Designs, Codes and Cryptography

, Volume 75, Issue 1, pp 71–80 | Cite as

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 

Mathematics Subject Classification

51E20 11T06 11T71 05B05 


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Mathematical Sciences CenterTsinghua UniversityBeijingChina

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