Optical image encryption using improper Hartley transforms and chaos

We propose a new method for image encryption using improper Hartley transform and chaos theory. Improper Hartley transform is a Hartley transform in which the phase between the two Fourier transforms is a fractional multiple of π/2. This fractional order is called fractional parameter and serves as a key in the image encryption and decryption process. Four types of chaos functions have been used. These functions are the logistic map, the tent map, the Kaplan-Yorke map and the Ikeda map. Random intensity masks have been generated using these chaotic functions and are called chaotic random intensity masks. The image is encrypted by using improper Hartley transform and two chaotic random intensity masks. The mean square error has been calculated. The robustness of the proposed technique in terms of blind decryption has been tested. The computer simulations are presented to verify the validity of the proposed technique.     

Chaos based multiple image encryption using multiple canonical transforms

We propose a new method for multiple image encryption using linear canonical transforms and chaotic maps. Three linear canonical transforms and three chaotic maps are used in the proposed technique. The three linear canonical transforms that have been used are the fractional Fourier transform, the extended fractional Fourier transform and the Fresnel transform. The three chaotic maps that have been used are the tent map, the Kaplan-Yorke map and the Ikeda map. These chaotic maps are used to generate the random phase masks and these random phase masks are known as chaotic random phase masks. The mean square error and the signal to noise ratio have been calculated. Robustness of the proposed technique to blind decryption has been evaluated. Optical implementation of the technique has been proposed. Experimental and simulations results are presented to verify the validity of the proposed technique.     

Bifurcations in globally coupled map lattices

The dynamics of globally coupled map lattices can be described in terms of a nonlinear Frobenius-Perron equation in the limit of large system size. This approach allows for an analytical computation of stationary states and their stability. The bifurcation behavior of coupled tent maps near the chaotic band merging point is presented. Furthermore, the time-independent states of coupled logistic equations are analyzed. The bifurcation diagram of the uncoupled map carries over to the map lattice. The analytical results are supplemented with numerical simulations     

Classification of one-dimensional chaotic models

Classifying chaotic maps using the relation between a map and its conjugate basic map with uniform invariant distribution is suggested. It is shown that every symmetric one-dimensional chaotic map with two monotonic branches is topologically equivalent to a tent map or a Bernoulli shift. An algorithm for finding a function conjugating two maps is formulated.     

A novel image encryption method based on total shuffling scheme

In this paper, a novel image encryption method based on skew tent chaotic map and permutation-diffusion architecture is proposed. In the proposed method, the P-box is chosen as the same size of plain-image, which shuffles the positions of pixels totally. The keystream generated by skew tent chaotic map is related to the plain-image. Statistical analysis, information entropy analysis, and sensitivity analysis to plaintext and key on the proposed scheme are provided in this paper. It can be seen that this algorithm is efficient and reliable, with high potential to be adopted for network security and secure communications.     

一类可以产生独立同分布密钥流的混沌系统

构造了一类与帐篷映射拓扑同构的混沌系统, 并根据拓扑共轭变换关系给出了此类混沌系统产生独立、均匀分布密钥流序列的采样规则. 理论证明和数值模拟, 均验证了结论的有效性. 本文为产生独立同分布密钥流提供了更多的非线性系统选择. 实验结果证明利用本文定理产生的密钥流能够通过美国信息技术管理改革法案的随机数检测标准(FIPS PUB 140-2)和美国国家标准与技术研究院安全检测标准(NIST SP800-22), 符合密钥流的选取标准. 关键词： 独立同分布 混沌系统 帐篷映射 拓扑同构     

Investigation of classical and fractional Bose–Einstein condensation for harmonic potential

In this study, classical and fractional Gross–Pitaevskii (GP) equations were solved for harmonic potential and repulsive interactions between the boson particles using the Homotopy Perturbation Method (HPM) to investigate the ground state dynamics of Bose–Einstein Condensation (BEC). The purpose of writing fractional GP equations is to consider the system in a more realistic manner. The memory effects of non-Markovian processes involving long-range interactions between bosons with the restriction of the ergodic hypothesis and the effect of non-Gaussian distributions of bosons in the condensation can be taken into account with time fractional and space fractional GP equations, respectively. The obtained results of the fractional GP equations differ from the results of the classical one. While the Gauss distribution describing the homogeneous, reversible and unitary system is obtained from the classical GP equation, the probability density of the solution function of fractional GP equations is non-conserved. This situation describes the inhomogeneous, irreversible and non-unitary systems.     

