http://hdl.handle.net/123456789/4542 2026-04-15T18:50:03Z http://hdl.handle.net/123456789/4681 On new multivariate cryptosystems with nonlinearity gap Ustimenko, V. The pair of families of bijective multivariate maps of kind Fn and Fn −1 on affine space Kn over finite commutative ring K given in their standard forms has a nonlinearity gap if the degree of Fn is bounded from above by independent constant d and degree of F −1 is bounded from below by c n, c > 1. We introduce examples of such pairs with invertible decomposition Fn = G1 nG2 n . . . Gk n, i.e. the decomposition which allows to compute the value of F n−1 in given point p = (p1, p2, . . . , pn) in a polynomial time O(n 2 ). The pair of families Fn, F ′ n of nonbijective polynomial maps of affine space Kn such that composition FnF ′ n leaves each element of K∗n unchanged such that deg(Fn) is bounded by independent constant but deg(F ′ n ) is of an exponential size and there is a decom- position G1 nG2 n . . . Gk n of Fn which allows to compute the reimage of vector from F(K∗n ) in time 0(n 2 ). We introduce examples of such families in cases of rings K = Fq and K = Zm. Ustimenko V. On new multivariate cryptosystems with nonlinearity gap / V. Ustimenko // Algebra and Discrete Mathematics. - 2017. - Vol. 23. - Number 2. - Рp. 331 - 348 2017-01-01T00:00:00Z http://hdl.handle.net/123456789/4678 On groups with biprimary subgroups of even order Sokhor, I. We investigate groups in which maximal sub- groups of even order are primary or biprimary. We also research soluble groups with restriction on a number of prime devisors of some proper subgroup orders. We give applications of received results to cofactors of proper subgroups. Sokhor I. On groups with biprimary subgroups of even order / I.Sokhor // Algebra and Discrete Mathematics. - 2017. - Vol. 23. - Number 2. - Рp. 312 - 330 2017-01-01T00:00:00Z http://hdl.handle.net/123456789/4675 A note on Hall S-permutably embedded subgroups of finite groups Sinitsa, D.A. Let G be a finite group. Recall that a subgroup A of G is said to permute with a subgroup B if AB = BA. A subgroup A of G is said to be S-quasinormal or S-permutable in G if A permutes with all Sylow subgroups of G. Recall also that HsG is the S-permutable closure of H in G, that is, the intersection of all such S-permutable subgroups of G which contain H. We say that H is Hall S-permutably embedded in G if H is a Hall subgroup of the S-permutable closure HsG of H in G. We prove that the following conditions are equivalent: (1) every subgroup of G is Hall S-permutably embedded in G; (2) the nilpotent residual GN of G is a Hall cyclic of square-free order subgroup of G; (3) G = D ⋊ M is a split extension of a cyclic subgroup D of square-free order by a nilpotent group M, where M and D are both Hall subgroups of G. Sinitsa D.A. A note on Hall S-permutably embedded subgroups of finite groups / D.A.Sinitsa // Algebra and Discrete Mathematics. - 2017. - Vol. 23. - Number 2. - Рp.305- 311 2017-01-01T00:00:00Z http://hdl.handle.net/123456789/4669 Profinite closures of the iterated monodromy groups associated with quadratic polynomials Samoilovych, I. In this paper we describe the profinite closure of the iterated monodromy groups arising from the arbitrary post- critically finite quadratic polynomial. Samoilovych I. Profinite closures of the iterated monodromy groups associated with quadratic polynomials / I. Samoilovych // Algebra and Discrete Mathematics. - 2017. - Vol. 23. - Number 2. - Рp. 285 - 304 2017-01-01T00:00:00Z