Skip to main content
Log in

Latin Squares over Quasigroups

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

We propose a construction that allows generating large families of Latin squares, i.e., Cayley tables of finite quasigroups. This construction generalizes proper families of functions over Abelian groups introduced by Nosov and Pankratiev. We also show that all quasigroups generated by the original construction contain at least one subquasigroup, while the generalized construction generates quasigroups free of subquasigroups.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. C. Shannon, ‘‘Communication theory of secrecy systems,’’ Bell Syst. Tech. J. 28, 656–715 (1949).

    Article  MathSciNet  Google Scholar 

  2. M. M. Glukhov, ‘‘On appplications of quasigroups in cryptography,’’ Appl. Discrete Math., No. 2, 28–32 (2008).

  3. V. A. Artamonov, S. Chakrabarti, S. Gangopadhyay, and S. K. Pal, ‘‘On Latin squares of polynomially complete quasigroups and quasigroups generated by shifts,’’ Quasigroups Rel. Syst. 21, 117–130 (2013).

    MathSciNet  MATH  Google Scholar 

  4. V. A. Artamonov, S. Chakrabarti, and S. K. Pal, ‘‘Characterizations of highly non-associative quasigroups and associative triples,’’ Quasigroups Rel. Syst. 25, 1–19 (2017).

    MathSciNet  MATH  Google Scholar 

  5. A. V. Galatenko, A. E. Pankratiev, and S. B. Rodin, ‘‘Polynomially complete quasigroups of prime order,’’ Algebra Logic 57, 327–335 (2018).

    Article  MathSciNet  Google Scholar 

  6. A. V. Galatenko and A. E. Pankratiev, ‘‘The complexity of decision of polynomial completeness of finite quasigroups,’’ Discrete Math. 30 (4), 3–11 (2018).

    Google Scholar 

  7. V. A. Nosov, ‘‘Construction of classes of latin squares in a boolean database,’’ Intell. Syst. 4, 307–320 (1999).

    Google Scholar 

  8. V. A. Nosov and A. E. Pankratiev, ‘‘Latin squares over Abelian groups,’’ J. Math. Sci. 149, 1230–1234 (2008).

    Article  MathSciNet  Google Scholar 

  9. V. A. Nosov, ‘‘Construction of a parametric family of Latin squares in a vector database,’’ Intell. Syst. 8, 517–528 (2004).

    Google Scholar 

  10. T. Kepka, ‘‘A note on simple quasigroups,’’ Acta Univ. Carolinae, Math. Phys. 19, 59–60 (1978).

    Article  Google Scholar 

  11. V. A. Nosov and A. E. Pankratiev, ‘‘A generalization of the Feistel cipher,’’ in Proceedings of the International Conference Maltsev Readings, Novosibirsk, 2015, p. 59.

  12. N. A. Piven, ‘‘Investigation of quasigroups generated by proper families of Boolean functions of order 2,’’ Intell. Syst. Theory Appl. 22, 21–35 (2018).

    Google Scholar 

Download references

ACKNOWLEDGMENTS

The authors are grateful to professor V.A. Artamonov for fruitful discussions.

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to A. V. Galatenko, V. A. Nosov or A. E. Pankratiev.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Galatenko, A.V., Nosov, V.A. & Pankratiev, A.E. Latin Squares over Quasigroups. Lobachevskii J Math 41, 194–203 (2020). https://doi.org/10.1134/S1995080220020079

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080220020079

Keywords and phrases:

Navigation