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An Alternate Decomposition of an Integer for Faster Point Multiplication on Certain Elliptic Curves

  • Young-Ho Park
  • Sangtae Jeong
  • Chang Han Kim
  • Jongin Lim
Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2274)

Abstract

In this paper the Gallant-Lambert-Vanstone method is reexamined for speeding up scalar multiplication. Using the theory of μ- Euclidian algorithm, we provide a rigorous method to reduce the theoretical bound for the decomposition of an integer k in the endomorphism ring of an elliptic curve. We then compare the two different methods for decomposition through computational implementations.

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References

  1. 1.
    Ian Blake, Gadiel Seroussi and Nigel Smart, ‘Elliptic Curves in Cryptography’, London Mathematical Society Lecture Note Series. 265, Cambridge University Press, (1999).Google Scholar
  2. 2.
    G. Cornacchia, “Su di un metodo per la risoluzione in numeri interi dell’ equazione Σhn=0C h x n−h y h = P”, Giornale di Matematiche di Battaglini, 46, (1908),33–90.Google Scholar
  3. 3.
    R. Gallant, R. Lambert and S. Vanstone, “Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms”, Advances in Cryptology-Crypto 2001, LNCS 2139, Springer-Verlag (2001), 190–200.CrossRefGoogle Scholar
  4. 4.
    K. Ireland and M. Rosen, ‘A classical introduction to modern number theory’, Graduate Texts in Mathematics, vol 84, Springer-Verlag, (1982).Google Scholar
  5. 5.
    N. Koblitz, “CM-curves with good cryptographic properties”, Advances in Cryptology-Crypto’ 91, LNCS 576, Springer-Verlag (1992), 279–287.CrossRefGoogle Scholar
  6. 6.
    V. Müller,“ Fast multiplication in elliptic curves over small fields of characteristic two”, Journal of Cryptology, 11 (1998), 219–234.zbMATHCrossRefGoogle Scholar
  7. 7.
    W. Meier and O. Staffelbach, “Efficient multiplication on certain nonsupersingular elliptic curves”, Advances in Cryptology-Crypto’92, Springer-Verlag (1992), 333–344.Google Scholar
  8. 8.
    S. Pohlig, M. Hellman,“An improved algorithm for computing logarithms over GF(p) its cryptographic significance,”, IEEE Trans. Inform. Theory, 24 (1978), 106–110.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    D. Shanks, “Five number theoretic algorithms” In Proc. 2nd Manitoba Conference on Numerical Mathematics (1972), 51–70.Google Scholar
  10. 10.
    B. Vallée,“Une approche géométrique des algorithmes de réduction des réseaux en petite dimension”, (1986) Thése, Université de Caen.Google Scholar
  11. 11.
    N. Smart, “Elliptic curve cryptosystems over small fields of odd characteristic”, Journal of Cryptology, 12 (1999), 141–145.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    J. Solinas, “An improved algorithm for arithmetic on a familyof elliptic curves”, Advances in Cryptology-Crypto’ 97, LNCS 1294, Springer-Verlag (1997), 357–371.CrossRefGoogle Scholar
  13. 13.
    J. Solinas, “Efficient arithmetic on Koblitz curves”, Design, Codes and Cryptography, 19 (2000), 195–249.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    I. Stewart and D. Tall, “Algebraic Number Theory”, Chapman and Hall, Halsted Press, (1979).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Young-Ho Park
    • 1
  • Sangtae Jeong
    • 2
  • Chang Han Kim
    • 3
  • Jongin Lim
    • 1
  1. 1.CISTKorea Univ.SeoulKorea
  2. 2.Dept. of Math.Seoul National Univ.SeoulKorea
  3. 3.Dept. of CAMISSemyung Univ.JechonKorea

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