一种改进的高性能Lorenz系统构造及其应用∗

摘要

Lorenz system is one of the most typical chaotic system models, and it has been well studied and widely applied. In order to obtain more complex structure and dynamic behavior of chaotic attractor for Lorenz system, improving the Lorenz system has become one of the important tasks in chaotic dynamical system. Therefore, an improved Lorenz system with complex dynamic behavior is proposed and used to protect image information security. Based on the existing various improvement Lorenz systems, firstly, a new Lorenz chaotic system is constructed by increasing the control parameters and modifying nonlinear expression in the existing Lorenz chaotic system; secondly, the mathematical properties of dissipation, symmetry, and stability in the proposed Lorenz system, which are similar to those in the existing Lorenz system, Bao system, Tee system and Y system, are investigated by modern differential dynamical system, and the experimental results of Lyapunov index and random sequence correlation of five different Lorenz systems show that the proposed Lorenz system has a more complex structure and chaotic dynamic behavior;finally, the discrete pseudo random sequences generated by five different Lorenz chaotic systems are used for scrambling the pixel position and diffusing the pixel value to protect image information security. The analyses of correlation and statistic histogram entropy of adjacent pixels, anti-differential attack and key sensitivity of the encrypted image, indicate that the improved Lorenz system proposed in this paper has much better potential advantages than other existing improved Lorenz system in image encryption application.%Lorenz系统是一种最具有代表性、典型性的混沌模型之一，一直被众多学者深入研究和广泛应用。为了获取结构和动力学行为更为复杂的混沌吸引子，不断改善Lorenz系统已成为混沌动力系统研究中的重要课题之一。为此，本文提出了一个具有复杂系统动力学行为的改进的Lorenz系统，并将其用于图像信息安全保护。在现有各种改进的Lorenz系统的基础上，首先通过增加Lorenz系统的控制参数和改变非线性项相结合的方法构造出一种新的Lorenz混沌系统；其次采用微分动力系统方法深入研究该系统并获得与Lorenz系统、Bao系统、Tee系统和Y系统等具有相似的耗散性、对称性、稳定性，以及更加复杂的混沌特性和动力学行为，同时分析该系统所产生随机序列具有良好的相关性和复杂性；最后将其所产生的离散伪随机序列用于图像置乱和扩散加密，通过对图像加密结果的相邻像素相关性分析、灰度空间相关特性不确定性分析、抗差分攻击以及密钥敏感性测试，表明本文所构造的改进的Lorenz系统应用于图像加密能获得相对较高的安全性。