Band 18, Heft 3

We consider the relations between the level of affinity and other cryptographic parameters of Boolean functions such as nonlinearity, the order of correlation, and algebraic immunity. It is proved that the problem of finding the level of affinity for certain class of functions is an NP-hard problem.

We suggest an algorithm to test numbers of the form for primality, where , k is an odd positive integer, is a prime number, i = 1,…, n , and . The algorithm makes use of the Lucas functions. The algorithm suggested is of complexity Ô (log 2 N ).

We investigate the complexity of circuits constructing complete and complete bipartite graphs with the use of two operations of glueing vertices. These operations are the operations of identification of a pair of vertices with removal of loops and multiple edges. The first operation is applied to pairs of vertices in one graph, the second operation is applied to pairs of vertices in two graphs which have no common elements. The initial graph of the construction is the simplest graph consisting of two vertices connected by an edge. The number of operations performed on graphs is considered as the complexity of such a construction. Upper bounds for the complexity of construction of complete and complete bipartite graphs have been obtained previously. In this paper, we obtain lower bounds which give a possibility to find the order of the asymptotics of the complexity.

We consider the class M of deterministic binary two-tape automata for which those states of the automaton where reading from the additional second tape is implemented permit to continue the calculation only in the case where the read symbol is fixed in advance for all automata. For the automata of the class M , the so-called generalised minimisation problem is considered. This problem consists in finding all automata which are minimal in the number of states in any class of equivalent automata of M . In the paper, the decidability of the generalised minimisation problem in M is proved and a procedure to solve this problem is described and analysed.

We prove the asymptotic normality of the number of absent s -chains of identical outcomes in the equiprobable polynomial scheme under the condition that the number of trials n and the number of outcomes N tend to infinity in such a way that .

We construct a combinatorial representation of an octahedral polyhedron (crystal), find its geometrical parameters, and show that the parameters of such a polyhedron and its mirror image coincide and are simply measured. Using the combinatorial representation of a diamond, we are able to construct an algorithm which finds a round cut diamond of maximum radius embedded into an octahedral diamond and determines how to place this cut diamond in the octahedral one. This results in creating a technological process to cut round diamonds of maximum value from octahedral ones.