Three-Dimensional Composite Chaotic System for Image Encryption

doi:10.3778/j.issn.1002-8331.2301-0104

Abstract

Abstract: Chaos is an important subject of nonlinearity, which has been widely used in information security, physics and economics. It is meaningful to construct new chaotic systems continuously. In view of this situation, a class of three-dimensional trigonometric complex chaotic system containing the number of iteration [t] is constructed. The dynamic behavior of the system is tested and analyzed. By analyzing the bifurcation diagram, Lyapunov, derivative of periodic points and spectral entropy of the system, it is determined that it has strong chaotic characteristics and large chaotic interval. At the same time, it is proved that the chaotic system satisfies the Devaney definition of chaos. Through 0-1 test and NIST test, the chaotic sequence randomly generated by the system is determined to have good quality. Based on this system, an efficient image encryption scheme is constructed, which is simple to implement and fast to encrypt. And further through the simulation test, index coefficients of encryption results are calculated and analyzed, and the results are very close to the theoretical optimal value. Among them, the key space is huge, up to 10252. The results are compared with other methods proposed recently, which can prove that the image encryption algorithm has strong security and reliability.

Key words: three-dimensional chaotic system, Devaney chaos, image encryption, key space

摘要： 混沌是非线性学的一个重要学科，在信息安全、物理学和经济学等领域得到了广泛应用，持续构造新的混沌系统是有意义的。针对这一情况，构造了一类包含迭代次数[t]的三维三角函数复合混沌系统，对系统的动力学行为进行测试和分析，通过对系统分岔图、Lyapunov指数、周期点导数、谱熵等进行分析，确定其具有较强的混沌特性和较大的混沌区间，同时证明了混沌系统满足Devaney混沌定义，通过对系统随机生成的混沌序列进行0-1测试和NIST测试确定其混沌序列质量良好。因此基于该系统的混沌序列对图像进行加密，实现简单，加密速度快，同时对加密结果的各项指标系数进行了计算和分析，测试结果均非常接近理论最优值。其中，密钥空间很大，可达10252。与最近的其他方法进行对比，结果表明，基于该系统的加密方案可以具有较强的安全性和可靠性。

关键词: 三维混沌系统, Devaney混沌, 图像加密, 密钥空间

YU Wanbo, HUANG Rongrong, WANG Wenjin. Three-Dimensional Composite Chaotic System for Image Encryption[J]. Computer Engineering and Applications, 2023, 59(20): 85-93.

于万波, 黄蓉荣, 王文晋. 三维复合混沌系统及其在图像加密的应用[J]. 计算机工程与应用, 2023, 59(20): 85-93.

