Constructing permutation polynomials over finite fields

TL;DR

This paper introduces new methods for constructing permutation polynomials over finite fields, extending previous theorems and solving an open problem, with potential applications in cryptography and coding theory.

Contribution

The paper presents novel constructions of permutation polynomials over finite fields using linearized polynomials, elementary symmetric polynomials, and linear translators, extending and generalizing prior results.

Findings

Constructed permutation polynomials using linearized polynomials.

Generalized results of Coulter, Henderson, and Matthews.

Solved an open problem posed by Zieve in 2010.

Abstract

In this paper, we construct several new permutation polynomials over finite fields. First, using the linearized polynomials, we construct the permutation polynomial of the form $\sum_{i=1}^k(L_{i}(x)+\gamma_i)h_i(B(x))$ over ${\bf F}_{q^{m}}$, where $L_i(x)$ and $B(x)$ are linearized polynomials. This extends a theorem of Coulter, Henderson and Matthews. Consequently, we generalize a result of Marcos by constructing permutation polynomials of the forms $x h(\lambda_{j}(x))$ and $xh(\mu_{j}(x))$, where $\lambda_{j}(x)$ is the $j$-th elementary symmetric polynomial of $x, x^{q}, ..., x^{q^{m-1}}$ and $\mu_{j}(x)=\textup{Tr}_{{\bf F}_{q^{m}}/{\bf F}_{q}}(x^{j})$. This answers an open problem raised by Zieve in 2010. Finally, by using the linear translator, we construct the permutation polynomial of the form $L_1(x)+L_{2}(\gamma)h(f(x))$ over ${\bf F}_{q^{m}}$, which extends a result of Kyureghyan.