New constructions of permutation polynomials of the form $x^rh\left( x^{q-1}\right)$ over ${\mathbb F}_{q^2}$

Abstract

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 $r, k>0, s, t$ 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 Ding et al. (SIAM J Discret Math 29:79–92, 2015), Gupta and Sharma (Finite Fields Appl 41:89–96, 2016), Li and Helleseth (Cryptogr Commun 9:693–705, 2017), Li et al. (New permutation trinomials constructed from fractional polynomials, arXiv: 1605.06216v1, 2016), Li et al. (Finite Fields Appl 43:69–85, 2017) and Zha et al. (Finite Fields Appl 45:43–52, 2017) over finite field with even characteristic.