In the paper, it is proved that almost all quasigroups are strongly polynomially complete, i.e., are not isotopic to quasigroups that are not polynomially complete.

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 work presents the results of an experimental study of some properties of low-order finite Abelian groups from the standpoint of the applicability of such groups in cryptographic applications.

Definition. H is called a hyperbolic product of the Gi, i ∈ I, if max|w|=n min lw ≤ cn for some constant c (the minimum to be taken over all possible representations (1)). Here and below the length |w| of a word (or element) is understood in the sense of syllable length, that is, the metric on the free product [3]. By hyperbolic factors we shall mean the subgroups which are images of the free factors Gi under the natural homomorphism F → H. An element of a hyperbolic product which has infinite order and is not conjugate to an element of a hyperbolic factor will be called free. We consider the sequence of powers of an arbitrary free element x ∈ H and denote by x̄n a shortest element in the conjugacy class (xn)H . Let us say that a hyperbolic product H = 〈F |R〉 is non-degenerate if it contains at least one free element and if for any free element x ∈ H the lengths of the elements x̄n are not bounded above as n→∞. The Cayley graph C(H) is formed over the productH = 〈F |R〉 as over a group with a system of generators A consisting of all the elements of the factors: A = ⋃ iGi.