On the constructions of $n$-cycle permutations

Any permutation polynomial is an $n$-cycle permutation. When $n$ is a specific small positive integer, one can obtain efficient permutations, such as involutions, triple-cycle permutations and quadruple-cycle permutations. These permutations have important applications in cryptography and coding theory. Inspired by the AGW Criterion, we propose criteria for $n$-cycle permutations, which mainly are of the form $x^rh(x^s)$. We then propose unified constructing methods including recursive ways and a cyclotomic way for $n$-cycle permutations of such form. We demonstrate our approaches by constructing three classes of explicit triple-cycle permutations with high index and two classes of $n$-cycle permutations with low index.