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Characterizations of Partially Bent and Plateaued Functions over Finite Fields

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Arithmetic of Finite Fields (WAIFI 2018)

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

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Abstract

Partially bent and plateaued functions over finite fields have significant applications in cryptography, sequence theory, coding theory, design theory and combinatorics. They have been extensively studied due to their various desirable cryptographic properties. In this paper, we study on characterizations of partially bent and plateaued functions over finite fields, with the aim of clarifying their structure. We first redefine the notion of partially bent functions over any finite field \({\mathbb {F}}_q\), with q a prime power, and then provide a few characterizations of these functions in terms of their derivatives, Walsh power moments and autocorrelation functions. We next characterize partially bent (vectorial) functions over \({\mathbb {F}}_p\), with p a prime, by means of their derivatives and Walsh power moments. We finally characterize plateaued functions over \({\mathbb {F}}_p\) in terms of their Walsh power moments, derivatives and autocorrelation functions.

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Acknowledgment

The authors would like to thank the anonymous reviewers of WAIFI-2018 for their valuable comments and suggestions. The third author is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK), program no: BİDEB 2214/A.

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Authors and Affiliations

  1. Department of Mathematics, University of Paris VIII, Saint-Denis, France

    Sihem Mesnager

  2. LAGA, UMR 7539, CNRS, University of Paris VIII, Saint-Denis, France

    Sihem Mesnager & Ahmet Sınak

  3. LAGA, UMR 7539, CNRS, University of Paris XIII, Villetaneuse, France

    Sihem Mesnager & Ahmet Sınak

  4. Telecom ParisTech, Paris, France

    Sihem Mesnager

  5. Department of Mathematics, Middle East Technical University, Ankara, Turkey

    Ferruh Özbudak

  6. Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey

    Ferruh Özbudak & Ahmet Sınak

  7. Department of Mathematics and Computer Sciences, Necmettin Erbakan University, Konya, Turkey

    Ahmet Sınak

Authors
  1. Sihem Mesnager
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  2. Ferruh Özbudak
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  3. Ahmet Sınak
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Correspondence to Ahmet Sınak .

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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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Mesnager, S., Özbudak, F., Sınak, A. (2018). Characterizations of Partially Bent and Plateaued Functions over Finite Fields. 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_12

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  • DOI: https://doi.org/10.1007/978-3-030-05153-2_12

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