Solution of Belousov's problem

@article{Akivis2000SolutionOB,
  title={Solution of Belousov's problem},
  author={M. Akivis and V. V. Goldberg},
  journal={arXiv: Group Theory},
  year={2000}
}
The authors prove that a local $n$-quasigroup defined by the equation x_{n+1} = F (x_1, ..., x_n) = [f_1 (x_1) + ... + f_n (x_n)]/[x_1 + ... + x_n], where f_i (x_i), i, j = 1, ..., n, are arbitrary functions, is irreducible if and only if any two functions f_i (x_i) and f_j (x_j), i \neq j, are not both linear homogeneous, or these functions are linear homogeneous but f_i (x_i)/x_i \neq f_j (x_j)/x_j. This gives a solution of Belousov's problem to construct examples of irreducible $n… 
On irreducible n-ary quasigroups with reducible retracts
  • D. Krotov
  • Computer Science, Mathematics
    Eur. J. Comb.
  • 2008
On connection between reducibility of an n-ary quasigroup and that of its retracts
TLDR
It is shown that all n-ary quasigroups of order 5 or 7 whose all binary retracts are isotopic to Z"5 or Z"7 are reducible for n>=4.
Quasigroup associativity and biased expansion graphs
We present new criteria for a multary (or polyadic) quasigroup to be isotopic to an iterated group operation. The criteria are consequences of a structural analysis of biased expansion graphs. We
Associativity in multiary quasigroups: the way of biased expansions
A multiary (polyadic, n-ary) quasigroup is an n-ary operation which is invertible with respect to each of its variables. A biased expansion of a graph is a kind of branched covering graph with an
GENERALIZATION OF ISOMORPHISM THEOREMS GROUPS TO PARTIAL GROUPS
In this paper, we define a new structure analogous to group, called partial group. This structure concerns the partial stability by the composition inner law. We generalize the three isomorphism
On some quasigroup cryptographical primitives
Using Vojvoda approach [34] we demonstrate that cryptographical primitives proposed in [26] are vulnerable relative to chosen ciphertext attack and chosen plaintext attack. We develop proposed in
Quasigroups in cryptology
  • V. Shcherbacov
  • Mathematics, Computer Science
    Comput. Sci. J. Moldova
  • 2009
TLDR
This work reviews some known published applications of quasi- groups in cryptology and gives a review of the quasigroup, stream- cipher, secret-sharing system, and zero knowledge protocol applications.
Quasigroup based crypto-algorithms
TLDR
Modifications of Markovski quasigroup based crypto-algorithm have been proposed, some of which are based on the systems of orthogonal n-ary groupoids.

References

SHOWING 1-10 OF 33 REFERENCES
Varieties I:
Summary. For a complex polynomial, f: (C" + 1, 0)-.~ (C, 0), with a singular set of complex dimension s at the origin, we define a sequence of varieties -the L6 varieties, Af ), of f at 0. The
Sur les équations du second ordre à $n$ variables analogues à l’équation de Monge-Ampère
© Bulletin de la S. M. F., 1899, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http://smf. emath.fr/Publications/Bulletin/Presentation.html) implique l’accord
Sur les équations du second ordre à n variables
  • analogues à l’équation de Monge–Ampère, Bull. Soc. Math. France 27
  • 1899
On the question of reducibility of principal parastrophies of n-quasigroups (Russian)
  • Mat. Issled., Vyp
  • 1990
Irreducible n-quasigroups on finite sets of composite order
  • Mat. Issled. Vyp
  • 1979
Irreducible n-quasigroups on finite sets of composite order (Russian)
  • Mat. Issled., Vyp
  • 1979
Reducibility and uniform reducibility in certain classes of ngroupoids , Mat
  • Issled . Vyp .
  • 1979
Glukhov , On the question of reducibility of principal parastrophies of nquasigroups ( Russian )
  • Mat . Issled . , Vyp .
  • 1976
Reducible (n + 1)-webs
  • group (n + 1)-webs, and (2n + 2)hedral (n+1)-webs of multidimensional surfaces (Russian), Sibirsk. Mat. Zh. 17 (1976), no. 1, 44–57. (English transl. in: Siberian Math. J. 17
  • 1976
...
1
2
3
4
...