
==== Front
iScience
iScience
iScience
2589-0042
Elsevier

S2589-0042(24)01993-X
10.1016/j.isci.2024.110768
110768
Article
Face privacy protection scheme by security-enhanced encryption structure and nonlinear dynamics
Xie Zhiyu 12
Lin Yiting 12
Liu Tengyu 12
Wen Heping wenheping@uestc.edu.com
123∗
1 University of Electronic Science and Technology of China, Zhongshan Institute, Zhongshan 528402, China
2 School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
∗ Corresponding author wenheping@uestc.edu.com
3 Lead contact

20 8 2024
20 9 2024
20 8 2024
27 9 1107687 4 2024
4 7 2024
15 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
Summary

This article proposes a secure communication enhancement scheme based on face detection and chaotic partition permutation. The scheme uses edge detection technology to detect facial information, which is then used as an encryption object. The hash value of the plaintext image is extracted as the secret key to the chaotic sequence generated by the chaotic system. Then a series of encryption operations are performed on the face image to obtain the final ciphertext image. In this article, two chaotic systems are used to generate pseudo-random chaotic sequences for different encryption steps. The initial key is computed by combining the hash function of the image and external parameters. The experimental results and security analysis show that the algorithm has excellent encryption effectiveness and security performance against various typical attacks.

Graphical abstract

Highlights

• Facial recognition technology is used to identify facial images used to encrypt

• Local encryption can improve the efficiency of encryption algorithms

• The encryption algorithm driven by chaos sequences improves security

• The proposed security enhancement structure can resist cryptographic attacks

Applied sciences; Computer science; Engineering

Subject areas

Applied sciences
Computer science
Engineering
Published: August 20, 2024
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pmcIntroduction

With the rapid development of network technology, massive image information is rapidly generated and shared through the network.1,2,3 At the same time, unauthorized access and misuse issues have emerged. In recent years, encryption technology has received significant attention as an important privacy protection technique.4,5,6 Current image encryption technologies7,8,9 are typically combined with other techniques to enhance encryption security, such as chaos theory,10,11,12 biometric encoding,13,14,15 Fourier transform,16,17,18 and so on.19,20,21 Most existing image encryption schemes primarily focus on encrypting the entire image.22,23,24 By fully utilizing the unpredictability and sensitivity of chaos, image encryption algorithms can provide strong protection against potential attacks or unauthorized access, making them the preferred solution in the field of image security.25,26,27

Looking at the current international situation, research on chaotic digital image encryption28,29,30 has become a hot topic. With the popularization of multimedia information, local encryption schemes for specific sensitive information have gradually become the current mainstream security communication solution.31,32 In recent years, many scholars have devoted themselves to research in this field and have achieved good results. In 2022, Gao et al.33 introduced a new method for encrypting facial images. This method only encrypts the face region in the image, and the remaining regions remain unchanged. The experimental results show that the method has high computational efficiency and strong security. In 2023, Du et al.34 proposed a new multi-face image encryption scheme based on non-adjacent dynamic coupling mapping lattice, confirming that the algorithm has high security. In 2024, Li et al.35 proposed a ciphertext face recognition system based on secure inner product protocols and analyzed its correctness, security, and time expenditure, all of which yielded good results. From these studies, it can be seen that most secure communication algorithms based on face recognition have achieved satisfactory results, significantly advancing the development of information security technology and leading to the emergence of more encryption algorithms. Although these studies have yielded satisfactory results, as encryption systems become increasingly complex and data volumes grow exponentially, reducing data storage space and transmission bandwidth is becoming increasingly crucial. Therefore, proposing a face image encryption algorithm constructed using chaotic mappings becomes an especially important and urgent task to defend against various illegal attacks.

In response to the above problem, this article proposes a scheme to protect the privacy of portrait information in images using chaotic partition scrambling technology. First, the original image containing faces is subjected to face detection using edge detection recognition technology, and the detection results are used for further encryption. Then, the feature values of the original image are processed to obtain the key for generating the chaotic system and the chaotic sequence required for encryption. By using the generated chaotic sequence for partition permutation, irregular diffusion, and bidirectional parallel confusion, the final ciphertext is obtained. The experimental results show that the algorithm has excellent encryption effect and good encryption efficiency, and the proposed image encryption algorithm can securely resist various attacks.

