Permutation rational functions over quadratic extensions of finite fields

Abstract

Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over Fq2, whose numerators are so-called q-quadratic polynomials. To this end, we will first determine the exact number of zeros of a special q-quadratic polynomial in Fq2, by calculating character sums related to quadratic forms of Fq2/Fq. Then given some rational function, we can demonstrate whether it induces a permutation of Fq2.