Diffusion Operator Research Articles

Multiple-image encryption scheme based on a new 2D hyperchaotic map with blurred pixels

It is known that chaotic, especially hyperchaotic system can be suitable for the application in image encryption owing to itself characteristics. While currently, certain improved chaotic or hyperchaotic systems are confronted with the security issue of encryption due to their less complex dynamical behaviors. To address the problem well, we introduce a novel two-dimensional (2D) crossed hyperchaotic map which is based on the logistic map and the infinite collapse map. The analysis of phase diagram and Lyapunov exponential spectrum demonstrate that the given system can exhibit extensive hyperchaotic behavior and good traversal properties. Moreover, the growing use of digital images has prompted demand for multi-image encryption scheme. For this reason, based on the given 2D crossed hyperchaotic map, a multiple image encryption (MIE) scheme that employs a cross-plane with the operation of simultaneous permutation and diffusion to modify the values of its positions and pixels across multiple images is proposed. A pixel blur preprocessing technique is introduced such that the efficiency of key calculation and the speed and safety of information encryption are greatly improved. Eventually, some simulation examples and security analysis reveal that the put forward encryption scheme is able to keep out kinds of attacks such as the selective plaintext attacks and data loss.

A two-parameter extended logistic chaotic map for modern image cryptosystems

In this paper, a novel Extended Logistic Chaotic Map (ELCM) with two control parameters is proposed. Overcoming the major drawback of the standard Logistic and other existing chaotic maps, the ELCM not only has infinite chaotic range as well as good ergodicity, but also has a simple structure just like the Logistic map, which greatly facilitates its practical implementation and becomes very suitable for today's real-time applications. Moreover, a new color image encryption scheme based on chaos concept, DNA (Deoxyribonucleic Acid) encoding and Convolutional Neural Network (CNN) is also presented. To perform a chaotic scrambling, the ELCM is however used at different stages of the encryption process. In addition, a pre-trained AlexNet CNN is used to generate a public key. After XORing with the secret key, the latter is used, on the one hand, to generate the initials values and the control parameters of the ELCM. On the other hand, it will be split into two keys in order to generate a random grayscale image, which will then be XORed with the three components (i.e., R, G and B) of the color plaintext image. Afterward, a permutation operation, DNA encoding, diffusion operation as well as bit reversion operation are then applied to the R, G and B components. The performance evaluation demonstrates that the ELCM not only has an infinite chaotic parameter range, but also exhibits a high chaotic complexity. Besides, experimental results as well as security analysis confirm that with NPCR of 99.6287%, the designed color image encryption scheme achieves an average entropy of 7.9975 and a near zero correlation (-0.0027). Furthermore, the proposed encryption scheme is in fact of higher security level in comparison to other schemes recently presented in the literature, thus making it highly suitable for today's real-time applications.

Real-time chaotic video encryption based on multi-threaded parallel confusion and diffusion

Due to the strong correlation among adjacent pixels, image encryption schemes typically perform multiple rounds of confusion and diffusion to protect images against various attacks. This is time-consuming and cannot meet the real-time requirements for video encryption. Existing works, therefore, realize video encryption by simplifying encryption process or selectively encrypting specific pixels, resulting in lower security compared to image encryption. This paper proposes a real-time chaotic video encryption strategy based on parallel computing. It splits video frame into sub-frames, creates a dedicated set of threads to concurrently perform confusion and diffusion operations on their respective sub-frames, and efficiently outputs encrypted frames. To assess its performance, two cryptosystems are implemented using different chaotic systems. Encryption speed evaluation demonstrates a significant acceleration in byte generation, confusion, and diffusion phases, enabling real-time encryption and decryption on different X86 platforms. The average encryption time is less than 42 ms, despite performing five rounds of confusion and diffusion operations on each frame. Statistical and security analysis prove that the deployed cryptosystems exhibit exceptional statistical properties and provide resistance to different attacks. Moreover, our method is adopted to implement a multi-user, real-time mobile video secure communication system using embedded systems, validating the feasibility of the proposed strategy for various application scenarios.

