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A Super-Set of Patterson-Wiedemann Functions – Upper Bounds and Possible Nonlinearities

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Arithmetic of Finite Fields (WAIFI 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10064))

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Abstract

Constructing Boolean functions on odd number of variables with nonlinearity exceeding the bent concatenation bound is one of the most difficult combinatorial problems in the domain of Boolean functions and it has deep implications to coding theory and cryptology. After demonstration of such functions by Patterson and Wiedemann in 1983, for more than three decades the efforts have been channelized in obtaining the instances only. For the first time, in this paper, we try to explore non-trivial upper bounds on nonlinearity of such functions which are invariant under several group actions. In fact, we consider much larger sets of functions than what have been considered so far and obtain tight upper bounds on the nonlinearity in several cases. To support our claims, we present computational results for functions on n variables where n is an odd composite integer, \(9\le n\le 39\). In particular, our results for \(n = 15\) and 21 are of immediate interest given recent research results in this domain. Not only the upper bounds, we also identify what are the nonlinearities that can actually be achieved above the bent concatenation bound for such class of functions.

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Notes

  1. 1.

    In fact we also consider the cases where n is an odd composite integer such as \(n=9\), 25, or 27.

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Author information

Authors and Affiliations

  1. Department of Computer Engineering, Balıkesir University, 10145, Balıkesir, Turkey

    Selçuk Kavut

  2. Applied Statistics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata, 700108, India

    Subhamoy Maitra

  3. Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey

    Ferruh Özbudak

  4. Department of Mathematical Sciences, Aalborg University, Aalborg, Denmark

    Ferruh Özbudak

Authors
  1. Selçuk Kavut
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  2. Subhamoy Maitra
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  3. Ferruh Özbudak
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Corresponding author

Correspondence to Selçuk Kavut .

Editor information

Editors and Affiliations

  1. University of Rennes, Rennes, France

    Sylvain Duquesne

  2. University of Leuven, Leuven, Belgium

    Svetla Petkova-Nikova

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Kavut, S., Maitra, S., Özbudak, F. (2016). A Super-Set of Patterson-Wiedemann Functions – Upper Bounds and Possible Nonlinearities. In: Duquesne, S., Petkova-Nikova, S. (eds) Arithmetic of Finite Fields. WAIFI 2016. Lecture Notes in Computer Science(), vol 10064. Springer, Cham. https://doi.org/10.1007/978-3-319-55227-9_16

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  • DOI: https://doi.org/10.1007/978-3-319-55227-9_16

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-55226-2

  • Online ISBN: 978-3-319-55227-9

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