Enumerating permutation polynomials

We consider the problem of enumerating polynomials over Fq, that have certain coefficients prescribed to given values and permute certain substructures of Fq. In particular, we are interested in the group of N-th roots of unity and in the submodules of Fq. We employ the techniques of Konyagin and Pappalardi to obtain results that are similar to their results in Konyagin and Pappalardi (2006) [8]. As a consequence, we prove conditions that ensure the existence of low-degree permutation polynomials of the mentioned substructures of Fq.