Diffusion Operator Research Articles

The circuit realization of a fifth‐order multi‐wing chaotic system and its application in image encryption

SummaryIn this paper, a new fifth‐order eight‐wing chaotic system is proposed on the basis of a five‐dimensional four‐wing chaotic system. The resulting new system is used as a basis for introducing a method that can control both the number of wings of the system and the direction of the change in the number of wings. Add a rounding function with a cosine function or a sine function to the system equation. The number of wings of the control system is controlled by changing the parameter values of the rounding function. This method can significantly increase the complexity of the system when applied in multiple directions of the system equations. And this method can achieve the purpose of controlling the wing number of the whole system by changing only one parameter. By observing the phase diagram and bifurcation diagram of the system, it is found that it has more complex chaotic properties and rich dynamical behavior and there are coexisting attractors. The chaotic properties of the system are confirmed by observing the Poincaré cross section and the 0–1 test results of the system. Afterward, the simulation circuit of the fifth‐order system is built in the simulation software Multisim. And the obtained phase diagram is consistent with the results obtained in MATLAB which confirmed the realizability of the system. An FPGA hardware implementation of this fifth‐order eight‐wing system has also been completed at the end of the paper to demonstrate the implementability of the system. And the results are highly consistent with those obtained in the MATLAB software.The fifth‐order chaotic system is combined with the dislocation algorithm and diffusion algorithm to apply to the image encryption system. The encryption system contains a variety of encryption algorithms for secondary encryption of images including Arnold dislocation and repeat‐free scrambling algorithm. The diffusion algorithm includes the addition of the modulus length diffusion and multiplicative operation diffusion based on the GF(257) domain. After a series of performance analysis, the encryption system is proved to have good secrecy effect and great application value in the image encryption neighborhood.

From random walks on networks to nonlinear diffusion.

Mathematical models of motility are often based on random-walk descriptions of discrete individuals that can move according to certain rules. It is usually the case that large masses concentrated in small regions of space have a great impact on the collective movement of the group. For this reason, many models in mathematical biology have incorporated crowding effects and managed to understand their implications. Here, we build on a previously developed framework for random walks on networks to show that in the continuum limit, the underlying stochastic process can be identified with a diffusion partial differential equation. The diffusion coefficient of the emerging equationis, in general, density dependent, and can be directly related to the transition probabilities of the random walk. Moreover, the relaxation time of the stochastic process is directly linked to the diffusion coefficient and also to the network structure, as it usually happens in the case of linear diffusion. As a specific example, we study the equivalent of a porous-medium-type equationon networks, which shows similar properties to its continuum equivalent. For this equation, self-similar solutions on a lattice and on homogeneous trees can be found, showing finite speed of propagation in contrast to commonly used linear diffusion equations. These findings also provide insights into reaction-diffusion systems with general diffusion operators, which have appeared recently in some applications.

Fast quantum image encryption scheme based on multilayer short memory fractional order Lotka-Volterra system and dual-scale triangular map

The Caputo fractional order Lotka-Volterra system is time-consuming in practical applications, since its starting point is fixed. To tackle this problem, a short memory fractional order Lotka-Volterra system (SMFrLVS) is proposed, where the chaotic attractor of the short memory fractional order Lotka-Volterra system is achieved by the predictor-corrector method. Then, a multilayer fractional order Lotka-Volterra system with short memory (MSMFrLVS) is introduced, whose chaotic behaviors are explored via Poincare sections and frequency power spectra. A quantum image encryption algorithm is proposed by combining MSMFrLVS with quantum dual-scale triangular map. A quantum circuit of the dual-scale triangular map is designed with ADDER-MOD2n. At the permutation stage, the plaintext image is transformed into quantum form with the generalized quantum image representation model. The resulting quantum image is divided into sub-blocks and scrambled by the quantum dual-scale triangular map. Subsequently, the intra and the inter permutation operations on bit-planes are realized by sorting pseudo-random sequence and by quantum Gray code, respectively. At the diffusion stage, the initial values of the MSMFrLVS are generated with a plaintext correlation mechanism. The ciphertext image can be acquired by carrying out three-level diffusion operations. It is demonstrated that the proposed quantum image encryption algorithm performs better than some typical image encryption algorithm in terms of security, robustness, computational complexity and encryption speed.

