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Symmetric and Asymmetric Cryptography on the Special Linear Cracovian Quasigroup

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Intelligent Information and Database Systems (ACIIDS 2022)

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Abstract

We propose symmetric and public key cryptographic protocols on the non-associative special linear Cracovian quasigroup. The \(\textrm{SL}_n(Z)\) Cracovian quasigroup is the set of \(n\times n\) matrices with determinant one and entries in the ring of integers Z with the non-associative multiplication. The strength of the proposed symmetric cipher is based on the NP-hardness of the non-negative matrix factorization problem. We define the totient function for a given matrix from the special linear group over the ring Z/mZ of integers modulo m as the order of a cyclic subgroup generated by this matrix. Defined in this way totient function allows us to construct the public key cryptographic protocol on the special linear group \(\textrm{SL}_n(Z_m)\) and generalize it to the Cracovian quasigroup.

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Author information

Authors and Affiliations

  1. University of Opole, Opole, Poland

    Zbigniew Lipiński

  2. Military University of Land Forces, Wrocław, Poland

    Jolanta Mizera-Pietraszko

Authors
  1. Zbigniew Lipiński
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  2. Jolanta Mizera-Pietraszko
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Correspondence to Jolanta Mizera-Pietraszko .

Editor information

Editors and Affiliations

  1. Wrocław University of Science and Technology, Wrocław, Poland

    Ngoc Thanh Nguyen

  2. Vietnam National University, Ho Chi Minh City, Ho Chi Minh City, Vietnam

    Tien Khoa Tran

  3. Al-Farabi Kazakh National University, Almaty, Kazakhstan

    Ualsher Tukayev

  4. National University of Kaohsiung, Kaohsiung, Taiwan

    Tzung-Pei Hong

  5. Wrocław University of Science and Technology, Wrocław, Poland

    Bogdan Trawiński

  6. University of Newcastle, Newcastle, NSW, Australia

    Edward Szczerbicki

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Lipiński, Z., Mizera-Pietraszko, J. (2022). Symmetric and Asymmetric Cryptography on the Special Linear Cracovian Quasigroup. In: Nguyen, N.T., Tran, T.K., Tukayev, U., Hong, TP., Trawiński, B., Szczerbicki, E. (eds) Intelligent Information and Database Systems. ACIIDS 2022. Lecture Notes in Computer Science(), vol 13758. Springer, Cham. https://doi.org/10.1007/978-3-031-21967-2_52

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  • DOI: https://doi.org/10.1007/978-3-031-21967-2_52

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-21966-5

  • Online ISBN: 978-3-031-21967-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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