Planar functions and perfect nonlinear monomials over finite fields
The study of finite projective planes involves planar functions, namely, functions \(f:\mathbb {F}_q\rightarrow \mathbb {F}_q\) such that, for each \(a\in \mathbb {F}_q^*\), the function \(c\mapsto f(c+a)-f(c)\) is a bijection on \(\mathbb {F}_q\). 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 \(\mathbb {F}_q\) of the form \(c\mapsto c^t\), under the assumption that \(q\ge (t-1)^4\). This resolves two conjectures of Hernando, McGuire and Monserrat. Our arguments also yield a new proof of a conjecture of Segre and Bartocci about monomial hyperovals in finite Desarguesian projective planes.
planar functions perfect nonlinear functions monomial hyperovals 51e20 11t06 11t71 05b05