The main innovations and contributions of this article are as follows:(1) Most existing image encryption schemes mainly focus on encrypting the entire image. This often leads to loss of encryption performance due to processing redundant information. In order to enhance the security and efficiency of the encryption system, this algorithm encrypts facial images. In this article, facial recognition technology is used to encrypt facial images with a focus, without considering protecting redundant information. Encrypting facial images locally enhances the security and efficiency of the encryption system.

(2) Traditional encryption algorithm is relatively simple compared with traditional permutation algorithms and encoding algorithms. The algorithm dynamically adjusts the permutation algorithm based on chaotic sequence to achieve excellent encryption effect in the shortest possible time. The final ciphertext is obtained by using irregular diffusion and sequence bidirectional staggered confusion. Experimental results show that the algorithm effectively enhances the security of image encryption and improves the security and robustness of the encryption algorithm, making it more resistant to attacks and ensuring the confidentiality of transmitted images.

(3) Many existing encryption algorithm structures are not reasonable. If there is no relevant plaintext or ciphertext feedback, they are easily susceptible to known-plaintext or chosen-plaintext attacks. This secure image encryption scheme uses a dynamic feedback mechanism to update encryption keys based on encrypted data. Building upon our current foundation of cryptanalysis research,36,37,38 it enhances security and the ability to resist attacks such as chosen-plaintext and chosen-ciphertext attacks.

The rest of the article is organized as follows. Section related works contains the related works. Section the proposed face image privacy protection scheme discusses the encryption algorithm. Section results and analysis presents the results and analysis. The last section concludes the article.

Related works

The adopted chaotic system

Chaotic systems have properties such as strong ergodicity, randomness, and output uncertainty. Integrating chaotic systems into encryption schemes can enhance security by expanding the key space and improving computational efficiency. Using multiple chaotic systems further introduces complex nonlinear characteristics, thereby increasing the complexity and challenge for potential attackers.

4D-AHS

Applying controllers λ1x4,λ2x4,λ3x4 to the state equation of 3D chaotic system, Υand setting x4=-τx1, the following adaptive 4D autonomous hyperchaotic system39 used in this article can be constructed.(Equation 1) {x1=α(x2−x1)+λ1x4x2=βx1−x1x3+λ2x4x3=−Υx3+x1x2+λ3x4x4=−τx1

where α,β,Υ,τ,λ1,λ2,λ3 are the control parameters of the system. When α=35,β=35,Υ=3,τ=5,λ1=1,λ2=0.2 and λ3=0.3, the system exhibits hyperchaotic behavior.

2D-LSCM

The security and efficiency of chaotic image encryption depend heavily on the properties of the chaotic sequence. In this study, a 2D discrete chaotic system40 is chosen for image encryption due to its simplicity, ease of implementation and superior performance. The definition of the system is as follows:(Equation 2) {xi=sin(π(4rxi−1(1−xi−1)+(1−r)sin(πyi−1)))yi=sin(π(4ryi−1(1−yi−1)+(1−r)sin(πxi)))

where r is the system control parameter.

SHA256

The SHA256 algorithm is a one-way hash function used in cryptography and information security. Its primary purpose is to transform messages of any length into shorter, fixed-length hash values to ensure data integrity and security. This algorithm is widely used in data integrity verification, password security, and digital signatures. Unlike other encryption schemes, the SHA256 algorithm does not use a single hash value as a key.

