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Generalized Walsh transforms of symmetric and rotation symmetric Boolean functions are linear recurrent

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Abstract

Exponential sums of symmetric Boolean functions are linear recurrent with integer coefficients. This was first established by Cai, Green and Thierauf in the mid nineties. Consequences of this result has been used to study the asymptotic behavior of symmetric Boolean functions. Recently, Cusick extended it to rotation symmetric Boolean functions, which are functions with good cryptographic properties. In this article, we put all these results in the general context of Walsh transforms and some of its generalizations (nega–Hadamard transform, for example). Precisely, we show that Walsh transforms, for which exponential sums are just an instance, of symmetric and rotation symmetric Boolean functions satisfy linear recurrences with integer coefficients. We also provide a closed formula for the Walsh transform and nega–Hadamard transform of any symmetric Boolean functions. Moreover, using the techniques presented in this work, we show that some families of rotation symmetric Boolean functions are not bent when the number of variables is sufficiently large and provide asymptotic evidence to a conjecture of Stănică and Maitra.

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Acknowledgements

The authors would like to thank the anonymous reviewers for the thorough reading and for their detailed and useful comments that improved the paper. This paper was written during a pleasant visit of P. S. to the Department of Mathematics of the University of Puerto Rico in Spring of 2017. This author thanks the institution for hospitality.

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Authors and Affiliations

  1. Department of Mathematics, University of Puerto Rico, San Juan, PR, 00936, USA

    Francis N. Castro & Luis A. Medina

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

    Pantelimon Stănică

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  1. Francis N. Castro
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  2. Luis A. Medina
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  3. Pantelimon Stănică
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Correspondence to Luis A. Medina.

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Castro, F.N., Medina, L.A. & Stănică, P. Generalized Walsh transforms of symmetric and rotation symmetric Boolean functions are linear recurrent. AAECC 29, 433–453 (2018). https://doi.org/10.1007/s00200-018-0351-5

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  • DOI: https://doi.org/10.1007/s00200-018-0351-5

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