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On generating invertible circulant binary matrices with a prescribed number of ones

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Abstract

We study the problem how to efficiently generate circulant binary matrices with a prescribed number of ones which are invertible over \(\mathbb {Z}_{2}\). A natural method to generate such matrices consists of two steps. Firstly, a circulant binary matrix with the prescribed number of ones is generated. Afterwards, it is tested for invertibility and if needed the process is repeated. To increase the efficiency of the process, we are interested in generating the matrices directly, without the need for the additional invertibility testing. We propose algorithms which fulfill this task for a wide range of parameters. Furthermore, we propose algorithms to construct matrices S and Q in the QC-LDPC McEliece cryptosystem. Matrices S and Q have to be composed of blocks of circulant matrices and they have to be invertible. In addition, S has to be dense and Q has to have a prescribed number of ones in a row. To avoid known attacks on the QC-LDPC McEliece cryptosystem, our algorithms generate S and Q with blocks of an odd size.

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Acknowledgements

The authors wish to thank the anonymous referees for the conference ArcticCrypt 2016 for remarks which helped to improve the quality of the paper.

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Authors and Affiliations

  1. Faculty of Electrical Engineering and Information Technology, Slovak University of Technology in Bratislava, Ilkovičova 3, 81219, Bratislava, Slovakia

    Tomáš Fabšič, Otokar Grošek & Pavol Zajac

  2. Slovak Academy of Sciences, Mathematical Institute, Štefánikova 49, 814 73, Bratislava, Slovakia

    Karol Nemoga

Authors
  1. Tomáš Fabšič
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  2. Otokar Grošek
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  3. Karol Nemoga
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  4. Pavol Zajac
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Corresponding author

Correspondence to Tomáš Fabšič.

Additional information

This article is part of the Topical Collection on Recent Trends in Cryptography

This work is a partial result of the Research and Development Operational Programme for the project International centre of excellence for research of intelligent and secure information-communication technologies and systems, ITMS 26240120039, co-funded by the ERDF.

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Fabšič, T., Grošek, O., Nemoga, K. et al. On generating invertible circulant binary matrices with a prescribed number of ones. Cryptogr. Commun. 10, 159–175 (2018). https://doi.org/10.1007/s12095-017-0239-4

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  • DOI: https://doi.org/10.1007/s12095-017-0239-4

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