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Designs, Codes and Cryptography

Erschienen in:

01.04.2015

Planar functions and perfect nonlinear monomials over finite fields

verfasst von: Michael E. Zieve

Erschienen in: Designs, Codes and Cryptography | Ausgabe 1/2015

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Abstract

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.

Vorheriger Artikel Hardness of learning problems over Burnside groups of exponent 3

Nächster Artikel Perfect codes in the discrete simplex

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Titel: Planar functions and perfect nonlinear monomials over finite fields
verfasst von: Michael E. Zieve
Publikationsdatum: 01.04.2015
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 1/2015
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-013-9890-8

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