On The Constructions Of N-cycle Permutations

Permutation polynomials play an important role in the theory of finite fields.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.We first obtain the basic properties of n-cycle permutations.Then,inspired by the AGW Criterion,we propose criteria for n-cycle permutations of the form xrh(xs).We then propose unified constructing methods to construct n-cycle permutations,also including recursive ways and a cyclotomic way for n-cycle permutations of such form.We demonstrate our methods by constructing explicit n-cycle permutations,where most of them are new both at levels of permutation property and cycle property.From unified constructing methods,we construct three classes of explicit triple-cycle permutations with high index.From the cyclotomic perspective,we use piecewise method to construct two classes of n-cycle permutations with low index.After that,we also research the properties of n-cycle permutations of the form g(xqi-x+δ)+x,g(x)+g0(λ(x)),h(ψ(x))φ(x)+g(ψ(x)).These constructions enrich the theory of permutation polynomials and provide more choices for devices with limited resources when decoding or decryption.