A chaotic image encryption scheme based on multi-directional confusion and diffusion operations

A careful perusal of the literature regarding the image encryption algorithms indicates that many research articles based on pixel swapping operations in the horizontal and vertical orientations, have been published. This study ventures to develop a yet another sophisticated algorithm by extending the swapping operations along the diagonal–antidiagonal directions. Besides, unlike the previous studies, the confusion and diffusion operations over the pixels of the given input plain image have been crafted in a parallel fashion. This work has employed a 4D chaotic system to spawn the streams of random numbers. The initial values and the system parameters have been made dependent over the given plain image in order to inject the plaintext sensitivity in the proposed cipher. As a modus operandi of the proposed encryption algorithm, a non-intersecting pair of row and column is selected randomly with the same arbitrary length and swapped/XORed according to the value dictated by the random numbers. The same operations have been carried out for the non-intersecting pair of diagonal and anti-diagonal. The reason for selecting the non-intersecting pair stems from the fact that intersecting pairs cause to lose the precious pixels data which questions the potential invertibility feature of the proposed algorithm. To demonstrate the improved efficiency, ability to withstand the varied attacks by the cryptanalysis savvy, and prospects for some real-world application, the algorithm has been subjected to a comprehensive set of evaluation metrics.

A novel hyperchaotic system with fast and slow attractors

A novel hyperchaotic system with only one equilibrium point is proposed by introducing an additional state feedback. The dynamic behaviors of this system are investigated by means of the Lyapunov exponent, Poincaré map, and bifurcation diagrams, i.e., this new hyperchaotic system possesses two positive Lyapunov exponents within a large range. Interestingly, the chaotic attractor of the system is special with two slow and fast attractors, that is, the system is a combination of slow and fast parts. The bound estimation of this system is also investigated. In addition, the image cryptology method is performed. In the image encryption part, an algorithm is suggested to provide confusion and diffusion operations on the image. The results show that, by employing the hyperchaotic system, the original image information can be completely hidden after being encrypted and can be decrypted efficiently.

Triple-Coupling Cellular Automata and Their Application to Image Encryption

Cellular automaton (CA) as a discrete and deterministic dynamic system has been widely used in cryptography. In this paper, a novel triple-coupling CA model is proposed. We first apply the triple-coupling model to 1D CA to design a high-quality pseudo-random number generator (PRNG). Numerical simulation and analyses show that triple-coupling CA exhibits more complex evolutionary behavior than uncoupling CA. The pseudo-random sequence generated by triple-coupling 1D CA can pass all NIST SP800-22 tests. Subsequently, the triple-coupling model is also applied to 2D CA, and a novel image encryption scheme based on this model is proposed. Thanks to the triple-coupling model, the diffusion operation can be completed between different color channels of the color image. Experimental simulations and extensive cryptanalysis show that the image encryption algorithm using the triple-coupling model has superior security and high efficiency.

Counter-current imbibition and non-linear diffusion in fractured porous media: Analysis of early- and late-time regimes and application to inter-porosity flux

The displacement of non-aqueous phase liquid (NAPL) in permeable porous rock by water saturating the surrounding fractures is studied. In many situations of practical interest, capillarity is the dominant driving force. Using the global pressure concept, it is shown that the water saturation is driven by a generic non-linear diffusion equation governing the matrix block counter-current spontaneous imbibition. In addition, in most cases, the saturation-dependent diffusion coefficient vanishes at the saturation end points, that renders the driving equation highly singular.In this paper, two exact asymptotic solutions valid for short and long times are presented, under the assumption that the conductivity vanishes as a power-law of both phase saturations at the extreme values of the fluid saturation.Focusing on the late-time domain, the asymptotic solution is derived using an Ansatz that is written under the form of a power-law time decay of the NAPL saturation. In the spatial domain, this solution is an eigenvector of the non-linear diffusion operator driving the saturation, for a problem with Dirichlet boundary conditions. If the diffusion coefficient varies as a power law of the NAPL saturation, the spatial variations of the solution are given analytically for a one-dimensional porous medium corresponding to parallel fracture planes. The analytical solution is in very good agreement with results of numerical simulations involving various realistic sets of input transport parameters.Generalization to the case of two- or three-dimensional matrix blocks of arbitrary shape is proposed using a similar Ansatz, solution of a non-linear eigenvalue problem. A fast converging algorithm based on a fixed-point sequence starting from a suitable first guess was developed. Comparisons with full-time simulations for several typical block geometries show an excellent agreement.These results permit to set-up an analytical formulation generalizing linear single-phase representation of matrix-to-fracture exchange term. It accounts for the non-linearity of the local flow equations using the power-law dependence of the conductivity for low NAPL saturation. The corresponding exponent can be predicted from the input conductivity parameters. Similar findings are also presented and validated numerically for two- or three-dimensional matrix blocks. That original approach paves the way to research leading to a more faithful description of matrix-to-fracture exchanges when considering a realistic fractured medium composed of a population of matrix blocks of various size and shapes.