The proposed face image privacy protection scheme

This article uses two chaotic systems and the SHA256 hash function to generate distinct chaotic keys, as opposed to traditional algorithms. The 4D-AHS is used for chaos-driven three-dimensional partition permutation, while the 2D-LSCM generates the chaotic sequence for diffusion and confusion. This improves the algorithm’s security and makes code cracking more difficult. The flowchart of algorithm design is shown in Figure 1.Figure 1 The proposed algorithm flowchart

Face recognition

To begin the face detection process, the RGB image must first be converted to grayscale for processing, as demonstrated in Figure 2.Figure 2 Image RGB conversion

Next, the image is subjected to mean filtering operations to weaken the detailed parts in the image, avoiding the impact of jagged edges and redundant information on the formation of later connected domains. As can be seen in Figure 3, mean filtering reduces information redundancy in the image.Figure 3 Image mean filtering

As shown in Figure 4, the image is binarized to reduce the computational load of the subsequent morphological operations.Figure 4 Image binarization

Meanwhile, considering the diversity of skin colors, it is difficult to form large connected regions in the facial region after binarizing images of people with darker skin colors. To address this issue, this study adopts a morphological boundary extraction method. By using a sufficiently large structuring element, large closed connected domains can be formed. As can be seen in Figure 5, the boundaries are successfully extracted, forming a relatively large connected domain in the face region.Figure 5 Morphological boundary extraction

In addition, Figures 6 and 7 show that after thickening the boundaries and filling the holes, the black blocks in the eye area are eliminated, thus enlarging the facial connected domain.Figure 6 Border bolding

Figure 7 Image hole filling

As shown in Figure 8, this step uses vertically elongated structural elements for vertical closure operations. Because elements such as the face, neck, hair, and clothing have vertical distribution characteristics, it is easy to separate these adjacent elements when performing morphological boundary extraction, which adversely affects the determination of connected domains. Therefore, in this study, vertically elongated structural elements are used for vertical closure operations to reconnect the upper and lower regions of the face.Figure 8 Vertical image closure

Next, use horizontally elongated structural elements for horizontal erosion operations. This is because there are many connected domains below the head region which can easily interfere with the determination of the largest connected domain. As this part is mostly vertically distributed, horizontal erosion can segment these large connected domains. It is important to note that the degree of segmentation should not be too large to ensure that the previous closure operation remains meaningful. Figure 9 shows that after the vertical closure operation, the connected domains are connected and the facial connected domain continues to expand. Subsequently, the horizontal erosion reduces the connected domains at the bottom of the image while preserving the size of the facial connected domain as much as possible.Figure 9 Image lateral corrosion

After a number of operations, it is found that factors such as the background clutter in Figure 10 can also generate numerous connected domains that may interfere with the final judgment and therefore need to be eliminated.Figure 10 Eliminate redundant connected domain of image

After several rounds of screening, the largest connected region among the remaining regions is selected and surrounded by a rectangular box as a result of face detection, as shown in Figure 11.Figure 11 Maximum connectivity domain of intercepted image

The final face detection image is shown in Figure 12.Figure 12 Face detection effect

Chaotic initial value disturbance and sequence preprocessing

After performing feature extraction on the plaintext image P using the SHA256 function, a fixed-length character-based 64-bit hexadecimal number is obtained. To facilitate computation, each digit is converted to a decimal number. Each digit is represented by the variable Hash, respectively, denoted as Hash1, Hash2, Hash3 ⋯ Hash64.

The image hashes are processed mathematically to obtain the initial key parameters. The hash values of the images, represented by variables a and b, are used as the initial values for 2D-LSCM. Variables c and d are used as control parameters for 2D-LSCM and iteration counts for 4D-AHS, respectively, after further processing by a and b.

However, it is important to note that in order to further encrypt the key, 2D-LSCM operates as follows:(Equation 3) {R1=2D_LSCM(a+0.1,b+0.1,c,3×H×W)R2=2D_LSCM(a+0.2,b+0.2,c,3×H×W)

where H represents the height of the image and W represents the width of the image; 3×H×W represents the length of the generated chaotic sequence; R1 and R2 are the initial chaotic sequences generated by the chaotic system through iteration. Then, further processing is performed on the obtained chaotic sequence following the steps below:(Equation 4) {Sj=(R1×10000)mod255Sk=(R2×10000)mod255

where Sj and Sk are used in irregular diffusion and bidirectional parallel confusion, respectively.

