Planar functions and perfect nonlinear monomials over finite fields

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.

10 pages

References in corpus (1)

• On the Classification of Exceptional Planar Functions over ${\mathbb{F}}_p$

Cited by in corpus (10)

• Low-degree planar monomials in characteristic two
• Semifields, relative difference sets, and bent functions
• Towards the full classification of exceptional scattered polynomials
• Planar functions over fields of characteristic two
• The apparent structure of dense Sidon sets
• Exceptional Scattered Polynomials
• Dembowski-Ostrom polynomials and Dickson polynomials
• $C$-differentials, multiplicative uniformity and (almost) perfect $c$-nonlinearity
• Exceptional planar polynomials
• Characterizations of a Class of Planar Functions over Finite Fields