Skip to main content
SpringerLink
Log in

Polynomially Complete Quasigroups of Prime Order

  • Published:
Algebra and Logic Aims and scope

We formulate a polynomial completeness criterion for quasigroups of prime order, and show that verification of polynomial completeness may require time polynomial in order. The results obtained are generalized to n-quasigroups for any n ≥ 3. In conclusion, simple corollaries are given on the share of polynomially complete quasigroups among all quasigroups, and on the cycle structure of row and column permutations in Cayley tables for quasigroups that are not polynomially complete.

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

Access this article

Log in via an institution

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. Techn. J., 28, No. 4, 656-715 (1949).

    Article  MathSciNet  Google Scholar 

  2. M. M. Glukhov, “Some applications of quasigroups in cryptography,” Prikl. Diskr. Mat., No. 2(2), 28-32 (2008).

  3. S. Markovski, D. Gligoroski, and V. Bakeva, “Quasigroup string processing: Part 1,” Proc. Maked. Acad. Sci. Arts Math. Tech. Sci., 20, Nos. 1/2, 13-28 (1999).

  4. S. Markovski and V. Kusacatov, “Quasigroup string processing: Part 2,” Proc. Maked. Acad. Sci. Arts Math. Tech. Sci., 21, Nos. 1/2, 15-32 (2000).

  5. V. Shcherbacov, “Quasigroup based crypto-algorithms,” arXiv:1201.3016 [math.GR].

  6. G. Horváth, C. L. Nehaniv, and Cs. Szabó, “An assertion concerning functionally complete algebras and NP-completeness,” Theor. Comput. Sci., 407, Nos. 1-3, 591-595 (2008).

    Article  MathSciNet  Google Scholar 

  7. V. A. Artamonov, S. Chakrabarti, S. Gangopadhyay, and S. K. Pal, “On Latin squares of polynomially complete quasigroups and quasigroups generated by shifts,” Quasigroups Relat. Syst., 21, No. 2, 117-130 (2013).

  8. V. A. Artamonov, S. Chakrabarti, and S. K. Pal, “Characterization of polynomially complete quasigroups based on Latin squares for cryptographic transformations,” Discr. Appl. Math., 200, 5-17 (2016).

  9. V. A. Artamonov, S. Chakrabarti, and S. K. Pal, “Characterizations of highly non-associative quasigroups and associative triples,” Quasigroups Relat. Syst., 25, No. 1, 1-19 (2017).

  10. A. V. Galatentko, A. E. Pankrat’ev, and S. B. Rodin, “Polynomially complete quasigroups of prime order,” Intellekt. Sist. Teor. Pril., 20, No. 3, 194-198 (2016).

    Google Scholar 

  11. S. V. Yablonskii, Introduction to Discrete Mathematics [in Russian], 6th ed., Vysshaya Shkola, Moscow (2010).

  12. D. Lau, Function Algebras on Finite Sets. A Basic Course on Many-Valued Logic and Clone Theory, Springer Monogr. Math., Springer, Berlin (2006).

  13. V. B. Kudryavtsev, Functional Systems [in Russian], MGU, Moscow (1982).

  14. D. E. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3d ed., Addison-Wesley, Bonn (1998).

  15. H. J. Ryser, “Permanents and systems of distinct representatives,” in Combin. Math. Appl., Proc. Conf. Univ. North Carolina, 1967 (1969), pp. 55-70.

Download references

Author information

Authors and Affiliations

  1. Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991, Russia

    A. V. Galatentko, A. E. Pankrat’ev & S. B. Rodin

Authors
  1. A. V. Galatentko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. E. Pankrat’ev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. B. Rodin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Galatentko.

Additional information

Supported by RFBR and by Department of Science and Technology of Government of India, project No. 15-51-45031.

Translated from Algebra i Logika, Vol. 57, No. 5, pp. 509-521, September-October, 2018.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Galatentko, A.V., Pankrat’ev, A.E. & Rodin, S.B. Polynomially Complete Quasigroups of Prime Order. Algebra Logic 57, 327–335 (2018). https://doi.org/10.1007/s10469-018-9505-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10469-018-9505-6

Keywords

Search

Navigation