Improved Weil and Tate Pairings for Elliptic and Hyperelliptic Curves

  • Kirsten Eisenträger
  • Kristin Lauter
  • Peter L. Montgomery
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3076)

Abstract

We present algorithms for computing the squared Weil and Tate pairings on elliptic curves and the squared Tate pairing on hyperelliptic curves. The squared pairings introduced in this paper have the advantage that our algorithms for evaluating them are deterministic and do not depend on a random choice of points. Our algorithm to evaluate the squared Weil pairing is about 20% more efficient than the standard Weil pairing. Our algorithm for the squared Tate pairing on elliptic curves matches the efficiency of the algorithm given by Barreto, Lynn, and Scott in the case of arbitrary base points where their denominator cancellation technique does not apply. Our algorithm for the squared Tate pairing for hyperelliptic curves is the first detailed implementation of the pairing for general hyperelliptic curves of genus 2, and saves an estimated 30% over the standard algorithm.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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
  2. 2.
    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, Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Boneh, D., Franklin, M.: Identity-based encryption from the Weil pairing. SIAM J. Comput. 32(3), 586–615 (electronic) (2003)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cantor, D.G.: Computing in the Jacobian of a hyperelliptic curve. Math. Comp. 48(177), 95–101 (1987)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Duursma, I.M., Lee, H.-S.: Tate Pairing Implementation for Hyperelliptic Curves y2 = xp − x + d. In: Laih, C.-S. (ed.) ASIACRYPT 2003. LNCS, vol. 2894, pp. 111–123. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Eisenträger, K., Lauter, K., Montgomery, P.L.: Fast elliptic curve arithmetic and improved Weil pairing evaluation. In: Joye, M. (ed.) CT-RSA 2003. LNCS, vol. 2612, pp. 343–354. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Frey, G., Rück, H.-G.: A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves. Math. Comp. 62(206), 865–874 (1994)MATHMathSciNetGoogle Scholar
  8. 8.
    Galbraith, S., Harrison, K., Soldera, D.: Implementing the Tate Pairing. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 324–337. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Joux, A.: The Weil and Tate pairings as building blocks for public key cryptosystems (survey). In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 20–32. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  10. 10.
    Miller, V.S.: Short programs for functions on curves (1986) (unpublished manuscript)Google Scholar
  11. 11.
    Silverman, J.: The Arithmetic of Elliptic Curves. GTM 106, Springer (1986)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Kirsten Eisenträger
    • 1
  • Kristin Lauter
    • 2
  • Peter L. Montgomery
    • 2
  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  2. 2.Microsoft ResearchRedmondUSA

Personalised recommendations