Low-degree planar polynomials over finite fields of characteristic two

TL;DR

This paper classifies low-degree planar polynomials over finite fields of characteristic two, showing they are composed of monomials with degrees that are powers of two, and explores their implications for algebraic curves and cryptography.

Contribution

It provides a complete classification of low-degree planar polynomials over characteristic two fields, linking polynomial structure to algebraic curve properties and extending to exceptional cases.

Findings

Polynomials of degree ≤ q^{1/4} inducing planar functions have monomials with degrees as powers of two.

Complete classification of exceptional planar polynomials over finite fields.

New partial results on the classification of almost perfect nonlinear functions.

Abstract

Planar functions are mappings from a finite field $\mathbb{F}_q$ to itself with an extremal differential property. Such functions give rise to finite projective planes and other combinatorial objects. There is a subtle difference between the definitions of these functions depending on the parity of $q$ and we consider the case that $q$ is even. We classify polynomials of degree at most $q^{1/4}$ that induce planar functions on $\mathbb{F}_q$, by showing that such polynomials are precisely those in which the degree of every monomial is a power of two. As a corollary we obtain a complete classification of exceptional planar polynomials, namely polynomials over $\mathbb{F}_q$ that induce planar functions on infinitely many extensions of~$\mathbb{F}_q$. The proof strategy is to study the number of $\mathbb{F}_q$-rational points of an algebraic curve attached to a putative planar function.~Our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear~functions.