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Erschienen in: Cryptography and Communications 1/2017

01.07.2016

Bent functions linear on elements of some classical spreads and presemifields spreads

verfasst von: Kanat Abdukhalikov, Sihem Mesnager

Erschienen in: Cryptography and Communications | Ausgabe 1/2017

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Abstract

Bent functions are maximally nonlinear Boolean functions with an even number of variables. They have attracted a lot of research for four decades because of their own sake as interesting combinatorial objects, and also because of their relations to coding theory, sequences and their applications in cryptography and other domains such as design theory. In this paper we investigate explicit constructions of bent functions which are linear on elements of spreads. After presenting an overview on this topic, we study bent functions which are linear on elements of presemifield spreads and give explicit descriptions of such functions for known commutative presemifields. A direct connection between bent functions which are linear on elements of the Desarguesian spread and oval polynomials over finite fields was proved by Carlet and the second author. Very recently, further nice extensions have been made by Carlet in another context. We introduce oval polynomials for semifields which are dual to symplectic semifields. In particular, it is shown that from a linear oval polynomial for a semifield one can get an oval polynomial for transposed semifield.

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Fußnoten
1
Recall that the general partial spreads class \(\mathcal {P}\mathcal {S}\), introduced by Dillon, equals the union of \(\mathcal {PS}^{-}\) and \(\mathcal {PS}^{+}\). Dillon has applied the construction to the Desarguesian spread and deduced the subclass of \(\mathcal {PS}^{-}\) denoted by \(\mathcal {PS}_{ap}\) whose elements are constant on the elements of the Desarguesian spread. Functions f of the class \(\mathcal {PS}_{ap}\) are given in bivariate form as \(f(x,y)=g(xy^{2^{m}-2})\) where \(x,y\in \mathbb {F}_{2^{m}}\) and g is any balanced Boolean function on \(\mathbb {F}_{2^{m}}\) which vanishes at 0.
 
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Metadaten
Titel
Bent functions linear on elements of some classical spreads and presemifields spreads
verfasst von
Kanat Abdukhalikov
Sihem Mesnager
Publikationsdatum
01.07.2016
Verlag
Springer US
Erschienen in
Cryptography and Communications / Ausgabe 1/2017
Print ISSN: 1936-2447
Elektronische ISSN: 1936-2455
DOI
https://doi.org/10.1007/s12095-016-0195-4

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