On Constructing Families of Pairing-Friendly Elliptic Curves with Variable Discriminant

  • Robert Dryło
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7107)


In [10] Freeman, Scott and Teske consider three types of families: complete, sparse and complete with variable discriminant. A general method for constructing complete families is due to Brezing and Weng. In this note we generalize this method to construct families of the latter two types. As an application, we find variable-discriminant families for a few embedding degrees, which improve the previous best ρ-values of families given in [10].


Elliptic Curve Elliptic Curf Minimal Polynomial Variable Discriminant Complete Family 
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  1. 1.
    Barreto, P.S.L.M., Lynn, B., Scott, M.: Constructing Elliptic Curves with Prescribed Embedding Degrees. In: Cimato, S., Galdi, C., Persiano, G. (eds.) SCN 2002. LNCS, vol. 2576, pp. 257–267. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Barreto, P.S.L.M., 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
  3. 3.
    Boneh, D., Franklin, M.: Identity-Based Encryption from the Weil Pairing. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 213–229. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    Boneh, D., Lynn, B., Shacham, H.: Short Signatures from the Weil Pairing. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 514–532. Springer, Heidelberg (2001); Full version: J. Cryptol. 17, 297–319 (2004)CrossRefGoogle Scholar
  5. 5.
    Bisson, G., Satoh, T.: More Discriminants with the Brezing-Weng Method. In: Chowdhury, D.R., Rijmen, V., Das, A. (eds.) INDOCRYPT 2008. LNCS, vol. 5365, pp. 389–399. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Brezing, F., Weng, A.: Elliptic curves suitable for pairing based cryptography. Des. Codes Cryptogr. 37, 133–141 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cocks, C., Pinch, R.G.E.: Identity-based cryptosystems based on the Weil pairing (2001) (unpublished manuscript)Google Scholar
  8. 8.
    Cha, J.C., Cheon, J.H.: An Identity-Based Signature from Gap Diffie-Hellman Groups. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 18–30. Springer, Heidelberg (2002)Google Scholar
  9. 9.
    Freeman, D.: Constructing Pairing-Friendly Elliptic Curves with Embedding Degree 10. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 452–465. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Freeman, D., Scott, M., Teske, E.: A taxonomy of pairing-friendly elliptic curves. J. Cryptol. 23, 224–280 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Joux, A.: A one round protocol for tripartite DiffieHellman. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 385–393. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  12. 12.
    Galbraith, S., McKee, J., Valença, P.: Ordinary abelian varieties having small embedding degree. Finite Fields Appl. 13, 800–814 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kachisa, E.J., Schaefer, E.F., Scott, M.: Constructing Brezing-Weng Pairing-Friendly Elliptic Curves Using Elements in the Cyclotomic Field. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 126–135. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Miyaji, A., Nakabayashi, M., Takano, S.: New explicit conditions of elliptic curves traces for FR-reduction. IEICE Trans. Fundam. E84-A, 1234–1243 (2001)Google Scholar
  15. 15.
    Sakai, R., Ohgishi, K., Kasahara, M.: Cryptosystems based on pairings. In: 2000 Symposium on Cryptography and Information Security, SCIS 2000, Okinawa, Japan (2000)Google Scholar
  16. 16.
    Scott, M., Barreto, P.S.L.M.: Generating more MNT elliptic curves. Des. Codes Cryptogr. 38, 209–217 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Silverman, J.: The Arithmetic of Elliptic Curves. Springer, Berlin (1986)CrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Robert Dryło
    • 1
    • 2
  1. 1.Institute of MathematicsPolish Academy of SciencesWarszawaPoland
  2. 2.Instytut MatematykiUniwersytet Humanistyczno-Przyrodniczy w KielcachKielcePoland

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