New families of permutation trinomials constructed by permutations of $\mu_{q+1}$

TL;DR

This paper introduces new classes of permutation trinomials over finite fields, constructed via permutations of roots of unity, advancing algebraic tools for applications in coding theory and cryptography.

Contribution

It presents novel permutation trinomials in finite fields derived from permutations of roots of unity, expanding the known classes of such polynomials.

Findings

New permutation trinomials constructed in $\

Permutation polynomials based on permutations of $(q+1)$-th roots of unity.

Applications potential in coding theory and cryptography.

Abstract

Permutation polynomials are of particular significance in several areas of applied mathematics, such as Coding theory and Cryptography. Many recent constructions are based on the Akbary-Ghioca-Wang (AGW) criterion. Along this line of research, we provide new classes of permutation trinomials in $\mathbb{F}_{q^2}$ of the form $f(x)=x^r h(x^{q-1})$, by studying permutations of the set of $(q+1)$-th roots of unity, which look like monomials on the sets of suitable partitions.