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arXiv · 1609.06696

Counting fixed points and rooted closed walks of the singular map $x \mapsto x^{x^n}$ modulo powers of a prime

Joshua Holden·Pamela A. Richardson·Margaret M. Robinson
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Abstract

The "self-power" map $x \mapsto x^x$ modulo $m$ and its generalized form $x \mapsto x^{x^n}$ modulo $m$ are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use $p$-adic methods, primarily $p$-adic interpolation, Hensel's lemma, and lifting singular points modulo $p$, to count fixed points and rooted closed walks of equations related to these maps when $m$ is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacobian matrix.

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Joshua Holden, Pamela A. Richardson, Margaret M. Robinson. 2016-09-21. Counting fixed points and rooted closed walks of the singular map $x \mapsto x^{x^n}$ modulo powers of a prime. https://doi.org/10.1134/s2070046620010021

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