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On a Class of Permutation Polynomials and Their Inverses

Ruikai Chen, Sihem Mesnager

Abstract

We introduce a class of permutation polynomial over $\mathbb F_{q^n}$ that can be written in the form $\frac{L(x)}{x^{q+1}}$ or $\frac{L(x^{q+1})}x$ for some $q$-linear polynomial $L$ over $\mathbb F_{q^n}$. Specifically, we present those permutation polynomials explicitly as well as their inverses. In addition, more permutation polynomials can be derived in a more general form.

On a Class of Permutation Polynomials and Their Inverses

Abstract

We introduce a class of permutation polynomial over that can be written in the form or for some -linear polynomial over . Specifically, we present those permutation polynomials explicitly as well as their inverses. In addition, more permutation polynomials can be derived in a more general form.
Paper Structure (4 sections, 11 theorems, 69 equations)

This paper contains 4 sections, 11 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Two Kinds of Permutation Polynomials with Inverses
  3. A More General Form of Permutation Polynomials
  4. Conclusions

Key Result

Theorem 1

If $\frac{L(x)}{x^{q+1}}$ is a permutation polynomial of $\mathbb F_{q^n}$, then so is $\frac{L^\prime(x^{q+1})}{x}$. The converse is true provided that $n$ is odd.

Theorems & Definitions (23)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Proposition 3
  • proof
  • Lemma 4
  • Example 5
  • Theorem 6
  • proof
  • ...and 13 more