A Novel Method for the Initial-Condition Estimation of a Tent Map

Based on the connection between the tent map and the saw tooth map or Bernoulli map, a novel method for the initial-condition estimation of the tent map is presented. In the method, firstly the symbolic sequence generated from the tent map is converted to the forms obtained from the saw tooth map and Bernoulli map, and then the relationship between the symbolic sequence and the initial condition of the tent map can be obtained from the initial-condition estimation equations, which can be easily obtained, hence the estimation of the tent map can be achieved finally. The method is computationally simple and the error of the estimator is less than 1/2N. The method is verified by software simulation.     

Lyapunov exponents for multi-parameter tent and logistic maps

The behaviour of logistic and tent maps is studied in cases where the control parameter is dependent on iteration number. Analytic results for global Lyapunov exponent are presented in the case of the tent map and numerical results are presented in the case of the logistic map. In the case of a tent map with N control parameters, the fraction of parameter space for which the global Lyapunov exponent is positive is calculated. The case of bi-parameter maps of period N are investigated.     

Optical image encryption using fractional Fourier transform and chaos

We propose a new method for image encryption using fractional Fourier transform and chaos theory. Random phase masks are generated using iterative chaos functions. The input image is combined with the first random phase mask at the object plane and is then transformed using the fractional Fourier transform. After the first fractional Fourier transform, the second random phase mask, again generated by using the chaos functions, is used at the fractional plane. The second fractional Fourier transform operation is then carried out to obtain the encrypted image. Three types of chaos functions have been used: the logistic map, the tent map and the Kaplan–Yorke map. The mean square error and the signal-to-noise ratio between the decrypted image and the input image for the correct order and the incorrect order of the fractional Fourier transform have been calculated. The computer simulations are presented to verify the validity of the proposed technique.     

Double image encryption based on discrete multiple-parameter fractional Fourier transform and chaotic maps

A double image encryption method is proposed by utilizing discrete multiple-parameter fractional Fourier transform and chaotic maps. One of the two original images scrambled by one chaotic map is encoded into the amplitude of a complex signal with the other original image as its phase. The complex signal multiplied by another chaotic random phase mask is then encrypted by discrete multiple-parameter fractional Fourier transform. The parameters in chaotic map and discrete multiple-parameter fractional Fourier transform serve as the keys of this encryption scheme. Numerical simulations have been done to demonstrate the performance of this algorithm.     

Fractional standard map

Properties of the phase space of the standard map with memory are investigated. This map was obtained from a kicked fractional differential equation. Depending on the value of the map parameter and the fractional order of the derivative in the original differential equation, this nonlinear dynamical system demonstrates attractors (fixed points, stable periodic trajectories, slow converging and slow diverging trajectories, ballistic trajectories, and fractal-like structures) and/or chaotic trajectories. At least one type of fractal-like sticky attractors in the chaotic sea was observed.     

Digital image watermarking using gyrator transform and chaotic maps

We propose a new method for digital image watermarking using gyrator transform and chaotic maps. Four chaotic maps have been used in the proposed technique. The four chaotic maps that have been used are the logistic map, the tent map, the Kaplan-Yorke map and the Ikeda map. These chaotic maps are used to generate the random phase masks and these random phase masks are known as chaotic random phase masks. A new technique has been proposed to generate the single chaotic random phase mask by using two chaotic maps together with different seed values. The watermark encoding method in the proposed technique is based on the double random phase encoding method. The gyrator transform and two chaotic random phase masks are used to encode the input image. The mean square error, the peak signal-to-noise ratio and the bit error rate have been calculated. Robustness of the proposed technique has been evaluated in terms of the chaotic maps, the number of the chaotic maps used to generate the CRPM, the rotation angle of the gyrator transform and the seed values of the chaotic random phase masks. Optical implementation of the technique has been proposed. The computer simulations are presented to verify the validity of the proposed technique.     

Detecting chaos in fractional-order nonlinear systems using the smaller alignment index

The identification between chaos and ordered states in fractional-order chaotic systems is a challenge as well as a hot topic due to the complex of fractional calculus. In this paper, the smaller alignment index (SALI) is developed to detect chaos in the fractional-order chaotic systems by introducing the fractional-order tangent systems. Numerical simulations are carried out based on the fractional-order simplified Lorenz system and the fractional-order Hénon map, which are continuous chaotic system and discrete chaotic system, respectively. It shows that the proposed method is effective for distinguishing chaos and order in different kinds of fractional-order chaotic systems.     

Gyrator transform-based optical image encryption,using chaos

We propose a new method for image encryption, using gyrator transform and chaos theory. Random phase masks are generated using chaos functions and are called as chaotic random phase masks. In the proposed technique, the image is encrypted using gyrator transform and two chaotic random phase masks. Three types of chaos functions have been used to generate the chaotic random phase masks. These chaos functions are the logistic map, the tent map and the Kaplan-Yorke map. The computer simulations are presented to verify the validity of the proposed technique. The mean square errors have been calculated. The robustness of the proposed technique to blind decryption in terms of rotation angle and the seed values of the chaotic random phase mask have been evaluated. The optical implementation of the encryption and the decryption technique has been proposed.     

