Reversed Dickson polynomials

We investigate fixed points and cycle types of permutation polynomials and complete permutation polynomials arising from reversed Dickson polynomials of the first kind and second kind over ${\mathbb{F}}_p$. We also study the permutation behaviour of reversed Dickson polynomials of the first kind and second kind over ${\mathbb{Z}}_m$. Moreover, we prove two special cases of a conjecture on the permutation behaviour of reversed Dickson polynomials over ${\mathbb{F}}_p$.