多涡卷保守混沌系统的构建及在图像加密中的应用

安新磊; 李治甫; 薛睿; 熊丽; 张莉

doi:10.11999/JEIT250432

多涡卷保守混沌系统的构建及在图像加密中的应用

doi: 10.11999/JEIT250432 cstr: 32379.14.JEIT250432

安新磊^1, ,,
李治甫¹,
薛睿¹,
熊丽²,
张莉³

1.
兰州交通大学数理学院兰州 730070
2.
河西学院物理与机电工程学院张掖 734000
3.
兰州工业学院基础学科部兰州 730050

基金项目: 国家自然科学基金(62461022)，甘肃省自然科学基金(23JRRA861, 24JRRA291)，甘肃省重点研发计划-工业类项目资助(24YFGA040)，甘肃省高校教师创新基金项目(2025A-234)，甘肃省基础研究创新群体(25JRRA805)

详细信息

作者简介:
安新磊：男，教授，研究方向为微分方程动力学及其应用、视频及图像加密和人工智能

李治甫：男，研究生，研究方向为视频及图像加密和人工智能

薛睿：女，研究生，研究方向为视频及图像加密和人工智能

熊丽：女，教授，研究方向为非线性电路与系统及混沌保密通信

张莉：女，副教授，研究方向为视频及图像加密和人工智能

通讯作者:
安新磊　anxin1983@163.com

中图分类号: TP309.7; TN918.91
计量
- 文章访问数: 18
- HTML全文浏览量: 7
- PDF下载量: 1
- 被引次数: 0
出版历程
- 收稿日期: 2025-05-19
- 修回日期: 2025-09-28
- 网络出版日期: 2025-10-11

Construction of Multi-Scroll Conservative Chaotic System and Its Application in Image Encryption

AN Xinlei^{1
, ,},
LI Zhifu¹,
XUE Rui¹,
XIONG Li²,
ZHANG Li³

1.
School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China
2.
School of Physics and Electromechanical Engineering, Hexi University, Zhangye 734000, China
3.
The Basic Courses Department of Lanzhou Institute of Technology, Lanzhou 730050, China

Funds: The National Natural Science Foundation of China (62461022), The Natural Science Foundation of Gansu (23JRRA861, 24JRRA291), Gansu Provincial Key Research and Development Project (24YFGA040), Innovation Fund Project for University Teachers in Gansu Province (2025A-234), The Foundation for Innovative Fundamental Research Group Project of Gansu Province (G25JRRA805)

摘要

摘要: 鉴于当前保守混沌系统中多涡卷等复杂动力学行为的研究尚不充分，该文构建了五维保守超混沌系统，通过哈密顿能量函数的调控实现多涡卷保守混沌流的可控生成，进而探究其在图像加密领域的应用。基于哈密顿能量函数的驻点分析，该文揭示了涡卷结构的形成机制：系统运动轨迹沿哈密顿能量等值面演化，增加哈密顿能量函数的驻点可诱导多方向涡卷保守混沌流的形成。复杂度分析结果进一步验证了多涡卷保守混沌系统具有显著提升的谱熵复杂度。基于上述研究，该文设计了一种融合人脸检测技术的图像加密算法，实现针对人脸区域的信息加密。仿真结果表明该算法具备良好的安全性能。
- 广义哈密顿系统 /
- 哈密顿能量函数 /
- 多稳态 /
- 保守系统 /
- 图像加密
Abstract: Objective Existing conservative chaotic systems often suffer from structural simplicity and weak nonlinear characteristics, and research on complex dynamical behaviors such as multi-scroll structures remains limited, constraining their potential in engineering applications. To address security risks in face image transmission and the inefficiency of traditional global encryption methods, this study constructs a conservative chaotic system with multi-scroll characteristics, investigates its complex dynamical behavior, and designs a face-detection-based selective image encryption algorithm targeting sensitive regions. The work explores the practical application of conservative chaotic systems in image encryption. Methods A five-dimensional conservative hyperchaotic system is constructed on the basis of the generalized Hamiltonian system, and the controlled generation of multi-scroll chaotic flows is achieved through modulation of the Hamiltonian energy function. The Hessian matrix is used to analyze the stationary points of the Hamiltonian energy function, thereby revealing the relationship between scroll structures and stationary points. The spatial distribution of multi-scroll chaotic flows is further characterized by energy isosurfaces. The complex dynamical behaviors of the proposed system are investigated using Lyapunov exponent spectra and phase diagrams, while the sequence complexity is evaluated with the SE complexity algorithm. On this basis, an image encryption algorithm integrated with face detection technology is designed. The algorithm applies a diffusion–scrambling strategy to selectively encrypt facial regions. The security performance is evaluated through multiple indicators, including key space, pixel correlation, and information entropy. Results and Discussions Analysis of stationary points in the Hamiltonian energy function revealed a positive correlation between their number and scroll generation. Extreme points primarily drive scroll formation, whereas saddle points define transition zones, indicating that the scroll structure can be effectively regulated through the Hamiltonian energy function. The Lyapunov exponent spectrum of the multi-scroll conservative chaotic system is distributed symmetrically about the x-axis and exhibits an integer Lyapunov dimension, fully confirming the system’s volume-conserving property. Under different initial conditions, the system demonstrates diverse coexistence behaviors, including phase trajectories of varying types and scales. Complexity evaluation further showed that the multi-scroll conservative chaotic system achieves markedly higher spectral entropy complexity, supporting its potential for image encryption applications. Experimental validation demonstrated that the proposed algorithm can accurately detect faces and selectively encrypt sensitive regions, thereby avoiding the computational inefficiency of indiscriminate global encryption. Moreover, the algorithm exhibited strong performance across multiple security metrics. Conclusions A conservative chaotic system is constructed on the basis of the generalized Hamiltonian system, and its complex dynamical behavior and application in image encryption are investigated. The study provides theoretical references for the generation of multi-scroll conservative chaotic flows and offers practical guidance for the application of image encryption technology.
- Generalized Hamiltonian system /
- Hamiltonian energy function /
- Multistability /
- Conservative system /
- Image encryption