Analysis of a class of completely non-local elliptic diffusion operators

This work explores the possibility of developing the analog of some classic results from elliptic PDEs for a class of fractional ODEs involving the composition of both left- and right-sided Riemann-Liouville (R-L) fractional derivatives, Da+αDb-β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${D^\\alpha _{a+}}{D^\\beta _{b-}}$$\\end{document}, 1<α+β<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<\\alpha +\\beta <2$$\\end{document}. Compared to one-sided non-local R-L derivatives, these composite operators are completely non-local, which means that the evaluation of Da+αDb-βu(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${D^\\alpha _{a+}}{D^\\beta _{b-}}u(x)$$\\end{document} at a point x will have to retrieve the information not only to the left of x all the way to the left boundary but also to the right up to the right boundary, simultaneously. Therefore, only limited tools can be applied to such a situation, which is the most challenging part of the work. To overcome this, we do the analysis from a non-traditional perspective and eventually establish elliptic-type results, including Hopf’s Lemma and maximum principles. As α→1-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\rightarrow 1^-$$\\end{document} or α,β→1-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha ,\\beta \\rightarrow 1^-$$\\end{document}, those operators reduce to the one-sided fractional diffusion operator and the classic diffusion operator, respectively. For these reasons, we still refer to them as “elliptic diffusion operators", however, without any physical interpretation.

Transient Dynamics of a Fractional Fisher Equation

We investigate the transient dynamics of the Fisher equation under nonlinear diffusion and fractional operators. Firstly, we investigate the effects of the nonlinear diffusivity parameter in the integer-order Fisher equation, by considering a Gaussian distribution as the initial condition. Measuring the spread of the Gaussian distribution by u(0,t)−2, our results show that the solution reaches a steady state governed by the parameters present in the logistic function in Fisher’s equation. The initial transient is an anomalous diffusion process, but a power law cannot describe the whole transient. In this sense, the main novelty of this work is to show that a q-exponential function gives a better description of the transient dynamics. In addition to this result, we extend the Fisher equation via non-integer operators. As a fractional definition, we employ the Caputo fractional derivative and use a discretized system for the numerical approach according to finite difference schemes. We consider the numerical solutions in three scenarios: fractional differential operators acting in time, space, and in both variables. Our results show that the time to reach the steady solution strongly depends on the fractional order of the differential operator, with more influence by the time operator. Our main finding shows that a generalized q-exponential, present in the Tsallis formalism, describes the transient dynamics. The adjustment parameters of the q-exponential depend on the fractional order, connecting the generalized thermostatistics with the anomalous relaxation promoted by the fractional operators in time and space.

Conservative correction procedures utilizing artificial dissipation operators

The conservative correction procedure of Abgrall [1] is studied from the perspective of filter-based artificial dissipation methods, which motivates the ability to tailor the behavior of the method in both physical and spectral space. Compared to the original formulation, employing diffusion operators biases the correction towards smaller scales and better controls discretization errors when seeking to enforce auxiliary conservation relations. Effective entropy-stable regularization of sharp gradients is furthermore shown to be attainable. Calculations of the Sod shock tube problem as governed by the one-dimensional Euler equations are used to highlight the utility of considering alternate filters within the original correction framework, where the notion of entropy conservation/stability is leveraged for improving non-linear scheme robustness.

