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On Symmetry and Differential Properties of Generalized Boolean Functions

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Book cover Arithmetic of Finite Fields (WAIFI 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11321))

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Abstract

In this paper we investigate various differential properties of generalized Boolean functions defined on \({\mathbb F}_2^n\) with values in \({\mathbb Z}_{2^k}\), \(k\ge 2\). We characterize linear structures for the generalized Boolean functions in terms of their binary expansion components, and find all symmetric generalized bent functions. Next, we show that there are no symmetric balanced functions defined on \({\mathbb F}_2^n\) with values in a group of order \(2^k, k\ge 2\), a contrast to the classical case for \(k=1\), commonly known as the bisection of binomial coefficients. Further, we characterize the avalanche features of a generalized Boolean function in terms of differentials. Lastly, we show that a partially gbent function is plateaued.

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Acknowledgement

This paper was started while the second and fourth named authors visited the third named author at the Institute of Algebra and Geometry, of Otto von Guericke University Magdeburg. They thank the host and the institute for hospitality and excellent working conditions.

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Authors and Affiliations

  1. Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, 93943–5216, USA

    Thor Martinsen & Pantelimon Stănică

  2. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, 4040, Linz, Austria

    Wilfried Meidl

  3. Institute of Algebra and Geometry, Faculty of Mathematics, Otto von Guericke University Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Alexander Pott

Authors
  1. Thor Martinsen
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  2. Wilfried Meidl
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  3. Alexander Pott
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  4. Pantelimon Stănică
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Corresponding author

Correspondence to Pantelimon Stănică .

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Editors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Lilya Budaghyan

  2. Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Mexico, Mexico

    Francisco Rodríguez-Henríquez

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Martinsen, T., Meidl, W., Pott, A., Stănică, P. (2018). On Symmetry and Differential Properties of Generalized Boolean Functions. In: Budaghyan, L., Rodríguez-Henríquez, F. (eds) Arithmetic of Finite Fields. WAIFI 2018. Lecture Notes in Computer Science(), vol 11321. Springer, Cham. https://doi.org/10.1007/978-3-030-05153-2_11

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  • DOI: https://doi.org/10.1007/978-3-030-05153-2_11

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-05152-5

  • Online ISBN: 978-3-030-05153-2

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