A public-key cryptosystem and a digital signature system based on the Lucas function analogue to discrete logarithms

  • Peter Smith
  • Christopher Skinner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 917)


Since 1975 many new cryptosystems have been based on elementary number theory, but until now it has not been recognised that they have been just as much grounded in the process of exponentiation. Lucas functions can be used to replace exponentiation to produce alternative cryptosystems that are not susceptible to attacks which rely on the fact that multiplication is closed under exponentiation, since Lucas functions do not exhibit this closure.


Elliptic Curf Discrete Logarithm Message Block Elementary Number Theory High Order Linear 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Bac]
    E. Bach, “Intractable Problems in Number Theory”, Advances in Cryptology: Proceedings of CRYPTO '88, Springer-Verlag, Berlin, 1989, pp 105–122.Google Scholar
  2. [Buc]
    J. Buchmann & Loho & Zayer, “An Implementation of the general number field sieve”, Advances in Cryptology: Proceedings of CRYPTO '93, Springer-Verlag, Berlin, 1994, pp 159–165.Google Scholar
  3. [Den]
    T. Denny and B. Dodson and A.K. Lenstra A. K. Manasse and M.S. Manasse, “On the factorization of RSA-120”, Advances in Cryptology: Proceedings of CRYPTO '93, Springer-Verlag, Berlin, 1994, pp 166–174.Google Scholar
  4. [Elg]
    T. El Gamal, “A public-key cryptosystem and a signature scheme based on discrete logarithms”, IEEE Transactions on Information Theory 31 (1985), pp 469–472.Google Scholar
  5. [Elg1]
    T. El. Gamal and B. Kaliski, Dr. Dobb's Journal, letter to the editor, 18, No.5, May 1993, p10.Google Scholar
  6. [Gor]
    D.M. Gordon, “Discrete Logarithms in GF(p) using the Number Field Sieve”, Siam J Disc Math 6 No. 1, Feb 1993, pp 124–138.Google Scholar
  7. [Leh]
    D. H. Lehmer, “An extended theory of Lucas' functions”, Annals of Math., 31 (1930) pp 419–448.Google Scholar
  8. [Luc]
    F. E. A. Lucas, “Théorie des fonctions numériques simplement périodiques”, American Jnl Math, 1 (1878) pp 184–240, 289–321.Google Scholar
  9. [Men]
    A. J. Menezes, T. Okamoto, S. A. Vanstone, “Reducing Elliptic Curve Logarithms to Logarithms in a Finite Field”, IEEE Transactions on Information Theory 39 (1993), pp 1639–1646.Google Scholar
  10. [Pom]
    C. Pomerance, “Fast Factorization & Discrete Logarithm Algorithms”, Discrete Algorithms & Complexity, W. Rheinboldt et al, Proceedings 1986 Kyoto Conference on Algorithms and Complexity.Google Scholar
  11. [Smi]
    P. Smith and M. Lennon, “LUC: A new public-key system”, Proceedings of IFIP/Sec '93, Elsevier Science Publications, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Peter Smith
    • 1
  • Christopher Skinner
    • 2
  1. 1.LUC Encryption Technology, LtdHerne BayNew Zealand
  2. 2.LUC Encryption Technology, LtdWoollahraAustralia

Personalised recommendations