On Various Families of Twisted Jacobi Quartics

  • Jérôme Plût
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7118)

Abstract

We provide several results on some families of twisted Jacobi quartics. We give new addition formulæ for two models of twisted Jacobi quartic elliptic curves, which represent respectively 1/6 and 2/3 of all elliptic curves, with respective costs 7M + 3S + D a and 8M + 3 S + D a . These formulæ allow addition and doubling of points, except for points differing by a point of order two.

Furthermore, we give an intrinsic characterization of elliptic curves represented by the classical Jacobi quartic, by the action of the Frobenius endomorphism on the 4-torsion subgroup. This allows us to compute the exact proportion of elliptic curves representable by various models (the three families of Jacobi quartics, plus Edwards and Huff curves) from statistics on this Frobenius action.

Keywords

Conjugacy Class Elliptic Curve Elliptic Curf Asymptotic Probability Intrinsic Characterization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [AG11]
    Ahmadi, O., Granger, R.: On isogeny classes of edwards curves over finite fields. Arxiv preprint arXiv:1103.3381 (2011)Google Scholar
  2. [BBJ+08]
    Bernstein, D.J., Birkner, P., Joye, M., Lange, T., Peters, C.: Twisted Edwards Curves. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 389–405. Springer, Heidelberg (2008), doi:10.1007/978-3-540-68164-9_26CrossRefGoogle Scholar
  3. [BJ03]
    Billet, O., Joye, M.: The Jacobi Model of an Elliptic Curve and Side-Channel Analysis. In: Fossorier, M., Høholdt, T., Poli, A. (eds.) AAECC 2003. LNCS, vol. 2643, pp. 34–42. Springer, Heidelberg (2003), doi:10.1007/3-540-44828-4_5CrossRefGoogle Scholar
  4. [BL07]
    Bernstein, D.J., Lange, T.: Inverted Edwards Coordinates. In: Boztaş, S., Lu, H.-F. (eds.) AAECC 2007. LNCS, vol. 4851, pp. 20–27. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. [CH11]
    Castryck, W., Hubrechts, H.: The distribution of the number of points modulo an integer on elliptic curves over finite fields (Preprint, 2011)Google Scholar
  6. [DIK06]
    Doche, C., Icart, T., Kohel, D.R.: Efficient Scalar Multiplication by Isogeny Decompositions. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T. (eds.) PKC 2006. LNCS, vol. 3958, pp. 191–206. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. [Edw07]
    Edwards, H.M.: A normal form for elliptic curves. Bulletin-American Mathematical Society 44(3), 393–422 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. [FW10]
    Feng, R., Wu, H.: On the isomorphism classes of legendre elliptic curves over finite fields. Arxiv preprint arXiv:1001.2871 (2010)Google Scholar
  9. [Gek06]
    Gekeler, E.-U.: The distribution of group structures on elliptic curves over finite prime fields. Documenta Mathematica 11, 119–142 (2006)MathSciNetMATHGoogle Scholar
  10. [HWCD08]
    Hisil, H., Wong, K.K.-H., Carter, G., Dawson, E.: Twisted Edwards Curves Revisited. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 326–343. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. [HWCD09]
    Hisil, H., Wong, K.K.-H., Carter, G., Dawson, E.: Faster group operations on elliptic curves. In: Proceedings of the Seventh Australasian Conference on Information Security, AISC 2009, vol. 98, pp. 7–20. Australian Computer Society, Inc., Darlinghurst (2009)Google Scholar
  12. [JTV10]
    Joye, M., Tibouchi, M., Vergnaud, D.: Huff’s Model for Elliptic Curves. In: Hanrot, G., Morain, F., Thomé, E. (eds.) ANTS-IX. LNCS, vol. 6197, pp. 234–250. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  13. [LS01]
    Liardet, P.-Y., Smart, N.P.: Preventing SPA/DPA in ECC Systems using the Jacobi Form. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 391–401. Springer, Heidelberg (2001), doi:10.1007/3-540-44709-1_32CrossRefGoogle Scholar
  14. [Mor09]
    Morain, F.: Edwards curves and cm curves. Arxiv preprint arXiv:0904.2243 (2009)Google Scholar
  15. [Nat00]
    National Institute of Standards and Technology. FIPS PUB 186-2: Digital Signature Standard (DSS) (January 2000)Google Scholar
  16. [Ono94]
    Ono, T.: Variations on a theme of Euler: quadratic forms, elliptic curves, and Hopf maps. Plenum. Pub. Corp. (1994)Google Scholar
  17. [RFS10]
    Farashahi, R.R., Shparlinski, I.: On the number of distinct elliptic curves in some families. Designs, Codes and Cryptography 54, 83–99 (2010), doi:10.1007/s10623-009-9310-2MathSciNetCrossRefMATHGoogle Scholar
  18. [Ser65]
    Serre, J.-P.: Zeta and L functions. In: Proc. Conf. on Arithmetical Algebraic Geometry, Purdue Univ., pp. 82–92. Harper & Row, New York (1965)Google Scholar
  19. [Sil86]
    Silverman, J.H.: The arithmetic of elliptic curves. Springer, Heidelberg (1986)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jérôme Plût
    • 1
  1. 1.Université de Versailles-Saint-Quentin-en-YvelinesVersaillesFrance

Personalised recommendations