# Computer Sciences

## The key exchange protocol based on non-commutative elements of Clifford algebra

Many of the asymmetric cryptography protocols are based on operations performed on commutative algebraic structures, which are vulnerable to quantum attacks. The development of algorithms in non-commutative structures makes it possible to strengthen these protocols. Cryptography is a branch of mathematics that solves the problem of transmitting information through unsafe channels. For this, information is encrypted, so it cannot be used without first decrypting it.

## The search for minimal edge 1-extension of an undirected colored graph

Let $G=(V, \alpha, f)$ be a colored graph with a coloring function $f$ defined on its vertices set $V$. Colored graph $G^*$ is an edge $1$-extension of a colored graph $G$ if $G$ could be included into each subgraph taking into consideration the colors. These subgraphs could be built from $G^*$ by removing one of the graph's edges. Let colored edge $1$-extension $G^*$ be minimal if $G^*$ has as many vertices as the original graph $G$ and it has the minimal number of edges among all edge $1$-extensions of graph $G$.

## Modeling the reliability of the onboard equipment of a mobile robot

Mobile robots with complex onboard equipment are investigated in this article. It is shown that their onboard equipment, for providing the required reliability parameters, must have fault-tolerant properties. For designing such equipment it is necessary to have an adequate model of reliability parameters evaluation. The approach, linked to the creation of the model, based on parallel semi-Markov process apparatus, is considered. At the first stage of modeling, the lifetime of the single block in a complex fault-recovery cycle is determined.

## Resource optimization in management of technical maintenance of automation and remote control systems

The article considers the topical problem of resource optimization during the maintenance process of technical systems. In order to illustrate the methodology (URRAN) developed by Russian Railways JSC to solve this problem, automation and remote control systems are used as an example. Due to a total lack of resources and the infinite needs, the company faced the challenge of improving the efficiency of maintenance and repair of railway automation and remote control systems.

## Construction of 3D solid vertebral models using convolutional neural networks

The quality of solving the problem of biomechanical modeling largely depends on the created solid-state model of the biological object under study. Building a model based on computed tomography data for a particular patient is possible both in manual mode (software packages for processing medical images) and using automated tools for building a model (image segmentation), which significantly speeds up the process of creating a solid model, in contrast to the manual mode.

## Generation of colored graphs with isomorphism rejection

In the article we consider graphs whose vertices or edges are colored in a given number of colors — vertex and edge colorings. The study of colorings of graphs began in the middle of the 19th century, but the main attention is paid to proper colorings, in which the restriction applies that the colors of adjacent vertices or edges must be different. This paper considers colorings of graphs without any restrictions. We study the problem of generating all non-isomorphic vertex and edge $k$-colorings of a given graph without direct checking for isomorphism.

## Mathematical and computer simulation of the electrophysical properties of a multicellular structure exposed to nanosecond electrical pulses

The article presents mathematical and computer models which allow to study the electrophysical properties (permittivity, impedance) of a multicellular structure exposed to nanosecond electrical pulses. The paper proposes a simulation approach that includes complex use of the classical theory of describing the electrodynamic properties of dispersed systems and the effective medium theory. We describe cell geometry using Gielis equations, which allow us to take account of the irregular shapes of cell membranes.

## Imitation tools for automated systems used in small arms testing

Currently, various imitators (imitation tools) are widely used to test various equipment and evaluate its parameters. The main goal of developing new imitation tools in the field of creating automated systems used in testing small arms is to accelerate and cheapen the development and implementation of such systems. The article discusses the general structure of information-measuring systems used in testing small arms based on acoustic and optical blocking devices, as well as on the basis of a video camera.

## The construction of all nonisomorphic minimum vertex extensions of the graph by the method of canonical representatives

A graph $G = (V, \alpha)$ is called \textit{primitive} if there exists a natural $k$ such that between any pair of vertices of the graph $G$ there is a route of length $k$. This paper considers undirected graphs with exponent 2. A criterion for the primitivity of a graph with the exponent 2 and a necessary condition are proved. A graph is primitive with the exponent 2 if and only if its diameter is 1 or 2, and each of its edges is included in a triangle.

## Method of Markovian summation for study the repeated flow in queueing tandem M|GI|∞ → GI|∞

The paper presents a mathematical model of queueing tandem M|GI|∞ → GI|∞ with feedback. The service times at the first stage are independent and identically distributed (i.i.d.) with an arbitrary distribution function B1(x). Service times at the second stage are i.i.d. with an arbitrary distribution function B2(x). The problem is to determine the probability distribution of the number of repeated customers (r-flow) during fixed time period.