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Automorphisms of Finite Quasi-Groups without Sub-Quasi-Groups

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Abstract

Finite quasi-groups without sub-quasi-groups are considered. It is shown that polynomially complete quasi-groups with this property are quasi-primal. The case in which the automorphism groups act transitively on these quasi-groups is considered. Quasi-groups of prime-power order defined on an arithmetic vector space over a finite field are also studied. Necessary conditions for a multiplication in this space given in coordinate form to determine a quasi-group are found. The case of a vector space over the two-element field is considered in more detail. A criterion for a multiplication given in coordinate form by Boolean functions to determine a quasi-group is obtained. Under certain assumptions, quasi-groups of order 4 determined by Boolean functions are described up to isotopy. Polynomially complete quasi-groups are important in that the problem of solving polynomial equations is NP-complete in such quasi-groups. This property suggests using them for protecting information, because cryptographic transformations are based on quasi-group operations. In this context, an important role is played by quasi-groups containing no sub-quasi-groups.

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ACKNOWLEDGMENTS

The author is very grateful to the referees for many useful comments.

Funding

This work was supported by the Russian–Indian QGSEC project.

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Authors and Affiliations

  1. Department of Mechanics and Mathematics, Moscow State University, 119991, Moscow, Russia

    V. A. Artamonov

  2. All-Russian Academy of International Trade, 119285, Moscow, Russia

    V. A. Artamonov

  3. Academy of National Economy under the Government of the Russian Federation, 119571, Moscow, Russia

    V. A. Artamonov

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  1. V. A. Artamonov
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Correspondence to V. A. Artamonov.

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For the 70th birthday of Prof. A.I. Generalov

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Artamonov, V.A. Automorphisms of Finite Quasi-Groups without Sub-Quasi-Groups. Vestnik St.Petersb. Univ.Math. 53, 122–130 (2020). https://doi.org/10.1134/S106345412002003X

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  • DOI: https://doi.org/10.1134/S106345412002003X

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