A new chaotic signal based on deep learning and its application in image encryption

Zhao Zhi-Peng; Zhou Shuang; Wang Xing-Yuan

doi:10.7498/aps.70.20210561

Abstract

To improve the security of image encryption in singular chaotic systems, an encryption algorithm based on deep-learning is proposed in this paper. To begin with, the chaos sequence is generated by using a hyperchaotic Lorenz system, prior to creating new chaotic signals based on chaotic characteristics obtained from he simulations of the powerful complex network structure of long-short term memory artificial neural network (LSTM-ANN). Then, dynamic characteristics of the new signals are analyzed with the largest Lyapunov exponent, 0-1 test, power spectral analysis, phase diagrams and NIST test. In the end, the new signals are applied to image encryption, the results of which verify the expected increased difficulty in attacking the encrypted system. This is attributable to the differences of the new signals generated using the proposed method from the original chaotic signals, as well as arises from the high complexity and nonlinearity of the system. Considering its ability to withstand common encryption attacks, it is hence reasonable to conclude that the proposed method exhibits higher safety and security than other traditional methods.

Keywords:

Authors and contacts

1.
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China
2.
School of Information Science and Technology, Dalian Maritime University, Dalian 116026, China

Corresponding author: Zhou Shuang, zhoushuang@cqnu.edu.cn ; Wang Xing-Yuan, xywang@dlmu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61672124), the Password Theory Project of the 13th Five-Year Plan National Cryptography Development Fund, China (Grant No. MMJJ20170203), the Liaoning Province Science and Technology Innovation Leading Talents Program Project, China (Grant No. XLYC1802013), the Key R&D Projects of Liaoning Province, China (Grant No. 2019020105-JH2/103), the Jinan City’ 20 Universities’ Funding Projects-Introducing Innovation Team Program, China (Grant No. 2019GXRC031), and the Science and Technology Research Program of Chongqing Municipal Education Commission, China (Grant No. KJQN201900529)

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Cited By

图 1 超混沌Lorenz相图三维投影

Figure 1. Three dimensional projection of hyperchaotic Lorenz phase diagram.

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图 2 LSTM训练模型示意

Figure 2. LSTM training model diagram.

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图 3 混沌图像加密算法流程图(I)

Figure 3. Chaotic image encryption flow chart algorithm (I).

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图 4 混沌图像加密算法流程图(Ⅱ)

Figure 4. Chaotic image encryption flow chart algorithm (Ⅱ).

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图 5 $ \left\{ {y_i'} \right\} $LSTM模型训练过程

Figure 5. LSTM model training process of $ \left\{ {y_i'} \right\} $.

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图 6 LSTM模型预测

Figure 6. Forecast of LSTM model.

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图 7 截取的$ \left\{ {{y_i}} \right\} $

Figure 7. Part of $ \left\{ {{y_i}} \right\} $.

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图 8 Wolf方法求$ \left\{ {y_i'} \right\} $最大Lyapunov指数过程图

Figure 8. Using wolf method to find the largest Lyapunov exponent process of $ \left\{ {y_i'} \right\} $.

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图 9 新混沌信号0-1测试图

Figure 9. 0-1 test image of the new chaotic signal.

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图 10 超混沌Lorenz系统混沌信号0-1测试图

Figure 10. 0-1 test image of the hyperchaotic Lorenz signal.

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图 11 新混沌信号功率谱图

Figure 11. The spectrum image of the new chaotic signal

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图 12 超混沌Lorenz信号功率谱图

Figure 12. Spectrum image of hyperchaotic Lorenz signal.

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图 13 序列$ \left\{ {y_i'} \right\} $的相图

Figure 13. Phase diagram of $ \left\{ {y_i'} \right\} $.

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图 14 序列$ \left\{ {{y_i}} \right\} $的相图

Figure 14. Phase diagram of $ \left\{ {{y_i}} \right\} $.

