动力系统混沌化及其在图像加密中的应用研究

发布时间：2018-05-30 21:55

本文选题：混沌反控制 + 超混沌系统　；参考：《广东工业大学》2017年博士论文

【摘要】：混沌作为20世纪物理学里辉煌的科学奇迹之一,引起了国内外科学家们的广泛关注。混沌反控制(混沌化)是混沌理论研究的重要课题,连续时间系统反控制取得了一些相关结果,但尚未形成具有普适性的系统理论,如何设计具有多个正李氏指数的超混沌系统依然是一个研究热点课题。同时,混沌系统具有初始值高度敏感性、有界性、伪随机性等特点,使得混沌系统在保密通信和信息加密设计上有显著优势,适合于流密码设计和应用,因而如何设计基于混沌系统的密码算法也是信息安全的重要研究方向。为此,本论文进一步研究了动力系统混沌反控制方法和基于混沌系统的信息加密应用,主要工作如下:1.研究了基于QR正交分解和SVD正交分解方法的两种李氏指数(李雅普诺夫指数)计算的改进算法。离散时间系统李氏指数的特征值算法,理论上使用该算法计算李氏指数时,迭代次数k越大,计算结果越准确,并且极限也是存在的。然而,在实际的数值计算中,因计算机有限精度等原因,计算结果会出现数值溢出、无法识别、计算结果不准确等问题,可以表述为如下三个方面:(1)迭代次数k不能过大,否则仿真结果将出现NaN或Inf的错误提示;(2)随着迭代次数k增加,所有的计算结果将趋近于最大李氏指数;(3)如果迭代次数k太小,将使得计算结果不准确。最后,通过几个实例来说明改进算法的可行性和有效性。2.探讨了一些高维超混沌系统的反控制理论和设计方法。首先,给出李氏指数算法定义的理论分析,进而讨论了正李氏指数与雅可比矩阵正实部特征根之间的定性关系,即受控系统配置越多具有正实部的特征根,则受控系统具有正李氏指数的个数也越多。因而,可通过设计受控系统雅可比矩阵具有正实部的特征根个数来解决正李氏指数的配置问题。最后,给出了两个高维超混沌反控制例子。3.提出了构造具有n-2个正李氏指数n维超混沌系统(无简并超混沌系统)的平均特征值准则法。该方法由以下四个步骤组成:(1)设计一个渐近稳定的标称系统并对其作相似变换,用一致有界的控制器和控制矩阵对相似变换后的标称系统进行混沌反控制,得到一个轨道全局有界的受控系统。(2)通过控制器和控制矩阵的闭环极点配置,使得受控系统在两类鞍焦平衡点处对应的具有正实部的特征值个数分别满足n-1和n-2。(3)在控制器的一个周期内,所有雅可比矩阵所对应的平均特征值的正实部个数满足n-2。(4)平均特征值正实部的最小值大于某个给定的阈值Th。最后,给出了若干典型实例,验证该方法的可行性和有效性。4.提出了具有多个控制器的高维无简并超混沌系统构造方法。首先,设计n维耗散的线性系统,经过变换矩阵的相似变换。其次,施加主控制器,并寻找主控制器的最佳控制位置和控制参数,使得在主控制器的一个周期内,受控系统雅克比矩阵对应的平均特征值的正实部个数为n-2,且平均特征值的最小正实部满足某个给定的阈值条件。然后施加多个非主控制器,并预先给定控制参数和固定控制位置,从而生成轨道全局有界的受控系统。最后,给出了构造具有多个控制器的高维无简并超混沌系统两个例子。5.设计了一个新的三维混沌动力系统,并根据拓扑马蹄理论,利用计算机辅助证明该受控系统庞加莱截面映射存在一个不变紧集,使其拓扑半共轭于2移位动力系统。同时,基于该混沌系统设计了图像加密方案。首先,根据混沌系统生成的伪随机序列对图像像素位置进行置乱。然后,利用混沌流密码对图像像素进行多轮加密,从而达到双重加密效果。最后,给出了统计分析来说明该加密方案的安全性和可行性。6.研究了一种基于分数阶超混沌系统的图像加密算法。首先,一个已知明文图像用来对原始明文进行像素混淆。然后利用分数阶超混沌系统生成的伪随机序列对混淆后的图像进行加密。根据加密算法,得到了经过置乱和扩散后的加密图像,该加密方案具有较好的安全性和加密效果。仿真实验验证了该加密方案的可行性,并给出一些统计测试说明该方案的安全性。
[Abstract]:Chaos, as one of the brilliant scientific wonders of physics in the twentieth Century, has aroused wide attention of scientists at home and abroad. Chaos anti control (chaos) is an important subject in the study of chaos theory. The inverse control of continuous time system has obtained some related results, but it has not yet formed a universal system theory and how to design a number of positive. The hyperchaotic system of Li's exponent is still a hot topic of research. At the same time, the chaotic system has the characteristics of high sensitivity, boundedness and pseudo randomness of the initial value, which makes the chaotic system have remarkable advantages in the design of secure communication and information encryption. It is suitable for the design and application of stream ciphers, so how to design the cipher based on the chaotic system The algorithm is also an important research direction of information security. Therefore, this thesis further studies the chaotic anti control method of power system and the information encryption application based on chaotic systems. The main work is as follows: 1. the improved algorithm for calculating two kinds of Liapunov exponents (Lyapunov exponents) based on QR orthogonal decomposition and SVD orthogonal decomposition method is studied. The eigenvalue algorithm of the Li's exponent of the scattered time system, when the algorithm is used to calculate the Li's exponent, the greater the number of iterations is k, the more accurate the calculation results are, and the limit also exists. However, in the actual numerical calculation, the numerical overflow will appear because of the limited accuracy of the computer and so on, the results are inaccurate and the results are inaccurate. The following three aspects can be expressed as following three aspects: (1) the number of iterations can not be too large, otherwise the result of the simulation will appear the error of NaN or Inf; (2) as the number of iterations increases, all calculation results will be close to the maximum Li's index; (3) if the number of iterations is too small, the calculation results will be inaccurate. Finally, through several examples .2. discusses the anti control theory and design