An Improved Method of Multiplication on Certain Elliptic Curves

  • Young-Ho Park
  • Sangho Oh
  • Sangjin Lee
  • Jongin Lim
  • Maenghee Sung
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2274)


The Frobenius endomorphism is known to be useful in efficient implementation of multiplication on certain elliptic curves. In this note a method to minimize the length of the Frobenius expansion of integer multiplier, ellipticc urves defined over small finite fields, is introduced. It is an optimization of previous works by Solinas and Müller. Finally, experimental results are presented and compared with curves recommended in standards by time-performance of multiplication.


  1. 1.
    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
  2. 2.
    P. Gaudry, F. Hess and N. Smart, “Constructive and destructive facets of Weil descent on elliptic curves”, to appear J. Cryptology.Google Scholar
  3. 3.
    R. Gallant, R. Lambert, and S. Vanstone, “Improving the parallelized Pollard lambda search on binary anomalous curves”, Math. of Com., 69 (2000), 1699–1705.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    N. Koblitz, “CM-curves with good cryptographic properties”, In Advances in Cryptology, CRYPTO 91, LNCS 576, Springer-Verlag (1992), 279–287.Google Scholar
  5. 5.
    W. Meier, O. Staffelbach, “Efficient multiplication on certain nonsupersingular elliptic curves”, Advances in Cryptology, Crypto’92, 333–344.Google Scholar
  6. 6.
    V. Müller, “Fast multiplication on elliptic curves over small fields of characteristic two”, Journal of Cryptology, 11 (1998), 219–234.zbMATHCrossRefGoogle Scholar
  7. 7.
    D. Shanks, “Five number theoretic algorithms” In Proc. 2nd Manitoba Conference on Numerical Mathematics (1972), 51–70.Google Scholar
  8. 8.
    N.P. Smart, “Elliptic curve cryptosystems over small fields of odd characteristic”, Journal of Cryptology, 12 (1998), 141–151.CrossRefMathSciNetGoogle Scholar
  9. 9.
    J. Solinas, “Efficient arithmetic on Koblitz curves”, Design, Codes and Cryptography, 19 (2000), 195–249.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    I. Stewart, D. Tall, “Algebraic Number Theory”, Chapman and Hall, Halsted Press, 1979.Google Scholar
  11. 11.
    B. Vallée,“Une approche géométrique des algorithmes de réduction des réseaux en petite dimension”, (1986) Thése, Université de Caen.Google Scholar
  12. 13.
    M. Wiener and R. Zuccherato, ‘Faster Attacks on Elliptic Curve Cryptosystems’, contribution to IEEE P1363.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Young-Ho Park
    • 1
  • Sangho Oh
    • 1
  • Sangjin Lee
    • 1
  • Jongin Lim
    • 1
  • Maenghee Sung
    • 2
  1. 1.CISTKorea UniversitySeoulKorea
  2. 2.KISASeoulKorea

Personalised recommendations