Impact of synergistic interfacial modification on the electrochemical performance of LiNi0.5Mn1.5O4 cathode materials

Developing sophisticated lithium-ion batteries with high energy and power density requires using high-voltage positive electrodes. Due to its three-dimensional lithium-ion diffusion and greater nominal operating voltage, spinel LiNi0.5Mn1.5O4 has emerged as one of lithium-ion batteries' most viable cathode materials. Electrolyte breakdown, Mn dissolution, and rapid cathode-electrolyte interface (CEI) degradation in lithium-ion cells are exacerbated by the high operating voltage of LNMO. Consequently, the long-term cycling of LNMO is hampered by such adverse side effects, making the commercialization of such a battery impractical. Here, we document the enhancement in the electrochemical performance of LNMO by surface modification utilizing a combination of Al2O3 coating and Graphene enveloping employing a facile wet synthesis technique. The presence of highly crystalline spherical secondary microspheres consisting of primary nanoparticles of disordered LiNi0.5Mn1.5O4, the surface modification with Al2O3, and the subsequent graphene wrapping were all confirmed by structural and surface analysis techniques. The fabricated cells containing the enhanced cathode material (LNMO-Al-GO) were cycled at a C/10 rate for 100 cycles in a voltage window of 3.5–4.9 V, providing a specific discharge capacity of 134.7 ± 3.8 mAhg−1. Delivering a capacity retention of 97.7 ± 3.9% compared to the unmodified LNMO sample (84.7 ± 5.3%). Ex-situ XRD, Electrochemical Impedance Spectroscopy (EIS), and Differential Scanning Calorimetry (DSC) investigations reveal that the alumina coating protects the cathode by acting as a hydrogen fluoride (H.F.) scavenger and minimizes unfavorable phase formations at the CEI, inhibiting Mn3+ dissolution and enhancing cyclability.

Well‐posedness for a system of diffusion–reaction equations with noncoercive diffusion

We prove that there is a unique solution for a system of diffusion–reaction equations, which occur when simulating microbiological growth at the pore scale with a high enough spatial resolution. Moreover, we show that the solution depends continuously on initial data. The diffusion for each component of the system is either coercive on , only elliptic on a subset (and zero elsewhere), or zero everywhere. This yields a noncoercive diffusion operator for the system of partial differential equations. The reaction is assumed to be Lipschitz continuous.

Fully decoupled and high-order linearly implicit energy-preserving RK-GSAV methods for the coupled nonlinear wave equation

This paper is concerned with high-order accurate, linearly implicit and energy-preserving schemes for the coupled nonlinear wave equation. To this end, a novel auxiliary variable approach proposed in a recent paper (Ju et al., 2022) is applied to reformulate the governing equation in an equivalent system, which possesses a modified energy that consists of primary functional and quadratic functional. Subsequently, we utilize the extrapolation strategy/prediction–correction technique for treating the nonlinear terms of the equivalent system to obtain a linearized energy-preserving system. Then, a series of fully decoupled high-order linear energy-preserving schemes are constructed by using the symplectic Runge–Kutta (RK) methods. We show that the modified energy functional can be precisely conserved under certain circumstances for the coefficients of the symplectic RK methods, and the unique solvability of the proposed time-stepping scheme is analyzed. In numerical implementations, a class of linear algebraic equations with constant-coefficient matrices need to be solved at each time step and only less computational costs are required. Extensive numerical experiments of the coupled nonlinear wave equation with local/nonlocal diffusion operators are provided to illustrate the energy conservation law and the computational efficiency of the schemes in long-time simulations.

An efficient image encryption scheme integrating chaotic keystream generator with S-box and triangular block scrambling

A novel image encryption scheme has been developed to effectively meet the growing need for image security and privacy. This scheme combines a chaotic keystream generator with S-box and triangular block scrambling to provide efficient encryption. The keystream generator consists of a two-dimensional coupled map lattice (2DCML), linear interval transformation (LIT) and an S-box. By using a 2DCML, two initial chaotic sequences are generated and used to construct an S-box. Then, a final chaotic sequence is obtained after undergoing LIT and S-box processing. The proposed algorithm applies two rounds of scrambling operations and two rounds of substitutions. During the scrambling processes, global scrambling and triangular block scrambling are combined. Diffusion operations use an S-box while incorporating a key policy linked with image content to enhance resistance against chosen-plaintext attacks. The simulation outcomes, coupled with the security analysis, substantiate the efficacy and robust encryption performance of the proposed image encryption scheme.

