Dickson Polynomials of the Second Kind that Permute $\mathbb{Z}_m$
Longjiang Qu, Cunsheng Ding · SIAM Journal on Discrete Mathematics · 2014
In this paper, we investigate the permutation property of the Dickson polynomials $E_n(x, a)$ of the second kind over $\mathbb{Z}_m$. Due to a known result, it suffices to consider permutation polynomials $E_n(x, a)$ over $\mathbb{Z}_{p^t}$, where $p$ is a prime and $t$ is a positive integer. We identify all permutation polynomials of $E_n(x, a)$ over $\mathbb{Z}_{p^t}$ for (I) $p=2$ and (II) $p$ is odd and $a$ is a square over $\mathbb{Z}_p$. For odd $p$ and nonsquares $a$ in $\mathbb{Z}_p$, we determine a large class (if not all) of permutation polynomials $E_n(x, a)$ over $\mathbb{Z}_{p^t}$. A conjecture is also presented in this paper. If this conjecture is true, then all Dickson permutation polynomials $E_n(x, a)$ of the second kind over $\mathbb{Z}_m$ are determined.