Features of maximal period polynomial generators over the Galois ring

For a polynomial mapping over the Galois ring $R=\mathrm{GR}(q^n,p^n)$ with the cardinality $q^n$ and characteristic $p^n$, the maximal length of a cycle equals $q(q-1)p^{n-2}$. In this paper, we present an algorithm for constructing the system of representatives of all maximal length cycles and an algorithm for constructing an element in a cycle of maximal length for a polynomial substitution $f\in R[x]$. The complexity of the first algorithm equals $d(q-1)q^{n-1}$ multiplication operations and $d(q-1)q^{n-1}$ addition operations in $R$, the complexity of the second algorithm equals $dq$ multiplication operations and $dq$ addition operations in $R$ where $d=\deg(f)$.