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Tate Pairing Computation on Jacobi’s Elliptic Curves

  • Sylvain Duquesne
  • Emmanuel Fouotsa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7708)

Abstract

We propose for the first time the computation of the Tate pairing on Jacobi intersection curves. For this, we use the geometric interpretation of the group law and the quadratic twist of Jacobi intersection curves to obtain a doubling step formula which is efficient but not competitive compared to the case of Weierstrass curves, Edwards curves and Jacobi quartic curves. As a second contribution, we improve the doubling and addition steps in Miller’s algorithm to compute the Tate pairing on the special Jacobi quartic elliptic curve Y 2 = dX 4 + Z 4. 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 all the curves with quartic twists. In particular for the doubling step in Miller’s algorithm, we obtain a theoretical gain between 6% and 21%, depending on the embedding degree and the extension field arithmetic, with respect to Weierstrass curves [6] and Jacobi quartic curves [23].

Keywords

Jacobi quartic curves Jacobi intersection curves Tate pairing Miller function group law geometric interpretation birational equivalence 

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References

  1. 1.
    Arene, C., Lange, T., Naehrig, M., Ritzenthaler, C.: Faster computation of the Tate pairing. Journal of Number Theory 131(5), 842–857 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Billet, O., Joye, M.: The Jacobi Model of an Elliptic Curve and Side-Channel Analysis. In: Fossorier, M.P.C., Høholdt, T., Poli, A. (eds.) AAECC 2003. LNCS, vol. 2643, pp. 34–42. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Boneh, D., Franklin, M.: Identity-based encryption from the Weil pairing. SIAM Journal of Computing 32(3), 586–615 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chudnovsky, D.V., Chudnovky, G.V.: Sequences of numbers generated by addition in formal groups and new primality and factorization tests. Advances in Applied Mathematics 7(2), 385–434 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Costello, C., Hisil, H., Boyd, C., Nieto, J.G., Wong, K.K.-H.: Faster Pairings on Special Weierstrass Curves. In: Shacham, H., Waters, B. (eds.) Pairing 2009. LNCS, vol. 5671, pp. 89–101. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Costello, C., Lange, T., Naehrig, M.: Faster Pairing Computations on Curves with High-Degree Twists. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 224–242. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Das, M.P.L., Sarkar, P.: Pairing Computation on Twisted Edwards Form Elliptic Curves. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 192–210. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Duquesne, S., Frey, G.: Background on pairings. In: Cohen, H., Frey, G. (eds.) Handbook of Elliptic and Hyperelliptic Curves Cryptography, pp. 115–124. Chapman and Hall/CRC (2005)Google Scholar
  9. 9.
    Dutta, R., Barua, R., Sarkar, P.: Pairing-based cryptography: A survey. Cryptology ePrint Archive, Report 2004/064 (2004)Google Scholar
  10. 10.
    Feng, R., Nie, M., Wu, H.: Twisted Jacobi Intersections Curves. In: Kratochvíl, J., Li, A., Fiala, J., Kolman, P. (eds.) TAMC 2010. LNCS, vol. 6108, pp. 199–210. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Freeman, D., Scott, M., Teske, E.: A taxonomy of pairing-friendly elliptic curves. Journal of Cryptology 23(2), 224–280 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Frey, G., Müller, M., Rück, H.: The Tate Pairing and the Discrete Logarithm applied to Elliptic Curve Cryptosystems. IEEE Transactions on Information Theory 45(5), 1717–1719 (1999)zbMATHCrossRefGoogle Scholar
  13. 13.
    Galbraith, S.D.: Pairings. In: Seroussi, G., Blake, I., Smart, N. (eds.) Advances in Elliptic Curve Cryptography, pp. 193–213. Cambridge University Press (2005)Google Scholar
  14. 14.
    Galbraith, S.D., McKee, J.F., Valenca, P.C.: Ordinary abelian varieties having small embedding degree. Finite Fields Applications 13, 800–814 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Hisil, H., Wong, K.K., Carter, G., Dawson, E.: Faster group operations on elliptic curves. In: Australasian Information Security Conference (AISC), Wellington, New Zealand, vol. 98, pp. 7–19 (2009)Google Scholar
  16. 16.
    Hisil, H., Wong, K.K.-H., Carter, G., Dawson, E.: Jacobi Quartic Curves Revisited. In: Boyd, C., Nieto, J.G. (eds.) ACISP 2009. LNCS, vol. 5594, pp. 452–468. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  17. 17.
    Ionica, S., Joux, A.: Another Approach to Pairing Computation in Edwards Coordinates. In: Chowdhury, D.R., Rijmen, V., Das, A. (eds.) INDOCRYPT 2008. LNCS, vol. 5365, pp. 400–413. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  18. 18.
    Joux, A.: A One-Round Protocol for Tripartite Diffie-Hellman. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 385–394. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  19. 19.
    Koblitz, N., Menezes, A.: Pairing-Based Cryptography at High Security Levels. In: Smart, N.P. (ed.) Cryptography and Coding 2005. LNCS, vol. 3796, pp. 13–36. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  20. 20.
    Menezes, A., Okamoto, T., Vanstone, S.: Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Transactions on Information Theory 39(5), 1639–1646 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Merriman, J.R., Siksek, S., Smart, N.P.: Explicit 4-descents on an elliptic curve. Acta Arithmetica 77, 385–404 (1996)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Miller, S.V.: The Weil pairing, and its efficient calculation. Journal of Cryptology 17(4), 235–261 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Wang, H., Wang, K., Zhang, L., Li, B.: Pairing Computation on Elliptic Curves of Jacobi Quartic Form. Chinese Journal of Electronics 20(4), 655–661 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Sylvain Duquesne
    • 1
  • Emmanuel Fouotsa
    • 2
  1. 1.IRMAR, UMR CNRS 6625Université Rennes 1Rennes cedexFrance
  2. 2.Département de MathématiquesUniversité de Yaoundé 1, Faculté des SciencesYaoundéCameroun

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