Volume 16, Issue 6

We consider the effect of transition of distributions of sums of independent identically distributed non-negative integer-valued random variables from one lattice to another in the framework of the generalised allocation scheme.

We consider a two-stage scheme of allocating particles into cells. At the first stage, N 0 initial particles are independently and equiprobably allocated into N 1 cells of the first layer. At the second stage, these N 1 cells are taken as particles which are independently and equiprobably allocated into N 2 cells of the second layer. We present conditions under which the distribution of the number of cells of the second layer containing exactly r particles converges to the Poisson law.

We study problems of existence of consistent sequences of tests in a sequence of finite spaces. We also touch upon problems of existence of sequence of tests close in some sense to consistent ones. These problems are closely related to properties of some sets in Tikhonov products.

We consider the set of all forests consisting of N rooted trees such that the roots (and the corresponding trees) are labelled by the numbers 1, . . . , N , and the remaining n vertices of the forest are labelled by the numbers 1, . . . , n . Under the assumption that the uniform distribution is defined on this set and n , N → ∞, we prove local limit theorems for the distributions of the random variables equal to the number of trees of a given size and the maximum size of a tree, which permit to estimate the corresponding local probabilities with accuracy of known order, including the probability of large deviations.

We study the structure of equationally closed classes. We prove a theorem on representation of the graph of a function in an equationally closed class in the form of a union of the sets of values of special vector functions. For any k ≥ 2 we establish the equational generability of any equationally closed class in P k by the set of all its k -place functions. We find all equationally precomplete classes in P k and prove a criterion of equational completeness. Some results are extended from equationally closed classes to positively closed classes.

We introduce the notion of the depth of branching of a circuit of functional elements and consider classes of circuits of branching depth bounded by a constant. For these classes of circuits over various bases we obtain lower and upper bounds for complexity of a linear Boolean function. We construct infinitely decreasing sequences of measures of complexity for a fixed base and growing branching depth and for a fixed branching depth but varying base.

In the paper, we prove the α -completeness of finite systems of function of k -valued logic for k = 3, 4 containing all permutations of the symmetric group S k on the set E k = {0, 1, . . . , k −1}, the operation of addition modulo k , and k certain binary operation. This result is extended to some other systems of functions which are obtained by replacing the operation of addition by some quasi-group operation.

We obtain estimates of complexity and depth of Boolean inverter circuits in normal and polynomial bases of finite fields. In particular, we show that it is possible to construct a Boolean inverter circuit in the normal basis of the field GF (2 n ) whose complexity is at most ( λ ( n − 1) + (1 + o (1)) λ ( n )/ λ ( λ ( n ))) M ( n ) and the depth is at most ( λ ( n − 1) + 2) D ( n ), where M ( n ), D ( n ) are the complexity and the depth, respectively, of the circuits for multiplication in this basis and λ ( n ) = ⌊log 2 n ⌋.

We study properties of two schemes of public distribution of a key suggested by V. M. Sidelnikov which are based on a noncommutative operation. As a noncommutative operation we consider the operations belonging to the family suggested by M. A. Cherepnev, namely, the operations in the rings of integers of cyclotomic fields based on the power residue symbol. In the paper, a cryptanalysis of both schemes is performed for a particular noncommutative operation. We show that for an arbitrary operation of the mentioned above family the first scheme in not resistant. For the second scheme, we prove a theorem on the equivalence of its breaking to a solution of some problem of a computational algebraic number theory.