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On the existence and non-existence of some classes of bent–negabent functions

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Abstract

In this paper we investigate different questions related to bent–negabent functions. We first take an expository look at the state-of-the-art research in this domain and point out some technical flaws in certain results and fix some of them. Further, we derive a necessary and sufficient condition for which the functions of the form \({\mathbf{x}}\cdot \pi ({\mathbf{y}})\oplus h({\mathbf{y}})\) [Maiorana–McFarland (\({\mathcal {M}}\))] is bent–negabent, and more generally, we study the non-existence of bent–negabent functions in the \({\mathcal {M}}\) class. We also identify some functions that are bent–negabent. Next, we continue the recent work by Mandal et al. (Discrete Appl Math 236:1–6, 2018) on rotation symmetric bent–negabent functions and show their non-existence in larger classes. For example, we prove that there is no rotation symmetric bent–negabent function in \(4p^k\) variables, where p is an odd prime. We present the non-existence of such functions in certain classes that are affine transformations of rotation symmetric functions. Keeping in mind the existing literature, we correct here some technical issues and errors found in other papers and provide some novel results.

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Acknowledgements

The authors are grateful to the reviewers for their helpful comments and suggestions which have highly improved the manuscript. The work was started during an enjoyable visit of the third-named author to ISI-Kolkata in the Spring of 2019. This author would like to thank the host institution for the excellent working conditions.

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Authors and Affiliations

  1. CARAMBA, INRIA, 54600, Nancy, Grand Est., France

    Bimal Mandal

  2. Applied Statistics Unit, Indian Statistical Institute, Kolkata, India

    Subhamoy Maitra

  3. Department of Applied Mathematics, Naval Postgraduate School, Monterey, 93943, USA

    Pantelimon Stănică

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  1. Bimal Mandal
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  2. Subhamoy Maitra
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  3. Pantelimon Stănică
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Correspondence to Pantelimon Stănică.

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Mandal, B., Maitra, S. & Stănică, P. On the existence and non-existence of some classes of bent–negabent functions. AAECC 33, 237–260 (2022). https://doi.org/10.1007/s00200-020-00444-w

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  • DOI: https://doi.org/10.1007/s00200-020-00444-w

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