Diffusion Operator Research Articles

An Image Encryption Algorithm Based on an Epidemic Spreading Model

An intrinsic similarity was noted between epidemic spreading models and diffusion operations in cryptography. In this paper, a new image encryption algorithm is proposed based on an epidemic spreading model. The notable feature of the model is the evolution of the network node state. We utilize the evolution of an epidemic spreading model, namely the susceptible-infectious-recovered (SIR) model, to drive the encryption progress by abstracting pixels to the network nodes. In the new cryptosystem, the pixels in plain-text are considered as susceptible individuals, permutation and diffusion are seamlessly integrated rather than having two independent stages, and the encryptions are performed in a two-dimensional and nonlinear manner. Simulation results demonstrate its high encryption speed and excellent encryption security.

Multi Chaotic System to Generate Novel S-Box for Image Encryption

A novel method on the basis of multi chaos theory is suggested in the presented study. Also, the study used two different dimensions to generate S-Box to get a strong cipher that is difficult to break. The suggested image cryptosystem includes an identical (decryption and encryption) process, which involves a single keystream generator, shifting process (based on 3D Lorenz map) related diffusion operations, and generate S-Box (based on 2D Henon map) that related confusion operation. The comparative analysis and the simulate test show that the suggested image cryptosystem has a few properties, like high-sensitivity, fast encryption/decryption, large keyspace, excellent statistical properties related to the ciphertext, and so on. The suggested cryptosystem is considered as an alternative for practical secure communications.

Essential m-dissipativity for Possibly Degenerate Generators of Infinite-dimensional Diffusion Processes

First essential m-dissipativity of an infinite-dimensional Ornstein-Uhlenbeck operator N, perturbed by the gradient of a potential, on a domain mathcal {F}C_b^{infty } of finitely based, smooth and bounded functions, is shown. Our considerations allow unbounded diffusion operators as coefficients. We derive corresponding second order regularity estimates for solutions f of the Kolmogorov equation alpha f-Nf=g, alpha in (0,infty ), generalizing some results of Da Prato and Lunardi. Second, we prove essential m-dissipativity for generators (L_{Phi },mathcal {F}C_b^{infty }) of infinite-dimensional degenerate diffusion processes. We emphasize that the essential m-dissipativity of (L_{Phi },mathcal {F}C_b^{infty }) is useful to apply general resolvent methods developed by Beznea, Boboc and Röckner, in order to construct martingale/weak solutions to infinite-dimensional non-linear degenerate stochastic differential equations. Furthermore, the essential m-dissipativity of (L_{Phi },mathcal {F}C_b^{infty }) and (N,mathcal {F}C_b^{infty }), as well as the regularity estimates are essential to apply the general abstract Hilbert space hypocoercivity method from Dolbeault, Mouhot, Schmeiser and Grothaus, Stilgenbauer, respectively, to the corresponding diffusions.

A multi-splitting method to solve 2D parabolic reaction–diffusion singularly perturbed systems

In this paper we design and analyze a numerical method to solve a type of reaction–diffusion 2D parabolic singularly perturbed systems. The method combines the central finite difference scheme on an appropriate piecewise uniform mesh of Shishkin type to discretize in space, and the fractional implicit Euler method together with a splitting by directions and components of the reaction–diffusion operator to integrate in time. We prove that the method is uniformly convergent of first order in time and almost second order in space. The use of this time integration technique has the advantage that only tridiagonal linear systems must be solved to obtain the numerical solution at each time step; because of this, our method provides a remarkable reduction of computational cost, in comparison with other implicit methods which have been previously proposed for the same type of problems. Full details of the uniform convergence are given only for systems with two equations; nevertheless, our ideas can be easily extended to systems with an arbitrary number of equations as it is shown in the numerical experiences performed. The numerical results show in practice the qualities of our proposal.

A double scrambling-DNA row and column closed loop image encryption algorithm based on chaotic system.

