  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 $x^rh\left(x^{q-1}\right)$ over
$\mathbb{F}_{q^2}$, where $q=2^k$, $h(x)=1+x^s+x^t$ and $r, s, t, k>0$ 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 $\mathbb{F}_{q^2}$ into that of showing the corresponding
fractional 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, we
reduce 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 known
methods. In particular, we demonstrate our method by constructing many new
explicit classes of permutation polynomials of the form
$x^rh\left(x^{q-1}\right)$ over $\mathbb{F}_{q^2}$. 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.
