Vol. 12, No. 1, 2019

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Statistics for fixed points of the self-power map

Matthew Friedrichsen and Joshua Holden

Vol. 12 (2019), No. 1, 63–78

The map xxx modulo p is related to a variation of the ElGamal digital signature scheme in a similar way as the discrete exponentiation map, but it has received much less study. We explore the number of fixed points of this map by a statistical analysis of experimental data. In particular, the number of fixed points can in many cases be modeled by a binomial distribution. We discuss the many cases where this has been successful, and also the cases where a good model may not yet have been found.

self-power map, exponential equation, ElGamal digital signatures, fixed point, random map, number theory
Mathematical Subject Classification 2010
Primary: 11Y99
Secondary: 11-04, 11T71, 94A60, 11A07, 11D99
Received: 22 April 2017
Revised: 31 January 2018
Accepted: 14 February 2018
Published: 31 May 2018

Communicated by Anant Godbole
Matthew Friedrichsen
Department of Mathematics
Tufts University
Medford, MA
United States
Joshua Holden
Department of Mathematics
Rose-Hulman Institute of Technology
Terre Haute, IN
United States