Testing permutation polynomials

The simple test for determining whether an arbitrary polynomial is a permutation polynomial, by producing its list of values, is considered, and it is found that off-the-shelf techniques from computer algebra improve the running time slightly, without requiring any new insights into the problem. A probabilistic variant of the Hermite test that reduces its running time is given. A criterion for permutation polynomials is then examined, and a probabilistic test whose number of operations is essentially linear in the input size is then given. Exceptional polynomials, which are closely related to permutation polynomials, are also considered, and a random polynomial-time test for these is described