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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.
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关键词:
- Lorenz系统 /
- Lyapunov指数 /
- 图像加密 /
- 相关性分析
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.-
Keywords:
- Lorenz system /
- Lyapunov index /
- image encryption /
- correlation analysis
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[2] Pareek N K,Patidar V,Sud K K 2006 Image Vision Comput. 24 9
[3] Rhouma R, Belghith S 2008 Phys. Lett. A 372 38
[4] Singh N, Sinha A 2009 Opt. Commun. 282 6
[5] Cui L, Ou Q L, Zhang H Q 2010 Commun. Technol. 5 43 (in Chinese) [崔力, 欧青立, 张红强 2010 通信技术 5 43]
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[7] L J H, Chen G R 2002 Int. J. Bifur. Chaos 12 659
[8] Qi G Y, Chen G R, Du S Z 2005 Physica A 352 295
[9] Elwakil A S, Özoğuz S, Kennedy P M 2002 IEEE Trans. Circ. Syst. 49 4
[10] Panchev S, Spassova T, Vitanov N K 2007 Chaos, Soliton. Fract. 33 5
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[15] Tee L S, Salleh Z 2013 J. Math. 2013 820946
[16] Li C B, Wang H K, Huang X M 2009 Microcomput. Inform. 25 8 (in Chinese) [李春彪, 王翰康, 黄新民 2009 微计算机信息 25 8]
[17] Pan S Q, Liu Y J, Zhu C X 2013 Comput. Engin. Appl. 49 7 (in Chinese) [庞寿全, 刘永建, 朱从旭 2013 计算机工程与应用 49 7]
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[23] Luo X W, Wang C H, Wan Zhao 2014 Nonlinear Anal. Model. Control 19 2
[24] Li C G, Chen G R 2005 Chaos, Soliton. Fract. 5 23
[25] Liu Y J 2010 Ph. D. Dissertation (Guangzhou: South China University of Teehnology) (in Chinese) [ 刘永建 2010 博士学位论文(广州: 华南理工大学)]
[26] Grassberger P, Procaccia I 1983 Physica D 9 189
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[29] Guo J B, Xu X Z, Shi Q H, Hu T H 2013 Acta Phys. Sin. 62 110508 (in Chinese) [郭静波, 徐新智, 史启航, 胡铁华 2013 62 110508]
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[31] Chen X J, Li Z, Bai B M, Pan W, Chen Q H 2011 J. Electron. Inform. Technol. 33 5 (in Chinese) [陈小军, 李赞, 白宝明, 潘玮, 陈清华 2011 电子与信息学报 33 5]
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[1] Lorenz E N 1963 J. Atmos. Sci. 20 113
[2] Pareek N K,Patidar V,Sud K K 2006 Image Vision Comput. 24 9
[3] Rhouma R, Belghith S 2008 Phys. Lett. A 372 38
[4] Singh N, Sinha A 2009 Opt. Commun. 282 6
[5] Cui L, Ou Q L, Zhang H Q 2010 Commun. Technol. 5 43 (in Chinese) [崔力, 欧青立, 张红强 2010 通信技术 5 43]
[6] Chen G R, Ueta T 1999 Int. J. Bifur. Chaos 9 1465
[7] L J H, Chen G R 2002 Int. J. Bifur. Chaos 12 659
[8] Qi G Y, Chen G R, Du S Z 2005 Physica A 352 295
[9] Elwakil A S, Özoğuz S, Kennedy P M 2002 IEEE Trans. Circ. Syst. 49 4
[10] Panchev S, Spassova T, Vitanov N K 2007 Chaos, Soliton. Fract. 33 5
[11] Wang Z, Mao P W 2008 J. Dyn. Contr. 6 16 (in Chinese) [王震, 毛鹏伟 2008 动力学与控制学报 6 16]
[12] Zhang Z Z, Shan Z P, Liu Y M 2013 Appl. Math. Mech. 34 12 (in Chinese) [张转周, 陕振沛, 刘衍民 2013 应用数学和力学 34 12]
[13] Huang G Q, Zhou Y Z 2013 J. Inform. Comput. Sci. 10 4763
[14] El-Sayed A M A, Salman S M 2013 Electon. J. Math. Anal. Appl. 1 2
[15] Tee L S, Salleh Z 2013 J. Math. 2013 820946
[16] Li C B, Wang H K, Huang X M 2009 Microcomput. Inform. 25 8 (in Chinese) [李春彪, 王翰康, 黄新民 2009 微计算机信息 25 8]
[17] Pan S Q, Liu Y J, Zhu C X 2013 Comput. Engin. Appl. 49 7 (in Chinese) [庞寿全, 刘永建, 朱从旭 2013 计算机工程与应用 49 7]
[18] Celikovsky S, Chen G 2002 Int. J. Bifur. Chaos 12 8
[19] Sun K H, Sprott J C 2009 Int. J. Bifur. Chaos 19 4
[20] Bao B C 2013 An Introduction to Chaotic Circuits (Beijing: Science Press) p41 (in Chinese) [包伯成 2013 混沌电路导论(北京:科学出版社) 第41页]
[21] Bao B C, Liu Z, Xu J P 2009 J. Systems Eng. Electron. 20 6
[22] Yuan D 2009 J. Anyang Teach. Coll. 2 26 (in Chinese) [袁地 2009 安阳师范学院学报 2 26]
[23] Luo X W, Wang C H, Wan Zhao 2014 Nonlinear Anal. Model. Control 19 2
[24] Li C G, Chen G R 2005 Chaos, Soliton. Fract. 5 23
[25] Liu Y J 2010 Ph. D. Dissertation (Guangzhou: South China University of Teehnology) (in Chinese) [ 刘永建 2010 博士学位论文(广州: 华南理工大学)]
[26] Grassberger P, Procaccia I 1983 Physica D 9 189
[27] Ramasubramanian K, Sriram M S 2000 Physica D 139 72
[28] Han M, Wei R 2010 Syst. Eng. Electron. 32 11 (in Chinese) [韩敏, 魏茹 2010 系统工程与电子技术 32 11]
[29] Guo J B, Xu X Z, Shi Q H, Hu T H 2013 Acta Phys. Sin. 62 110508 (in Chinese) [郭静波, 徐新智, 史启航, 胡铁华 2013 62 110508]
[30] Wang Y X, Weng Y F, Zheng D L 2006 J. Beijing Technol. Bus. Univ. (Nat. Sci. Ed.) 24 2 (in Chinese) [王云雄, 翁贻方, 郑德玲 2006 北京工商大学学报(自然科学版) 24 2 ]
[31] Chen X J, Li Z, Bai B M, Pan W, Chen Q H 2011 J. Electron. Inform. Technol. 33 5 (in Chinese) [陈小军, 李赞, 白宝明, 潘玮, 陈清华 2011 电子与信息学报 33 5]
[32] Cao G H, Hu K 2013 J. Beijing Univ. Aeron. Astron. 39 1 (in Chinese) [曹光辉, 胡凯 2013 北京航空航天大学学报 39 1]
[33] Zhu C X, Sun K H 2012 Acta Phys. Sin. 61 120503 (in Chinese) [朱从旭, 孙克辉 2012 61 120503]
[34] Li H N, Guo C F 2012 J. Data Acquisi. Process. 27 1 (in Chinese) [李慧娜, 郭超峰 2012 数据采集与处理 27 1]
[35] Wang J, Jiang G P 2011 Acta Phys. Sin. 60 060503 (in Chinese) [王静, 蒋国平 2011 60 060503]
[36] Wang X Y, Xie Y X 2011 Chin. Phys. B 20 8
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