Planar functions and perfect nonlinear monomials over finite fields

The study of finite projective planes involves planar functions, namely, functions $$f:\mathbb {F}_q\rightarrow \mathbb {F}_q$$f:Fq→Fq such that, for each $$a\in \mathbb {F}_q^*$$a∈Fq∗, the function $$c\mapsto f(c+a)-f(c)$$c↦f(c+a)-f(c) is a bijection on $$\mathbb {F}_q$$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 $$\mathbb {F}_q$$Fq of the form $$c\mapsto c^t$$c↦ct, under the assumption that $$q\ge (t-1)^4$$q≥(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.

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