离散分数阶随机变换与加权直方图交叉置乱的双图像加密算法

为了实现对两幅图像进行同步加密,降低传输负载并提高密文的抗明文攻击能力,提出了离散分数阶随机变换与加权像素混沌置乱的双图像加密算法。将2个分阶参数引入到Tent映射中,设计了新的Tent映射;根据明文像素值,构建加权像素直方图模型,联合位外部密钥,生成改进的Tent映射的初值;再利用初值对分数阶Tent映射进行迭代,输出2组随机序列,对2幅明文进行位置交叉混淆,获取2个置乱密文;基于DWT(discrete wavelet transform)技术,对2个置乱密文进行稀疏表示;根据混沌序列,定义随机循环矩阵,联合稀疏表示,获取2个置乱密文对应的测量矩阵。根据随机掩码与调制相位掩码,建立数据融合模型,将2个测量矩阵组合为复合矩阵;基于离散分数阶随机变换,对复合图像进行扩散,获取密文。测试数据显示:与已有的多图像加密方案相比,该算法的抗明文攻击能力与用户响应值更理想,密文的NPCR、UACI值分别达到了99.83%、34.57%。该算法具有较高的加密安全性,能够有效抵御网络中的外来攻击,确保图像安全传输。     

Discretizing constant curvature surfaces via loop group factorizations: The discrete sine- and sinh-Gordon equations

The sine- and sinh-Gordon equations are the harmonic map equations for maps of the (Lorentz) plane into the 2-sphere. Geometrically they correspond to the integrability equations for surfaces of constant Gauss and constant mean curvature. There is a well-known dressing action of a loop group on the space of harmonic maps. By discretizing the vacuum solutions we obtain via the dressing action completely integrable discretizations (in both variables) of the sine- and sinh-Gordon equations. For the sine-Gordon equation we get Hirota's discretization. Since we work in a geometric context we also obtain discrete models for harmonic maps into the 2-sphere and discrete models of constant Gauss and mean curvature surfaces.     

级联混沌及其动力学特性研究

初值敏感性是混沌的本质,混沌的随机性来源于其对初始条件的高度敏感性,而Lyapunov指数又是这种初值敏感性的一种度量.本文的研究发现,混沌系统的级联可明显提高级联混沌的Lyapunov指数,改善其动力学特性.因此,本文研究了混沌系统的级联和级联混沌对动力学特性的影响,提出了混沌系统级联的定义及条件,从理论上证明了级联混沌的Lyapunov指数为各个级联子系统Lyapunov指数之和;适当的级联可增加系统参数、扩展混沌映射和满映射的参数区间,由此可提高混沌映射的初值敏感性和混沌伪随机序列的安全性.以Logistic映射、Cubic映射和Tent映射为例,研究了Logistic-Logistic级联、Logistic-Cubic级联和Logistic-Tent级联的动力学特性,验证了级联混沌动力学性能的改善.级联混沌可作为伪随机数发生器的随机信号源,用以产生初值敏感性更高、安全性更好的伪随机序列.     

A finite difference method with non-uniform timesteps for fractional diffusion and diffusion-wave equations

An implicit finite difference method with non-uniform timesteps for solving fractional diffusion and diffusion-wave equations in the Caputo form is presented. The non-uniformity of the timesteps allows one to adapt their size to the behaviour of the solution, which leads to large reductions in the computational time required to obtain the numerical solution without loss of accuracy. The stability of the method has been proved recently for the case of diffusion equations; for diffusion-wave equations its stability, although not proven, has been checked through extensive numerical calculations.     

混沌背景下信号的盲分离

混沌信号与确定性小信号叠加生成的混合信号是一更高维的混沌信号,因而不能用一般的混沌信号噪声抑制的方法进行分离.提出了一种这类信号盲分离的方法.在重构未知的混沌信号的动力方程时,充分利用混沌吸引子的几何特性,并且限定动力映射为原混沌吸引子所在流形的内部映射,从而保证了重构的动力系统方程对应于原混沌信号,而不是同样具有混沌特性的混合信号.然后利用重构的动力方程,借用混沌信号中的噪声抑制思想,估计出原混沌信号对应的轨道,实现信号分离.通过对Lorenz系统中谐波信号、Henon映象中自回归过程,以及脑电信号中谐波信号进行提取的数值实验,验证了信号盲分离方法的有效性和可行性. 关键词： 混沌 非线性 信号处理 盲分离