On the use of nonlinear anisotropic diffusion filters for seismic imaging using the full waveform

Nonlinear anisotropic diffusion filters have been introduced in the field of image processing for image denoising and image restoration. They are based on the solution of partial differential equations involving a nonlinear anisotropic diffusion operator. From a mathematical point of view, these filters enjoy attractive properties, such as minimum–maximum principle, and an inherent decomposition of the images in different scales. We investigate in this study how these filters can be applied to help solving data-fitting inverse problems. We focus on seismic imaging using the full waveform, a well known nonlinear instance of such inverse problems. In this context, we show how the filters can be applied directly to the solution space, to enhance the structural coherence of the parameters representing the subsurface mechanical properties and accelerate the convergence. We also show how they can be applied to the seismic data itself. In the latter case, the method results in an original low-frequency data enhancement technique making it possible to stabilize the inversion process when started from an initial model away from the basin of attraction of the global minimizer. Numerical results on a 2D realistic synthetic full waveform inversion case study illustrate the interesting properties of both approaches.

Dynamic DNA coding multi-image encryption based on compound chaos

We present a dynamic DNA coding multi-image encryption algorithm based on integer lifting wavelet transform and compound chaos. In the encryption process, the low-frequency sub-bands of four plaintexts are extracted to get a combined image containing the multiple plaintext image information. With the help of a chaotic system, the pixel positions and pixel values of the combined image are modified by the permutation and the diffusion operations. Then we divide the image into equal-sized blocks, and the dynamic DNA coding method is used to complete the final encryption. To enhance security, we employ the RSA algorithm to protect the keys used in our encryption algorithm. Simulation results and security analysis demonstrate the robustness and the security.

High-quality visually secure image cryptosystem using improved Chebyshev map and 2D compressive sensing model

In this paper, we propose an improved Chebyshev map and a visually meaningful image encryption (VMIE) algorithm based on two-dimensional compressive sensing (2DCS). First, the adoption of 2DCS compresses the plain image, wherein the measurement matrix is processed by the simulated annealing algorithm to improve the quality of the reconstructed image. Second, a diffusion operation is performed on the sample data. Furthermore, a matrix encoding matrix is employed to insert the cryptographic image into the host image to obtain a visually meaningful steganographic image. The experimental results and comprehensive analyses show that the proposed scheme has feasible performance and stands up to various attacks. Eventually, the comparison with existing related schemes demonstrates that the proposed image encryption system has the advantages of visual security and reconstruction quality.

Vorticity equation on surfaces with arbitrary topology embedded in three-dimensional Euclidean space

We derive the vorticity equation for an incompressible fluid on a two-dimensional surface with an arbitrary topology, embedded in three-dimensional Euclidean space and arising from a first integral of the flow, by using a tailored Clebsch parameterization of the velocity field. In the inviscid limit, we identify conserved surface energy and enstrophy and obtain the corresponding noncanonical Hamiltonian structure. We then discuss the formulation of the diffusion operator on the surface by examining two alternatives. In the first case, we follow the standard approach for Navier–Stokes equations on a Riemannian manifold and calculate the diffusion operator by requiring that flows corresponding to Killing fields of the Riemannian metric are not subject to dissipation. For an embedded surface, this leads to a diffusion operator, including derivatives of the stream function across the surface. In the second case, using an analogy with the Poisson equation for the Newtonian gravitational potential in general relativity, we construct a diffusion operator taking into account the Ricci scalar curvature of the surface. The resulting vorticity equation is two-dimensional, and the corresponding diffusive equilibria minimize dissipation under the constraint of curvature energy.

