http://dspace.nbuv.gov.ua:80/handle/123456789/150387 2026-04-22T18:36:46Z http://dspace.nbuv.gov.ua:80/handle/123456789/152794 On the flag geometry of simple group of Lie type and multivariate cryptography Ustimenko, V. We propose some multivariate cryptosystems based on finite BN-pair G defined over the fields Fq. We convert the adjacency graph for maximal flags of the geometry of group G into a finite Tits automaton by special colouring of arrows and treat the largest Schubert cell Sch isomorphic to vector space over Fq on this variety as a totality of possible initial states and a totality of accepting states at a time. The computation (encryption map) corresponds to some walk in the graph with the starting and ending points in Sch. To make algorithms fast we will use the embedding of geometry for G into Borel subalgebra of corresponding Lie algebra. We also consider the notion of symbolic Tits automata. The symbolic initial state is a string of variables tα∈Fq, where roots α are listed according Bruhat's order, choice of label will be governed by special multivariate expressions in variables tα, where α is a simple root. Deformations of such nonlinear map by two special elements of affine group acting on the plainspace can produce a computable in polynomial time nonlinear transformation. The information on adjacency graph, list of multivariate governing functions will define invertible decomposition of encryption multivariate function. It forms a private key which allows the owner of a public key to decrypt a ciphertext formed by a public user. We also estimate a polynomial time needed for the generation of a public rule. 2015-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/152793 On two windows multivariate cryptosystem depending on random parameters Romańczuk-Polubiec, U.; Ustimenko, V. The concept of multivariate bijective map of an affine space Kn over commutative Ring K was already used in Cryptography. We consider the idea of nonbijective multivariate polynomial map Fn of Kn into Kn represented as ''partially invertible decomposition'' F(1)nF(2)n…F(k)n, k=k(n), such that knowledge on the decomposition and given value u=F(v) allow to restore a special part v′ of reimage v. We combine an idea of ''oil and vinegar signatures cryptosystem'' with the idea of linguistic graph based map with partially invertible decomposition to introduce a new cryptosystem. The decomposition will be induced by pseudorandom walk on the linguistic graph and its special quotient (homomorphic image). We estimate the complexity of such general algorithm in case of special family of graphs with quotients, where both graphs form known families of Extremal Graph Theory. The map created by key holder (Alice) corresponds to pseudorandom sequence of ring elements. The postquantum version of the algorithm can be obtained simply by the usage of random strings instead of pseudorandom. 2015-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/152792 On subgroups of finite exponent in groups Artemovych, O.D. We investigate properties of groups with subgroups of finite exponent and prove that a non-perfect group G of infinite exponent with all proper subgroups of finite exponent has the following properties: (1) G is an indecomposable p-group, (2) if the derived subgroup G′ is non-perfect, then G/G′′ is a group of Heineken-Mohamed type. We also prove that a non-perfect indecomposable group G with the non-perfect locally nilpotent derived subgroup G′ is a locally finite p-group. 2015-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/152791 Type of a point in Universal Geometry and in Model Theory Plotkin, B.; Plotkin, E.; Zhitomirski, G. The paper is devoted to relations between model theoretic types and logically geometric types. We show that the notion of isotypic algebras can be equally defined through MT-types and LG-types. 2015-01-01T00:00:00Z