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基于新的五维多环多翼超混沌系统的图像加密算法

庄志本 李军 刘静漪 陈世强

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基于新的五维多环多翼超混沌系统的图像加密算法

庄志本, 李军, 刘静漪, 陈世强

Image encryption algorithm based on new five-dimensional multi-ring multi-wing hyperchaotic system

Zhuang Zhi-Ben, Li Jun, Liu Jing-Yi, Chen Shi-Qiang
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  • 本文提出了一种基于新的五维多环多翼超混沌系统的数字图像加密方法. 首先, 将明文图像矩阵和五条混沌序列分别通过QR分解法分解成一个正交矩阵和一个上三角矩阵, 将混沌系统产生的五条混沌序列分别通过LU分解法分解成一个上三角矩阵和一个下三角矩阵, 分别将两个上三角矩阵和一个下三角矩阵相加, 得到五个离散后的混沌序列; 其次, 将明文图像矩阵分解出来的正交矩阵与五个混沌序列分解出来的五个正交矩阵相乘, 同时把明文图像矩阵分解出来的上三角矩阵中的元素通过混沌序列进行位置乱, 再将操作后的两个矩阵相乘; 最后, 将相乘后的矩阵通过混沌序列进行比特位位置乱, 再用混沌序列与其进行按位“异或”运算, 得到最终加密图像. 理论分析和仿真实验结果表明该算法的密钥空间远大于10200, 密钥敏感性强, 能够有效地抵御统计分析和灰度值分析的攻击, 对数字图像的加密具有很好的加密效果.
    The complex structure of hyperchaos and its complex dynamic behavior have a good application prospect in the fields of image encryption, digital watermarking and information security. Therefore, it has become very important to generate chaotic attractors with multi-vortex and multi-winged multi-rings with complex topologies. In this paper, we propose a new five-dimensional hyperchaotic system capable of generating multi-ring and multi-wing, and carry out theoretical analysis and numerical simulation experiments on some basic dynamic characteristics of the chaotic system. Such as equilibrium point, dissipation, Lyapunov exponent, bifurcation diagram, phase diagram and so on. In the process of encryption, first, we decompose the plaintext image matrix and the five chaotic sequences into an orthogonal matrix and an upper triangular matrix by QR decomposition. The five chaotic sequences generated by the chaotic system are respectively decomposed into an upper triangular matrix and a lower triangular matrix by the LU decomposition method. The upper triangular matrix decomposed by the QR decomposition method and the lower triangular matrix decomposed by the LU decomposition method are respectively added to obtain five discrete chaotic sequences. At the same time, the five discrete chaotic sequences are added to the upper triangular matrix decomposed by the LU decomposition method to obtain the final five discrete chaotic sequences. Secondly, the orthogonal matrix decomposed by the plaintext image matrix is multiplied by five orthogonal matrices decomposed by five chaotic sequences. At the same time, the elements in the upper triangular matrix decomposed by the plaintext image matrix are chaotically arranged by the chaotic sequence, and then the two matrices after the operation are multiplied. Finally, the multiplied matrix is chaotically placed on the bit by a chaotic sequence. Then use the chaotic sequence to perform a bitwise XOR operation to obtain the final encrypted image. The theoretical analysis and simulation results show that the algorithm has large key space and strong key sensitivity. It can effectively resist the attacks of statistical analysis and gray value analysis, and has good encryption effect on digital image encryption. This image encryption algorithm using a combination of conventional encryption and chaotic encryption does not have a defined plaintext ciphertext mapping relationship.
      通信作者: 陈世强, chensq8808@126.com
    • 基金项目: 其它-湖北民族学院博士启动基金项目(MY2018B014)
      Corresponding author: Chen Shi-Qiang, chensq8808@126.com
    [1]

    王平, 冯勇, 孙黎霞, 韩凤玲 2002 控制理论与应用 21 1Google Scholar

    Wang P, Feng Y, Sun L X, Han F L 2002 Control Theory & Appl. 21 1Google Scholar

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    禹思敏 2005 物理学报 54 1500Google Scholar

    Yu S M 2005 Acta Phys.Sin. 54 1500Google Scholar

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    Karthikeyan R, Serdar C, Peiman N, Abdul J M, Sajad J, Anitha K 2018 Eur. Phys. J. Plus 133 354Google Scholar

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    贾美美, 蒋浩刚, 李文静 2019 物理学报 68 130503Google Scholar