HTML全文

图 1 式(1)系统在三维空间中的相轨迹图

下载: 全尺寸图片幻灯片

图 2 保守混沌流 (红色表示极小值点，黄色表示鞍点)

下载: 全尺寸图片幻灯片

图 3 哈密顿能量等值面

下载: 全尺寸图片幻灯片

图 4 Lyapunov指数谱

下载: 全尺寸图片幻灯片

图 5 不同初值下的相轨迹

下载: 全尺寸图片幻灯片

图 6 SE复杂度分析

下载: 全尺寸图片幻灯片

图 7 图像的人脸检测结果

下载: 全尺寸图片幻灯片

图 8 加密算法流程图

下载: 全尺寸图片幻灯片

图 9 图像加密与解密测试结果

下载: 全尺寸图片幻灯片

图 10 不同密钥的解密结果

下载: 全尺寸图片幻灯片

图 11 直方图分析结果

下载: 全尺寸图片幻灯片

图 12 相关性分析结果

下载: 全尺寸图片幻灯片

图 13 噪声测试

下载: 全尺寸图片幻灯片

表 1 密钥空间对比结果

算法	本文	文献 [3]	文献[22]	文献[23]
密钥空间	$ > {2^{498}}$	${2^{478}}$	${2^{390}}$	${2^{249}}$

下载: 导出CSV

表 2 不同通道的相关系数计算结果

图像	方向	明文图像			密文图像
图像	方向	R	G	B	R	G	B
图像01	水平	0.9914	0.9898	0.9888	–0.0002	–0.0018	–0.0021
	垂直	0.9934	0.9920	0.9914	–0.0032	0.0012	–0.0001
	对角	0.9855	0.9827	0.9812	0.0018	0.0014	–0.0034
图像02	水平	0.9953	0.9897	0.9801	–0.0003	0.0023	–0.0005
	垂直	0.9922	0.9834	0.9692	–0.0006	0.0040	–0.0014
	对角	0.9886	0.9760	0.9552	0.0009	0.0011	0.0006
图像03	水平	0.9793	0.9869	0.9888	–0.0005	0.0028	0.0037
	垂直	0.9774	0.9840	0.9869	0.0015	0.0030	0.0017
	对角	0.9632	0.9746	0.9790	0.0038	0.0003	0.0012

下载: 导出CSV

表 3 密文图像相关系数比较结果

算法	水平	垂直	对角	平均值
本文	0.0025	0.0003	0.0014	0.0014
文献[2]	0.0029	0.0026	0.0031	0.0029
文献[24]	0.0066	0.0026	0.0075	0.0056
文献[27]	0.0105	0.0052	–0.0023	0.0045

下载: 导出CSV

表 4 信息熵计算结果

图像	明文图像			密文图像
图像	R	G	B	R	G	B
图像01	7.6223	7.4306	7.4413	7.9977	7.9982	7.9980
图像02	6.4791	6.0049	5.9409	7.9993	7.9992	7.9993
图像03	7.6081	7.5927	7.6069	7.9989	7.9992	7.9992

