A reduction of integer factorization to modular tetration

Let $a,k\in \mathbb{N}$. For the $k-1$-th iterate of the exponential function $x\mapsto a^x$, also known as tetration, we write $${}^{k}a:=a^{a^{{\mathrm{.}}^{{\mathrm{.}}^{{\mathrm{.}}^a}}}}\mathrm{.}$$ In this paper, we show how an efficient algorithm for tetration modulo natural numbers $N$ may be used to compute the prime factorization of $N$. In particular, we prove that the problem of computing the squarefree part of integers is deterministically polynomial-time reducible to modular tetration.