Introduction

CSSE

Computer Systems Science & Engineering

0267-6192

Tech Science Press

USA

36652

10.32604/csse.2023.036652

Article

A Cross-Plane Color Image Encryption Algorithm Based on 1D-SLM

Wang

Xiaohong

Huiqing

Yuying

Huang

Shuzhen

sdgc_whq@suet.edu.cn School of Artificial Intelligence, Shandong Vocational and Technical University of Engineering, Jinan, 250200, China

*Corresponding Author: Shuzhen Huang. Email: sdgc_whq@suet.edu.cn

2023

1712023

46 1 1093 1108 08102022 15112022

2023

Wang et al.

This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

With the rapid development of 5G technology, it has become fast and easy for people to transmit information on the Internet. Digital images can express information more intuitively, so transmitting information through images has excellent applications. This paper uses a new chaotic system called 1D-Sin-Logistic-Map (1D-SLM). 1D-SLM has two control parameters, which can provide larger parameter space, and the parameter space in the chaotic state is continuous. Through Lyapunov exponent analysis (LE), bifurcation diagrams analysis, spectral entropy analysis (SE), and 0-1 test, it is verified that 1D-SLM has complex dynamic behavior and is very suitable for cryptography. Compared with other 1D chaotic systems, the 1D-SLM has a larger Lyapunov exponent (LE) and spectral entropy (SE). For color image encryption algorithms, only relying on chaotic mapping is not enough to ensure security. So combined with 1D-SLM, we design a color image encryption algorithm, which is implemented by plane expansion, which reduces the correlation between the three channels of color images. The experimental results show that the proposed cross-plane color image encryption algorithm is safe and resistant to common attack methods.

Chaos theory chaotic system 1D-SLM image encryption information security

1Introduction

Digital images can express information more intuitively, so the transmission of information through images has great applications [1–3]. The ensuing transmission process brings many security risks. Many image protection algorithms have been proposed nowadays [4–8]. Chaos and cryptography have many similarities and connections [9–11]. A chaotic system can generate a pseudo-random key stream with good performance, which is unpredictable and very suitable for cryptography [12–14].

In order to design efficient and secure cryptosystems, image encryption algorithms combined with chaos have gradually become a hot field [15,16]. Kang et al. used two identical 4D chaotic systems to couple into an 8D chaotic system. The system has eight initial values and eight control parameters. The key stream of the cryptosystem is generated by this 8D chaotic system, and combined with 2D-VMD, a new image encryption method is proposed [17]. Yu et al. proposed a 6D fractional-order chaotic system called 6D-FMHNN. They studied this system’s coexistence attractor characteristics and applied it to image encryption. The designed algorithm is secure and sensitive to secret keys [18]. Although high-dimensional chaotic systems have more control parameters and can provide a larger parameter space for cryptosystems, their keystream generation efficiency is not high [19–21]. The low-dimensional chaotic system has smaller control parameters, and its key stream generation is efficient. However, the low-dimensional chaotic system has fewer control parameters. The cryptosystem is challenging to resist brute force attacks. This paper designs a new one-dimensional chaotic system called 1D-SLM. This chaotic system has two control parameters, an improvement on the Logistic map. The parameters corresponding to the chaotic region of 1D-SLM are continuous, and the chaotic region is wide, which is very suitable for the design of cryptographic systems.

For color image encryption algorithms, only relying on chaotic mapping is not enough to ensure security [22]. Zhou et al. used a one-dimensional chaotic system to generate multiple different key streams, using different signals to encrypt the three channels of the color image separately. The proposed encryption algorithm is simple and efficient [23]. Wang et al. used the hyperchaotic Lorenz system to generate the key stream of the cryptographic system. In order to enhance the security of the algorithm, they introduced the matrix semi-tensor product diffusion strategy. The encryption algorithm showed promising results on grayscale images. They extended this algorithm to color image encryption [24]. These algorithms encrypt the three channels of the plaintext image simultaneously, ignoring the correlation between the three channels. The attacker can obtain all the plaintext information by deciphering the content of one channel [25,26]. This paper proposes an encryption algorithm for the cross-plane, which treats the three channels as a whole, and the three channels interact with each other during encryption. This design structure increases the algorithm’s security, can resist common attack methods, and ensures the security of color images during transmission.