下载: 导出CSV

表 5 密文图像信息熵比较结果

算法	本文	文献[2]	文献[24]	文献[25]
信息熵值	7.9974	7.9972	7.9972	7.9969

下载: 导出CSV

表 6 NPCR和UACI 测试结果

图像	NPCR(%)				UACI(%)
图像	R	G	B	平均值	R	G	B	平均值
图像01	99.6109	99.6012	99.6147	99.6089	33.4502	33.4978	33.4271	33.4584
图像02	99.6201	99.5972	99.6028	99.6067	33.4975	33.4756	33.4872	33.4867
图像03	99.6117	99.5932	99.6287	99.6112	33.4260	33.4966	33.4313	33.4513

下载: 导出CSV

表 7 密文图像NPCR和UACI 比较结果

算法	本文	文献[6]	文献[16]	文献[26]
NPCR(%)	99.5968	99.6287	99.5900	99.5500
UACI(%)	33.4524	33.4512	33.4900	33.0200

下载: 导出CSV

参考文献(30)

[1]	LORENZ E N. Deterministic nonperiodic flow[J]. Journal of the Atmospheric Sciences, 1963, 20(2): 130–141. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.
[2]	LIU Siyang, AN Xinlei, WANG Yue, et al. Design of a new multi-wing chaotic system and its application in color image encryption[J]. Optik, 2023, 290: 171334. doi: 10.1016/j.ijleo.2023.171334.
[3]	ZHOU Nanrun, HU Longlong, HUANG Zhiwen, et al. Novel multiple color images encryption and decryption scheme based on a bit-level extension algorithm[J]. Expert Systems with Applications, 2024, 238: 122052. doi: 10.1016/j.eswa.2023.122052.
[4]	DENG Quanli, WANG Chunhua, YANG Gang, et al. Discrete memristive delay feedback rulkov neuron model: Chaotic dynamics, hardware implementation, and application in secure communication[J]. IEEE Internet of Things Journal, 2025, 12(13): 25559–25567. doi: 10.1109/JIOT.2025.3558963.
[5]	DENG Quanli, WANG Chunhua, SUN Yichuang, et al. Delay difference feedback memristive map: Dynamics, hardware implementation, and application in path planning[J/OL]. IEEE Transactions on Circuits and Systems I: Regular Papers, 2025. doi: 10.1109/tcsi.2025.3571961.
[6]	GAO Xinyu, MOU Jun, BANERJEE S, et al. Color-gray multi-image hybrid compression–encryption scheme based on BP neural network and knight tour[J]. IEEE Transactions on Cybernetics, 2023, 53(8): 5037–5047. doi: 10.1109/tcyb.2023.3267785.
[7]	ZHOU Shuang, QIU Yuyu, QI Guoyuan, et al. A new conservative chaotic system and its application in image encryption[J]. Chaos, Solitons & Fractals, 2023, 175: 113909. doi: 10.1016/j.chaos.2023.113909.
[8]	LIU Xilin, TONG Xiaojun, ZHANG Miao, et al. A highly secure image encryption algorithm based on conservative hyperchaotic system and dynamic biogenetic gene algorithms[J]. Chaos, Solitons & Fractals, 2023, 171: 113450. doi: 10.1016/j.chaos.2023.113450.
[9]	HÉNON M and HEILES C. The applicability of the third integral of motion: Some numerical experiments[J]. The Astronomical Journal, 1964, 69(1): 73–79. doi: 10.1086/109234.
[10]	SPROTT J C. Some simple chaotic flows[J]. Physical Review E, 1994, 50(2): R647–R650. doi: 10.1103/physreve.50.r647.
[11]	仓诗建. 基于哈密顿能量函数的混沌系统构造与动力学分析[D]. [博士论文], 天津大学, 2018. doi: 10.27356/d.cnki.gtjdu.2018.000273. CANG Shijian. Construction and dynamics analysis of chaotic systems using Hamiltonian method[D]. [Ph. D. dissertation], Tianjin University, 2018. doi: 10.27356/d.cnki.gtjdu.2018.000273.
[12]	DONG Enzeng, YUAN Mingfeng, DU Shengzhi, et al. A new class of Hamiltonian conservative chaotic systems with multistability and design of pseudo-random number generator[J]. Applied Mathematical Modelling, 2019, 73: 40–71. doi: 10.1016/j.apm.2019.03.037.
[13]	QI Guoyuan, HU Jianbing, and WANG Ze. Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic, chaos and strong chaos[J]. Applied Mathematical Modelling, 2020, 78: 350–365. doi: 10.1016/j.apm.2019.08.023.
[14]	JI’E Musha, YAN Dengwei, SUN Shuqi, et al. A simple method for constructing a family of Hamiltonian conservative chaotic systems[J]. IEEE Transactions on Circuits and Systems I: Regular Papers, 2022, 69(8): 3328–3338. doi: 10.1109/tcsi.2022.3172313.
