Pairing Calculation on Supersingular Genus 2 Curves

  • Colm Ó hÉigeartaigh
  • Michael Scott
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4356)


In this paper we describe how to efficiently implement pairing calculation on supersingular genus 2 curves over prime fields. We find that, contrary to the results reported in [8], pairing calculation on supersingular genus 2 curves over prime fields is efficient and a viable candidate for the practical implementation of pairing-based cryptosystems. We also show how to eliminate divisions in an efficient manner when computing the Tate pairing, assuming an even embedding degree, and how this algorithm is useful for curves of genus greater than one.


Tate pairing hyperelliptic curves pairing computation 


  1. 1.
    Barreto, P.S.L.M., Galbraith, S.D., Ó hÉigeartaigh, C., Scott, M.: Pairing computation on supersingular abelian varieties. Cryptology ePrint Archive, Report, 2004/375 (2004), Available from
  2. 2.
    Barreto, P.S.L.M., Kim, H.Y., Lynn, B., Scott, M.: Efficient algorithms for pairing-based cryptosystems. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 354–368. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  3. 3.
    Barreto, P.S.L.M., Lynn, B., Scott, M.: On the selection of pairing-friendly groups. In: Matsui, M., Zuccherato, R.J. (eds.) SAC 2003. LNCS, vol. 3006, pp. 17–25. Springer, Heidelberg (2004)Google Scholar
  4. 4.
    Blake, I.F., Seroussi, G., Smart, N.P.: Advances in elliptic curve cryptography. Cambridge (2005)Google Scholar
  5. 5.
    Boneh, D., Franklin, M.: Identity-based encryption from the weil pairing. SIAM Journal of Computing 32(3), 586–615 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cantor, D.G.: Computing in the jacobian of a hyperelliptic curve. Mathematics of Computation 48(177), 95–101 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Choie, Y., Jeong, E., Lee, E.: Supersingular hyperelliptic curves of genus 2 over finite fields. Journal of Applied Mathematics and Computation 163(2), 565–576 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Choie, Y., Lee, E.: Implementation of tate pairing on hyperelliptic curves of genus 2. In: Lim, J.-I., Lee, D.-H. (eds.) ICISC 2003. LNCS, vol. 2971, pp. 97–111. Springer, Heidelberg (2004)Google Scholar
  9. 9.
    Duursma, I., Lee, H.-S.: Tate pairing implementation for hyperelliptic curves y 2 = x p − x + d. In: Laih, C.-S. (ed.) ASIACRYPT 2003. LNCS, vol. 2894, pp. 111–123. Springer, Heidelberg (2003)Google Scholar
  10. 10.
    Frey, G., Lange, T.: Fast bilinear maps from the tate-lichtenbaum pairing on hyperelliptic curves. In: Hess, F., Pauli, S., Pohst, M. (eds.) Algorithmic Number Theory VII. LNCS, vol. 4076, pp. 466–479. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  11. 11.
    Frey, G., Rück, H.-G.: A remark concerning m-divisibility and the discrete logarithm problem in the divisor class group of curves. Mathematics of Computation 62(206), 865–874 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Galbraith, S., Harrison, K., Soldera, D.: Implementing the tate pairing. In: Fieker, C., Kohel, D.R. (eds.) Algorithmic Number Theory Symposium – ANTS V. LNCS, vol. 2369, pp. 324–337. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Galbraith, S.D., Paterson, K.G., Smart, N.P.: Pairings for cryptographers. Cryptology ePrint Archive, Report 2006/165 (2006),
  14. 14.
    Granger, R., Page, D., Smart, N.P.: High security pairing-based cryptography revisited. In: Hess, F., Pauli, S., Pohst, M. (eds.) Algorithmic Number Theory Symposium – ANTS VII. LNCS, vol. 4076, pp. 480–494. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Ó hÉigeartaigh, C.: Speeding up pairing computation (2005),
  16. 16.
    Hu, L., Dong, J.-W., Pei, D.-Y.: Implementation of cryptosystems based on tate pairing. Journal of Computer Science and Technology 20(2), 264–269 (2005)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Kobayashi, T., Aoki, K., Imai, H.: Efficient algorithms for tate pairing. IEICE Transactions Fundamentals, E89-A(1) (January 2006)Google Scholar
  18. 18.
    Lange, T.: Formulae for arithmetic on genus 2 hyperelliptic curves. Applicable Algebra in Engineering, Communication and Computing 15(5), 295–328 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lenstra, A.K., Verheul, E.R.: Selecting cryptographic key sizes. Journal of Cryptology 14(4), 255–293 (2001)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Miller, V.S.: Short programs for functions on curves. Unpublished manuscript (1986),
  21. 21.
    Miller, V.S.: The weil pairing and its efficient calculation. Journal of Cryptology 17(4), 235–261 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Miyamoto, Y., Doi, H., Matsuo, K., Chao, J., Tsuji, S.: A fast addition algorithm of genus two hyperelliptic curve. In: Symposium on Cryptography and Information Security – SCIS 2002, pp. 497–502.Google Scholar
  23. 23.
    Ribenboim, P.: Classical Theory of Algebraic Numbers. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  24. 24.
    Scott, M.: Miracl (multiprecision integer and rational arithmetic c/c++ library). Available from
  25. 25.
    Scott, M.: Faster identity based encryption. Electronics Letters 40(14), 861 (2004)CrossRefGoogle Scholar
  26. 26.
    Scott, M.: Computing the tate pairing. In: Menezes, A.J. (ed.) CT-RSA 2005. LNCS, vol. 3376, pp. 293–304. Springer, Heidelberg (2005)Google Scholar
  27. 27.
    Scott, M.: Scaling security in pairing-based protocols. Cryptology ePrint Archive, Report 2005/139 (2005),
  28. 28.
    Scott, M., Barreto, P.: Compressed pairings. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 140–156. Springer, Heidelberg (2004)Google Scholar
  29. 29.
    Smith, P., Skinner, C.: A public-key cryptosystem and a digital signature system based on the lucas function analogue to discrete logarithms. In: Safavi-Naini, R., Pieprzyk, J.P. (eds.) ASIACRYPT 1994. LNCS, vol. 917, pp. 357–364. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  30. 30.
    Solinas, J.: Generalized mersenne numbers. Technical Report CORR 99-39, University of Waterloo (1999), Available from

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Colm Ó hÉigeartaigh
    • 1
  • Michael Scott
    • 1
  1. 1.School of Computing, Dublin City University, Ballymun, Dublin 9Ireland

Personalised recommendations