New Constructions of Permutation Polynomials of the Form $x^rh\left(x^{q-1}\right)$ over $\mathbb{F}_{q^2}$

Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation trinomials of the form xh ( x ) over Fq2 , where q = 2 , h(x) = 1+x+x and r, s, t, k > 0 are integers. Their methods are essentially usage of a multiplicative version of AGW Criterion because they all transformed the problem of proving permutation polynomials over Fq2 into that of showing the corresponding fractional polynomials permute a smaller set μq+1, where μq+1 := {x ∈ Fq2 : x q+1 = 1}. Motivated by these results, we characterize the permutation polynomials of the form xh ( x ) over Fq2 such that h(x) ∈ Fq[x] is arbitrary and q is also an arbitrary prime power. Using AGW Criterion twice, one is multiplicative and the other is additive, we reduce the problem of proving permutation polynomials over Fq2 into that of showing permutations over a small subset S of a proper subfield Fq, which is significantly different from previously known methods. In particular, we demonstrate our method by constructing many new explicit classes of permutation polynomials of the form xh ( x ) over Fq2 . Moreover, we can explain most of the known permutation trinomials, which are in [6, 13, 14, 16, 20, 29], over finite field with even characteristic. MSC: 06E30, 11T06, 94A60 Index Terms Finite Fields, Permutation Polynomials, Rational Function, AGW Criterion