Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dembowski-Ostrom polynomials and Dickson polynomials

Sartaj Ul Hasan, Mohit Pal

TL;DR

A complete classification of Dembowski-Ostrom polynomials from the composition of Dickson poynomials of arbitrary kind and monomials over algebraic curves over finite flelds is given.

Abstract

We give a complete classification of Dembowski-Ostrom polynomials from the composition of Dickson polynomials of arbitrary kind and monomials over finite fields. Moreover, by using a variant of the Weil bound for the number of points of affine algebraic curves over finite fields, we discuss the planarity of the obtained Dembowski-Ostrom polynomials. Dembowski-Ostrom polynomials play a crucial role in coding theory and cryptography.

Dembowski-Ostrom polynomials and Dickson polynomials

TL;DR

A complete classification of Dembowski-Ostrom polynomials from the composition of Dickson poynomials of arbitrary kind and monomials over algebraic curves over finite flelds is given.

Abstract

We give a complete classification of Dembowski-Ostrom polynomials from the composition of Dickson polynomials of arbitrary kind and monomials over finite fields. Moreover, by using a variant of the Weil bound for the number of points of affine algebraic curves over finite fields, we discuss the planarity of the obtained Dembowski-Ostrom polynomials. Dembowski-Ostrom polynomials play a crucial role in coding theory and cryptography.

Paper Structure

This paper contains 7 sections, 7 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. DO polynomials from the polynomial $\mathfrak{D}_{k,2}$
  3. DO polynomials from the Polynomial $\mathfrak{D}_{k,3}$
  4. DO polynomials from the polynomial $\mathfrak{D}_{k,4}$
  5. The case $m \geq 5$
  6. Discussion on planarity
  7. The complete list of DO polynomials

Key Result

Theorem 2.1

Let $q$ be a power of odd prime $p$ and $a\in\mathbb{F}_q^*$. The polynomial $\mathfrak D_{k,2}$ is DO polynomial over $\mathbb{F}_q$ if and only if one of the following holds.

Theorems & Definitions (11)

  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • proof
  • Lemma 6.1
  • Lemma 6.2
  • ...and 1 more