A study of hydrodynamic viscous fluid flow parameters change regularities in case of a conical diffuser

Studies of patterns of changes in hydrodynamic parameters of the viscous incompressible fluid in a conical diffuser were conducted. The specificity of the viscous liquid flow in a conical diffuser is that the kinetic energy of the flow, depending on the opening angle, is converted into pressure energy. Depending on Reynolds numbers and diffuser opening angles, the velocity vector field is stationary. With an increase in the Reynolds number, the symmetry of the flow relative to the axis of the diffuser is broken. A general solution to the approximate Navier-Stokes equations is given, based on the diffuser opening angle and the Reynolds number. A method for integrating the boundary value problem has been developed, and the patterns of velocity changes across the diffuser length at a parabolic distribution of velocities in the inlet section are obtained. By integrating partial differential equations that match all boundary conditions, the solution to the boundary value problem can be found. Graphs of changes in radial and axial velocities along the length and with a fixed value of the opening angle are shown; the flow pattern and the transition of a single-mode flow to multimode regimes are obtained. For a fixed opening angle and Reynolds number, the conditions for flow separation from a fixed wall are derived, where the flow velocity changes the sign. A mixing process is observed in the multi-mode region, which is accompanied by numerous pulsation phenomena and an unstable diffuser operation, where the resulting solutions are inappropriate. Based on the results of the studies obtained, it is possible to correctly design a conical diffuser, namely, under the condition of non-separated flow, to choose the opening angle and its length.

A novel fast image encryption scheme based on a new one-dimensional compound sine chaotic system

In the paper, a new one-dimensional (1-D) compound Sine chaotic system (CSCS) is first proposed. Then new chaotic maps are generated by the CSCS. And four novel generated maps are used for the illustration about the CSCS. Secondly, the results of performance analysis show that the four maps have large Lyapunov exponents and high complexity. Thirdly, a high-efficiency image encryption scheme is proposed by employing two of the four new produced chaotic maps. In the proposed encryption scheme, the simplest diffusion operation is used. And we use a variety of scrambling operations, such as Zigzag transform, Magic confusion and the row confusion. In addition, to increase key space and in order to improve the ability to resist two kinds of attacks, namely the known plaintext attack and the selected plaintext attack, the control parameters and the initial values of the two new chaotic systems are generated based on the SHA-256 function. Finally, compared to other schemes, simulation tests show that our scheme not only has higher security but also faster encryption speed.

Quantum Color Image Encryption Scheme Based on Geometric Transformation and Intensity Channel Diffusion

A quantum color image encryption algorithm based on geometric transformation and intensity channel diffusion was designed. Firstly, a plaintext image was transformed into a quantum state form using the quantum image representation based on HSI color space (QIRHSI) representation as a carrier. Next, a pseudo-random sequence was generated using the generalized logistic map, and the pixel positions permuted multiple two-point swap operations. Immediately afterward, the intensity values were changed by an intensity bit-plane cross-swap and XOR, XNOR operations. Finally, the intensity channel of the above image was diffused in combination with the pseudo-confusion sequence as produced by the quantum logistic map to perform a diffusion operation on the intensity bit-plane to obtain the ciphertext image. Numerical simulations and analyses show that the designed algorithm is implementable and robust, especially in terms of outstanding performance and less computational complexity than classical algorithms in terms of security perspective.

A Novel Grayscale Image Encryption Scheme Based on the Block-Level Swapping of Pixels and the Chaotic System.

Hundreds of image encryption schemes have been conducted (as the literature review indicates). The majority of these schemes use pixels as building blocks for confusion and diffusion operations. Pixel-level operations are time-consuming and, thus, not suitable for many critical applications (e.g., telesurgery). Security is of the utmost importance while writing these schemes. This study aimed to provide a scheme based on block-level scrambling (with increased speed). Three streams of chaotic data were obtained through the intertwining logistic map (ILM). For a given image, the algorithm creates blocks of eight pixels. Two blocks (randomly selected from the long array of blocks) are swapped an arbitrary number of times. Two streams of random numbers facilitate this process. The scrambled image is further XORed with the key image generated through the third stream of random numbers to obtain the final cipher image. Plaintext sensitivity is incorporated through SHA-256 hash codes for the given image. The suggested cipher is subjected to a comprehensive set of security parameters, such as the key space, histogram, correlation coefficient, information entropy, differential attack, peak signal to noise ratio (PSNR), noise, and data loss attack, time complexity, and encryption throughput. In particular, the computational time of 0.1842 s and the throughput of 3.3488 Mbps of this scheme outperforms many published works, which bears immense promise for its real-world application.

