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Digital signature schemes based on Lucas functions

  • Patrick Horster
  • Markus Michels
  • Holger Petersen
Chapter
Part of the IFIP Advances in Information and Communication Technology book series (IFIPAICT)

Abstract

In 1993 Lennon and Smith proposed to use Lucas functions instead of the exponentiation function as a one-way function in cryptographic mechanisms. Recently Smith and Skinner presented an ElGamal signature scheme based on Lucas functions.

In this paper we point out a serious weakness in this approach and present our version of an ElGamal signature scheme based on Lucas functions. Furthermore, we outline how to apply the ideas of the Meta-ElGamal signature scheme to Lucas functions. As a result we get various new signature schemes. In contradiction to a conjecture by Smith and Skinner the security of the schemes isn’t increased: It can be proved that a variant of the signature schemes based on Lucas functions can be universally forged if a related signature scheme in GF(p) can be universally forged. We further outline how the Meta signature scheme can be described in an elliptic curve environment and mention some other possible extensions.

Keywords

Cryptography digital signatures Lucas functions elliptic curves 

References

  1. T. ElGamal, (1984), Cryptography and logarithms over finite fields, Ph. D. thesis, Stanford University, CA., UMI Order No. DA 8420519, 119 pages.Google Scholar
  2. P. Horster, M. Michels, H. Petersen, (1994), Meta-ElGamal signature scheme for one message block, Proc. of the Workshop IT-Security, Vienna, Sep. 22–23, 1994, R. Oldenbourg Wien München, 1995, pp. 66–81.Google Scholar
  3. P. Horster, M. Michels, H. Petersen, (1994), Meta-ElGamal signature scheme, Proc. of the 2nd ACM Conference on Computer and Communications Security, Fairfax, Virginia, Nov. 2–4, 1994, pp. 96–107.Google Scholar
  4. P. Horster, M. Michels, H. Petersen, (1994), Meta Message Recovery and Meta blind signature schemes based on the discrete logarithm problem and some applications, Lecture Notes in Computer Science 917, Advances in Cryptology: Proc. Asiacrypt ‘84, Berlin: Springer Verlag, 1995, pp. 224–37.Google Scholar
  5. D. E. Knuth, (1981), The art of computer programming, Vol. 2: Seminumerical algorithms, 2nd Edition, Addison-Wesley, Reading, MA.Google Scholar
  6. C.-S. Laih, F.-K. Tu, W.-C. Tai, (1994), Remarks on LUC public key system, Electronics Letters, Vol. 30, No. 2, pp. 123–4.CrossRefGoogle Scholar
  7. C.-S. Laih, F.-K. Tu, W.-C. Tai, (1995), On the security of Lucas function, Information Processing Letters, Vol. 53, pp. 243–7.zbMATHMathSciNetCrossRefGoogle Scholar
  8. F. E. A. Lucas, (1878), Theorie des fonctions numeriques simplement periodiques, American Journal Mathematics, Vol. 1, pp. 184–240 and 289–321.zbMATHCrossRefGoogle Scholar
  9. D. H. Lehmer, (1930), An extended theory of Lucas’ functions, Annals of Mathematics (2), Vol. 31, pp. 419–48.zbMATHMathSciNetGoogle Scholar
  10. A. J. Menezes, M. Qu, S. A. Vanstone, (1995), Standard for RSA, Diffie-Hellman and related public-key cryptography, Part 6: Elliptic curve systems (Draft 3), Working Draft, IEEE P1363 Standard, January, 42 pages.Google Scholar
  11. A. Miyaji, (1992), Elliptic curves over F„ suitable for cryptosystems, Lecture Notes in Computer Science 718, Advances in Cryptology: Proc. Asiacrypt ‘82, Berlin: Springer Verlag, 1993, pp. 224–37.Google Scholar
  12. W. B. Müller, W. Nöbauer, (1981), Some remarks on public key cryptosystems, Studia Sci. Math. Hung., Vol. 16, pp. 71–6.zbMATHGoogle Scholar
  13. W. B. Müller, R. Nöbauer, (1985), Cryptanalysis of the Dickson-scheme, Lecture Notes in Computer Science 219, Advances in Cryptology: Proc, Eurocrypt’85, Berlin: Springer Verlag, 1986, pp. 50–61.Google Scholar
  14. S. Murphy, (1994), Comment: Remarks on LUC public key system, Electronics Letters, Vol. 30, No. 7, pp. 558–9.CrossRefGoogle Scholar
  15. K. Nyberg, R. A. Rueppel, (1994), Message recovery for signature schemes based on the discrete logarithm problem, 21 July 1994, to appear in Design, Codes and Cryptography, Kluwer Academic Publishers, Boston, 15 pages.Google Scholar
  16. H. Postl, (1988), Fast evaluation of Dickson Polynomials, Contributions to General Algebra 6, Verlag Hölder-Pichler-Tempsky, Wien–Verlag B. G. Teubner, Stuttgart, pp. 223–5.Google Scholar
  17. D. Shanks, (1971), Class number, a theory of factorisation and genera, Proceedings Symposia in Pure Mathematics (20), Providence: American Mathematical Society, pp. 415–40.Google Scholar
  18. P. Smith, C. Skinner, (1994), A public key cryptosystem and a digital signature scheme based on Lucas functions analogue to discrete logarithms, Lecture Notes in Computer Science 917, Advances in Cryptology: Proc. Asiacrypt ‘84, Berlin: Springer Verlag, 1995, pp. 357–64.Google Scholar
  19. P. Smith, M. Lennon, (1993), LUC: A new public-key system, Proc. of IFIP/SEC ‘83, Elsevier Science Publishers, 1994, pp. 97–110.Google Scholar
  20. H. Williams, (1982), A p + 1 method of factoring, Mathematics on Computation, Vol. 39, pp. 225–34.zbMATHGoogle Scholar
  21. S.-M. Yen, C.-S. Laih, (1995), Fast algorithms for the LUC digital signature computation, IEE Proc.-Comput. Digit. Tech., Vol. 142, No. 2, pp. 165–9.CrossRefGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 1995

Authors and Affiliations

  • Patrick Horster
    • 1
  • Markus Michels
    • 1
  • Holger Petersen
    • 1
  1. 1.Theoretical Computer Science and Information SecurityUniversity of Technology Chemnitz-ZwickauChemnitzGermany

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