Petr Fedoseev

The security of sensitive data is a crucial issue in the information age. While the existing encryption protocols cannot always guarantee the required level of security due to the rapidly increasing computational capability of attackers, developing new cryptographically strong encryption techniques is of great importance in modern computer science. One of the advanced approaches in the field of cryptography is chaos-based encryption. In this study, we propose an efficient algorithm for arbitrary multimedia data encryption using the novel finite-difference scheme with adaptive symmetry based on the Gingerbreadman chaotic map. In the experimental part of the study, we use several analysis techniques to prove the presence of chaos in the dynamics of the reported discrete map and investigate the dependence between system dynamics and symmetry coefficient. Parametric chaotic sets and the largest Lyapunov exponent plots are given to evaluate the dynamics of the investigated finite-difference model. NIST statistical tests were applied to assess the properties of the developed pseudo-random numbers generator, and correlation analysis was performed to evaluate the secrecy of the encrypted image. It is experimentally shown, that varying the symmetry coefficient can significantly increase the keyspace for the encryption algorithm based on the symmetric Gingerbreadman map. The results of this study can be used to develop encryption software, including secure text messengers or stream data ciphers.

The numerical solution to boundary differential problems is a crucial task in modern applied mathematics. Usually, implicit integration methods are applied to solve this class of problems due to their high numerical stability and convergence. The known shortcoming of implicit algorithms is high computational costs, which can become unacceptable in the case of numerous right-hand side function calls, which are typical when solving boundary problems via the shooting method. Meanwhile, recently semi-implicit numerical integrators have gained major interest from scholars, providing an efficient trade-off between computational costs, stability, and precision. However, the application of semi-implicit methods to solving boundary problems has not been investigated in detail. In this paper, we aim to fill this gap by constructing a semi-implicit boundary problem solver and comparing the performance of explicit, semi-implicit, semi-explicit, and implicit methods using a set of linear and nonlinear test boundary problems. The novel blinking solver concept is introduced to overcome the main shortcoming of the semi-implicit schemes, namely, the low convergence on exponential solutions. The numerical stability of the blinking semi-implicit solver is investigated and compared with existing methods by plotting the stability regions. The performance plots for investigated methods are obtained as a dependence between global truncation error and estimated computation time. The experimental results confirm the assumption that semi-implicit numerical methods can significantly outperform both explicit and implicit solvers while solving boundary problems, especially in the proposed blinking modification. The results of this study can be efficiently used to create software for solving boundary problems, including partial derivative equations. Constructing semi-implicit numerical methods of higher-accuracy orders is also of interest for further research.

Discrete chaotic maps are a mathematical basis for many useful applications. One of the most common is chaos-based pseudorandom number generators (PRNGs), which should be computationally cheap and controllable and possess necessary statistical properties, such as mixing and diffusion. However, chaotic PRNGs have several known shortcomings, e.g., being prone to chaos degeneration, falling in short periods, and having a relatively narrow parameter range. Therefore, it is reasonable to design novel simple chaotic maps to overcome these drawbacks. In this study, we propose a novel fractal chaotic tent map, which is a generalization of the well-known tent map with a fractal function introduced into the right-hand side. We construct and investigate a PRNG based on the proposed map, showing its high level of randomness by applying the NIST statistical test suite. The application of the proposed PRNG to the task of generating surrogate data and a surrogate testing procedure is shown. The experimental results demonstrate that our approach possesses superior accuracy in surrogate testing across three distinct signal types—linear, chaotic, and biological signals—compared to the MATLAB built-in randn() function and PRNGs based on the logistic map and the conventional tent map. Along with surrogate testing, the proposed fractal tent map can be efficiently used in chaos-based communications and data encryption tasks.

The paper offers metrics to evaluate the image processing performance and video processing programs implemented on the MIPS SIMD Architecture (MSA). These metrics were also used to analyze and present key sets for the GCC compiler. As a result of the analysis, which was performed earlier, numerical representations of each parameter of four algorithms (CABAC, CAVLC, FDCT-IDCT, and Motion Estimation), used in the H.264 video compression standard, were calculated, and 5952 key sets were obtained, respectively. The weight coefficients were determined for each of the key parameters according to the Saati hierarchy analysis methodology. After that, using weight functions, 372 recommended sets of keys were determined, the target functions of which are maximal for each of the 4 algorithms considered.

