Tests for Permutation Functions

Let q be a finite field with q elements, ƒ q(x) a rational function over q, and q the domain of definition of ƒ. Consider three notions of "permutation functions": ƒ is a permutation on q, or on , or ƒ is injective on . 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.