The computational complexity of recognizing permutation functions

Abstract

Let\(\mathbb{F}_q \) be a finite field withq elements and\(f \in \mathbb{F}_q \left( x \right)\) a rational function over\(\mathbb{F}_q \). No polynomial-time deterministic algorithm is known for the problem of deciding whetherf induces a permutation on\(\mathbb{F}_q \). The problem has been shown to be in co-R\( \subseteq \)co-NP, and in this paper we prove that it is inR\( \subseteq \)NP and hence inZPP, and it is deterministic polynomial-time reducible to the problem of factoring univariate polynomials over\(\mathbb{F}_q \). Besides the problem of recognizing prime numbers, it seems to be the only natural decision problem inZPP unknown to be inP. A deterministic test and a simple probabilistic test for permutation functions are also presented.