In this paper, a dynamic update algorithm of double scrambling-DNA row and column closed loop based on chaotic system is proposed. The classical scrambling and diffusion structure are used in the whole process. In the scrambling stage, a new pixel reconstruction method is proposed by combining the Hilbert curve with Knuth-Durstenfeld shuffle algorithm to overcome the shortcoming of nearby storage of Hilbert curve. This method reconstructs the pixel matrix of one-dimensional vector according to the Hilbert curve coding method, and achieves good scrambling effect, while reducing its time complexity and space complexity. In the diffusion stage, combining the plaintext row, the ciphertext row and the key row, and taking advantage of the parallel computing power and high storage density of the DNA encoding, the existing block diffusion operation is improved, and the two-round diffusion of the DNA encoding is proposed. When the last line of ciphertext is generated, the first line of ciphertext is updated and the closed-loop dynamic update of the encryption system is realized. Finally, SHA-256 is used to give the secret key and calculate the initial value of the chaotic system. The simulation results show that the “double scrambling-DNA row and column closed loop dynamic” update algorithm proposed in this paper can effectively improve the efficiency of information transmission and have high security.

A novel multiple grayscale image encryption method based on 3D bit-scrambling and diffusion

Since a large number of digital images are shared widely in our daily life, the demand and need for multiple-image encryption algorithms are increasing to ensure the security of the shared images. To present a new perspective to this request, a secure multiple-image encryption method based on 3D bit-scrambling and diffusion operations is proposed for grayscale images in this paper. A three-dimensional bit array is created by extending the eight bit planes of each grayscale image to a third dimension. This 3D bit array is pseudo-randomly scrambled by changing the order of the rows, columns, and layers. The scrambled bit planes are then used to reconstruct the shuffled images. A controlled diffusion operation is also applied to the shuffled and merged images. Both scrambling and diffusion parameters are obtained separately by iterating the chaotic Henon map twice. The initial values of the Henon map are plain images dependent. Several performance analyses including key space and sensitivity, histogram, correlation, entropy, differential, data loss, noise, and running time are carried out to test the proposed method’s resistance against various cryptanalytic attacks. Simulation results show that the proposed method can encrypt multiple images effectively and securely.

Secret image encryption based on chaotic system and elliptic curve cryptography

The advent of internet has been a boon to society. It has helped in information retrieval and is a great help in communication. Nowadays, communication comprises digital information in images, videos, GIFs, and many other formats. Amongst the digital data, images are one of the most transmitted data. Therefore, the security of image data has gained the attention of researchers. This paper addresses the drawbacks of Abdelfatah's method, a block-based image encryption scheme based on a chaotic system and elliptic curve cryptography. One of the issues is caused when multiple pixels are grouped to form large integers, which turns out to be more than the modulo prime of the elliptic curve leading to incorrect decryption. Another issue is to transfer extra information for the number of pixels that are grouped. In order to overcome, an algorithm is proposed that uses the largest 512 bits elliptic curve prime modulo along with confusion and diffusion operation using a chaotic system. Security and statistical analysis are conducted to substantiate the strength of the proposed enciphering method.

Gaussian Amplitude Amplification for Quantum Pathfinding.

We study an oracle operation, along with its circuit design, which combined with the Grover diffusion operator boosts the probability of finding the minimum or maximum solutions on a weighted directed graph. We focus on the geometry of sequentially connected bipartite graphs, which naturally gives rise to solution spaces describable by Gaussian distributions. We then demonstrate how an oracle that encodes these distributions can be used to solve for the optimal path via amplitude amplification. And finally, we explore the degree to which this algorithm is capable of solving cases that are generated using randomized weights, as well as a theoretical application for solving the Traveling Salesman problem.

