCC primarily publishes research papers, but comprehensive and timely survey articles on a topic inside the scope of the journal are also welcome.