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Algebraic Structure of Permutational Polynomials over $\mathbb{F}_{q^n}$

Pingzhi Yuan

Abstract

In this paper, we propose a new algebraic structure of permutation polynomials over $\mathbb{F}_{q^n}$. As an application of this new algebraic structure, we give some classes of new PPs over $\mathbb{F}_{q^n}$ and answer an open problem in Charpin and Kyureghyan.

Algebraic Structure of Permutational Polynomials over $\mathbb{F}_{q^n}$

Abstract

In this paper, we propose a new algebraic structure of permutation polynomials over . As an application of this new algebraic structure, we give some classes of new PPs over and answer an open problem in Charpin and Kyureghyan.

Paper Structure

This paper contains 4 sections, 14 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Some lemmas and propositions
  3. Proof of Theorem 1.2
  4. PPs of the shape $G(x) + \gamma{\rm Tr}(H(x))$

Key Result

Theorem 1.1

1. Let $\{\omega_1, \omega_2, \dots, \omega_n\}$ be any given basis of ${{\mathbb F}} _{q^n}$ over ${{\mathbb F}} _{q}$, and let $L(x) = \sum_{i=0}^{n-1}a_ix^{q^i}$ be a linear polynomial over ${{\mathbb F}} _{q^n}$. Then there are n elements $\theta_1, \theta_2, \dots, \theta_n\in {{\mathbb F}} _{q Moreover, $L(x)$ is a permutation polynomial if and only if $\{\theta_1, \theta_2, \dots, \theta_n\

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Lemma 2.1
  • Proposition 2.1
  • proof
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 16 more