Constructing permutation polynomials from permutation polynomials of subfields

In this paper we study the permutational property of polynomials of the form $f\left(L\right(x\left)\right)+k\left(L\right(x\left)\right)\cdot M\left(x\right)\in F_{q^n}\left[x\right]$ over the finite field $F_{q^n}$, where $L,M\in F_q\left[x\right]$ are q-linearized polynomials and $k\in F_{q^n}\left[x\right]$ satisfies a generic condition. We specialize in the case where $L\left(x\right)$ is the linearized q-associate of $g_{t,a}\left(x\right)=(x^n-1)/(x^t-a)$, t is a divisor of n and $a\in F_q$ satisfies $a^{n/t}=1$. This unifies many recent explicit constructions and provides new explicit constructions of permutation polynomials and their inverses. Moreover, we introduce a new algorithmic method to produce many permutation polynomials of $F_{q^n}$ from permutations of $F_{q^t}$, by simply solving a system of independent equations of the form ${\mathrm{Tr}}_{q^n/q^t}\left({\delta }^{i-1}{a}_i\right)=c_i$, where the $a_i$'s are the coefficients of f. In fact, the same method can be applied to construct complete mappings of $F_{q^n}$ from complete mappings of $F_q$.