Counting fixed points and rooted closed walks of the singular map $x \mapsto x^{x^n}$ modulo powers of a prime
This work addresses theoretical and cryptographic interests in number theory, but it is incremental as it builds on existing methods for specific maps.
The paper tackled the problem of counting fixed points and rooted closed walks for the singular self-power map modulo prime powers, achieving results through p-adic methods and a new lifting technique for singular solutions.
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.