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More Discriminants with the Brezing-Weng Method

  • Gaetan Bisson
  • Takakazu Satoh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5365)

Abstract

The Brezing-Weng method is a general framework to generate families of pairing-friendly elliptic curves. Here, we introduce an improvement which can be used to generate more curves with larger discriminants. Apart from the number of curves this yields, it provides an easy way to avoid endomorphism rings with small class number.

Keywords

Pairing-friendly curve generation Brezing-Weng method 

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References

  1. 1.
    Sakai, R., Ohgishi, K., Kasahara, M.: Cryptosystems based on pairing. In: Proceedings of the Symposium on Cryptography and Information Security (2000); ref. C20Google Scholar
  2. 2.
    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
  3. 3.
    Boneh, D., Franklin, M.: Identity-based encryption from the Weil pairing. SIAM Journal of Computing 32(3), 586–615 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Menezes, A., Okamoto, T., Vanstone, S.: Reducing elliptic curve logarithms in a finite field. IEEE Transactions on Information Theory 39(5), 1639–1646 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cocks, C., Pinch, R.: Identity-based cryptosystems based on the Weil pairing (Unpublished manuscript, 2001)Google Scholar
  6. 6.
    Brezing, F., Weng, A.: Elliptic curves suitable for pairing based cryptography. Design, Codes and Cryptography 37(1), 133–141 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abhandlungen aus dem mathematischen Seminar der hamburgischen Universität 14, 197–272 (1941)CrossRefzbMATHGoogle Scholar
  8. 8.
    Heilbronn, H.: On the class-number in imaginary quadratic fields. Quarterly Journal of Mathematics 5, 150–160 (1934)CrossRefzbMATHGoogle Scholar
  9. 9.
    Bateman, P., Horn, R.: Primes represented by irreducible polynomials in one variable. In: Proceedings of Symposia in Pure Mathematics, vol. 3, pp. 119–132. American Mathematical Society (1965)Google Scholar
  10. 10.
    Schinzel, A., Sierpinski, W.: Sur certaines hypothèses concernant les nombres premiers. Acta Arithmetica 4, 185–208 (1958)MathSciNetGoogle Scholar
  11. 11.
    Freeman, D., Scott, M., Teske, E.: A taxonomy of pairing-friendly elliptic curves. Cryptology ePrint Archive, Report 2006/372 (2006)Google Scholar
  12. 12.
    Galbraith, S.: Constructing isogenies between elliptic curves over finite fields. The London Mathematical Society Journal of Computation and Mathematics 2, 118–138 (1999)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Enge, A.: The complexity of class polynomial computation via floating point approximations. ArXiv preprint, cs.CC/0601104 (2006)Google Scholar
  14. 14.
    Siegel, C.: Über die Classenzahl quadratischer Zahlkörper. Acta Arithmetica 1, 83–86 (1935)zbMATHGoogle Scholar
  15. 15.
    Berndt, B., Evans, R., Williams, K.: Gauss and Jacobi sums. John Wiley & Sons, Chichester (1998)zbMATHGoogle Scholar
  16. 16.
    Barreto, P., Naehrig, M.: Pairing-friendly elliptic curves of prime order. In: Preneel, B., Tavares, S. (eds.) SAC 2005. LNCS, vol. 3897, pp. 319–331. Springer, Heidelberg (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Gaetan Bisson
    • 1
    • 2
  • Takakazu Satoh
    • 3
  1. 1.LORIAVandoeuvre-lès-NancyFrance
  2. 2.Technische Universiteit EindhovenEindhovenThe Netherlands
  3. 3.Tokyo Institute of TechnologyTokyoJapan

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