A Simple and Fast Medical Image Encryption System Using Chaos-Based Shifting Techniques

Medical images such as brain MRI, ultrasound, and X-ray images are currently used by doctors to treat many serious diseases in most hospitals and clinics. These images play a major role in medical diagnosis. Diagnostic images with sensitive patient data are recorded and exchanged owing to the advancement of public networks. As a result, maintaining the privacy of sensitive patient data is becoming increasingly important. In medical imaging, the pixels are highly correlated. To reduce pixel correlation, a variety of complex encryption methods have been used in recent years, increasing the algorithm's computational complexity. This paper proposes an image encryption method based on chaos-based shifting operations. Security, convenience, and speed are all advantages of shifting activities. This method initially executes bit-plane flipping and permutation operations, then block-based flipping and diffusion operations to obtain the cipher images. The simulation and security analysis findings show that the proposed approach performs better.

A Color Image Encryption Algorithm Based on Double Fractional Order Chaotic Neural Network and Convolution Operation.

A color image encryption algorithm based on double fractional order chaotic neural network (CNN), interlaced dynamic deoxyribonucleic acid (DNA) encoding and decoding, zigzag confusion, bidirectional bit-level diffusion and convolution operation is proposed. Firstly, two fractional order chaotic neural networks (CNNs) are proposed to explore the application of fractional order CNN in image encryption. Meanwhile, spectral entropy (SE) algorithm shows that the sequence generated by the proposed fractional order CNNs has better randomness. Secondly, a DNA encoding and decoding encryption scheme with evolutionary characteristics is adopted. In addition, convolution operation is utilized to improve the key sensitivity. Finally, simulation results and security analysis illustrate that the proposed algorithm has high security performance and can withstand classical cryptanalysis attacks.

On the existence of global‐in‐time weak solutions and scaling laws for Kolmogorov's two‐equation model for turbulence

AbstractThis paper is concerned with Kolmogorov's two‐equation model for turbulence in involving the mean velocity u, the pressure p, an average frequency , and a mean turbulent kinetic energy k. We consider the system with space‐periodic boundary conditions in a cube , which is a good choice for studying the decay of free turbulent motion sufficiently far away from boundaries. In particular, this choice is compatible with the rich set of similarity transformations for turbulence. The main part of this work consists in proving existence of global weak solutions of this model. For this we approximate the system by adding a suitable regularizing r‐Laplacian and invoke existence result for evolutionary equations with pseudo‐monotone operators. An important point constitutes the derivation of pointwise a priori estimates for ω (upper and lower) and k (only lower) that are independent of the box size a, thus allow us to control the parabolicity of the diffusion operators.

Image Encryption using MVK Algorithm, El-Gamal and Chaotic Systems

Digital images are often transmitted in this modern era through various networks. Digital images are used in various fields like Medical, Military, etc. These images require high-security levels to prevent misusing and unauthorized access. There are several Image Encryption Techniques which are used to achieve image confidentiality; this paper presents a newer approach for Image Encryption which consists of 3 stages. MVK Algorithm is the first stage of encryption which is used to construct a permuted image and demolish the relationships among the adjacent pixels. El- Gamal Encryption Algorithm is the second stage of encryption which modifies permuted image; it is an alternative for RSA public key cipher. Third stage of encryption uses Lorenz system and Rossler system to achieve confusion and diffusion operations. To evaluate the effectiveness and security of the suggested strategy, a thorough analysis has been done. Visual and numerical results show that the proposed image cryptosystem can defend against a number of well-known attacks. Key Words: El-Gamal, Lorenz, Rossler.

JKO estimates in linear and non-linear Fokker–Planck equations, and Keller–Segel: $L^p$ and Sobolev bounds

We analyze some parabolic PDEs with different drift terms which are gradient flows in the Wasserstein space and consider the corresponding discrete-in-time JKO scheme. We prove with optimal transport techniques how to control the L^p and L^\infty norms of the iterated solutions in terms of the previous norms, essentially recovering well-known results obtained on the continuous-in-time equations. Then we pass to higher-order results, and in particular to some specific BV and Sobolev estimates, where the JKO scheme together with the so-called “five gradients inequality” allows us to recover some estimates that can be deduced from the Bakry–Émery theory for diffusion operators, but also to obtain some novel ones, in particular for the Keller–Segel chemotaxis model.