    Jia M M, Jiang H G, Li W J 2019 Acta Phys. Sin. 68 130503Google Scholar

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    Li Y X, Tang W K S, Chen G R 2005 Int. J. Bifurcation Chaos. 15 3367Google Scholar

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    彭再平, 王春华, 林愿, 骆小文 2014 物理学报 63 240506Google Scholar

    Peng Z P, Wang C H, Lin Y, Luo X W 2014 Acta Phys. Sin. 63 240506Google Scholar

    [7]

    刘杨 2015 博士学位论文 (哈尔滨: 哈尔滨工业大学)

    Liu Y 2015 Ph. D. Dissertation (Harbin: Harbin Institute of Technology) (in Chinese)

    [8]

    禹思敏 2018 新型混沌电路与系统的设计原理及其应用 (北京: 科学出版社) 第139—155页

    Yu S M 2018 Design Principles and Applications of New Chaotic Circuits and Systems (Beijing: Science Press) pp139– 155 (in Chinese)

    [9]

    Zhang L H, Liao X F, Wang X B 2005 Chaos, Solitons Fractals 24 759Google Scholar

    [10]

    Wong K, Kwor B, Law W 2008 Phys. Lett. A 372 2645Google Scholar

    [11]

    Zhang W, Yu H, Zhao Y L, Zhu Z L 2016 Signal Process. 118 36Google Scholar

    [12]

    Luo Y L, Zhou R L, Liu J X, Gao Y, Ding X M 2018 Nonlinear Dyn. 4 1

    [13]

    Ye G D, Pan C, Huang X L, Mei Q X 2018 Nonlinear Dyn. 20 18

    [14]

    Abanda Y, Tiedeu A 2016 IET Image Proc. 10 742Google Scholar

    [15]

    Zhang Y 2018 Inf. Sci. 255 31145

    [16]

    He Y, Zhang Y Q, Wang X Y 2018 Neural Comput. Appl. 10 1

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    Raza S F, Satpute V 2018 Nonlinear Dyn. 254 1

    [18]

    Ahmad J, Khan M A, Hwang S O, Khan J S 2017 Neural Comput. Appl. 28 953

    [19]

    Ahmad J, Khan M A, Ahmed F, Khan J S 2018 Neural Comput. Appl. 3 1

    [20]

    Sprott J C 1994 Phys. Rev. E 50 647Google Scholar

    [21]

    Enayatifar R, Abdullah A H, Isnin I F, Altameem A, Lee A 2017 Opt. Lasers Eng. 90 146Google Scholar

    [22]

    Liu H J, Wang X Y, Kadir A 2012 Appl. Soft Comput. 12 1457Google Scholar

  • 图 1  Lyapunov指数谱

    Fig. 1.  Lyapunov exponent diagram.

    图 2  系统(2)随$f$变化的分岔图

    Fig. 2.  Bifurcation diagram of system (2) variation with $f$.

    图 3  时间序列图 (a) $x\text- t$时间序列; (b) $y \text- t$时间序列; (c) $z \text- t$时间序列; (d) $w \text- t$时间序列; (e) $v \text- t$时间序列

    Fig. 3.  Time series diagram: (a) $x \text- t$ time series; (b) $y \text- t$ time series; (c) $z \text- t$ time series; (d) $w \text- t$ time series; (e) $v \text- t$time series

    图 4  三维相图 (a) $x \text- y \text- z$三维图; (b) $x \text- y \text- w$三维图; (c) $x \text- y \text- v$三维图; (d) $x \text- z \text- w$三维图; (e) $x \text- z \text- v$三维图; (f) $x \text- w \text- v$三维图; (g) $y \text- z \text- w$三维图; (h) $y \text- z \text- v$三维图; (i) $y \text- w \text- v$三维图; (j) $z \text- w \text- v$三维图

    Fig. 4.  Three-dimensional phase diagram: (a) $x \text- y \text- z$ Three-dimensional map; (b) $x \text- y \text- w$ Three-dimensional map; (c) $x \text- y \text- v$ Three-dimensional map; (d) $x \text- z \text- w$ Three-dimensional map; (e) $x \text- z \text- v$ Three-dimensional map; (f) $x \text- w \text- v$ Three-dimensional map; (g) $y \text- z \text- w$ Three-dimensional map; (h) $y \text- z \text- v$ Three-dimensional map; (i) $x \text- y \text- z$ Three-dimensional map; (j) $z \text- w \text- v$ Three-dimensional map.

