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 two-cycles of equations related to these maps when $m$ is a prime power.

中文翻译：

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计算奇异映射 $$x \mapsto {x^{{x^n}}}$$x↦xxn 素数的模幂

“自力”映射 $x \mapsto x^x$ modulo $m$ 及其广义形式 $x \mapsto x^{x^n}$ modulo $m$ 出于理论原因和潜力密码学的应用。在本文中，我们使用 $p$-adic 方法，主要是 $p$-adic 插值、Hensel 引理和提升奇异点模 $p$，当 $m $ 是主要权力。