References

[1] LI T Y，YORKE J A.Period three implies chaos[J].The American Mathematical Monthly，1975，82（10）：985-992.
[2] LORENZ E N.Deterministic nonperiodic flow[J].Journal of the Atmospheric Sciences，1963，20（2）：130-141.
[3] FRIDRICH J.Symmetric ciphers based on two-dimensional chaotic maps[J].International Journal of Bifurcation and chaos，1998，8（6）：1259-1284.
[4] LIU L F，MIAO S X.A new image encryption algorithm based on logistic chaotic map with varying parameter[J].Springerplus，2016，5（1）：289-301.
[5] WANG X Y，ZHANG Y Q，ZHAO Y Y.A novel image encryption scheme based on 2D logistic map and DNA sequence operations[J].Nonlinear Dynamics，2015，82（3）：1269-1280.
[6] HUA Z Y，ZHOU Y C.Image encryption using 2D logistic-adjusted-Sine map[J].Information Sciences，2016，339：237-253.
[7] CHEN G R，MAO Y B，CHUI C K.A symmetric image encryption scheme based on 3D chaotic cat maps[J].Chaos，Solitons and Fractals，2004，21（3）：749-761.
[8] BOUTEGHRINE B，TANOUGAST C，SADOUDI S.Novel image encryption algorithm based on new 3D chaos map[J].Multimedia Tools and Applications，2021，80：25583-25605.
[9] LIU C Y，DING Q.A color image encryption scheme based on a novel 3D chaotic mapping[J].Complexity，2020：3837209.
[10] 于万波.截面的几何形状决定三维函数的混沌特性[J].物理学报，2014，63（12）：26-34.
YU W B.Chaotic characteristics of three-dimensional function determined by cross-section geometric shape[J].Acta Physica Sinica，2014，63（12）：26-34.
[11] 于万波，王大庆.曲面迭代的混沌特性及其在人脸识别中的应用[J].计算机辅助设计与图形学学报，2015，27（12）：2264-2271.
YU W B，WANG D Q.Image chaotic characteristics and application in face recognition[J].Journal of Computer-Aided Design & Computer Graphics，2015，27（12）：2264-2271.
[12] ALAWIDA M，SAMSUDIN A，TEH J S，et al.Digital cosine chaotic map for cryptographic applications[J].IEEE Access，2019，7：150609-150622.
[13] YU W B，YU T.Analysis of chaotic characteristics of trigonometric function system[J].Modern Physics Letters B，2020，34（21）：2050210.
[14] GOTTWALD G A，MELBOURNE I.A new test for chaos in deterministic systems[J].Proceedings：Mathematical，Physical and Engineering Sciences，2004，460：603-611.
[15] BANKS J，BROOKS J，CAIRNS G，et al.On Devaney’s definition of chaos[J].Amer Math Monthly，1992，99（4）：332-334.
[16] ZHU H G，QI W T，GE J X，et al.Analyzing Devaney chaos of a sine-cosine compound function system[J].International Journal of Bifurcation and Chaos，2018，28（14）：21850176.
[17] GUO Y，JING S W，ZHOU Y Y，et al.An image encryption algorithm based on logistic-Fibonacci cascade chaos and 3D bit scrambling[J].IEEE Access，2020，8：9896-9912.
[18] CHEN W，GUO Y，JING S W.General image encryption algorithm based on deep learning compressed sensing and compound chaotic system[J].Acta Physica Sinica，2020，69（24）：240502.
[19] YASIR N，TARIQ S，DAWOOD S.A novel hybrid permutation substitution base colored image encryption scheme for multimedia data[J].Journal of Information Security and Applications，2021，59：2214-2126.
[20] CHAI X L，FU X L，GAN Z H，et al.A color image cryptosystem based on dynamic DNA encryption and chaos[J].Signal Processing，2019，155：44-62.
[21] YAVUZ E.A new parallel processing architecture for accelerating image encryption based on chaos[J].Journal of Information Security and Applications，2021，63：2214-2126.
[22] LUO Y L，TANG S B，LIU J X，et al.Image encryption scheme by combining the hyper-chaotic system with quantum coding[J].Optics and Lasers in Engineering，2020，124：105836.
[23] WANG X Y，WANG S W，WEI N，et al.A novel chaotic image encryption scheme based on Hash function and cyclic shift[J].IETE Technical Review，2019，36（1）：39-48.
[24] WANG X Y，ZHANG Y Q，XUE M B.A novel chaotic image encryption scheme using DNA sequence operations[J].Optics and Lasers in Engineering，2015，73：53-61.
[25] LIU Y，WANG J H，GONG L H.Image encryption algorithm based on chaotic system and dynamic S-boxes composed of DNA sequences[J].Multimedia Tools and Applications，2016，75（8）：4363-4382.
[26] CHEN Y，XIE S C，ZHANG J Z.A novel double image encryption algorithm based on coupled chaotic system[J].Physica Scripta，2022，97（6）：065207.
[27] CHONG J，XIE S C，ZHANG J Z，et al.Block color image encryption algorithm based on elementary cellular automata and DNA sequence operations[J].Journal of Electronic Imaging，2021，30（4）：043025.
[28] ZHANG X Q，YANG X C.Color image encryption algorithm based on 3D spiral transform and radia diffusion[J].Physica Scripta，2022，97（9）：095210.
[29] RAN B，ZHANG T S，WANG L H，et al.Image security based on three-dimensional chaotic system and random dynamic selection[J].Entropy，2022，24（7）：958.
[30] BOUSSIF M，ALOUI N，CHERIF A.Images encryption algorithm based on the quaternion multiplication and the XOR operation[J].Multimedia Tools and Applications，2022，78（24）：35493-35510.