Binary Huff Curves

  • Julien Devigne
  • Marc Joye
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6558)

Abstract

This paper describes the addition law for a new form for elliptic curves over fields of characteristic 2. Specifically, it presents explicit formulæ for adding two different points and for doubling points. The case of differential point addition (that is, point addition with a known difference) is also addressed. Finally, this paper presents unified point addition formulæ; i.e., point addition formulæ that can be used for doublings. Applications to cryptographic implementations are discussed.

Keywords

Elliptic curves Huff curves binary fields cryptography 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bernstein, D.J.: Batch binary edwards. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 317–336. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Bernstein, D.J., Lange, T., Farashahi, R.R.: Binary Edwards curves. In: Oswald, E., Rohatgi, P. (eds.) CHES 2008. LNCS, vol. 5154, pp. 244–265. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Blake, I.F., Seroussi, G., Smart, N.P. (eds.): Advances in Elliptic Curve Cryptography. London Mathematical Society Lecture Note Series, vol. 317. Cambridge University Press, Cambridge (2005)MATHGoogle Scholar
  4. 4.
    Chudnovsky, D.V., Chudnovsky, G.V.: Sequences of numbers generated by addition in formal groups and new primality and factorization tests. Advances in Applied Mathematics 7(4), 385–434 (1986)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Explicit-formulas database (EFD), http://www.hyperelliptic.org/EFD/
  6. 6.
    Gaudry, P., Lubicz, D.: The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines. Finite Fields and Applications 15, 246–260 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Huff, G.B.: Diophantine problems in geometry and elliptic ternary forms. Duke Math. J. 15, 443–453 (1948)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Izu, T., Takagi, T.: Exceptional procedure attack on elliptic curve cryptosystems. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 224–239. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    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
  10. 10.
    Koblitz, N.: Elliptic curve cryptosystems. Mathematics of Computation 48, 203–209 (1987)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kocher, P., Jaffe, J., Jun, B.: Differential power analysis. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 388–397. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  12. 12.
    Lenstra Jr., H.W.: Factoring integers with elliptic curves. Annals of Mathematics 126(2), 649–673 (1987)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    López, J., Dahab, R.: Fast multiplication on elliptic curves over GF(2m) without precomputation. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 316–327. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  14. 14.
    Menezes, A., Okamoto, T., Vanstone, S.: Reducing elliptic curve discrete logaritms to a finite field. IEEE Transactions on Information Theory 39(5), 1639–1646 (1993)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Miller, V.S.: Use of elliptic curves in cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986)Google Scholar
  16. 16.
    Montgomery, P.L.: Speeding up the Pollard and elliptic curve methods of factorization. Mathematics of Computation 48(177), 243–264 (1987)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    National Institute of Standards and Technology: Recommended elliptic curves for federal government use (July 1999), http://csrc.nist.gov/CryptoToolkit/dss/ecdsa/NISTReCur.pdf
  18. 18.
    Peeples Jr., W.D.: Elliptic curves and rational distance sets. Proc. Am. Math. Soc. 5, 29–33 (1954)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Silverman, J.H.: The Arithmetic of Elliptic Curves. In: Graduate Texts in Mathematics, vol. 106, ch. III. Springer, Heidelberg (1986)Google Scholar
  20. 20.
    Stam, M.: On montgomery-like representationsfor elliptic curves over GF(2k). In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 240–253. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  21. 21.
    Stein, W.A., et al.: Sage Mathematics Software (Version 4.5.1). The Sage Development Team (2010), http://www.sagemath.org
  22. 22.
    Yen, S.M., Joye, M.: Checking before output not be enough against fault-based cryptanalysis. IEEE Transactions on Computers 49(9), 967–970 (2000)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Julien Devigne
    • 1
  • Marc Joye
    • 1
  1. 1.Technicolor, Security & Content Protection LabsCesson-Sévigné CedexFrance

Personalised recommendations