Positivity-preserving entropy stable schemes for the 1-D compressible Navier-Stokes equations: High-order flux limiting

This is the second of two companion papers, in which we develop and analyze a new class of positivity-preserving, entropy stable spectral collocation schemes of arbitrary order of accuracy for the 1-D compressible Navier-Stokes equations. The key distinctive property of the proposed methodology is that it is proven to guarantee the pointwise positivity of thermodynamic variables for compressible viscous flows. The new schemes are constructed by combining a positivity-violating entropy stable method of arbitrary order of accuracy and a novel first-order positivity-preserving entropy stable method developed in the companion paper [1]. This first-order scheme is discretized on the same Legendre-Gauss-Lobatto points used for the high-order counterpart. As a result, no interpolation or restriction is required between high- and low-order elements which are coupled by using the simultaneous approximation term (SAT) penalty method. To provide positivity preservation and excellent discontinuity-capturing properties, the Navier-Stokes equations are regularized by adding artificial dissipation in the form of the Brenner-Navier-Stokes diffusion operator. The high- and low-order schemes are combined by using a limiting procedure, so that the resultant scheme provides conservation, guarantees pointwise positivity of thermodynamic variables, preserves the design–order of accuracy of the corresponding high-order baseline spectral collocation method for smooth solutions, and satisfies the discrete entropy inequality, thus facilitating a rigorous L2-stability proof for the symmetric form of the discretized Navier-Stokes equations. The proposed methodology is very general and can be extended to multiple dimensions and other summation-by-parts SAT-type schemes.

Positivity-preserving entropy stable schemes for the 1-D compressible Navier-Stokes equations: First-order approximation

This is the first of two companion papers that introduce a new class of positivity-preserving, entropy stable schemes for the 1-D compressible Navier-Stokes equations. In this paper, we develop and analyze a novel first–order finite volume scheme that is discretized on Legendre-Gauss-Lobatto grids used for high-order spectral collocation methods. The scheme is constructed by regularizing the Navier-Stokes equations by adding artificial dissipation in the form of the Brenner-Navier-Stokes diffusion operator. The key distinctive property of the proposed scheme is that it is proven to satisfy the discrete entropy inequality and to guarantee the pointwise positivity of density, temperature, pressure, and internal energy for 1–D compressible viscous flows, while using high–order spectral collocation summation-by-parts operators for discretizing the Navier–Stokes viscous terms. To eliminate time step stiffness caused by the high-order approximation of the viscous terms, we develop two new limiters that bound the magnitude of velocity and temperature gradients and preserve the entropy stability and positivity properties of the baseline scheme. Numerical results demonstrating positivity-preserving and discontinuity-capturing capabilities of the new scheme are presented for viscous and inviscid flows with nearly vacuum regions and very strong shocks and contact discontinuities.

Multi-Image Encryption Algorithm for 2D and 3D Images Based on Chaotic System

In order to accommodate multiple types of image encryption, a multi-image encryption algorithm for 2D and 3D images is designed in this paper. After recording the type and number of images, the pixels/coordinates of multiple images are stored in a cube block and are subjected to confusion and diffusion operations. The confusion step uses the random length sequence position swapping method to swap a row (column) vector of variable length with another row (column) vector of the same length at a random position. The diffusion is done by Exclusive OR to combine pixels/coordinates at different locations with different chaotic matrices. Finally, the cipher images are output one by one. Experimental simulations and security analysis verify the effectiveness and security of the algorithm.

The Maximization of the First Eigenvalue for a Two-Phase Material

We study the optimal arrangement of two conductive materials in order to maximize the first eigenvalue of the corresponding diffusion operator with Dirichlet conditions. The amount of the highest conductive composite is assumed to be limited. Since this type of problems has no solution in general, we work with a relaxed formulation. We show that the problem has some good concavity properties which allow us to get some uniqueness results. By proving that it is related to the minimization of the energy for a two-phase material studied in [4] we also obtain some smoothness results. As a consequence, we show that the unrelaxed problem has never solution. The paper is completed with some numerical results corresponding to the convergence of the gradient algorithm.

