Volume 34, Issue 3

We examine relationships between the nonlinearity parameters of mappings f : V n → V m of binary vector spaces V n = GF(2) n , V m = GF(2) m , diffusion properties of imprimitivity systems of the translation group Vn+ $\begin{array}{} \displaystyle V_{n}^{+} \end{array}$ of space V n , and also (for m = n and f ∈ S ( V n )) transitivity and primitivity properties of the groups 〈 W + , f 〉, where W + is the translation group of the subspace W < V n . It is shown that, in some methods of cryptoanalysis of block cipher algorithms, in fact, insufficient diffusion of blocks of the imprimitivity system of the group Vn+ $\begin{array}{} \displaystyle V_{n}^{+} \end{array}$ is used.

The paper studies mappings F of binary vector spaces of large dimensions. The mappings are assumed to be represented by deep branching superpositions of local non-linear mappings of low-dimensional spaces. We propose and investigate the methods for construction of probabilistic linear and differential relations connecting the arguments and the values of a mapping F . Relation selection is based on optimization not the exact probability of satisfying these relations, but some approximation of it since it is easier to estimate. We prove theorems on exact values of the probability of satisfying the relations obtained, identify the shortcomings and features of the proposed approach to the relation construction and illustrate them by a number of examples. We discuss the role of the developed theory for cryptographic synthesis.

A total edge irregular k -labeling of a graph G = ( V , E ), ∂ : V ∪ E → {1, 2, 3, ⋯, k } is a labeling of vertices and edges of G in such a way that the weights of all edges are distinct. A total edge irregularity strength of graph G , denoted by tes( G ) is defined as the minimal k for which a graph G has a totally irregular total k -labeling. Analogously we can define total vertex irregularity strength of graph G , denoted by tvs( G ). In this paper, we provide the no-hole total (both edge and vertex) irregularity strength for some well known planar graphs.

Let { Z n , n = 0, 1, 2, …} be a critical branching process in a random environment, and { S n , n = 0, 1, 2, …} be its associated random walk. It is known that if the increments of this random walk belong (without centering) to the domain of attraction of a stable law, then there exists a regularly varying at infinity sequence a 1 , a 2 , … such that conditional distributions P(Snan≤x|Zn>0),x∈(−∞,+∞), $$\begin{array}{} \displaystyle \mathbf{P}\bigg(\frac{S_{n}}{a_{n}}\leq x\Bigm| Z_{n} \gt 0\bigg),\quad x\in(-\infty,+\infty), \end{array}$$ converge weakly to the distribution of strictly positive proper random variable. In this paper we add to this result the description of the asymptotic behavior of the probability P(Zn>0,Sn≤φ(n)), $$\begin{array}{} \displaystyle \mathbf{P}(Z_{n} \gt 0, S_{n}\leq \varphi(n)), \end{array}$$ where φ ( n ) → ∞ for n → ∞ in such a way that φ ( n ) = o ( a n ).