Fixed Points and Two-Cycles of the Discrete Logarithm

  • Joshua Holden
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2369)


We explore some questions related to one of Brizolis: does every prime p have a pair (g, h) such that h is a fixed point for the discrete logarithm with base g? We extend this question to ask about not only fixed points but also two-cycles. Campbell and Pomerance have not only answered the fixed point question for sufficiently large p but have also rigorously estimated the number of such pairs given certain conditions on g and h. We attempt to give heuristics for similar estimates given other conditions on g and h and also in the case of two-cycles. These heuristics are well-supported by the data we have collected, and seem suitable for conversion into rigorous estimates in the future.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Manuel Blum and Silvio Micali. How to generate cryptographically strong sequences of pseudorandom bits. SIAM J. Comput., 13(4):850–864, 1984.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cristian Cobeli and Alexandru Zaharescu. An exponential congruence with solutions in primitive roots. Rev. Roumaine Math. Pures Appl., 44(1):15–22, 1999.zbMATHMathSciNetGoogle Scholar
  3. 3.
    Rosario Gennaro. An improved pseudo-random generator based on discrete log. In M. Bellare, editor, Advances in Cryptology — CRYPTO 2000, pages 469–481. Springer, 2000.Google Scholar
  4. 4.
    Richard K. Guy. Unsolved Problems in Number Theory. Springer-Verlag, 1981.Google Scholar
  5. 5.
    Sarvar Patel and Ganapathy S. Sundaram. An efficient discrete log pseudo-random generator. In H. Krawczyk, editor, Advances in Cryptology — CRYPTO’ 98, pages 304–317. Springer, 1998.Google Scholar
  6. 6.
    Carl Pomerance. On fixed points for discrete logarithms. Talk given at the Central Section meeting of the AMS, Columbus, OH, September 22, 2001. Joint work with Mariana Campbell.Google Scholar
  7. 7.
    Wen Peng Zhang. On a problem of Brizolis. Pure Appl. Math., 11(suppl.):1–3, 1995.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Joshua Holden
    • 1
  1. 1.Department of MathematicsRose-Hulman Institute of TechnologyTerre HauteUSA

Personalised recommendations