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TR05-008 | 11th December 2004 00:00
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#### Recognizing permutation functions in polynomial time.

**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.