Some more constructions of $n-$cycle permutation polynomials

TL;DR

This paper develops new criteria and methods for constructing n-cycle permutation polynomials, including explicit forms and generalizations, with applications in cryptography and coding theory.

Contribution

It introduces novel criteria for constructing larger n-cycle permutation polynomials using linearized polynomials and generalizes existing forms with explicit examples.

Findings

Constructed explicit n-cycle permutation polynomials of specific forms.

Demonstrated quadruple and quintuple permutation polynomials with Boolean functions.

Built linear binomial triple-cycle permutation polynomials.

Abstract

$ n-$cycle permutation polynomials with small n have the advantage that their compositional inverses are efficient in terms of implementation. These permutation polynomials have significant applications in cryptography and coding theory. In this article, we propose criteria for the construction of $ n-$cycle permutation using linearized polynomial $ L(x) $ for larger $ n $. Furthermore, we investigate and generalize certain novel forms of $ n-$cycle permutation polynomials. Finally, we demonstrate our approach by constructing explicit $ n-$cycle permutation of the form $ L(x)+\gamma h(Tr_{q^{m}/q}(x)) $, and $ G(x)+\gamma f(x) $ with a Boolean function $ f(x) $. The polynomial $ x^{d}+\gamma f(x) $ with $ f(x) $ being a Boolean function is shown to be quadruple and quintuple permutation polynomials. Moreover, linear binomial triple-cycle permutation polynomials are constructed.