  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.
