Characterizations and constructions of triple-cycle permutations of the form $$x^rh(x^s)$$

Abstract

Let $${\mathbb {F}}_q$$ be the finite field with q elements and let f be a permutation polynomial over $${\mathbb {F}}_q$$ . Let $$S_q$$ denote the symmetric group on $${\mathbb {F}}_q$$ . In this paper, we mainly investigate some characterizations on the elements $$f \in S_q$$ of order 3, i.e., $$f\circ f\circ f=I$$ , where f is also called a triple-cycle permutation in the literature. Some explicit triple-cycle permutations are constructed.