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图 15 数字图像加密解密实验图　(a)鸟(256 × 256)原图; (b)鸟(256 × 256)加密图; (c)鸟(256 × 256)解密图; (d)辣椒(256 × 256)原图; (e)辣椒(256 × 256)加密图; (f)辣椒(256 × 256)解密图; (g)Lena(512 × 512)原图; (h) Lena(512 × 512)加密图; (i) Lena(512 × 512)解密图; (j) 液体泼洒(512 × 512)原图; (k) 液体泼洒(512 × 512)加密图; (l) 液体泼洒(512 × 512)解密图; (m)机场(1024 × 1024)原图; (n) 机场(1024 × 1024)加密图; (o) 机场(1024 × 1024)解密图; (p)飞机(1024 × 1024)原图; (q)飞机(1024 × 1024)加密图; (r)飞机(1024 × 1024)解密图

Figure 15. Experimental picture of digital image encryption and decryption: (a) Original bird image; (b) encrypted bird image; (c) decrypted bird image; (d) original pepper (256 × 256) image; (e) encrypted pepper (256 × 256) image; (f) decrypted pepper (256 × 256) image; (g) original Lena (512 × 512) image; (h) encrypted Lena (512 × 512) image; (i) decrypted Lena (512 × 512) image; (j) original splash (512 × 512) image; (k) encrypted splash (512 × 512) image; (l) decrypted splash (512 × 512) image; (m) original airport (1024 × 1024) image; (n) encrypted airport (1024 × 1024) image; (o) decrypted airport (1024 × 1024) image; (p) original airplane (1024 × 1024) image; (q) encrypted airplane (1024 × 1024) image; (r) decrypted airplane (1024 × 1024) image.

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图 16 密钥敏感性　(a)明文图像; (b)密文用$ x_0' $解密结果; (c)密文用$ \left\{ {{y_i}} \right\} $解密结果

Figure 16. Sensitivity of secret key: (a) Original image; (b) error key $ x_0' $ restoring diagram; (c) error key $ \left\{ {{y_i}} \right\} $ restoring diagram.

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图 17 加密解密图像直方图　(a)鸟(256 × 256)原图与明文直方图; (b)鸟(256 × 256)加密图和密图直方图; (c)辣椒(256 × 256)原图与明文直方图; (d)辣椒(256 × 256)加密图和密图直方图; (e) Lena(512 × 512)原图与明文直方图; (f) Lena(512 × 512)加密图和密图直方图; (g)水滴泼洒(512 × 512)原图与明文直方图; (h)水滴泼洒(512 × 512)加密图和密图直方图; (i)机场(1024 × 1024)原图与明文直方图; (j)机场(1024 × 1024)加密图和密图直方图; (k)飞机(1024 × 1024)原图与明文直方图; (l)飞机(1024 × 1024)加密图和密图直方图

Figure 17. Histograms of plain images and ciphered images: (a) Original image and histogram of bird (256 × 256); (b)cipher image and histogram of bird (256 × 256); (c) original image and histogram of pepper (256 × 256); (d) cipher image and histogram of pepper(256 × 256); (e) original image and histogram of Lena (512 × 512); (f) cipher image and histogram of Lena (512 × 512); (g) original image and histogram of splash (512 × 512); (h) cipher image and histogram of splash (512 × 512); (i) original image and histogram of airport (1024 × 1024); (j) cipher image and histogram of airport (1024 × 1024); (k) original image and histogram of airplane (1024 × 1024); (l) cipher image and histogram of airplane (1024 × 1024).

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图 18 Lena图相关性分析　(a)明文; (b)密图

Figure 18. Correlation coefficients of Lena: (a) Original image; (b) encrypted image

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图 19 抗攻击性检验　(a) 10%剪切; (b) 30%剪切; (c) 80%剪切; (d) 0.001高斯白噪声攻击; (e) 0.01高斯白噪声攻击; (f) 0.1高斯白噪声攻击

Figure 19. Anti attack test: (a) 10% data missed; (b) 30% date missed; (c) 80% data missed; (d) attack of 0.001 Gaussian white noise; (e) attack of 0.01 Gaussian white noise; (f) attack of 0.1 Gaussian white noise.

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图 20 全黑全白图加密解密图像

Figure 20. Encryption and decryption image of all black and all white image.

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表 1 新混沌时间序列NIST测试

Table 1. NIST test of the new chotic time series

统计测试	p 值	结果
单比特频率测试	0.8752	通过
块内频率测试	0.8523	通过
游程测试	0.6121	通过
块内最长1游程测试	0.0828	通过
二进制矩阵秩测试	0.1445	通过
离散傅里叶(谱)测试	0.8152	通过
非重叠模板匹配测试	0.3527	通过
重叠模板匹配测试	0.4305	通过
Maurer通用统计测试	0.4214	通过
线性复杂度测试	0.2341	通过
序列测试	0.3053	通过
近似熵测试	0.1568	通过
累加和测试	0.3257	通过
随机旅行测试	0.1523	通过
随机旅行变种测试	0.1057	通过

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表 2 超混沌Lorenz混沌序列NIST测试

Table 2. NIST test of the hyperchaotic Lorenz time series.

