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Permutation Polynomials of the form $L(X)+γTr_q^{q^3}(h(X))$ over finite fields with even characteristic

Xuan Pang, Danyao Wu, Pingzhi Yuan

Abstract

Permutation polynomials over finite fields have extensive applications in various areas. Particularly, permutation polynomials with simple forms are of great interest. In recent papers, several classes of permutation polynomials of the form $L(X)+Tr_q^{q^3}(h(X))$ have been constructed. This paper further investigates permutation polynomials of such form over $\mathbb{F}_{q^3}$. Unlike previous studies, we transform the problem of constructing univariate permutation polynomials over finite fields into that of constructing corresponding multivariate permutations over $\mathbb{F}_{q}$-vector spaces. Through this approach, we completely characterize a class of permutation polynomials of the form $L(X)+γTr_q^{q^3}(c_1X+c_2X^2+c_3X^3+c_4X^{q+2})$ over $\mathbb{F}_{q^3}$, where $q=2^m$, $L(X)=X^q+aX$ and $a,c_1,c_2,c_3,c_4,γ\in\mathbb{F}_q$ with $a^2+a+1\neq0$. Furthermore, using a similar method, we generalize several results from a recent work by Jiang, Li and Qu (2026).

Permutation Polynomials of the form $L(X)+γTr_q^{q^3}(h(X))$ over finite fields with even characteristic

Abstract

Permutation polynomials over finite fields have extensive applications in various areas. Particularly, permutation polynomials with simple forms are of great interest. In recent papers, several classes of permutation polynomials of the form have been constructed. This paper further investigates permutation polynomials of such form over . Unlike previous studies, we transform the problem of constructing univariate permutation polynomials over finite fields into that of constructing corresponding multivariate permutations over -vector spaces. Through this approach, we completely characterize a class of permutation polynomials of the form over , where , and with . Furthermore, using a similar method, we generalize several results from a recent work by Jiang, Li and Qu (2026).
Paper Structure (6 sections, 20 theorems, 56 equations, 1 table)

This paper contains 6 sections, 20 theorems, 56 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. PPs of the form $L(X)+\gamma{\rm Tr}_q^{q^3}(h(X))$ over $\mathbb{F}_{q^3}$
  4. PPs of the form $L(X)+\gamma{\rm Tr}_q^{q^3}(c_1X+c_2X^2+c_3X^3+c_4X^{q+2})$
  5. PPs of the form $X+\gamma{\rm Tr}_q^{q^3}(h(X))$ extended from Table \ref{['table1']}
  6. Conclusion

Key Result

Lemma 2.1

Lidl Let $f(X)\in\mathbb{F}_{q}[X]$. Then $f(X)$ is a PP of $\mathbb{F}_{q}[X]$ if and only if $cf(dX)+b$ is a PP of $\mathbb{F}_{q}[X]$ for any given $c,d\in\mathbb{F}_{q}^*$ and $b\in\mathbb{F}_q$.

Theorems & Definitions (34)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 24 more