Abstract:
Let $\mathbb{F}_q$ be a finite field and $f(x) \in \mathbb{F}_q(x)$ be a rational function over $\mathbb{F}_q$. The decision problem {\bf PermFunction} consists of deciding whether $f(x)$ induces a permutation on the elements of $\mathbb{F}_q$. That is, we want to decide whether the corresponding map $f : \mathbb{F}_q \mapsto \mathbb{F}_q$ defined by $a \mapsto f(a) $ is a bijective mapping or not. This problem was known to be in $\mathcal{ZPP}$ but not known to be in $\mathcal{P}$. We resolve the complexity of {\bf PermFunction} by giving a deterministic polynomial-time algorithm for this problem.