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Permutation polynomials over finite fields have been studied extensivelyrecently due to their wide applications in cryptography, coding theory,communication theory, among others. Recently, several authors have studiedpermutation trinomials of the form $x^rh\left(x^{q-1}\right)$ over$\mathbb{F}_{q^2}$, where $q=2^k$, $h(x)=1+x^s+x^t$ and $r, s, t, k>0$ areintegers. Their methods are essentially usage of a multiplicative version ofAGW Criterion because they all transformed the problem of proving permutationpolynomials over $\mathbb{F}_{q^2}$ into that of showing the correspondingfractional polynomials permute a smaller set $\mu_{q+1}$, where$\mu_{q+1}:=\{x\in\mathbb{F}_{q^2} : x^{q+1}=1\}$. Motivated by these results,we characterize the permutation polynomials of the form$x^rh\left(x^{q-1}\right)$ over $\mathbb{F}_{q^2}$ such that$h(x)\in\mathbb{F}_q[x]$ is arbitrary and $q$ is also an arbitrary prime power.Using AGW Criterion twice, one is multiplicative and the other is additive, wereduce the problem of proving permutation polynomials over $\mathbb{F}_{q^2}$into that of showing permutations over a small subset $S$ of a proper subfield$\mathbb{F}_{q}$, which is significantly different from previously knownmethods. In particular, we demonstrate our method by constructing many newexplicit classes of permutation polynomials of the form$x^rh\left(x^{q-1}\right)$ over $\mathbb{F}_{q^2}$. Moreover, we can explainmost of the known permutation trinomials, which are in [6, 13, 14, 16, 20, 29],over finite field with even characteristic.

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