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

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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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Notes

  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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Acknowledgments

The authors thank Prof. Claude Carlet for interesting discussions and the editors Prof. Cunsheng Ding and Prof. Zhengchun Zhou for their valuable comments.

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Authors and Affiliations

  1. Department of Mathematical Sciences, UAE University, PO Box, 15551, Al Ain, United Arab Emirates

    Kanat Abdukhalikov

  2. Department of Mathematics, University of Paris VIII, University of Paris XIII, LAGA, UMR 7539, CNRS, Paris, France

    Sihem Mesnager

  3. Télécom ParisTech, Paris, France

    Sihem Mesnager

Authors
  1. Kanat Abdukhalikov
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  2. Sihem Mesnager
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Correspondence to Sihem Mesnager.

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The first author was supported by UAEU grant 31S107.

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Abdukhalikov, K., Mesnager, S. Bent functions linear on elements of some classical spreads and presemifields spreads. Cryptogr. Commun. 9, 3–21 (2017). https://doi.org/10.1007/s12095-016-0195-4

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