More constructions of $n$-cycle permutations

TL;DR

This paper characterizes and constructs various types of $n$-cycle permutations, providing new explicit examples and criteria, which are useful in cryptography and coding theory.

Contribution

It introduces new criteria and explicit constructions for $n$-cycle permutations of several polynomial forms, expanding the known classes of such permutations.

Findings

New criteria for $n$-cycle permutations in different polynomial forms

Explicit constructions of triple-cycle permutations

Many new $n$-cycle permutations with the $n$-cycle property

Abstract

$n$-cycle permutations with small $n$ have the advantage that their compositional inverses are efficient in terms of implementation. They can be also used in constructing Bent functions and designing codes. Since the AGW Criterion was proposed, the permuting property of several forms of polynomials has been studied. In this paper, characterizations of several types of $n$-cycle permutations are investigated. Three criteria for $ n $-cycle permutations of the form $xh(\lambda(x))$, $ h(\psi(x)) \varphi(x)+g(\psi(x)) $ and $g\left( x^{q^i} -x +\delta \right) +bx $ with general $n$ are provided. We demonstrate these criteria by providing explicit constructions. For the form of $x^rh(x^s)$, several new explicit triple-cycle permutations are also provided. Finally, we also consider triple-cycle permutations of the form $x^t + c\rm Tr_{q^m/q}(x^s)$ and provide one explicit construction. Many of our constructions are both new in the $n$-cycle property and the permutation property.