# Some New Permutation Polynomials over Finite Fields

## TL;DR

This paper introduces new classes of permutation polynomials over finite fields, including complete permutation monomials and a novel characterization of o-polynomials, addressing an open problem in the field.

## Contribution

It constructs new permutation polynomials and provides a new characterization of o-polynomials, solving an open problem related to permutation polynomials of a specific form.

## Key findings

- New classes of permutation monomials constructed
- A new characterization of o-polynomials provided
- Solved an open problem on permutation polynomials of the form G(x)+ γ Tr(H(x))

## Abstract

In this paper, we construct a new class of complete permutation monomials and several classes of permutation polynomials. Further, by giving another characterization of o-polynomials, we obtain a class of permutation polynomials of the form $G(x)+ \gamma Tr(H(x))$, where G(X) is neither a permutation nor a linearized polynomial. This is an answer to the open problem 1 of Charpin and Kyureghyan in [P. Charpin and G. Kyureghyan, When does $G(x)+ \gamma Tr(H(x))$ permute $\mathbb{F}_{p^n}$?, Finite Fields and Their Applications 15 (2009) 615--632].