Chaos-driven three-dimensional partition permutation

Since permutation in existing encryption schemes only changes the positions of pixel values without changing the pixel values themselves, and does not process the color components of the RGB layers, this article proposes chaos-driven three-dimensional partition permutation based on chaotic sequence control that changes both the positions and values of pixels to achieve security enhancement. The specific operational flowchart is shown in Figure 13.Figure 13 Chaos-driven three-dimensional partition permutation flowchart

Step1: First, the plaintext image Image with a size of H×W is partitioned and converted into matrices with a number of rows and columns that is a multiple of 8. If the matrix does not have enough elements, zero padding is performed. The following steps outline the specific operations:(Equation 5) P(x,y,z)=Image

where x is W+W0; y is H+H0 ; W0 and H0 denote the number of ‘0’ to be added and z represents the dimensions of image I.

Step2: Preprocess P from a 3D array to a 5D array, the specific operations are as follows:(Equation 6) Pa(x,y,z,i,j)=P((i−1)×Px+1,(j−1)×Px+1:j×Px,z)

where x, y, and z are the rows, columns, and dimensions of a 5D array, respectively. i and j are the specific positions after blocking, i,j=1,2,… and Px is the number of unit blocks after partition.

Step3: Read the hash features of the image and compute the obtained key d, and after iterating the key through a chaotic system, obtain four chaotic sequences S1, S2, S3 and S4. Use mathematical methods to process and obtain the sequences S1′, S2′, S3′ and S4′, as follows:(Equation 7) {S1′=(S1×1010)mod(H×W)/L2S2′=(S2×1010)mod6S3′=(S3×1010)mod2S4′=(S4×1010)mod6

where L is the size of each block after partitioning the image.

Step4: As shown in Figure 14, the sequence S1′ obtained in Step 3 is used to scramble the partitioned image P.(Equation 8) {t=P(x,y,z,i)P(x,y,z,i)=P(x,y,z,S1’(i))P(x,y,z,S1(i))=t

where x=[1,2⋯H], y=[1,2⋯W], i=1,2⋯H×W(Px)2, z is the dimension of the matrix, and t is an intermediate variable. The matrix after partitioning is called Pa.Figure 14 Partition scrambling flowchart

Step5: Based on the sequence S2′, perform rotation encryption on the initially scrambled image to obtain the post-rotation encryption image matrix. Figure 15 shows the operation diagram, and the specific operation formula is as follows:• When S2′=1 rotate the image 90∘ clockwise;

(Equation 9) {Pa(k,l,z,i,j)=Pa(Px+1−l,k,z,i,j)Pa(Px+1−l,k,z,i,j)=Pa(Px+1−k,Px+1−l,z,i,j)Pa(Px+1−k,Px+1−l,z,i,j)=Pa(1,Px+1−k,z,i,j)Pa(l,Px+1−k,z,i,j)=Pa(k,l,z,i,j)

• When S2′=2 rotate the image 180∘ clockwise;

(Equation 10) {Pa(k,l,z,i,j)=Pa(Px+1,Px+1−l,z,i,j)Pa(Px+1−k,Px+1−l,z,i,j)=Pa(k,l,z,i,j)Pa(k,l+Px,z,i,j)=Pa(Px+1−k,Px/2+1−l,z,i,j)Pa(Px+1−k,Px/2+1−l,z,i,j)=Pa(k,l+Px,z,i,j)

• When S2′=3 rotate the image 270∘ clockwise;

(Equation 11) {Pa(k,l,z,i,j)=Pa(Px+1,Px+1−l,z,i,j)Pa(Px+1−k,Px+1−l,z,i,j)=Pa(k,l,z,i,j)Pa(k,l+Px,z,i,j)=Pa(Px+1−k,Px/2+1−l,z,i,j)Pa(Px+1−k,Px/2+1−l,z,i,j)=Pa(k,l+Px,z,i,j)

• When S2′=4 flip the image horizontally;

(Equation 12) {Pa(k,l,z,i,j)=Pa(k,Px+1−l,z,i,j)Pa(k,Px+1−l,z,i,j)=Pa(k,l,z,i,j)Pa(Px/2+k,l,z,i,j)=Pa(Px/2+k,Px+1−l,z,i,j)Pa(Px/2+k,Px+1−l,z,i,j)=Pa(Px/2+k,l,z,i,j)

• When S2′=5 flip the image vertically;

