#### 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
##### Abstract

The map $x↦{x}^{x}$ 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.

##### Keywords
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