Planar functions and perfect nonlinear monomials over finite fields

The study of finite projective planes involves planar functions, namely, functions f : Fq → Fq such that, for each a ∈ F ∗ q , the function c 7→ f(c + a) − f(c) is a bijection on Fq. 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 Fq of the form c 7→ c , under the assumption that q ≥ (t − 1). This implies two recent conjectures of Hernando, McGuire and Monserrat. Our arguments also yield a new proof of a conjecture of Segre and Bartocci from 1971 concerning monomial hyperovals in finite Desarguesian projective planes.