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Linear codes from weakly regular plateaued functions and their secret sharing schemes

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Abstract

Linear codes, the most significant class of codes in coding theory, have diverse applications in secret sharing schemes, authentication codes, communication, data storage devices and consumer electronics. The main objectives of this paper are twofold: to construct three-weight linear codes from plateaued functions over finite fields, and to analyze the constructed linear codes for secret sharing schemes. To do this, we generalize the recent contribution of Mesnager given in (Cryptogr Commun 9(1):71–84, 2017). We first introduce the notion of (non)-weakly regular plateaued functions over \({{\mathbb {F}}}_p\), with p being an odd prime. We next construct three-weight linear p-ary (resp. binary) codes from weakly regular p-ary plateaued (resp. Boolean plateaued) functions and determine their weight distributions. We finally observe that the constructed linear codes are minimal for almost all cases, which implies that they can be directly used to construct secret sharing schemes with nice access structures. To the best of our knowledge, the construction of linear codes from plateaued functions over \({{\mathbb {F}}}_p\), with p being an odd prime, is studied in this paper for the first time in the literature.

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Acknowledgements

The authors extend thanks to the Editor and anonymous reviewers for their valuable comments and suggestions, which improved the quality and presentation of the manuscript. 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 at University of Paris VIII, LAGA, UMR 7539, CNRS at University of Paris XIII and Telecom ParisTech, Paris, France

    Sihem Mesnager

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

    Ferruh Özbudak

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

    Ahmet Sınak

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

    Ahmet Sınak

  5. LAGA, UMR 7539, CNRS at Universities of Paris VIII and XIII, Paris, France

    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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The first version of this work [22] was presented at WCC-2017.

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”

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Mesnager, S., Özbudak, F. & Sınak, A. Linear codes from weakly regular plateaued functions and their secret sharing schemes. Des. Codes Cryptogr. 87, 463–480 (2019). https://doi.org/10.1007/s10623-018-0556-4

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  • DOI: https://doi.org/10.1007/s10623-018-0556-4

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