Bulletin of L.N. Gumilyov Eurasian National University. Mathematics, computer science, mechanics series

Analyzing the security of ZigBee communication in smart environments via SDR

Tue, 30 Sep 2025 00:00:00 +0000

The development of smart cities is accompanied by the widespread adoption of wireless
networks based on the ZigBee protocol, due to its energy efficiency and compatibility with Internet
of Things architectures. However, the active use of this protocol in an open radio environment
increases the risk of unauthorized radio-frequency interference. The study aims to experimentally
assess the vulnerability of ZigBee networks to targeted jamming using software-defined radio. The
paper presents the stages of preparing a test environment with real devices, identifying the active
data transmission channel, and generating tone interference using the SDR platform HackRF One
and the GNU Radio environment. The conducted experiment showed that, when affecting a specific
frequency, up to 95% of packets may be lost, rendering the network inoperable. The obtained
results confirm the critical vulnerability of the ZigBee protocol at the physical layer and highlight
the need to develop additional protection mechanisms for wireless IoT networks, especially within
urban infrastructure. The proposed methodology can be used to test the resilience of devices in
practical scenarios and to support the development of monitoring systems capable of detecting and
withstanding external attacks.

Optimal C(N)D-recovery of Functions from the Anisotropic Generalized Sobolev Class

Адилжан Утесов, Gulzhan Utesova — Tue, 30 Sep 2025 00:00:00 +0000

In this paper, the problem of recovering functions from an anisotropic generalized Sobolev class in the context of the computational (numerical) diameter is completely solved. The anisotropic generalized Sobolev class $W_2^{\omega_{r_1}, \ldots, \omega_{r_s}}$, which arises from the classification of periodic functions according to the rate of decay of their trigonometric Fourier coefficients, represents a finer scale of function characteristics than the anisotropic Sobolev class
$W_2^{r_1, \ldots, r_s}$ in the power scale. The paper presents two-sided estimates, up to multiplicative constants, for the approximation error of functions from the considered class by computational aggregates constructed from information given in the form of linear functionals.
In addition, based on trigonometric Fourier coefficients, the optimal computational aggregate is explicitly constructed. It is noted that the set of computational aggregates $(l^{(N)}, \varphi_N)$ used in the lower bounds is sufficiently wide, containing all partial sums of Fourier series with respect to various orthonormal systems, all possible finite convolutions with special kernels, as well as all finite approximation sums used in orthogonal widths, linear widths, and greedy algorithms. Furthermore, in the second part of the paper, the limiting error $\overline{\varepsilon}_N$ of numerical information in the form of trigonometric Fourier coefficients is obtained, which preserves the optimality of the computational aggregate and cannot be improved in order. In the third part, it is proved that all computational aggregates constructed using trigonometric Fourier coefficients do not have a limiting error greater than $\overline{\varepsilon}_N$.

Statistical Regularity in the Monty Hall Paradox Using Proprietary Random Algorithms

Tue, 30 Sep 2025 00:00:00 +0000

The article examines a classical problem in probability theory that has been widely discussed since the mid-1970s, known as the Monty Hall Paradox. It illustrates the discrepancy between the subjective perception of randomness and objective mathematical proofs, which are supported by appropriate computational and statistical experiments. A detailed logical analysis of the intuitive perception of the problem's solution is carried out and interpreted as another manifestation of cognitive dissonance, where even scientifically justified facts fail to change an individual's viewpoint formed within their established natural-scientific worldview. The mathematical justification of the player's optimal prize-selection strategy is presented in two probabilistic interpretations. Each of these interpretations aligns with well-known results of probability theory, which have once again been subjected to statistical verification through numerical experiments.

A distinctive feature of this study is that, in addition to the established theoretical and experimental results obtained using classical Monte Carlo methods (Mersenne Twister, PCG), quasi--Monte Carlo methods with refinements such as Sobol, Halton, Faure, and Niederreiter low-discrepancy sequences, as well as a linear congruential generator with local optimality properties, the computational procedures were also performed using previously unused random number generators. These include proprietary algorithms of a linear congruential generator in a non-improvable formulation and a quasi--Monte Carlo method based on ultra-economical Korobov lattices with small denominators $p$. In this context, the coordinate a_k/p of a lattice node of size p with denominator p is considered ``small'' within admissible limits, whereas in random algorithms there is no restriction on the length of decimal expansions relative to p.

The computations demonstrate that both of these randomness algorithms also confirm statistical regularity in convergence to the theoretical winning probabilities, both when the player keeps their initial choice and when they switch to the remaining unopened door, as the number of trials (games) increases. This result may also be interpreted inversely as confirmation of the quality of the proprietary randomness generators employed.

The article demonstrates the existence of statistical regularity in the Monty Hall game and serves as an illustrative example of the formation of probabilistic conclusions supported by statistical validation.