Geodesic
Classification of permutation fewnomials over simple finite fields
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIII, Tome 421 (2014), pp. 152-165

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We present a method for permutation trinomials and quadrinomials enumeration based on various symmetries and algebraic properties for search space reduction. Using this method, we enumerated all permutation trinomials and quadrinomials for the prime finite fields with orders up to 3000 and 500 respectively. Based on the enumeration results, we stated a hypothesis about permutation polynomials classification over prime finite fields. We evaluated randomness of such permutations. 
@article{ZNSL_2014_421_a12,
     author = {M. A. Rybalkin},
     title = {Classification of permutation fewnomials over simple finite fields},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {152--165},
     publisher = {mathdoc},
     volume = {421},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_421_a12/}
}
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M. A. Rybalkin. Classification of permutation fewnomials over simple finite fields. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIII, Tome 421 (2014), pp. 152-165. http://geodesic.mathdoc.fr/item/ZNSL_2014_421_a12/