Algebra and Logic

, Volume 57, Issue 5, pp 327–335 | Cite as

Polynomially Complete Quasigroups of Prime Order

  • A. V. Galatentko
  • A. E. Pankrat’ev
  • S. B. Rodin

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.


quasigroup Latin square polynomially complete quasigroup n-quasigroup permutation 


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  1. 1.
    C. Shannon, “Communication theory of secrecy systems,” Bell Syst. Techn. J., 28, No. 4, 656-715 (1949).MathSciNetCrossRefGoogle Scholar
  2. 2.
    M. M. Glukhov, “Some applications of quasigroups in cryptography,” Prikl. Diskr. Mat., No. 2(2), 28-32 (2008).Google Scholar
  3. 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).Google Scholar
  4. 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).Google Scholar
  5. 5.
    V. Shcherbacov, “Quasigroup based crypto-algorithms,” arXiv:1201.3016 [math.GR].Google Scholar
  6. 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).MathSciNetCrossRefGoogle Scholar
  7. 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).Google Scholar
  8. 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).Google Scholar
  9. 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).Google Scholar
  10. 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. 11.
    S. V. Yablonskii, Introduction to Discrete Mathematics [in Russian], 6th ed., Vysshaya Shkola, Moscow (2010).Google Scholar
  12. 12.
    D. Lau, Function Algebras on Finite Sets. A Basic Course on Many-Valued Logic and Clone Theory, Springer Monogr. Math., Springer, Berlin (2006).Google Scholar
  13. 13.
    V. B. Kudryavtsev, Functional Systems [in Russian], MGU, Moscow (1982).Google Scholar
  14. 14.
    D. E. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3d ed., Addison-Wesley, Bonn (1998).Google Scholar
  15. 15.
    H. J. Ryser, “Permanents and systems of distinct representatives,” in Combin. Math. Appl., Proc. Conf. Univ. North Carolina, 1967 (1969), pp. 55-70.Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • A. V. Galatentko
    • 1
  • A. E. Pankrat’ev
    • 1
  • S. B. Rodin
    • 1
  1. 1.Lomonosov Moscow State UniversityMoscowRussia

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