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The compositional inverses of three classes of permutation polynomials over finite fields

Danyao Wu, Pingzhi Yuan

Abstract

Recently, P. Yuan presented a local method to find permutation polynomials and their compositional inverses over finite fields. The work of P. Yuan inspires us to compute the compositional inverses of three classes of the permutation polynomials: (a) the permutation polynomials of the form $ax^q+bx+(x^q-x)^k$ over $\mathbb{F}_{q^2},$ where $a+b \in \mathbb{F}_q^*$ or $a^q=b;$ (b) the permutation polynomials of the forms $f(x)=-x+x^{(q^2+1)/2}+x^{(q^3+q)/2} $ and $f(x)+x$ over $\mathbb{F}_{q^3};$ (c) the permutation polynomial of the form $A^{m}(x)+L(x)$ over $\mathbb{F}_{q^n},$ where ${\rm Im}(A(x))$ is a vector space with dimension $1$ over $\mathbb{F}_{q}$ and $L(x)$ is not a linearized permutation polynomial.

The compositional inverses of three classes of permutation polynomials over finite fields

Abstract

Recently, P. Yuan presented a local method to find permutation polynomials and their compositional inverses over finite fields. The work of P. Yuan inspires us to compute the compositional inverses of three classes of the permutation polynomials: (a) the permutation polynomials of the form over where or (b) the permutation polynomials of the forms and over (c) the permutation polynomial of the form over where is a vector space with dimension over and is not a linearized permutation polynomial.

Paper Structure

This paper contains 5 sections, 14 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The compositional inverses of the PPs of the form $ax^q+bx+(x^q-x)^k$ over ${{\mathbb F}} _{q^2}$
  4. The compositional inverse of a class of CPP over ${{\mathbb F}} _{q^3}$
  5. The permutation polynomial of the form $A(x)^m +L(x)$ and its inverse over ${{\mathbb F}} _{q^n}$

Key Result

Lemma 1

akbary2011constructing Let $A, S$ and $\overline{S}$ be finite sets with $\sharp S =\sharp \overline{S}$, and let $f(x) : A\longrightarrow A$, $h(x): S\longrightarrow \overline{S}$, $\lambda(x): A\longrightarrow S,$ and $\overline{\lambda}(x):A\longrightarrow \overline{S}$ be maps such that $\overli

Theorems & Definitions (20)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Lemma 5
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 10 more