    图 5  二维平面相图 (a) $x \text- y$平面; (b) $x \text- z$平面; (c) $x \text- v$平面; (d) $y \text- z$平面; (e) $y \text- w$平面; (f) $z \text- w$平面; (g) $z \text- v$平面

    Fig. 5.  Two-dimensional plane phase diagram: (a) $x \text- y$ flat; (b) $x \text- z$ flat; (c) $x \text- v$ flat; (d) $y \text- z$ flat; (e) $y \text- w$ flat; (f) $z \text- w$ flat; (g) $z \text- v$ flat.

    图 6  数字图像加解密实验图 (a) lena原图; (b) lena加密图像; (c) lena解密图像; (d) baboon原图; (e) baboon加密图像; (f) baboon解密图像; (g) boat原图; (h) boat加密图像; (i) boat解密图像

    Fig. 6.  Digital image encryption and decryption experiment: (a) Original Lena image; (b) encrypted Lena image; (c) decrypted Lena image; (d) original baboon image; (e) encrypted baboon image; (f) decrypted baboon image; (g) original boat image.; (h) encrypted boat image; (i)decrypted boat image.

    图 7  明文图像和密文图像直方图 (a) lena明文直方图; (b) lena密文直方图; (c) baboon明文直方图; (d) baboon密文直方图; (e) boat明文直方图; (f) boat密文直方图

    Fig. 7.  Histogram of plaintext and ciphertext images (a) Plaintext Lena image histogram; (b) ciphertext Lena image histogram; (c) plaintext baboon image histogram; (d) ciphertext baboon image histogram; (e) plaintext boat image histogram; (f) ciphertext boat image histogram.

    图 8  密钥敏感性测试图 (a)明文图像; (b)密文${{{Y}}_1}$(密钥为${y_0}$); (c)密文${{{Y}}_2}$(密钥为${y_1}$); (d) ${{{Y}}_1}$正确解密结果; (e) ${{{Y}}_1}$${y_1}$错误解密结果; (f) ${{{Y}}_2}$${y_0}$错误解密结果

    Fig. 8.  Key sensitivity tests: (a) Plain-image; (b) cipher ${{{Y}}_1}$ with key ${y_0}$; (c) cipher ${{{Y}}_2}$ with key ${y_1}$; (d) right decrypted ${{{Y}}_1}$; (e) decrypted ${{{Y}}_1}$ with ${y_1}$; (f) decrypted${{{Y}}_2}$ with${y_0}$.

    图 9  baboon图像加密前后三个方向上的相关性分析图 (a), (b)对角相邻; (c), (d)水平相邻; (e), (f)垂直相邻;

    Fig. 9.  Correlation analysis chart in three directions before and after baboon image encryption: (a), (b) Diagonally adjacent; (c), (d) horizontally adjacent; (e), (f) vertically adjacent.

    图 10  抗剪切攻击能力分析图 (a)剪切前密文; (b)剪切后密文; (c)剪切前解密; (d)剪切后解密

    Fig. 10.  Anti-shear attack capability analysis chart: (a) Ciphertext before cutting; (b) ciphertext after cutting; (c) decrypted image before cutting; (d) decrypted image after cutting.

    图 11  抗噪声攻击能力分析图 (a)加噪前密文; (b)加噪后密文; (c)加噪前解密; (d)加噪后解密

    Fig. 11.  Anti-noise attack capability analysis chart: (a) Ciphertext before adding noise; (b) ciphertext after adding noise; (c) decrypted image before adding noise; (d) decrypted image after adding noise.

    表 1  明文图像与加密图像的信息熵分析表

    Table 1.  Information entropy analysis table of plain text and encrypted image.

    图像Lena图像Baboon图像Boat图像
    原图像7.46447.37137.1267
    密文图像7.99947.99947.9993
    文献[11]7.99927.9993
    文献[12]7.99947.9993
    文献[13]7.99717.9993
    文献[14]7.9960
    文献[15]7.99937.9993
    文献[16]7.99937.9992
    文献[17]7.9974
    下载: 导出CSV

    表 2  加密图像不动点比分析表

    Table 2.  Encrypted image fixed point ratio analysis table.

    图像总像素数不动点数不动点比
    Lena图像26214410150.39%
    Baboon图像26214410140.39%
    Boat图像2621449990.38%
    下载: 导出CSV

    表 3  灰度平均变化值分析表

    Table 3.  Grayscale average change value analysis table.

    图像Lena图像Baboon图像Boat图像
    灰度平均变化值73.193770.858974.8383
    下载: 导出CSV

    表 4  密钥敏感性测试结果表

    Table 4.  Key sensitivity test result table.

