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.