Invasive-Invaded Interaction Incorporating a Bramson Model with Density-Dependent Diffusion and a Non-Lipschitz Reaction

The primary objective of the presented study is to investigate the pairwise interaction dynamics between invasive and invaded species, considering a model characterized by a non-regular, non-Lipschitz type reaction, as well as non-homogeneous diffusion. To achieve this, we begin with the foundational model proposed by Bramson in 1988 and tailor it to account for density-dependent diffusion and the non-Lipschitz type reaction, rendering it more applicable to our specific ecological scenario. Subsequently, our newly developed model is subjected to different analyses to ascertain the existence and uniqueness of positive weak solutions. It is noteworthy that density-dependent diffusive operators exhibit a property known as “finite propagation”, which manifests as the existence of a propagating front in the ecological system. Furthermore, we delve into the problem domain by employing the concept of traveling waves to identify specific solutions. A key outcome of our investigation is as follows: When both species propagate at significantly different speeds in the context of traveling waves, the interaction between them is deemed unstable, resulting in oscillations in the concentration of the invaded species. Conversely, when both species propagate within a similar range of speeds, the dynamics of the system are predominantly governed by the invasive species.AMS Subject Classification: 35K55, 35K57, 35K59, 35K65

Real-time video encryption scheme based on multi-round confusion-diffusion architecture

With the rapid development of network and multimedia technologies, the widespread use of images in social networks, military applications, and other fields has led to substantial demands for image encryption. To protect images from potential threats, most protocols perform multiple rounds of confusion and diffusion until a satisfactory security level is attained. However, this strategy is time-consuming and unable to meet the requirements for real-time video encryption. Consequently, many works realize video encryption by simplifying the encryption process, e.g. employing a single round of confusion and diffusion to encrypt each frame. To enhance security to the level of image encryption while ensuring real-time performance, this paper proposes a real-time video encryption protocol based on multi-round confusion-diffusion architecture. It splits the frame into subframes, uses a set of threads to concurrently perform confusion and diffusion. The statistical and security analyses demonstrate the proposed protocol exhibits exceptional statistical properties and is resistant to various attacks. The encryption speed evaluation shows that our method significantly enhances computational efficiency, achieving latency-free encryption of [Formula: see text] 24FPS videos using the Intel Xeon Gold 6226R@2.9[Formula: see text]GHz CPU. Even with the execution of four rounds of confusion and six rounds of diffusion operations on each frame, the average encryption time remains below 30[Formula: see text]ms.

Remote sensing image encryption algorithm based on novel hyperchaos and an elliptic curve cryptosystem

Remote sensing images carry crucial ground information, often involving the spatial distribution and spatiotemporal changes of surface elements. To safeguard this sensitive data, image encryption technology is essential. In this paper, a novel Fibonacci sine exponential map is designed, the hyperchaotic performance of which is particularly suitable for image encryption algorithms. An encryption algorithm tailored for handling the multi-band attributes of remote sensing images is proposed. The algorithm combines a three-dimensional synchronized scrambled diffusion operation with chaos to efficiently encrypt multiple images. Moreover, the keys are processed using an elliptic curve cryptosystem, eliminating the need for an additional channel to transmit the keys, thus enhancing security. Experimental results and algorithm analysis demonstrate that the algorithm offers strong security and high efficiency, making it suitable for remote sensing image encryption tasks.