method of some high dimensional super chaotic systems. First, we give the theoretical analysis of the definition of the Li's exponent algorithm, and then discuss the qualitative relation between the positive real part of the positive lie index and the real part of the Jacobi matrix, that is, the more the controlled system has the positive real part. With the characteristic root, the number of the controlled system with the positive lie index is also more. Therefore, the number of eigenvalues of the positive real part of the Jacobi matrix of the controlled system can be solved by designing the number of eigenvalues of the positive real part of the system. Finally, two high dimensional super chaotic anti control examples.3. are given to construct n-2 positive lie index n-dimensional hyperchaos. This method consists of four steps: (1) design an asymptotically stable nominal system and make a similar transformation. A uniform bounded controller and a control matrix are used to control the nominal system after the similar transformation, and a global boundedness of the orbit is obtained. (2) through the closed loop pole configuration of the controller and the control matrix, the number of eigenvalues of the controlled system with the positive real part corresponding to the two kind of saddle coke equilibrium point satisfies n-1 and n-2. (3) in one period of the controller, and the positive real number of the average eigenvalues corresponding to all Jacobi matrices satisfies the average characteristic of the n-2. (4). The minimum value of the positive real part is greater than a given threshold Th.. A number of typical examples are given to verify the feasibility and effectiveness of the method..4. proposes a construction method for high dimensional hyperchaotic systems with multiple controllers. First, a linear system with n-dimensional dissipation is designed, and the transformation of the transformation matrix is transformed. Secondly, master control is applied. The optimal control position and control parameters of the master controller are found, so that the number of positive real parts of the average eigenvalue of the Jacobian matrix of the controlled system is n-2 within a period of the main controller, and the minimum positive real part of the average eigenvalue satisfies a given threshold bar. The control system with fixed control parameters and fixed control position is made to generate the controlled system of orbit global bounded. Finally, two examples of high dimensional non simple hyperchaos system with multiple controllers,.5., are given to design a new three-dimensional chaotic dynamic system, and the controlled system Poincare is proved by computer aided according to the topology horse's hoof theory. The cross section mapping has an invariant compact set, which makes its topology semi conjugate to the 2 shift power system. At the same time, the image encryption scheme is designed based on the chaotic system. First, the image pixel position is scrambled according to the pseudo random sequence generated by the chaotic system. Then, the image pixels are encrypted by the chaotic stream density code, thus achieving the double image pixels. Finally, a statistical analysis is given to illustrate the security and feasibility of the encryption scheme..6. research an image encryption algorithm based on fractional hyperchaos system. First, a known plaintext image is used to confuse the original plaintext in pixels. According to the encryption algorithm, the encrypted image after the scrambling and spreading is obtained. The encryption scheme has good security and encryption effect. The simulation experiment verifies the feasibility of the encryption scheme, and gives some statistical tests to illustrate the security of the scheme.
【学位授予单位】：广东工业大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：TP309.7;TP391.41

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