Dembowski-Ostrom polynomials and Dickson polynomials

TL;DR

This paper classifies Dembowski-Ostrom polynomials derived from Dickson polynomials and monomials over finite fields, and analyzes their planarity using algebraic geometry tools, with implications for coding theory and cryptography.

Contribution

It provides a complete classification of such polynomials and investigates their planarity properties using a Weil bound variant.

Findings

Classification of Dembowski-Ostrom polynomials from Dickson compositions

Identification of conditions for planarity of these polynomials

Implications for coding theory and cryptography

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.