Classification of permutation fewnomials over simple finite fields

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.