2Performance Analysis of the New Chaos System 2.1Existing One-Dimensional Chaotic Systems

Table 1 gives some one-dimensional chaotic systems, including the analytic formulas and control parameters. In Table 1, f0 is the initial value, fn is the iterative value. p and q are the control parameters.

Table 1 Existing one-dimensional chaotic systems

Name	Expressions	Parameters
Logistic map [27]	fn+1=p⋅fn(1−fn)	p
Cubic map [28]	fn+1=p⋅fn3+(1−p)fn	p
Sin map [29]	fn+1=psin⁡(πfn)	p
1D-SMCLM [30]	fn+1=psin⁡(q⋅sin⁡(πfn)⋅fn(1−fn)+1)	p and q

2.2New Chaos System 1D-Sin-Logistic-Map

The Logistic map has only one control parameter, the mapping range of the chaotic state is narrow, and the parameters are discontinuous. Therefore, we designed a new chaotic system named 1D-Sin-Logistic-Map (1D-SLM). The 1D-SLM is defined as,(1) fn+1=sin⁡(q⋅(1−p⋅fn⋅(1−fn))×1002+1).

In Eq. (1), f0 is the initial value, f0∈(0, 1) . fn is the iterative value, fn∈(0, 1) . p and q are control parameters, p∈(0,+∞) and q∈(0,+∞) .

2.3Bifurcation Diagram of 1D-SLM

The bifurcation diagram reflects the trajectory of the nonlinear dynamic system from the periodic motion state to the chaotic motion state. The Bifurcation diagrams of 1D-SLM are shown in Fig. 1 under different parameter spaces. The Bifurcation diagrams of existing one-dimensional chaotic systems in Table 1 are shown in Fig. 2. Compared with the Logistic map, Cubic map, Sin Map, and 1D-SMCLM, the 1D-SLM has a larger chaotic interval and more complex chaotic behavior.

Figure 1 Bifurcation diagrams of 1D-SLM Figure 2 Bifurcation diagrams of existing one-dimensional chaotic systems 2.4Lyapunov Exponents Analysis

Lyapunov exponents analysis is one of the most effective means of evaluating the dynamic behavior of nonlinear dynamical systems. The LE is defined as [31,32],(2) LE=limt→+∞1t∑i=1n−1ln⁡|f′(xi)|.

When the Lyapunov exponent is greater than 0, it means that in this parameter space, the system is in a chaotic state. The Lyapunov exponents of 1D-SLM are shown in Fig. 3. The Lyapunov exponents of existing one-dimensional chaotic systems are shown in Fig. 4. It can be seen from Figs. 3 and 4 that the 1D-SLM shows the chaotic global state. Therefore, 1D-SLM can produce a keystream with excellent performance. Compared with other 1D chaotic systems, the 1D-SLM has a larger Lyapunov exponent, and the parameter space in the chaotic state is continuous.

Figure 3 Lyapunov exponents analysis of 1D-SLM Figure 4 Lyapunov exponents analysis of existing one-dimensional chaotic systems 2.50–1 Test

The 0–1 test is a test algorithm that measures the presence of chaos in a time series. The 0–1 test of 1D-SLM is shown in Fig. 5. The 0–1 test of existing one-dimensional chaotic systems is shown in Fig. 6. Figs. 5 and 6 show the motion state of 1D-SLM is a Boolean motion state. The Logistic map, Cubic map, Sin map, and 1D-SMCLM exhibit regular motion states in the same parameters. It shows that 1D-SLM can provide sequences with more complex dynamical behavior.