    图像lena图像baboon图像boat图像
    指标NPCRUACINPCRUACINPCRUACI
    本文算法0.99640.33400.99620.33460.99580.3344
    文献[13]0.99620.33470.99600.3340
    文献[14]0.99570.3508
    文献[15]0.99610.33470.99610.3347
    文献[16]0.99620.33440.99610.3349
    下载: 导出CSV

    表 5  明文图像与密文图像相关系数测试结果表

    Table 5.  Plaintext image and ciphertext image correlation coefficient test result table.

    图像水平方向相关系数垂直方向相关系数对角线方向相关系数
    明文图像密文图像明文图像密文图像明文图像密文图像
    Lena0.9762–0.00840.96590.04610.94680.0131
    Baboon0.7204–0.00500.8264–0.00740.7046–0.0322
    boat0.96210.01060.82520.00870.8327–0.0423
    下载: 导出CSV
  • [1]

    王平, 冯勇, 孙黎霞, 韩凤玲 2002 控制理论与应用 21 1Google Scholar

    Wang P, Feng Y, Sun L X, Han F L 2002 Control Theory & Appl. 21 1Google Scholar

    [2]

    禹思敏 2005 物理学报 54 1500Google Scholar

    Yu S M 2005 Acta Phys.Sin. 54 1500Google Scholar

    [3]

    Karthikeyan R, Serdar C, Peiman N, Abdul J M, Sajad J, Anitha K 2018 Eur. Phys. J. Plus 133 354Google Scholar

    [4]

    贾美美, 蒋浩刚, 李文静 2019 物理学报 68 130503Google Scholar

    Jia M M, Jiang H G, Li W J 2019 Acta Phys. Sin. 68 130503Google Scholar

    [5]

    Li Y X, Tang W K S, Chen G R 2005 Int. J. Bifurcation Chaos. 15 3367Google Scholar

    [6]

    彭再平, 王春华, 林愿, 骆小文 2014 物理学报 63 240506Google Scholar

    Peng Z P, Wang C H, Lin Y, Luo X W 2014 Acta Phys. Sin. 63 240506Google Scholar

    [7]

    刘杨 2015 博士学位论文 (哈尔滨: 哈尔滨工业大学)

    Liu Y 2015 Ph. D. Dissertation (Harbin: Harbin Institute of Technology) (in Chinese)

    [8]

    禹思敏 2018 新型混沌电路与系统的设计原理及其应用 (北京: 科学出版社) 第139—155页

    Yu S M 2018 Design Principles and Applications of New Chaotic Circuits and Systems (Beijing: Science Press) pp139– 155 (in Chinese)

    [9]

    Zhang L H, Liao X F, Wang X B 2005 Chaos, Solitons Fractals 24 759Google Scholar

    [10]

    Wong K, Kwor B, Law W 2008 Phys. Lett. A 372 2645Google Scholar

    [11]

    Zhang W, Yu H, Zhao Y L, Zhu Z L 2016 Signal Process. 118 36Google Scholar

    [12]

    Luo Y L, Zhou R L, Liu J X, Gao Y, Ding X M 2018 Nonlinear Dyn. 4 1

    [13]

    Ye G D, Pan C, Huang X L, Mei Q X 2018 Nonlinear Dyn. 20 18

    [14]

    Abanda Y, Tiedeu A 2016 IET Image Proc. 10 742Google Scholar

    [15]

    Zhang Y 2018 Inf. Sci. 255 31145

    [16]

    He Y, Zhang Y Q, Wang X Y 2018 Neural Comput. Appl. 10 1

    [17]

    Raza S F, Satpute V 2018 Nonlinear Dyn. 254 1

    [18]

    Ahmad J, Khan M A, Hwang S O, Khan J S 2017 Neural Comput. Appl. 28 953

    [19]

    Ahmad J, Khan M A, Ahmed F, Khan J S 2018 Neural Comput. Appl. 3 1

    [20]

    Sprott J C 1994 Phys. Rev. E 50 647Google Scholar

    [21]

    Enayatifar R, Abdullah A H, Isnin I F, Altameem A, Lee A 2017 Opt. Lasers Eng. 90 146Google Scholar

    [22]

    Liu H J, Wang X Y, Kadir A 2012 Appl. Soft Comput. 12 1457Google Scholar

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出版历程
  • 收稿日期:  2019-09-05
  • 修回日期:  2019-11-21
  • 刊出日期:  2020-02-20

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