Permutation polynomials over residue class rings

Problems of finding inverse for a permutation polynomial over the ring $\mathbb Z_{p^k}$ for prime $p$ and any $k>1$ are studied. Necessary and sufficient conditions for two permutation polynomials to be inverse polynomials modulo prime power are found. Given a known inverse polynomial modulo $p^2$, a formula for inverse polynomial modulo $p^k$ is pointed. Given a pair of inverse polynomials modulo $p^k$, a method for constructing other such pairs is proposed.