Figure 5 0–1 test analysis of 1D-SLM Figure 6 0–1 test analysis of existing one-dimensional chaotic systems 2.6Spectral Entropy Analysis

Spectral entropy reflects the energy of the signal. The greater the spectral entropy, the greater the signal’s energy and the more complex the dynamic behavior of the signal. The spectral entropy of 1D-SLM and the Logistic map, Cubic map, Sin map, and 1D-SMCLM are shown in Fig. 7. It can be seen from Fig. 7 that compared with the chaotic sequence generated by the Logistic map, Cubic map, Sin map, and 1D-SMCLM, the chaotic sequence generated by 1D-SLM has more potent energy, which indicates that the key stream generated by 1D-SLM has more complex dynamic behavior.

Figure 7 Spectral entropy of 1D-SLM and existing one-dimensional chaotic systems 3Encryption Algorithm 3.1Key Generation

The plaintext image is P ( L×W ), the R channel of the plaintext image is denoted as PR ( L×W ), the G channel is denoted as PG ( L×W ), and the B channel is denoted as PB ( L×W ). Calculate the sum of the plaintext pixel values of each channel by(3) {k1=sum(PR)k2=sum(PG)k3=sum(PB).

The key of the cryptosystem is K1=k1/k2, K2=k2/k3, K3=k3/k1 .

3.2Cross-Plane for Color Images

Given the original initial value v1 and parameter p1 , q1 of 1D-SLM. According to the secret key in Section 3.1, the new initial value and parameter are,(4) v1=v1+K1, p1=p1+K2, q1=q1+K3.

The new initial value and parameters are brought into the 1D-SLM iteration to generate the key stream, discard the first 200 iterations of the initial value, denoted as X1 ( L×W ), definition(5) U=floor(|X1×1010|)mod6+1.

The rule for cross-plane for color images is(6) {P(Rl,w, Gl,w, Bl,w)=P(Rl,w, Gl,w, Bl,w), Ul,w=1P(Rl,w, Gl,w, Bl,w)=P(Rl,w, Bl,w, Gl,w), Ul,w=2P(Rl,w, Gl,w, Bl,w)=P(Bl,w, Gl,w, Rl,w), Ul,w=3P(Rl,w, Gl,w, Bl,w)=P(Bl,w, Rl,w, Gl,w), Ul,w=4P(Rl,w, Gl,w, Bl,w)=P(Gl,w, Rl,w, Bl,w), Ul,w=5P(Rl,w, Gl,w, Bl,w)=P(Gl,w, Bl,w, Rl,w), Ul,w=6.

3.3Zigzag Scrambling

Concatenate the plaintext processed by cross-planes in Section 3.2 into a new plaintext P ( L×3W ), in the order of PR , PG , and PB . Perform zigzag scrambling on the new plaintext P to obtain the scrambled matrix S. The zigzag scrambling is shown in Fig. 8.

Figure 8 Zigzag scrambling of 3 × 4 3.4Diffusion

Given the original initial value v2 and parameter p2 , q2 of 1D-SLM. According to the secret key in Section 3.1, the new initial value and parameter are,(7) v2=v2+K3, p2=p2+K1, q2=q2+K2.

The new initial value and parameters are brought into the 1D-SLM iteration to generate the key stream, discard the first 200 iterations of the initial value, denoted as X2 ( L×3W ), definition(8) T=floor(|X2×1010|)mod256.

The diffusion process is(9) {C[1, 1]=T[1, 1]⊕S[1, 1], C[1, u]=T[1, u]⊕S[1, u]⊕C[1, u−1], u=2:3W, C[v, 1]=T[v, 1]⊕S[v, 1]⊕C[v−1, 1], v=2:L, C[v, u]=T[v, u]⊕C[v−1, u]⊕S[v, u]⊕C[v, u−1].

Output the ciphertext C, and synthesize the ciphertext color image.

3.5Decryption Algorithm

The decryption algorithm is shown in Algorithm 1.

4Performance Analysis 4.1Simulation Experiments

The visual analysis of the proposed algorithm is shown in Figs. 9–11. Visual analysis shows that, visually, the proposed algorithm is safe.

