Fixed Points and Two-Cycles of the Discrete Logarithm
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.
- 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.Richard K. Guy. Unsolved Problems in Number Theory. Springer-Verlag, 1981.Google Scholar
- 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.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.Wen Peng Zhang. On a problem of Brizolis. Pure Appl. Math., 11(suppl.):1–3, 1995.Google Scholar