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On Existence (Based on an Arithmetical Problem) and Constructions of Bent Functions

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Cryptography and Coding (IMACC 2015)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 9496))

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Abstract

Bent functions are maximally nonlinear Boolean functions. They are wonderful creatures introduced by O. Rothaus in the 1960’s and studied firstly by J. Dillon since 1974. Using some involutions over finite fields, we present new constructions of bent functions in the line of recent Mesnager’s works. One of the constructions is based on an arithmetical problem. We discuss existence of such bent functions using Fermat hypersurface and Lang-Weil estimations.

The paper was presented as a part of an invited talk entitled “Bent functions and their connections to coding theory and cryptography” at the fifteenth International Conference on Cryptography and Coding, Oxford, United Kingdom (IMACC 2015) given by S. Mesnager.

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Notes

  1. 1.

    The Maiorana-McFarland completed class is the smallest class containing the class of Maiorana-McFarland which is globally invariant under the action of the general affine group and under the addition of affine functions.

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Acknowledgments

The first author thanks Jens Groth (Program Chair of the international conference IMACC 2015) for his nice invitation.

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Authors and Affiliations

  1. Department of Mathematics, University of Paris VIII, University of Paris XIII, LAGA, UMR 7539, CNRS, and Télécom ParisTech, Paris, France

    Sihem Mesnager

  2. Télécom ParisTech, UMR 5141, CNRS, Paris, France

    Gérard Cohen & David Madore

Authors
  1. Sihem Mesnager
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  2. Gérard Cohen
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  3. David Madore
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Correspondence to Sihem Mesnager .

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Editors and Affiliations

  1. University College London, Cambridge, United Kingdom

    Jens Groth

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Mesnager, S., Cohen, G., Madore, D. (2015). On Existence (Based on an Arithmetical Problem) and Constructions of Bent Functions. In: Groth, J. (eds) Cryptography and Coding. IMACC 2015. Lecture Notes in Computer Science(), vol 9496. Springer, Cham. https://doi.org/10.1007/978-3-319-27239-9_1

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  • DOI: https://doi.org/10.1007/978-3-319-27239-9_1

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  • Online ISBN: 978-3-319-27239-9

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