http://dspace.nbuv.gov.ua:80/handle/123456789/150399 Tue, 21 Apr 2026 01:26:08 GMT 2026-04-21T01:26:08Z http://dspace.nbuv.gov.ua:80/bitstream/id/563891/ http://dspace.nbuv.gov.ua:80/handle/123456789/150399 http://dspace.nbuv.gov.ua:80/handle/123456789/156037 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 cⁿ, c>1. We introduce examples of such pairs with invertible decomposition Fn=Gn¹Gn²…Gnk, i.e. the decomposition which allows to compute the value of Fⁿ⁻¹ in given point p=(p1,p2,…,pn) in a polynomial time O(n²). 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 decomposition Gn¹Gn²…Gnk of Fn which allows to compute the reimage of vector from F(K*ⁿ) in time 0(n²). We introduce examples of such families in cases of rings K=Fq and K=Zm. Sun, 01 Jan 2017 00:00:00 GMT http://dspace.nbuv.gov.ua:80/handle/123456789/156037 2017-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/156027 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. Sun, 01 Jan 2017 00:00:00 GMT http://dspace.nbuv.gov.ua:80/handle/123456789/156027 2017-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/156026 Generators and ranks in finite partial transformation semigroups Garba, G.U.; Imam, A.T. We extend the concept of path-cycle, to the semigroup Pn, of all partial maps on Xn={1,2,…,n}, and show that the classical decomposition of permutations into disjoint cycles can be extended to elements of Pn by means of path-cycles. The device is used to obtain information about generating sets for the semigroup Pn\Sn, of all singular partial maps of Xn. Moreover, we give a definition for the (m,r)-rank of Pn\Sn and show that it is n(n+1)/2. Sun, 01 Jan 2017 00:00:00 GMT http://dspace.nbuv.gov.ua:80/handle/123456789/156026 2017-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/156025 Nonuniqueness of semidirect decompositions for semidirect products with directly decomposable factors and applications for dihedral groups Daugulis, P. Nonuniqueness of semidirect decompositions of groups is an insufficiently studied question in contrast to direct decompositions. We obtain some results about semidirect decompositions for semidirect products with factors which are nontrivial direct products. We deal with a special case of semidirect product when the twisting homomorphism acts diagonally on a direct product, as well as with the case when the extending group is a direct product. We give applications of these results in the case of generalized dihedral groups and classic dihedral groups D2n. For D2n we give a complete description of semidirect decompositions and values of minimal permutation degrees. Sun, 01 Jan 2017 00:00:00 GMT http://dspace.nbuv.gov.ua:80/handle/123456789/156025 2017-01-01T00:00:00Z