Parallel numerical integration is a valuable tool used in many applications requiring high-performance numerical solvers, which is of great interest nowadays due to the increasing difficulty and complexity in differential problems. One of the possible approaches to increase the efficiency of ODE solvers is to parallelize recurrent numerical methods, making them more suitable for execution in hardware with natural parallelism, e.g., field-programmable gate arrays (FPGAs) or graphical processing units (GPUs). Some of the simplest and most popular ODE solvers are explicit Runge–Kutta methods. Despite the high implementability and overall simplicity of the Runge–Kutta schemes, recurrent algorithms remain weakly suitable for execution in parallel computers. In this paper, we propose an approach for parallelizing classical explicit Runge–Kutta methods to construct efficient ODE solvers with pipeline architecture. A novel technique to obtain parallel finite-difference models based on Runge–Kutta integration is described. Three test initial value problems are considered to evaluate the properties of the obtained solvers. It is shown that the truncation error of the parallelized Runge–Kutta method does not significantly change after its known recurrent version. A possible speed up in calculations is estimated using Amdahl’s law and is approximately 2.5–3-times. Block diagrams of fixed-point parallel ODE solvers suitable for hardware implementation on FPGA are given.

A numerical stability region is a valuable tool for estimating the practical applicability of numerical methods and comparing them in terms of stability. However, only a little information can be obtained from the stability regions when their shape is highly irregular. Such irregularity is inherent to many recently developed semi-implicit and semi-explicit methods. In this paper, we introduce a new tool for analyzing numerical methods called preference regions. This allows us to compare various methods and choose the appropriate stepsize for their practical implementation, such as stability regions, but imposes stricter conditions on the methods, and therefore is more accurate. We present a thorough stability and preference region analysis for a new class of composition methods recently proposed by F. Casas and A. Escorihuela-Tomàs. We explicitly show how preference regions, plotted for an arbitrary numerical integration method, complement the conventional stability analysis and offer better insights into the practical applicability of the method.

Composition is a powerful and simple approach for obtaining numerical integration methods of high accuracy order while preserving the geometric properties of a basic integrator. Adaptive step size control allows one to significantly increase the performance of numerical integration methods. However, there is a lack of efficient step size control algorithms for composition solvers due to some known difficulties in constructing a low-cost embedded local error estimator. In this paper, we propose a novel local error estimator based on a difference between the semi-implicit CD method and semi-explicit midpoint methods within a common composition scheme. We evaluate the performance of adaptive composition schemes with the proposed local error estimator, comparing it with the other state-of-the-art approaches. We show that composition ODE solvers with the proposed step size control algorithm possess higher numerical efficiency than known methods, by using a comprehensive set of nonlinear test problems.

This paper proposes a novel identification method for memristive devices using Knowm memristors as an example. The suggested identification method is presented as a generalized process for a wide range of memristive elements. An experimental setup was created to obtain a set of intrinsic I–V curves for Knowm memristors. Using the acquired measurements data and proposed identification technique, we developed a new mathematical model that considers low-current effects and cycle-to-cycle variability. The process of parametric identification for the proposed model is described. The obtained memristor model represents the switching threshold as a function of the state variables vector, making it possible to account for snapforward or snapback effects, frequency properties, and switching variability. Several tools for the visual presentation of the identification results are considered, and some limitations of the proposed model are discussed.

Multistep methods for the numerical solution of ordinary differential equations are an important class of applied mathematical techniques. This paper is motivated by recently reported advances in semi-implicit numerical integration methods, multistep and extrapolation solvers. Here we propose a novel type of multistep extrapolation method for solving ODEs based on the semi-implicit basic method of order 2. Considering several chaotic systems and van der Pol nonlinear oscillator as examples, we implemented a performance analysis of the proposed technique in comparison with well-known multistep methods: Adams–Bashforth, Adams–Moulton and the backward differentiation formula. We explicitly show that the multistep semi-implicit methods can outperform the classical linear multistep methods, providing more precision in the solutions for nonlinear differential equations. The analysis of stability regions reveals that the proposed methods are more stable than explicit linear multistep methods. The possible applications of the developed ODE solver are the long-term simulations of chaotic systems and processes, solving moderately stiff differential equations and advanced modeling systems.