Image encryption scheme based on blind signature and an improved Lorenz system

In this paper, a new improved Lorenz system (ImproLS) is designed with a better behavior due to a larger positive Lyapunov exponent. Then, an asymmetric image encryption scheme is presented based on blind signature and the ImproLS. First, the sender uses the public key from the receiver to produce the cipher keys in conjunction with the Rivest–Shamir–Adleman (RSA) algorithm. Then, a new mathematical model is established to convert both the plain keys and the cipher keys to generate the initial values for ImproLS. After doing preprocess to the secret plain image, its statistical characteristics is changed greatly. Then, by employing permutation and diffusion operations, the cipher image can be obtained and sent to the receiver and the signer. Second, the signer encrypts the digital signature image using Arnold transform in conjunction with the RSA algorithm, and applies the discrete wavelet transform (DWT) to the cipher image. The scrambled signature image is then embedded into the wavelet domain of cipher image blindly. Then, the cipher image containing signature is sent to the receiver for authentication. Third, the receiver uses the private key to check the signature and recover the cipher image. After comparing the decrypted images from both the sender and the signer, the receiver can judge the source of the secret plain image. Experimental results show that the information entropy is high and closer to the theoretical value. In addition, both unsigned and signed cipher images can effectively resist salt and pepper attack, and clipping attack.

On the convergence of the solutions to the integral SPH heat and advection–diffusion equations: Theoretical analysis and numerical verification

The convergence of the continuous Smoothed Particle Hydrodynamics (SPH) solution to the advection–diffusion equation is established in this paper. As a by-product of this result, the convergence of the SPH heat equation is also established. The most prevalent SPH approximations to diffusion operators are considered. The convergence is demonstrated using Fourier Analysis. First, the Fourier representation of the SPH approximation to the Hessian matrix is obtained. Next, the Fourier representations of the SPH diffusion operators are derived. Finally, the convergence of the SPH solution to the advection–diffusion equation (which includes, in particular, the case of the heat equation) is established in the Fourier space. The proof of convergence requires a certain condition over the Fourier transform of the SPH kernel to be satisfied. An example of kernel not satisfying this condition is numerically tested, leading to non-convergent solutions. To the authors’ knowledge, this is the first proof of convergence of an SPH equation involving the Laplacian operator. These analytical results are illustrated with a variety of numerical verification cases. In particular, the influence of the choice of the kernel on the nature of convergence is thoroughly studied.

A probabilistic framework for particle-based reaction–diffusion dynamics using classical Fock space representations

The modeling and simulation of stochastic reaction–diffusion processes is a topic of steady interest that is approached with a wide range of methods. At the level of particle-resolved descriptions, where chemical reactions are coupled to the spatial diffusion of individual particles, there exist comprehensive numerical simulation schemes, while the corresponding mathematical formalization is relatively underdeveloped. The aim of this paper is to provide a framework to systematically formulate the probabilistic evolution equation, termed chemical diffusion master equation (CDME), that governs particle-based stochastic reaction–diffusion processes. To account for the non-conserved and unbounded particle number of this type of open systems, we employ a classical analogue of the quantum mechanical Fock space that contains the symmetrized probability densities of the many-particle configurations in space. Following field-theoretical ideas of second quantization, we introduce creation and annihilation operators that act on single-particle densities and provide natural representations of symmetrized probability densities as well as of reaction and diffusion operators. These operators allow us to consistently and systematically formulate the CDME for arbitrary reaction schemes. The resulting form of the CDME further serves as the foundation to derive more coarse-grained descriptions of reaction–diffusion dynamics. In this regard, we show that a discretization of the evolution equation by projection onto a Fock subspace generated by a finite set of single-particle densities leads to a generalized form of the well-known reaction–diffusion master equation, which supports non-local reactions between grid cells and which converges properly in the continuum limit.