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Science China Information Sciences

, Volume 53, Issue 8, pp 1528–1538 | Cite as

Twisted Ate pairing on hyperelliptic curves and applications

  • FangGuo Zhang
Research Papers

Abstract

In this paper we show that the twisted Ate pairing on elliptic curves can be generalized to hyperelliptic curves, and give a series of variations of the hyperelliptic Ate and twisted Ate pairings. Using the hyperelliptic Ate pairing and twisted Ate pairing, we propose a new approach to speeding up the Weil pairing computation. For some hyperelliptic curves with high degree twist, computing Weil pairing by our approach may be faster than Tate pairing, Ate pairing, and all other known pairings.

Keywords

Ate pairing Weil pairing hyperelliptic curves pairing-based cryptosystems twisted curves 

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References

  1. 1.
    Miller V S. Short programs for functions on curves. Unpublished manuscript, 1986, [online]. Available: http://crypto.stanford.edu/miller/miller.pdf
  2. 2.
    Galbraith S D. Pairings. Ch. IX. In: Blake I F, Seroussi G, Smart N P, eds. Advances in Elliptic Curve Cryptography. Cambridge: Cambridge University Press, 2005Google Scholar
  3. 3.
    Hu L, Dong J, Pei D. An implementation of cryptosystems based on Tate pairing. J Comput Sci Tech, 2005, 20: 264–269CrossRefMathSciNetGoogle Scholar
  4. 4.
    Zhao C A, Zhang F, Huang J. Efficient Tate pairing computation using double-base chains. Sci China Ser F-Inf Sci, 2008, 51: 1096–1105zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Duursma I M, Lee H S. Tate pairing implementation for hyperelliptic curves y 2 = x px + d. In: ASIACRYPT 2003, LNCS 2894. Berlin: Springer, 2003. 111–123Google Scholar
  6. 6.
    Barreto P S L M, Galbraith S, ÓhÉigeartaigh C, et al. Efficient pairing computation on supersingular abelian varieties. Design Code Cryptography, 2007, 42: 239–271zbMATHCrossRefGoogle Scholar
  7. 7.
    Hess F, Smart N P, Vercauteren F. The Eta pairing revisited. IEEE Trans Inf Theory, 2006, 52: 4595–4602zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Matsuda S, Kanayama N, Hess F, et al. Optimised versions of the Ate and twisted Ate pairings. In: The 11th IMA International Conference on Cryptography and Coding, LNCS 4887. Berlin: Springer-Verlag, 2007. 302–312CrossRefGoogle Scholar
  9. 9.
    Zhao C A, Zhang F, Huang J. A note on the Ate pairing. Int J Inf Secur, 2008, 7: 379–382CrossRefGoogle Scholar
  10. 10.
    Lee E, Lee H S, Park C M. Efficient and generalized pairing computation on abelian varieties. IEEE Trans Inf Theory, 2009, 55: 1793–1803CrossRefGoogle Scholar
  11. 11.
    Vercauteren F. Optimal pairings. Preprint, 2008. Available at http://eprint.iacr.org/2008/096
  12. 12.
    Granger R, Hess F, Oyono R, et al. Ate pairing on hyperelliptic curves. In: Advance in Cryptology-EUROCRYPT’2007, LNCS 4515. Berlin: Springer-Verlag, 2007. 430–447CrossRefGoogle Scholar
  13. 13.
    Galbraith S D, Hess F, Vercauteren F. Hyperelliptic pairings. In: Pairing 2007, LNCS 4575. Berlin: Springer-Verlag, 2007. 108–131CrossRefGoogle Scholar
  14. 14.
    Freeman D. Constructing pairing-friendly genus 2 curves over prime fields with ordinary jacobians. In: Pairing 2007, LNCS 4575. Berlin: Springer-Verlag, 2007. 152–176CrossRefGoogle Scholar
  15. 15.
    Cocks C, Pinch R G E. Identity-based cryptosystems based on the Weil pairing, unpublished manuscript, 2001Google Scholar
  16. 16.
    Kawazoe M, Takahashi T. Pairing-friendly hyperelliptic curves with ordinary jacobians of type y 2 = x 5+ax. In: Pairing 2008, LNCS 5209. Berlin: Springer-Verlag, 2008. 164–177CrossRefGoogle Scholar
  17. 17.
    Koblitz N, Menezes A. Pairing-based cryptography at high security levels. In: Cryptography and Coding, LNCS 3796. Berlin: Springer-Verlag, 2005. 235–249CrossRefGoogle Scholar
  18. 18.
    Granger R, Page D, Smart N P. High security pairing-based cryptography revisited. In: Hess F, Pauli S, Pohst M, eds. ANTS-VII, LNCS 4076. Berlin: Springer, 2006. 480–494Google Scholar
  19. 19.
    Frey G, Rück H G. A remark concerning m-divisibility and the discrete logarithm problem in the divisor class group of curves. Math Comput, 1994, 62: 865–874zbMATHGoogle Scholar
  20. 20.
    Silverman J H. The Arithmetic of Elliptic Curves. New York: Springer-Verlag, 1986zbMATHGoogle Scholar
  21. 21.
    MAGMA Computational Algebra System, http://magma.maths.usyd.edu.au/magma
  22. 22.
    Bolza O. On binary sextics with linear transformations between themselves. Amer J Math, 1888, 10: 47–70CrossRefMathSciNetGoogle Scholar
  23. 23.
    Cardona G. On the number of curves of genus 2 over a finite field. Finite Fields Appl, 2003, 9: 505–526zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Duursma I M, Gaudry P, Morain F. Speeding up the discrete log computation on curves with automorphisms. In: AsiaCrypt’99, LNCS 1716. Berlin: Springer-Verlag, 1999. 103–121Google Scholar
  25. 25.
    Zhao C A, Zhang F. Reducing the complexity of the Weil pairing computation. Preprint, 2008. Available at http://eprint.iacr.org/2008/212

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.School of Information Science and TechnologySun Yat-Sen UniversityGuangzhouChina
  2. 2.Guangdong Key Laboratory of Information Security TechnologyGuangzhouChina

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