Three classes of permutation quadrinomials in odd characteristic (Q6542668)

Let \(p\) be a prime and \(q\) a power of \(p\). Let \(\mathbb{F}_q\) be the finite field with \(q\) elements and \(\mathbb{F}_q[x]\) the ring of polynomials in \(x\) over \(\mathbb{F}_q\). A polynomial \(f\in \mathbb{F}_q[x]\) is called a \textit{permutation polynomial} (PP) of \(\mathbb{F}_q\) if the associated mapping \(x\mapsto f(x)\) from \(\mathbb{F}_q\) to \(\mathbb{F}_q\) is a permutation of \(\mathbb{F}_q\).\N\NIn the paper under review, the authors construct three classes of permutation quadrinomials with Niho exponents of the form \[f(x)=\alpha_0x^r+\alpha_1x^{s_1(p^m-1)+r}+\alpha_2x^{s_2(p^m-1)+r}+\alpha_3x^{s_3(p^m-1)+r}\,\in \mathbb{F}_{p^n}[x],\] where \(p\) is an odd prime, \(n=2m\) is a positive even integer, and \[\displaystyle{(r,s_1,s_2,s_3)=\Big(1,\frac{-1}{p^k-2},1, \frac{p^k-1}{p^k-2}\Big), \Big(1,\frac{p^k+1}{p^k+2},1,\frac{1}{p^k+2}\Big)\,\,\,\text{and}\,\,\,(3,1,2,3)},\] respectively. The exponents of the first two classes are considered for the first time, and the third class covers all the permutation polynomials proposed by \textit{R. Gupta} in [Des. Codes Cryptography 88, No. 1, 223--239 (2020; Zbl 1428.11203)].