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Diffie-Hellman Oracles

  • Ueli M. Maurer
  • Stefan Wolf
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1109)

Abstract

This paper consists of three parts. First, various types of Diffie-Hellman oracles for a cyclic group G and subgroups of G are defined and their equivalence is proved. In particular, the security of using a subgroup of G instead of G in the Diffie-Hellman protocol is investigated. Second, we derive several new conditions for the polynomial-time equivalence of breaking the Diffie-Hellman protocol and computing discrete logarithms in G which extend former results by den Boer and Maurer. Finally, efficient constructions of Diffie-Hellman groups with provable equivalence are described.

Keywords

Public-key cryptography Diffie-Hellman protocol Discrete logarithms Elliptic curves 

References

  1. 1.
    L.M. Adleman and M.A. Huang, Primality testing and abelian varieties over finite fields, Lecture Notes in Mathematics, vol. 1512, Springer-Verlag, 1992.Google Scholar
  2. 2.
    E. Bach and J. Shallit, Factoring with cyclotomic polynomials, Math. Comp., vol. 52, pp. 201–219, 1989.CrossRefMathSciNetGoogle Scholar
  3. 3.
    D. Boneh and R.J. Lipton, Algorithms for black-box fields and their application to cryptography, preprint, 1995.Google Scholar
  4. 4.
    E.R. Canfield, P. Erdös and C. Pomerance, On a problem of Oppenheim concerning “Factorisatio Numerorum”, J. Number Theory, vol. 17, pp. 1–28, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    B. den Boer, Diffie-Hellman is as strong as discrete log for certain primes, Advances in Cryptology — CRYPTO’ 88, Lecture Notes in Computer Science, vol. 403, pp. 530–539, Berlin: Springer-Verlag, 1989.Google Scholar
  6. 6.
    W. Diffie and M.E. Hellman, New directions in cryptography, IEEE Transactions on Information Theory, vol. 22, no. 6, pp. 644–654, 1976.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    K.O. Geddes, S.R. Czapor and G. Labhan, Algorithms for computer algebra, Kluwer Academic Publisher, 1992.Google Scholar
  8. 8.
    K. Ireland and M. Rosen, A classical introduction to modern number theory, Springer-Verlag, 1982.Google Scholar
  9. 9.
    N. Koblitz, Elliptic curve cryptosystems, Math. Comp., vol. 48, pp. 203–209, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    G.-J. Lay and H.G. Zimmer, Constructing elliptic curves with given group order over large finite fields, preprint, 1994.Google Scholar
  11. 11.
    H.W. Lenstra, Jr., Factoring integers with elliptic curves, Annals of Mathematics, vol. 126, pp. 649–673, 1987.CrossRefMathSciNetGoogle Scholar
  12. 12.
    J.L. Massey, Advanced Technology Seminars Short Course Notes, pp. 6.66–6.68, Zürich, 1993.Google Scholar
  13. 13.
    U.M. Maurer, Towards the equivalence of breaking the Diffie-Hellman protocol and computing discrete logarithms, Advances in Cryptology — CRYPTO’ 94, Y. Desmedt (ed.), Lecture Notes in Computer Science, Berlin: Springer-Verlag, vol. 839, pp. 271–281, 1994.Google Scholar
  14. 14.
    U.M. Maurer and S. Wolf, On the complexity of breaking the Diffie-Hellman protocol, Tech. Rep. 244, Computer Science Department, ETH Zürich, April 1996. (Accessible at http://www.inf.ethz.ch/publications/isc.html)
  15. 15.
    K.S. McCurley, The discrete logarithm problem, in Cryptology and computational number theory, C. Pomerance (ed.), Proc. of Symp. in Applied Math., vol. 42, pp. 49–74, American Mathematical Society, 1990.Google Scholar
  16. 16.
    A.J. Menezes (ed.), Applications of finite fields, Kluwer Academic Publishers, 1992.Google Scholar
  17. 17.
    V. Miller, Uses of elliptic curves in cryptography, Advances in Cryptology — CRYPTO’ 85, Lecture Notes in Computer Science, Springer-Verlag, vol. 218, pp. 417–426, 1986.Google Scholar
  18. 18.
    S.C. Pohlig and M.E. Hellman, An improved algorithm for computing logarithms over GF(p) and its cryptographic significance, IEEE Transactions on Information Theory, vol. 24, no. 1, pp. 106–110, 1978.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    R. Schoof, Elliptic curves over finite fields and the computation of square roots mod p, Math. Comp., vol. 44, No. 170, pp. 483–494, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    S.A. Vanstone and R.J. Zuccherato, Elliptic curve cryptosystems using curves of smooth order over the ring Zn, Preliminary version, 1994.Google Scholar
  21. 21.
    S. Wolf, Diffie-Hellman and discrete logarithms, Thesis, March 1995.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Ueli M. Maurer
    • 1
  • Stefan Wolf
    • 1
  1. 1.Institute for Theoretical Computer ScienceETH ZürichZürichSwitzerland

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