Figure 9 Visual analysis of Lena with the size of 512 × 512 Figure 10 Visual analysis of Airplane with the size of 512 × 512 Figure 11 Visual analysis of Baboon with the size of 512 × 512 4.2Histogram Analysis

The histogram analysis of the proposed algorithm is shown in Fig. 12 [33]. Histogram analysis shows that the distribution of pixel values obtained by the proposed algorithm is uniform, and the algorithm has excellent security.

Figure 12 Histogram analysis 4.3Differential Attack Analysis

A secure algorithm is sensitive to plaintext, and NPCR and UACI are two metrics for detecting plaintext sensitivity [34].(10) { NPCR=∑i,j{ 1,c1(i, j)≠c2(i, j)0,c1(i, j)=c2(i, j) L×WUACI=1L×W∑i,j|c1(i, j)−c2(i, j)|255.

For a image with the size of 512 × 512, when the value of NPCR is [99.5893%, 100%] , and the value of UACI is [33.3730%, 33.5541%] , at this point, it indicates that the algorithm is sensitive to plaintext. The test results of NPCR and UACI are shown in Table 2. The results of the differential attack show that the algorithm is sensitive to plaintext.

Table 2 Differential attack analysis

Image		NPCR (%)	Pass (Y/N)	UACI (%)	Pass (Y/N)
Lena	R	99.6059	Y	33.4427	Y
	G	99.5914	Y	33.4700	Y
	B	99.6120	Y	33.4747	Y
Airplane	R	99.5960	Y	33.4068	Y
	G	99.6219	Y	33.4579	Y
	B	99.6158	Y	33.4020	Y
Baboon	R	99.6025	Y	33.4966	Y
	G	99.6070	Y	33.4701	Y
	B	99.6120	Y	33.5190	Y
Peppers	R	99.6082	Y	33.6462	Y
	G	99.5964	Y	33.4047	Y
	B	99.6067	Y	33.5091	Y

4.4NIST for Ciphertext and Plaintext

The NIST effectively checks whether the data has randomness [35–37]. The NIST test results of ciphertext and plaintext are shown in Table 3. NIST test results show that the ciphertext has terrific randomness.

Table 3 NIST for ciphertext and plaintext

Sub-tests	Pass (Y/N)	Ciphertext	Pass (Y/N)
Frequency	N	0.2932	Y
FFT	N	0.0027	Y
Block Frequency	N	0.0164	Y
Longest Run	N	0.0618	Y
Overlapping Template	N	0.1995	Y
Approximate Entropy	N	0.2589	Y
Rank	N	0.1137	Y
Non-Overlapping Template	N	0.5595	Y
Cumulative Sums	N	0.4145	Y
Random Excursions	N	0.3504	Y
Linear Complexity	N	0.3711	Y
Runs	N	0.2277	Y
Serial	N	0.1516	Y
Random Excursions Variant	N	0.5341	Y
Universal	N	0.2589	Y

4.5Adjacent Pixel Correlation Analysis

A secure encryption algorithm can reduce ciphertext’s horizontal, vertical, and diagonal adjacent pixel correlations. Otherwise, the algorithm can be easily cracked by statistical attacks. The adjacent pixel correlation analysis of the algorithm is shown in Fig. 13.

Figure 13 Correlation analysis of Baboon

The quantitative results of the correlation analysis of the proposed algorithm are shown in Table 4. The correlation analysis shows that the correlation of the plaintext image in the three directions is very high. The correlation of the ciphertext image in the three directions is very high, close to 0 (theoretical value). Therefore, the proposed encryption algorithm has good security.