A novel image encryption scheme with adaptive Fourier decomposition

This study creatively introduces a class of adaptive Fourier decomposition (AFD) techniques into the field of image encryption, where AFD and its variants are newly developed signal decomposition methods. The novelty of this article is twofold. First, we use three different AFD techniques, including Core AFD, Unwinding AFD, and Cyclic AFD, to generate two key streams through decomposing a source image. The two key streams generated by different AFD algorithms are different and aperiodic, which can effectively improve the security of encryption algorithm. Furthermore, this work also proposes a novel encryption method that implements the confusion and diffusion operations by index-sorted mapping method and pixel swapping operation based on two generated key streams, respectively. To ensure that the image is fully scrambled, both pixel-level and bit-level permutations are performed, which employs a novel bilateral sequence diffusion method to change the image pixel distribution based on pixel swap operations. The cryptographic performance of the scheme is evaluated through various security analyses, including key sensitivity, statistical, entropy, clipping attack, and differential attack analysis. Experimental results confirm that the proposed image encryption scheme ensures a high level of security and exhibits superior performance in resisting various attacks compared with several traditional and state-of-the-art image encryption methods.

Image encryption algorithm based on improved iterative chaotic map with infinite collapses and Gray code

The quick advancement of multimedia technology has led to the widespread utilization of digital images across various industries. Meanwhile, image security has become an urgent issue in need of resolution. To guarantee image security as well fulfill the requirements of real-time image cryptosystems, a novel image encryption algorithm is proposed based on the designed improved iterative chaotic map with infinite collapses (ICMIC), Gray code and semi-pixel level permutation and diffusion. Firstly, two chaotic sequences are produced using the improved ICMIC and then they are used to perform scrambling and diffusion operations. Secondly, the plain image is encoded using Gray code and then we convert each Gray code into two semi-pixels. Thirdly, the designed semi-pixel level permutation is used to scramble the semi-pixel vector. Finally, the scrambled semi-pixel vector is diffused using the designed dynamic semi-pixel level diffusion. The performance analyses illustrate that the proposed algorithm possesses robust security and high efficiency, making it ideally suited for application in real-time image cryptosystems.

Arbitrary-Lagrangian-Eulerian finite volume IMEX schemes for the incompressible Navier-Stokes equations on evolving Chimera meshes

In this article we design a finite volume semi-implicit IMEX scheme for the incompressible Navier-Stokes equations on evolving Chimera meshes. We employ a time discretization technique that separates explicit and implicit terms, accommodating the multi-scale nature of the governing equations, which involve both time scales of diffusion and advection operators. The finite volume approach for both explicit and implicit terms allows to encode into the nonlinear flux the velocity of displacement of the Chimera mesh via integration on moving cells. The numerical solution is then projected onto the physically meaningful solution manifold of non-solenoidal fields that stems from the energy equation. To attain second-order time accuracy, we employ semi-implicit IMEX Runge-Kutta schemes. These novel schemes are combined with a fractional-step method, thus the governing equations are eventually solved using a projection method to satisfy the divergence-free constraint of the velocity field. The implicit discretization of the viscous terms allows the CFL-type stability condition for the maximum admissible time step to be only defined by the relative fluid velocity referred to the movement of the frame and not depending on the viscous terms. Communication between different grid blocks is enabled through compact exchange of information from the fringe cells of one mesh block to the field cells of the other block. The continuity of the solution is recovered in one-shot during the solution of the arising algebraic systems by not involving neither direct discretization of the differential operators on fringe cells nor an iterative Schwartz-type method. Free-stream preservation property, i.e. compliance with the Geometric Conservation Law (GCL), is respected at the order of the scheme. The accuracy and capabilities of the new numerical schemes are proved through an extensive range of test cases, demonstrating ability to solve relevant benchmarks in the field of incompressible fluids.