Proper orthogonal decomposition based reduced-order modeling of flux-Limited gray thermal radiation

In this work, a proper orthogonal decomposition (POD) based reduced-order model (ROM) is developed to solve gray, flux-limited thermal radiation diffusion. We focus on the variable opacity radiation penetration benchmark posed by Olson, Auer, and Hall. The T−3 relationship for opacity in conjunction with high-temperature radiation penetrating an initially cold material produces a strong thermal radiation shock. This class of problems is particularly challenging for standard POD-based reduced-order modeling due to the nonlinearities presented by 1) the T4 source term, and 2) flux-limited diffusion operator. To address these challenges and develop a cost competitive ROM, we employ a “hyper-reduction” technique through discrete empirical interpolation (DEIM) and allow for adaptive reduced-order projections through principal interval decomposition (PID). Performance of the proposed methodology is quantified by comparing the cost savings and accuracy relative to a full-order computation. Reference solutions and snapshot data are obtained through a full-order calculation performed by the University of Chicago maintained astrophysics code, FLASH. For consistency and potential extensibility, the developed ROM is also implemented in FLASH. We find that in the initialization regime, where the thermal radiation wave is initially created by the warming of the material, this class of problems is highly reducible and suitable for POD-based ROMs. However, the strong convective nature of the wave propagation regime is less reducible and more challenging to create an efficient ROM.

Asymptotic Stability of the Relativistic Boltzmann Equation Without Angular Cut-Off

This paper is concerned with the relativistic Boltzmann equation without angular cutoff. We establish the global-in-time existence, uniqueness and asymptotic stability for solutions nearby the relativistic Maxwellian. We work in the case of a spatially periodic box. We assume the generic hard-interaction and soft-interaction conditions on the collision kernel that were derived by Dudyński and Ekiel-Je\(\dot{\text {z}}\)ewska (Comm. Math. Phys. 115(4):607–629, 1985) in [32], and our assumptions include the case of Israel particles (J. Math. Phys. 4:1163–1181, 1963) in [56]. In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator. The coercivity estimates that are needed rely crucially on the sharp asymptotics for the frequency multiplier that has not been previously established. We further derive the relativistic analogue of the Carleman dual representation for the Boltzmann collision operator. This resolves the open question of perturbative global existence and uniqueness without the Grad’s angular cut-off assumption.

Color Image Encryption Algorithm Based on Dynamic Block Zigzag Transformation and Six-Sided Star Model

As a result of the rise in network technology, information security has become particularly important. Digital images play an important role in network transmission. To improve their security and efficiency, a new color image encryption algorithm is proposed. The proposed algorithm adopts a classical scrambling–diffusion framework. In the scrambling stage, the dynamic block Zigzag transformation is designed by combining the chaotic sequence with the standard Zigzag transformation, which can dynamically select the transformation range and the number of times. It is used to scramble the pixel positions in the R, G, and B components. In the diffusion stage, the six-sided star model is established by combining the chaotic sequence and the six-sided star structure characteristics, which can store the 24 bits of the pixel in a defined order to realize bit-level diffusion operation. Experimental analyses indicate that our algorithm has the characteristics of high key sensitivity, large key space, high efficiency, and resistance to plaintext attacks, statistical attacks, etc.

An efficient chaos-based image encryption algorithm using real-time object detection for smart city applications

Currently, an increasing number of digital images are being generated by and for the general public and professionals. Simultaneously, the protection of these images is crucial for various emerging applications. Encryption is a popular technique for protecting image data confidentiality and privacy. There are multiple existing studies on image encryption, however, most of them are computationally overhead and do not achieve high security. Considering that time cost and security have become the most important factors, designing an efficient encryption algorithm for digital images are necessary. In this paper, a secure chaos-based image encryption algorithm for smart city applications is proposed. First, the You Only Look Once v3 object detection algorithm is used to quickly identify the objects of an image. Second, initialisation and generation of a key is performed with confusion and diffusion operations to get an encrypted object image. This algorithm for encryption has the advantage of strong security performance combined with speed, which is not available in the general encryption system. Experimental results show that this proposed algorithm outperforms other state-of-the-art schemes.

A meshfree point collocation method for elliptic interface problems

We present a meshfree generalized finite difference method for solving Poisson’s equation with a diffusion coefficient that contains jump discontinuities up to several orders of magnitude. To discretize the diffusion operator, we formulate a strong form method that uses a smearing of the discontinuity; and a conservative formulation based on locally computed Voronoi cells. Additionally, we propose a novel conservative formulation for enforcing Neumann boundary conditions that is compatible with the conservative formulation of the diffusion operator. Finally, we introduce a way to switch from the strong form to the conservative formulation to obtain a locally conservative and positivity preserving scheme. The presented numerical methods are benchmarked against four test cases of varying complexity and jump magnitude on point clouds with nodes that are not aligned to the discontinuity. Our results show that the new hybrid method that switches between the two formulations produces better results than the classical generalized finite difference approach for high jumps in diffusivity.