A reduction of integer factorization to modular tetration

TL;DR

This paper demonstrates that an efficient algorithm for computing tetration modulo natural numbers can be used to factor integers, establishing a polynomial-time reduction from integer factorization to modular tetration.

Contribution

It introduces a novel reduction showing that integer factorization can be solved via algorithms for modular tetration, linking these two computational problems.

Findings

Integer factorization reduces to modular tetration in polynomial time.

Computing the squarefree part of integers is also reducible to modular tetration.

The paper establishes a new connection between factorization and tetration algorithms.

Abstract

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 \[ ^k a:=a^{a^{.^{.^{.^{a}}}}}. \] 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.