A new image encryption based on hybrid heterogeneous time-delay chaotic systems

&lt;abstract&gt;&lt;p&gt;Chaos theory has been widely utilized in password design, resulting in an encryption algorithm that exhibits strong security and high efficiency. However, rapid advancements in cryptanalysis technology have rendered single system generated sequences susceptible to tracking and simulation, compromising encryption algorithm security. To address this issue, we propose an image encryption algorithm based on hybrid heterogeneous time-delay chaotic systems. Our algorithm utilizes a collection of sequences generated by multiple heterogeneous time-delay chaotic systems, rather than sequences from a single chaotic system. Specifically, three sequences are randomly assigned to image pixel scrambling and diffusion operations. Furthermore, the time-delay chaotic system comprises multiple hyperchaotic systems with positive Lyapunov exponents, exhibiting a more complex dynamic behavior than non-delay chaotic systems. Our encryption algorithm is developed by a plurality of time-delay chaotic systems, thereby increasing the key space, enhancing security, and making the encrypted image more difficult to crack. Simulation experiment results verify that our algorithm exhibits superior encryption efficiency and security compared to other encryption algorithms.&lt;/p&gt;&lt;/abstract&gt;

Analyzing the dynamics of fractional spatio-temporal $ \mathrm{SEIR} $ epidemic model

&lt;p&gt;In this work, we present a comprehensive analysis of the spatio-temporal $ \mathrm{SEIR} $ epidemic model of fractional order. The infection dynamics in the proposed fractional order model (FOM) are described by a system of partial differential equations (PDEs) within a time-fractional order and diffusion operator in one-dimensional space, considering that the total population is split into four compartments: Susceptible, exposed, infected, and recovered individuals denoted as $ \mathrm{S} $, $ \mathrm{E} $, $ \mathrm{I} $ and $ \mathrm{R} $, respectively. Our contributions commence by establishing the existence and uniqueness of positively bounded solutions for the proposed FOM. Moreover, we determined all equilibrium points (EPs) and investigated their local stability based on the basic reproduction number (BRN) $ \mathcal{R}_{0} $, which is calculated by the next-generation matrix (NGM) method. Additionally, we demonstrated global stability using an appropriate Lyapunov function with fractional LaSalle's invariance principle (LIP). Sensitivity analysis of the FOM parameters was discussed to identify the most critical parameters by which the volume of disease propagation can be measured. The theoretical findings were corroborated by numerical simulations of solutions that are displayed in 3D and 2D graphs. Graphical simulations highlight the effect of vaccination on infection severity. Changing the fractional order $ \alpha $ in the proposed FOM has an influence on the speed of convergence to the steady state as a result of the memory effect. Furthermore, vaccination emerges as an effective strategy for disease control.&lt;/p&gt;

A combined first and fractional order regularization method for mixed Poisson-white spike noisy image restoration

In this paper we study the problem of restoring images affected by mixed Poisson-Square Cauchy noise. A multi-convex variational model is derived by integrating infimal convolution likelihood into a process of joint maximum a posteriori estimation, where a modified four-directional fractional-order total variation is united with the first order total variation to characterize the image prior. To solve the proposed model numerically, a block coefficient descent based algorithm is derived, in which variable splitting and alternating direction minimization of multipliers are utilized along with anisotropic diffusion and additive operator splitting to gain efficiency and quality. The obtained numerical results are compared with results obtained from two other total variation regularized models with different fidelities. The numerical discuss confirms the flexibility and validity of the proposed model.

Generalized Invariance Principles for Stochastic Dynamical Systems and Their Applications

Investigating long-term behaviors of stochastic dynamical systems often requires to establish criteria that are able to describe delicate dynamics of the considered systems. In this article, we develop generalized invariance principles for continuous-time stochastic dynamical systems. Particularly, in a sense of probability one and by the developed semimartingale convergence theorem, we not only establish a local invariance principle, but also provide a generalized global invariance principle that allows the sign of the diffusion operator to be positive in some bounded region. We further provide an estimation for the time when a trajectory, initiating outside a particular bounded set, eventually enters it. Finally, we use several representative examples, including stochastic oscillating dynamics, to illustrate the practical usefulness of our analytical criteria in deciphering the stabilization or/and the synchronization dynamics of stochastic systems.