[15]	DENG Quanli, WANG Chunhua, SUN Yichuang, et al. Discrete memristive conservative chaotic map: Dynamics, hardware implementation, and application in secure communication[J]. IEEE Transactions on Cybernetics, 2025, 55(8): 3926–3934. doi: 10.1109/TCYB.2025.3565333.
[16]	刘思洋, 安新磊, 施倩倩, 等. 一类多涡卷Chua系统及其在图像加密中的应用[J]. 复杂系统与复杂性科学, 2024, 21(3): 85–92. doi: 10.13306/j.1672-3813.2024.03.012. LIU Siyang, AN Xinlei, SHI Qianqian, et al. A multi-scroll Chua system and its application in image encryption[J]. Complex Systems and Complexity Science, 2024, 21(3): 85–92. doi: 10.13306/j.1672-3813.2024.03.012.
[17]	ZHANG Jie, ZUO Jiangang, WANG Meng, et al. Design and application of multiscroll chaotic attractors based on a novel multi-segmented memristor[J]. Chaos, Solitons & Fractals, 2024, 181: 114676. doi: 10.1016/j.chaos.2024.114676.
[18]	YU Guofeng, FAN Chunlei, XI Jiale, et al. Design and FPGA implementation of nested grid multi-scroll chaotic system[J]. Journal of King Saud University-Computer and Information Sciences, 2024, 36(8): 102186. doi: 10.1016/j.jksuci.2024.102186.
[19]	ZHANG Lishuang, LI Zhijun, and PENG Yuexi. A hidden grid multi-scroll chaotic system coined with two multi-stable memristors[J]. Chaos, Solitons & Fractals, 2024, 185: 115109. doi: 10.1016/j.chaos.2024.115109.
[20]	DING Pengfei, ZHU Jingge, and ZHANG Juan. A four-dimensional no-equilibrium chaotic system with multi-scroll chaotic hidden attractors and its application in image encryption[J]. Physica Scripta, 2024, 99(10): 105211. doi: 10.1088/1402-4896/ad7237.
[21]	吉俄木沙. 保守混沌系统的复杂动力学研究与电路实现[D]. [博士论文], 西南大学, 2024. doi: 10.27684/d.cnki.gxndx.2024.000011. JI’E Musha. Complex dynamics research and circuit implementation of conservative chaotic systems[D]. [Ph. D. dissertation], Southwest University, 2024. doi: 10.27684/d.cnki.gxndx.2024.000011.
[22]	DEMIRTAŞ M. A novel multiple grayscale image encryption method based on 3D bit-scrambling and diffusion[J]. Optik, 2022, 266: 169624. doi: 10.1016/j.ijleo.2022.169624.
[23]	ZHANG Chenjun, FAN Haiju, ZHANG Mingzhu, et al. Plaintext-related image encryption scheme without additional plaintext based on 2DCS[J]. Optik, 2023, 272: 170312. doi: 10.1016/j.ijleo.2022.170312.
[24]	WANG Xingyuan, CHEN Xuan, FENG Sijia, et al. Color image encryption scheme combining cross-plane Zigzag scrambling and pseudo-random combination RGB component diffusion[J]. Optik, 2022, 269: 169933. doi: 10.1016/j.ijleo.2022.169933.
[25]	WU Xin, SHI Hang, JI’E Musha, et al. A novel image compression and encryption scheme based on conservative chaotic system and DNA method[J]. Chaos, Solitons & Fractals, 2023, 172: 113492. doi: 10.1016/j.chaos.2023.113492.
[26]	LONG Bofeng, CHEN Zhong, LIU Tongzhe, et al. A novel medical image encryption scheme based on deep learning feature encoding and decoding[J]. IEEE Access, 2024, 12: 38382–38398. doi: 10.1109/ACCESS.2024.3371888.
[27]	KUMAR K, ROY S, RAWAT U, et al. IEHC: An efficient image encryption technique using hybrid chaotic map[J]. Chaos, Solitons & Fractals, 2022, 158: 111994. doi: 10.1016/j.chaos.2022.111994.
[28]	HOSNY K M, ZAKI M A, HAMZA H M, et al. Privacy protection in surveillance videos using block scrambling-based encryption and DCNN-based face detection[J]. IEEE Access, 2022, 10: 106750–106769. doi: 10.1109/access.2022.3211657.
[29]	GAO Suo, WU Rui, WANG Xingyuan, et al. EFR-CSTP: Encryption for face recognition based on the chaos and semi-tensor product theory[J]. Information Sciences, 2023, 621: 766–781. doi: 10.1016/j.ins.2022.11.121.
[30]	ROTHE R, TIMOFTE R, and VAN GOOL L. Deep expectation of real and apparent age from a single image without facial landmarks[J]. International Journal of Computer Vision, 2018, 126(2/4): 144–157. doi: 10.1007/s11263-016-0940-3.