统计测试	p 值	结果
单比特频率测试	0.8815	通过
块内频率测试	0.7253	通过
游程测试	0.5986	通过
块内最长1游程测试	0.0823	通过
二进制矩阵秩测试	0.1263	通过
离散傅里叶(谱)测试	0.8164	通过
非重叠模板匹配测试	0.3580	通过
重叠模板匹配测试	0.5216	通过
Maurer通用统计测试	0.4418	通过
线性复杂度测试	0.5052	通过
序列测试	0.6015	通过
近似熵测试	0.1435	通过
累加和测试	0.4863	通过
随机旅行测试	0.3997	通过
随机旅行变种测试	0.2265	通过

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表 3 混沌信号统计参数对比

Table 3. Comparison of statistical parameters of chaotic signals.

信号来源	最大Lyapunov 指数	0-1 测试	功率谱分析	相图	NIST 测试
超混沌Lorenz	0.3381^[52]	0.7937	混沌	混沌	随机
深度学习	2.6002	0.9250	混沌	混沌	随机

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表 4 NPCR和UACI

Table 4. NPCR and UACI.

图片	改变$ P(256, 256) $
图片	NPCR/%	UACI/%
Bird ($ 256 \times 256 $)	99.56	33.35
Cameraman ($ 256 \times 256 $)	99.63	33.30
Pepper ($ 256 \times 256 $)	99.62	33.39
House ($ 256 \times 256 $)	99.63	33.37
Lena ($ 512 \times 512 $)	99.58	33.38
Airplane ($ 512 \times 512 $)	99.59	33.42
Tank ($ 512 \times 512 $)	99.61	33.49
Splash ($ 512 \times 512 $)	99.64	33.54
Truck ($ 512 \times 512 $)	99.60	33.47
Airport ($ 1024 \times 1024 $)	99.62	33.48
Airplane ($ 1024 \times 1024 $)	99.60	33.47

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表 5 NPCR和UACI的平均值与其他加密算法的比较

Table 5. The average of NPCR and UACI and comparison with other algorithms.

	本文平均值	文献[10]	文献[13]	文献[58]	文献[59]
NPCR/%	99.604	99.61	99.6641	99.61	99.6114
UACI/%	33.46	33.48	33.6124	33.46	33.4523

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表 6 图像相关系数

Table 6. Correlation coefficients of images.

图片	明文				密文
图片	水平	垂直	对角	反对角	水平	垂直	对角	反对角
Lena	0.9844	0.9668	0.9620	0.9790	–0.0016	0.0014	–0.0014	–0.0010
Bird	0.9889	0.9826	0.9713	0.9519	0.0114	0.0103	0.0104	–0.0031
Cameraman	0.9591	0.9335	0.9101	0.9377	–0.0099	0.0141	–0.0165	–0.0028
Pepper	0.9638	0.9585	0.9368	0.9339	–0.0127	–0.0014	–0.0079	0.0134
Airport	0.9066	0.9072	0.8446	0.8752	–0.0091	0.0088	–0.0014	–0.0054
Splash	0.9925	0.9850	0.9797	0.9507	0.0082	0.0016	0.0039	–0.0060
Airplane	0.9476	0.9664	0.9418	0.9299	–0.0278	–0.0105	0.0013	–0.0083
House	0.9549	0.9780	0.9399	0.9027	0.0141	–0.0115	–0.0176	0.0005
Tank	0.8678	0.8815	0.8414	0.7949	–0.0036	0.0003	–0.0098	–0.0065
Truck	0.9258	0.9561	0.9114	0.8194	0.0028	–0.0041	0.0018	0.0074

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表 7 密文图像相关系数比较

Table 7. Comparison of the correlation coefficients of images.

	Lena	文献[10]	文献[13]	文献[58]	文献[59]
水平	–0.0016	–0.00007	0.00047	–0.00251	–0.038118
垂直	0.0014	–0.0024	–0.03911	–0.00292	–0.029142
对角	–0.0014	0.0019	0.00305	–0.00156	0.002736

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表 8 信息熵

Table 8. Information entropy of images.