Table 4 Correlation coefficients analysis

Image		Plaintext			Ciphertext
Image		H	V	D	H	V	D
Lena	R	0.9797	0.9893	0.9696	−0.0001	−0.0012	−0.0019
	G	0.9689	0.9823	0.9554	0.0014	−0.0003	−0.0004
	B	0.9325	0.9574	0.9180	−0.0027	−0.0005	−0.00006
Airplane	R	0.9726	0.9568	0.9343	−0.00009	−0.0006	0.0017
	G	0.9576	0.9676	0.9324	0.0025	−0.0038	0.0005
	B	0.9639	0.9352	0.9145	−0.0010	−0.0015	0.0017
Baboon	R	0.9230	0.8659	0.8543	0.0008	−0.0013	0.0030
	G	0.8654	0.7650	0.7347	−0.0009	−0.0024	0.0025
	B	0.9072	0.8807	0.8397	−0.0016	0.0008	−0.0020
Peppers	R	0.9635	0.9663	0.9563	0.0022	−0.0013	−0.0006
	G	0.9810	0.9817	0.9685	0.0009	−0.0027	0.0022
	B	0.9663	0.9662	0.9476	−0.0013	0.0002	−0.0005

4.6R, G, B Correlation Analysis

A safe algorithm requires a slight correlation between adjacent pixels and a small correlation between the three channels. The correlation analysis between the three channels of the proposed algorithm is shown in Tables 5 and 6. The results of correlation analysis show that the proposed algorithm can reduce not only the correlation of adjacent pixels but also the correlation between different channels, and the attacker cannot obtain the information of the remaining channels from one channel.

Table 5 Plaintext correlation between R, G, and B

Image	(R, G)	(R, B)	(G, B)
Lena	0.8785	0.6763	0.9104
Airplane	0.9211	0.8410	0.9378
Baboon	0.3565	0.1236	0.8071
Peppers	0.2748	0.3951	0.8377

Table 6 Ciphertext correlation between R, G, and B

Image	(R, G)	(R, B)	(G, B)
Lena	−0.0013	0.0006	0.0031
Airplane	0.0005	0.0007	−0.0007
Baboon	0.0010	0.0004	−0.0041
Peppers	−0.0026	−0.0004	0.0006

4.7Information Entropy Analysis

The information entropy is defined as H=∑i=0255p(gi)log21p(gi).

Information entropy is an index to analyze the randomness of image information distribution. The higher the information entropy, the more chaotic the information.

The information entropy analysis of the proposed algorithm is shown in Table 7. The local information entropy reflects the degree of local confusion of the image [38]. When the value of local information entropy is between 7.9015 and 7.9034, it indicates that the local information of the image has good randomness. The local information entropy is shown in Table 7. The information entropy analysis results show that the information entropy of the ciphertext is close to the theoretical value, indicating that the algorithm has a good encryption effect.

Table 7 Information entropy analysis and Local information entropy

Image		IE of plaintext	IE of ciphertext	Local information entropy	Pass or No Pass
Lena	R	7.2530	7.9993	7.9021	Pass
	G	7.5951	7.9993	7.9026	Pass
	B	6.9685	7.9994	7.9022	Pass
Airplane	R	6.7177	7.9992	7.9019	Pass
	G	6.8055	7.9993	7.9030	Pass
	B	6.2139	7.9993	7.9026	Pass
Baboon	R	7.7066	7.9993	7.9025	Pass
	G	7.4752	7.9993	7.9029	Pass
	B	7.7522	7.9993	7.9018	Pass
Peppers	R	7.3388	7.9994	7.9023	Pass
	G	7.5183	7.9993	7.9031	Pass
	B	7.0583	7.9993	7.9023	Pass

4.8Key Analysis

The key space of a cryptographic system is at least greater than 2100 that to meet the conditions for resisting brute force attacks. The keys in this paper include v1 , p1 , q1 , v2 , p2 , q2 , K1 , K2 , K3 . If the computational precision of the computer is 1014 , the size of the key space of the proposed algorithm is, K=1014×1028×1028×1014×1028×1028×1014×1014=10168≈2558.

The designed algorithm is strong enough to resist brute-force attacks.

A highly sensitive key is a necessary condition for an encryption algorithm. This section tests the key sensitivity of the proposed algorithm, with the initial key set to v2=0.985612 , p2=10.36985 , q2=11.23654 . The key sensitivity analysis is shown in Fig. 14. Key sensitivity analysis shows that the key of the proposed algorithm is sensitive. Calculate the difference between the images in Fig. 14 using NPCR and UACI shown in Table 8 [35,36].

