Classification of Permutation Trinomials and Quadrinomials Over Prime Fields

Abstract

We present a method for enumerating the permutation trinomials and quadrinomials using various symmetries and algebraic properties to reduce the search space. Applying this method, we enumerated all permutation trinomials and quadrinomials for the prime finite fields with orders up to 3000 and 500, respectively. Using the enumeration results, we stated a conjecture about the classification of permutation polynomials over prime finite fields.