Advertisement

An improved algorithm for arithmetic on a family of elliptic curves

  • Jerome A. Solinas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1294)

Abstract

It has become increasingly common to implement discrete-logarithm based public-key protocols on elliptic curves over finite fields. The basic operation is scalar multiplication: taking a given integer multiple of a given point on the curve. The cost of the protocols depends on that of the elliptic scalar multiplication operation.

Koblitz introduced a family of curves which admit especially fast elliptic scalar multiplication. His algorithm was later modified by Meier and Staffelbach. We give an improved version of the algorithm which runs 50% faster than any previous version. It is based on a new kind of representation of an integer, analogous to certain kinds of binary expansions. We also outline further speedups using precomputation and storage.

Keywords

elliptic curves exponentiation public-key cryptography 

References

  1. 1.
    D. Gordon, “A survey of fast exponentiation methods” (to appear).Google Scholar
  2. 2.
    D. E. Knuth, Seminumerical Algorithms, Addison-Wesley, 1981, p. 272.Google Scholar
  3. 3.
    F. Morain and J. Olivos, “Speeding up the computations on an elliptic curve using addition-subtraction chains”, Inform. Theor. Appl. 24 (1990), pp. 531–543.zbMATHMathSciNetGoogle Scholar
  4. 4.
    A. Menezes, T. Okamoto and S. Vanstone, “Reducing elliptic curve logarithms to logarithms in a finite field”, Proc. 23rd Annual ACM Symp. on Theory of Computing (1991), pp. 80–89.Google Scholar
  5. 5.
    N. Koblitz, “CM curves with good cryptographic properties”, Proc. Crypto '91, Springer-Verlag, 1992, pp. 279–287.Google Scholar
  6. 6.
    D. W. Ash, I. F. Blake, and S. Vanstone, “Low complexity normal bases”, Discrete Applied Math. 25 (1989), pp. 191–210.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    T. Itoh, O Teechai, and S. Trojii, “A fast algorithm for computing multiplicative inverses in GF(2t)”, J. Soc. Electron. Comm. (Japan) 44 (1986), pp. 31–36.Google Scholar
  8. 8.
    E. Berlekamp, Algebraic Coding Theory, Aegean Park Press, 1984, pp. 36–44.Google Scholar
  9. 9.
    A. Menezes, P. van Oorschot, and S. Vanstone, Handbook of Applied Cryptography, CRC Press, 1997, pp. 107–109.Google Scholar
  10. 10.
    W. Meier and O. Staffelbach, “Efficient multiplication on certain non-supersingular elliptic curves”, Proc. Crypto '92, Springer-Verlag, 1993, pp. 333–344.Google Scholar
  11. 11.
    R. Reiter and J. Solinas, “Fast elliptic arithmetic on special curves”, NSA/R21 Informal Tech. Report, 1997.Google Scholar
  12. 12.
    K. Koyama and Y. Tsuruoka, “Speeding up elliptic cryptosystems by using a signed binary window method”, Proc. Crypto '92, Springer-Verlag, 1993, pp. 345–357.Google Scholar
  13. 13.
    R. Schroeppel, H. Orman, S. O'Malley, and O. Spatscheck, “Fast key exchange with elliptic curve systems”, Proc. Crypto '95, Springer-Verlag, 1995, pp. 43–56.Google Scholar
  14. 14.
    R. Schroeppel, H. Orman, S. O'Malley, and O. Spatscheck, “Fast key exchange with elliptic curve systems”, Univ. of Arizona. Comp. Sci. Tech. Report 95-03, 1995.Google Scholar
  15. 15.
    F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier, 1977, pp. 277–279.Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Jerome A. Solinas
    • 1
  1. 1.National Security AgencyFt. MeadeUSA

Personalised recommendations