Abstract
Let Fq be a finite field with q elements, ƒ ∈ Fq(x) a rational function over Fq, and D ⊆ Fq the domain of definition of ƒ. Consider three notions of "permutation functions": ƒ is a permutation on Fq, or on D, or ƒ is injective on D. For each of these, a random polynomial-time test is presented. For the image size of an arbitrary rational function, a fully polynomial-time randomized approximation scheme is given.
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