Volume 14, Issue 3

In this paper, we consider mappings of Cartesian powers S n of an arbitrary partially ordered set S into itself which possess the main properties of closures. For each partially ordered set, we describe the asymptotic behaviour of the logarithm of the number of such mappings as n → ∞.

In this paper for the linear congruent generator z N + 1 = G (z N ), N = 1,2, ... , where G( x ) = λx + c (mod W ), W = p F , p is a prime number, we find a non-trivial lower bound for the least non-zero wave number e L (λ), the fundamental characteristic introduced in the spectral test to check for randomness on the base of analysis of the frequence of occurrences of L -tuples ( t 1 , ... , t L ) in the sequence (z N ). The lower bound obtained is of the form W 1/L– δ , where δ is some variable explicitly depending on parameters which determine the factor λ. Under an appropriate choice of the parameters, δ can be made as small as desired. The factor 1/ L cannot be changed for a greater one. Such bounds are necessary in studying classes of multipliers that pass the spectral test.

We study the McEliece cryptosystem with u -fold use of binary Reed–Muller codes RM(r, m) . This modification of the McEliece cryptosystem was proposed by V. M. Sidelnikov in 1994 and combines high cryptographic security, transmission rate close to one, and moderate complexity of both enciphering and deciphering. For arbitrary values of the parameters u , r , and m we give an upper bound for the cardinality of the set of public keys of this cryptosystem and calculate its exact value in the case of u = 2 and r = 1.

We consider multi-valued logic functions represented by polarised polynomials. A polynomial is called polarised if each its variable can be polarised by a certain shift. We introduce the Shannon function which characterises the complexity of representations of multi-valued logic functions by polarised polynomials and obtain an exact estimate of the Shannon function for functions in one variable.

We study the time of simulation of Boolean circuits by three-tape Turing machine which uses one of the tapes to store the control program. We find that for any circuit S there exists a program P such that the simulation time T(P) for the circuit S satisfies the relation T(P) = O(L(S) log 2 L(S)) , where L(S) is the complexity of the circuit S . We demonstrate that this estimate is sharp.

The automaton implementation of a determinate function is a probabilistic automaton A 1 = ( S, Y, P s , λ( s )), where S is the Markov chain state set, P s is an m 1 × m 1 stochastic matrix, Y is the output alphabet of cardinality m 2 ≤ m 1 . The automaton implementation of a probabilistic function is a probabilistic automaton A 2 = ( S, Y, P s , P y ), where S , Y , P s are of the same sense as in A 1 , while P y is a stochastic m 1 × m 2 matrix. In this paper, we solve the problem of synthesis of generators of finite homogeneous Markov chains on the base of the analytical apparatus of polynomial functions over a Galois field. We suggest a method to calculate the coefficients of a polynomial in several variables which implements any mapping of the Galois field into itself. We study the case of implementing a finite automaton by a homogeneous computing structure defined over a Galois field; automaton mappings are implemented as polynomial functions over the Galois field. As the base polynomials, we use polynomial functions over the Galois field. As the base polynomials, we use polynomial functions over the Galois field where r = 2 n – 1, x, s, b i , a ij ∈ GF (2 n ). We give expressions of an automaton A 1 in the framework of a polynomial model over the field GF (2 n ) of the form M 1 = (, f 1 ( x, s ), f 2 ( s )), where is the discrete random variable which takes values µ ∈ GF (2 n ) determined by some probability vector = ( p 1 , . . . , pk 1 ) such that where B i are stochastic Boolean matrices and k 1 ≥ – m 1 + 1, and of an automaton M 2 = (, f 1 ( x, s ), f 2 ( s ), , f 3 ( x, s )), where is a discrete random variable which takes values µ′ ∈ GF (2 n ) determined by some probability vector = ( p 1 , . . . , pk 2 ) such that where B i are stochastic Boolean matrices and k 2 ≥ – m 1 + 1. The problem of representation of a discrete random variable over the field GF (2 n ) has been solved earlier.

We consider the problem of solving automata equations in one variable. We suggest an algorithm for determining whether a given equation has a solution. We introduce the notion of a boundedly non-determinate function. It is proved that if an automaton equation has a solution, then the set of all solutions of this equation is embedded into some boundedly non-determinate function which can be effectively constructed on the base of the initial equation.

We consider random triangulations of a disk with k holes and N triangles as N → ∞. The coefficient λ m , λ > 0, is assigned to a triangulation with the total number of boundary edges equal to m . In the case of two boundaries, we separate three domains of variation of the parameter λ , and in each of them find the limit joint distribution of boundary lengths. For a greater number of boundaries, we give an algorithm to calculate the generating functions for the number of multi-rooted triangulations depending of the number of triangles and the lengths of boundaries. In Appendix, we discuss the relation between multi-rooted triangulations and unrooted triangulations, and give analogues of limit distributions for unrooted triangulations.

In this paper, we find non-uniform bounds in the Poisson theorem. Let I 1 , ..., I n be indicators of independent random events. We set p k = P { I k = 1} = 1 − P { I k = 0}, 0 ≤ p k ≤ 1, k = 1, ..., n . Let Let b k be the jump of the distribution function B ( x ) at the point k . We also set Let be the jumps of the Poisson distribution function with parameter λ ≥ 0, and let be the corresponding distribution function. An example of the results obtained in the paper is formulated as follows. For λ = nP 1 and k ≥ 2 + λ, and for k > 1 + λ e