Fast Computation of Discrete Logarithms in GF (q)

  • Martin E. Hellman
  • Justin M. Reyneri


The Merkte-Adleman algorithm computes discrete logarithms in GF (q),the finite field with q elements, in subexponential time, when q is a prime number p. This paper shows that similar asymptotic behavior can be obtained for the logarithm problem when q = p m , in the case that m grows with p fixed. A method of partial precomputation, applicable to either problem, is also presented. The precomputation is particularly useful when many logarithms need to be computed for fixed values of p and m.


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  1. [1]
    S. Pohlig and M. Hellman, “An improved algorithm for computing logarithms over GF(p) and its cryptographic significance,” IEEE Trans. on Inform. Theory, vol. IT-24, pp. 106–110, Jan. 1978.Google Scholar
  2. [2]
    W. Diffie and M. E. Hellman, “New directions in cryptography,” IEEE Trans. on Inform. Theory, vol. IT-22, pp. 644–654, Nov. 1976.Google Scholar
  3. [3]
    R. L. Rivest, A. Shamir and L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems,” Communications of the ACM, vol. 21, no. 2, February 1978Google Scholar
  4. [4]
    R. Merkte, Secrecy, Authentication, and Public Key Systems, Ph.D. dissertation, Department of Electrical Engineering, Stanford University, June 1979.Google Scholar
  5. [5]
    S. C. Pohlig, Algebraic and Combinatoric Aspects of Cryptography, Ph.D. Dissertation, Department of Electrical Engineering, Stanford University, Oct. 1977Google Scholar
  6. [6]
    L. Adleman, “A subexponetial algorithm for the discrete logarithm with applications to cryptography,” Proceedings of the 20th Annual Symposium on Foundations of Computer Science, Oct. 29–31, 1979Google Scholar
  7. [7]
    S. Berkovits, J. Kowalchuk, and B. Schanning, “Implementing public-key scheme,” IEEE Commun. Mag., vol. 17, pp 2–3, May 1979Google Scholar
  8. [8]
    J. D. Dixon, “Asymptotically fast factorization of integers,” Math. Comp., vol. 38, no. 153, Jan. 1981Google Scholar
  9. [9]
    E.R. Berlekamp, Algebraic Coding Theory, New York: McGraw Hill, 1978Google Scholar
  10. [10]
    W. Feller, An Introduction to Probability Theory and Its Applications, 3rd ed. New York: Wiley, 1968.Google Scholar
  11. [11]
    R. Schroeppel, Private communicationGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • Martin E. Hellman
    • 1
  • Justin M. Reyneri
    • 1
  1. 1.Information Systems LaboratoryStanford UniversityStanfordUSA

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