图片	图片大小	信息熵
House	$ 256 \times 256 $	7.9969
Cameraman	$ 256 \times 256 $	7.9971
Bird	$ 256 \times 256 $	7.9968
Pepper	$ 256 \times 256 $	7.9971
Lena	$ 512 \times 512 $	7.9993
Splash	$ 512 \times 512 $	7.9993
Airplane	$ 512 \times 512 $	7.9993
Tank	$ 512 \times 512 $	7.9994
Truck	$ 512 \times 512 $	7.9993
Airport	$ 1024 \times 1024 $	7.9998
Airplane	$ 1024 \times 1024 $	7.9998

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表 9 全黑全白图的统计分析

Table 9. The statistical analysis of all-black image and all-white image.

	NPCR	UACI	信息熵	相关系数
	NPCR	UACI	信息熵	水平	垂直		对角
全黑图	0.9958	0.3332	7.9970	0.0016		0.0003	0.0036
全白图	0.9961	0.3351	7.9971	0.0070		0.0004	0.0070

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[1]	Wang X Y, Teng L, Qin X 2012 Signal Pro. 92 1101 Google Scholar
[2]	Liu W H, Sun K H, Zhu C X 2016 Optics Lasers Eng. 84 26 Google Scholar
[3]	Wu X J, Kan H C, Kurths J 2015 Appl. Soft Comput. 37 24 Google Scholar
[4]	Hua Z Y, Zhou Y C, Pun C M, Chen C. L. P 2015 Inf. Sci. 297 80 Google Scholar
[5]	郑洪英, 李琳, 肖迪 2021 信息网络安全 21 10 Google Scholar Zheng H Y, Li L, Xiao D 2021 Net Inf. Security 21 10 Google Scholar
[6]	Dharavathu K, Mosa A 2020 Int. J. Commun. Syst. 33 e4369
[7]	Zhao J F, Wang S Y, Chang Y X, Li X F 2015 Nonlinear Dyn. 80 1721 Google Scholar
[8]	Khan M 2015 Nonlinear Dyn. 82 527 Google Scholar
[9]	Chai X L, Gan Z H, Yuan K, Yang L Chen Y R 2017 Chin. Phys. B 26 020504 Google Scholar
[10]	Chai X L, Fu J Y, Zhang J T, Han D J, Gan Z H 2021 Neural Comput. Appl. 33 10271 Google Scholar
[11]	Ran Q W, Yuan L, Zhao T Y 2015 Opt. Commun. 348 43 Google Scholar
[12]	Kaur M, Kumar V 2018 Int. J. Bifurcat. Chaos 28 1850132 Google Scholar
[13]	Yasser I, Khalifa F, Mohamed M A, Samrah A S 2020 Complexity 2020 9597619 Google Scholar
[14]	Wu J H, Liao X F, Yang B 2017 Signal Pro. 141 109 Google Scholar
[15]	Wang X Y, Feng L, Zhao H Y 2019 Inf. Sci. 486 340 Google Scholar
[16]	Wang X Y, Li Z M 2019 Optics Lasers Engin. 115 107 Google Scholar
[17]	Zhang Y 2018 Multimed. Tools Appl. 77 21589 Google Scholar
[18]	Bansal R, Gupta S, Sharma G 2017 Multimed. Tools. Appl. 76 16529 Google Scholar
[19]	Hua Z Y, Zhou Y C, Huang H J 2019 Inf. Sci. 480 403 Google Scholar
[20]	Wang X Y, Yang J J 2021 Inf. Sci. 569 217 Google Scholar
[21]	Mandal M K, Kar M, Singh S K, Barnwal V K 2014 Secur. Commun. Netw. 7 2145 Google Scholar
[22]	Wang X Y, Gao S 2020 Inf. Sci. 539 195 Google Scholar
[23]	Wang M X, Wang X Y, Zhao T T, Zhang C, Xia Z Q, Yao N M 2021 Inf. Sci. 544 1 Google Scholar
[24]	Wang X Y, Wang T, Xu D H, Chen F 2014 Int. J. Modern Phys. B 28 1450023 Google Scholar
[25]	Zhou S, Wang X Y, Wang M X, Zhang Y Q 2020 Chaos Solitons & Fract. 141 110225 Google Scholar
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