Planar functions and perfect nonlinear monomials over finite fields

TL;DR

This paper classifies all monomial planar functions over finite fields of a certain size, confirming two conjectures and providing a new proof for a classical conjecture about hyperovals in finite projective planes.

Contribution

It completely characterizes monomial planar functions c --> c^t over finite fields for large q, resolving two open conjectures and offering a new proof of a longstanding conjecture.

Findings

All monomial planar functions c --> c^t are classified for q >= (t-1)^4.

Confirmed two conjectures of Hernando, McGuire, and Monserrat.

Provided a new proof of a conjecture on monomial hyperovals from 1971.

Abstract

The study of finite projective planes involves planar functions, namely, functions f : F_q --> F_q such that, for each nonzero a in F_q, the function c --> f(c+a) - f(c) is a bijection on F_q. Planar functions are also used in the construction of DES-like cryptosystems, where they are called perfect nonlinear functions. We determine all planar functions on F_q of the form c --> c^t, under the assumption that q >= (t-1)^4. This implies two conjectures of Hernando, McGuire and Monserrat. Our arguments also yield a new proof of a conjecture of Segre and Bartocci from 1971 about monomial hyperovals in finite Desarguesian projective planes.