Volume 21, Issue 2

Let R = GR ( q n , q n ) be a Galois ring of cardinality q n and characteristic p n , q = p r , p be a prime. We call a subset K ⊂ R a coordinate set if 0 ∈ K and for any a ∈ R there exists a unique ϰ ( a ) ∈ K such that a ≡ ϰ ( a ) (mod pR ). Let u be a linear recurring sequence of maximal period (MP LRS) over a ring R . Then any its term u ( i ) admits a unique representation in the form u ( i ) = w 0 ( i ) + pw 1 ( i ) + ⋯ + p n –1 w n –1 ( i ), w t ( i ) ∈ K , t ∈ {0, . . . , n – 1}. We pose the following conjecture: the sequence u can be uniquely reconstructed from the sequence w n –1 for any choice of the coordinate set K . It is proved that such a reconstruction is possible under some conditions on K . In particular, it is possible for any K if R = Z p n and for any Galois ring R if K is a p -adic (Teichmüller) coordinate set.

The pseudorandom number generator (PRNG) based on the transformation F c ( x ) = x + ( x 2 ∨ c ) (mod 2 n ) was suggested by Klimov and Shamir in 2002. The function F c ( x ) is transitive modulo 2 n if and only if either c ≡ 5 (mod 8) or c ≡ 7 (mod 8). We consider properties of the distribution of the pairs ( x i , F c ( x i )) for various c ∈ Z /2 n Z and demonstrate that their statistical properties are unsatisfactory, most notably for c ≥ 2 n /3 . We show that in the case n = 32, at most 9 distinct pairs ( x i , F c ( x i )) are needed to find the value of c with probability P ≥ 0.999.

We consider the problem of construction of a conditional diagnostic test for the read-once functions in an arbitrary basis with the use of queries of values of function at a point and queries to check an arbitrary subfunction for an identical equality to a constant. We show a connection of this problem to the checking test and prove its polynomial solvability for a wide class of Boolean functions.

We study properties of Boolean functions with a finite-length barrier and suggest a criterion for a function to possess a barrier. We introduce the notion of a Boolean function without prediction which describes certain positive cryptographic properties of the corresponding transformations of binary sequences. We suggest a criterion for a Boolean function to belong to the class of functions without prediction.

The problem of representation of topological models in the form of triangulation of cubic complexes is related to the character and behaviour of the triangulation in the n -dimensional space. As the base for analysis of these characteristics one can take the fundamental statistical information which gives the possibility to synthesise typical features of triangulation in model spaces of various dimensions. In this paper, we suggest an approach to solution of this problems in the case of analysing statistical features of triangulation in the space of dimension at most 4 with the use of binary coding.

The study of properties of irredundant complexes of faces is connected with the problem of minimisation of Boolean functions in the class of disjunctive normal forms (d.n.f.). In researches by S. V. Yablonskii, Yu. I. Zhuravlev, V. V. Glagolev, Yu. L. Vasilyev, A. A. Sapozhenko, on the base of construction and investigation of properties of particular Boolean functions, estimates of the maximum length and number of irredundant d.n.f. have been obtained. The author suggests a different approach to investigations of these objects based on constructing and estimating the cardinality of sets of irredundant complexes of faces. In this paper, with the use of the probabilistic approach, new methods of construction and estimation of characteristics of irredundant complexes of faces are suggested, which give a possibility to improve the known estimates. On the base of a method of construction of irredundant complexes of faces in a belt of the unit cube B n of width k , we obtain estimates of the maximum number of faces and the number of irredundant complexes for the faces of dimension k < (1/4 – ε ) n , where ε is as small as wished positive constant. By the optimal choice of the parameters we obtain for the logarithm of the number of irredundant complexes of faces the lower bound of order n 2 n with constant 1.355 · 2 –5 for the dimension of the faces k ≈ 0.0526 n . Because of equivalence of the problem of minimisation of Boolean functions in the class of d.n.f. and the problem of construction of complexes of faces covering subsets of vertices of the unit cube, the obtained results can be used for estimation of the maximum values of the length and the number of irredundant d.n.f.