Figure 14 Key sensitivity analysis

Table 8 NPCR and UACI of key sensitivity analysis

NPCR (%) and UACI (%)	Fig. 14a	Fig. 14b	Fig. 14c	Fig. 14d
Fig. 14a	0	32.2304	32.2352	32.2123
Fig. 14b	99.5924	0	33.4445	33.4585
Fig. 14c	99.6063	99.6138	0	33.4996
Fig. 14d	99.6081	99.6105	99.6200	0

4.9Robustness Analysis

The image will be interfered with by noise during transmission [39]. A safe algorithm should have good resistance to this attack. The robustness analysis of the algorithm is shown in Fig. 15. The PSNR is generally used to measure the restoring ability of images. The results of MSE are shown in Table 9. The results of PSNR are shown in Table 10.

Figure 15 Robustness analysis

Table 9 MSE of robustness analysis

Images	MSE	Images	MSE
Figs. 15a and 15d	11413	Fig. 15d and original image	2750
Figs. 15b and 15e	10789	Fig. 15e and original image	1461
Figs. 15c and 15f	10270	Fig. 15f and original image	305

Table 10 PSNR of robustness analysis

Images	PSNR (dB)	Images	PSNR (dB)
Figs. 15a and 15d	7.5565	Fig. 15d and original image	13.7367
Figs. 15b and 15e	7.8006	Fig. 15e and original image	16.4833
Figs. 15c and 15f	8.0148	Fig. 15f and original image	23.2832

4.10Time Analysis

Efficiency analysis is an important index to evaluate the practicability of an algorithm [40,41]. The efficiency analysis of the proposed algorithm is shown in Table 11. Experimental environment, Matlab R2019a, Windows 11, Intel i3-10105. Efficiency analysis shows that the proposed algorithm has excellent performance and is more practical.

Table 11 Time analysis

Algorithms	Size	Time (s)
Proposed	512 × 512 × 3	0.9856
Algorithm in Ref. [42]	512 × 512 × 3	1.4597
Algorithm in Ref. [43]	512 × 512 × 3	1.3053
Algorithm in Ref. [44]	512 × 512 × 3	1.2122
Algorithm in Ref. [45]	512 × 512 × 3	1.2271

4.11Comparative Analysis

In this section, the proposed algorithm is compared with some classical algorithms, and the comparison results are shown in Table 6. The values in Table 12 are averages. The comparison results show that our algorithm is more secure than the algorithm in Ref. [42], algorithm in Ref. [43], algorithm in Ref. [44], and algorithm in Ref. [45].

Table 12 Comparative analysis

Algorithms	IE	Horizontal	Vertical	Diagonal	Keyspace
Proposed	7.9993	0.0014	0.0007	0.0008	2⁵⁸⁸
Algorithm in Ref. [42]	7.5618	−0.0107	−0.0079	−0.0014	2²³²
Algorithm in Ref. [43]	7.9993	0.0034	−0.0008	−0.0010	2⁵¹²
Algorithm in Ref. [44]	7.9993	0.0038	0.0024	0.0020	2⁵¹²
Algorithm in Ref. [45]	7.9973	0.0060	0.0146	0.0384	2¹⁸⁹

5Conclusion

In this paper, a new chaotic system, 1D-SLM, is proposed. This system has two control parameters that give the cryptosystem a larger key space. It is verified that the key stream generated by 1D-SLM has strong energy and can generate a key stream with excellent performance that is very suitable for cryptography by the Lyapunov exponent, bifurcation graph analysis, and spectral entropy analysis. We propose a new image encryption algorithm based on 1D-SLM. This encryption algorithm combines the three channels of the color image to encrypt simultaneously. In other words, during the encryption process, the three channels of the color image are interactive. This design structure reduces the correlation between image channels. Through performance analysis, it is verified that the proposed algorithm has good performance. Comparative analysis shows that the proposed algorithm has excellent security.

Funding Statement

This research was supported by the National Natural Science Foundation of China (61802212).

Conflicts of Interest

The authors declare that they have no conflicts of interest to report regarding the present study.

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