New classes of permutation polynomials with coefficients 1 over finite fields

TL;DR

This paper introduces new classes of permutation polynomials with coefficients 1 over finite fields, constructed via fractional polynomials and a novel method, with some being EA-inequivalent to known polynomials and achieving maximum algebraic degree.

Contribution

The paper constructs multiple new classes of permutation polynomials with coefficients 1 over finite fields, including methods to generate EA-inequivalent polynomials with maximum algebraic degree.

Findings

Constructed four classes of fractional permutation polynomials over cyclic subgroups.

Derived three new classes of permutation polynomials with coefficients 1 over finite fields.

Proved existence of EA-inequivalent permutation polynomials with maximum algebraic degree.

Abstract

Permutation polynomials with coefficients 1 over finite fields attract researchers' interests due to their simple algebraic form. In this paper, we first construct four classes of fractional permutation polynomials over the cyclic subgroup of $ \mathbb{F}_{2^{2m}} $. From these permutation polynomials, three new classes of permutation polynomials with coefficients 1 over $ \mathbb{F}_{2^{2m}} $ are constructed, and three more general new classes of permutation polynomials with coefficients 1 over $ \mathbb{F}_{2^{2m}} $ are constructed using a new method we presented recently. Some known permutation polynomials are the special cases of our new permutation polynomials. Furthermore, we prove that, in all new permutation polynomials, there exists a permutation polynomial which is EA-inequivalent to known permutation polynomials for all even positive integer $ m $. This proof shows that EA-inequivalent permutation polynomials over $ \mathbb{F}_{q} $ can be constructed from EA-equivalent permutation polynomials over the cyclic subgroup of $ \mathbb{F}_{q} $. From this proof, it is obvious that, in all new permutation polynomials, there exists a permutation polynomial of which algebraic degree is the maximum algebraic degree of permutation polynomials over $ \mathbb{F}_{2^{2m}} $.