(Equation 13) {Pa(k,l,z,i,j)=Pa(Px+1−k,l,z,i,j)Pa(Px+1−k,l,z,i,j)=Pa(k,l,z,i,j)Pa(k,Px/2+l,z,i,j)=Pa(Px+1−k,Px/2+l,z,i,j)Pa(Px+1−k,Px/2+l,z,i,j)=Pa(k,Px/2+l,z,i,j)

where k, l are control parameters with specific values of 1,2,…,Px/2. z is the dimension of the array. i ,j are variables with specific values ranging from 1,2,…,m/Px; 1,2,…,n/Px.Figure 15 Partition rotation and resupination flowchart

Step6: Based on the sequence S3′, if the random sequence S3′=1, perform bitwise expansion on the image matrix after rotation encryption, and perform bit-level transformation with the chaotic sequence; if the random sequence S3′=0, take the difference of 255. The specific formula is:(Equation 14) {Pa(x,y,z,i,j)=255−Pa(x,y,z,i,j)ifS3’=0Pa(x,y,z,i,j)=bitxor(Pa(x,y,z,i,j),1)ifS3’=1

Step7: Select the appropriate RGB shuffling based on the random sequence S4′, as follows:• When S4′=1, the three-dimensional components of color are transformed from RGB to RBG. The G color channel is swapped with the B color channel and the data transformation is performed using the ratio specificity operation. The specific operation is as follows:

(Equation 15) {Pa(x,y,2,i,j)=bitxor(Pa(x,y,2,i,j),Pa(x,y,3,i,j))Pa(x,y,3,i,j)=bitxor(Pa(x,y,2,i,j),Pa(x,y,3,i,j))Pa(x,y,2,i,j)=bitxor(Pa(x,y,2,i,j),Pa(x,y,3,i,j))

• When S4′=2, the three-dimensional components of color are transformed from RGB to GRB. The G color channel is swapped with the R color channel and the data transformation is performed using the ratio specificity operation. The specific operation is as follows:

(Equation 16) {Pa(x,y,1,i,j)=bitxor(Pa(x,y,1,i,j),Pa(x,y,2,i,j))Pa(x,y,2,i,j)=bitxor(Pa(x,y,1,i,j),Pa(x,y,2,i,j))Pa(x,y,1,i,j)=bitxor(Pa(x,y,1,i,j),Pa(x,y,2,i,j))

• When S4′=3, the three-dimensional components of color are transformed from RGB to BGR. The B color channel is swapped with the R color channel and the data transformation is performed using the ratio specificity operation as follows:

(Equation 17) {Pa(x,y,1,i,j)=bitxor(Pa(x,y,1,i,j),Pa(x,y,2,i,j))Pa(x,y,3,i,j)=bitxor(Pa(x,y,1,i,j),Pa(x,y,2,i,j))Pa(x,y,1,i,j)=bitxor(Pa(x,y,1,i,j),Pa(x,y,2,i,j))

• when S4′=4, the three-dimensional components of color are transformed from RGB to BGR and then to BRG. Following the alignment of the B color channel and the R color channel, the G color channel and the R color channel are aligned, after which the data is converted by the following specific operations:

(Equation 18) {Pa(x,y,1,i,j)=bitxor(Pa(x,y,1,i,j),Pa(x,y,3,i,j))Pa(x,y,3,i,j)=bitxor(Pa(x,y,1,i,j),Pa(x,y,3,i,j))Pa(x,y,1,i,j)=bitxor(Pa(x,y,1,i,j),Pa(x,y,3,i,j))Pa(x,y,2,i,j)=bitxor(Pa(x,y,2,i,j),Pa(x,y,3,i,j))Pa(x,y,3,i,j)=bitxor(Pa(x,y,2,i,j),Pa(x,y,3,i,j))Pa(x,y,2,i,j)=bitxor(Pa(x,y,2,i,j),Pa(x,y,3,i,j))

• when S4′=5, the three-dimensional components of color are transformed from RGB to GRB and then to GBR. Following the alignment of the R color channel and the G color channel, the R color channel and the B color channel are aligned, and the data is converted by the following specific operations:

(Equation 19) {Pa(x,y,1,i,j)=bitxor(Pa(x,y,1,i,j),Pa(x,y,2,i,j))Pa(x,y,2,i,j)=bitxor(Pa(x,y,1,i,j),Pa(x,y,2,i,j))Pa(x,y,1,i,j)=bitxor(Pa(x,y,1,i,j),Pa(x,y,2,i,j))Pa(x,y,2,i,j)=bitxor(Pa(x,y,2,i,j),Pa(x,y,3,i,j))Pa(x,y,3,i,j)=bitxor(Pa(x,y,2,i,j),Pa(x,y,3,i,j))Pa(x,y,2,i,j)=bitxor(Pa(x,y,2,i,j),Pa(x,y,3,i,j))

• when S4′=0, the RGB order of the input image remains unprocessed.

Step8: To process the 5D array Pa back to a 3D array, the specific operations are as follows:(Equation 20) Cp((i−1)×Px+1:i×Px,(j−1)×Px+1:j×Px,:)=Pa(:,:,:,i,j)

Finally, the ciphertext image Cp is obtained after the chaos-driven three-dimensional partition permutation.

Irregular diffusion

To achieve the diffusion property of the encryption algorithm, we used a diffusion algorithm. The processing order is determined by the generated chaotic sequence, meaning that a pixel can be influenced by any pixel in other color planes. The specific operations are outlined below:(Equation 21) Cii,j,k={(Cpi,j,k+CpM,N,3+Ai,j,k)mod256ifi=1,j=1,k=1(Cpi,j,k+CiM,N,k−1+Ai,j,k)mod256ifi=1,j=1,k≠1(Cpi,j,k+CiM,j−1,k+Aij,k)mod256ifi=1,j≠1(Cpi,j,k+Cii−1,j,k+Aij,k)mod256ifi≠1

where Cp is the input color image and A is the chaotic matrix generated by the chaotic sequence Sj.

Sequence bidirectional staggered confusion

Existing confusion operations are typically processed sequentially on a pixel-by-pixel basis, which does not fully utilize computational parallelism. To address this issue, we have implemented a design that utilizes parallel processing. Figure 16 shows the order of the operations and Algorithm 1 introduces the specific operation process.Figure 16 Sequence bidirectional staggered confusion sketch

Algorithm 1 Sequence bidirectional staggered confusion

Input: The chaotic matrix S reshaped by the chaotic sequence Sk, intermediate ciphertext image Ci

Output: Final ciphertext image C

1: [M,N]= size (Ci);

2: for i←1 to M do

3: for ci←i+1 to N do

4:  I(i,ci)= mod (IC(i,ci)+EI(i,ci−1),256);

5: end for

6: for ri←i+1 to N do

7:  I(ri,i)= mod (IC(ri,i)+EI(ri−1,i),256);

8: end for

9: end for

10: C= mod (I+S,256);

Overall, the encryption process is hereby completed. And C is the final ciphertext image.

Results and analysis

Experimental platform

The experiment used a computer running the Windows 11 operating system and equipped with the MATLAB R2023b experimental software. The host is equipped with an 11th Gen Intel Core i7-11800H @ 2.30 GHz eight-core processor and 32 GB of memory. All images used in this article are from the USC-SIPI and Pexels database.

Statistical analysis

Histogram analysis

A sophisticated image encryption algorithm cleverly hides important information by manipulating the image distribution to simulate noise. After our processing, the selected image, as shown in Figure 17, has a histogram distribution that resembles noise after encryption. This effectively eliminates the potential for attackers to extract valuable information.Figure 17 Plaintext and ciphertext images and corresponding histograms

(A) Plain-images.

(B) Histograms of (A).

(C) Encryption results of (A).

(D) Histograms of (C).

(E) Decryption results of (C).

Correlation analysis

Disrupting the correlation between pixels is a crucial task for an image encryption algorithm. It effectively thwarts attackers’ decryption attempts that rely on pixel correlations. This article uses the ‘Lena’ image as an example to elucidate this process. We randomly selected 3000 pairs of adjacent pixels from both the plaintext and ciphertext. We then calculated the correlation coefficients for these adjacent pixels in horizontal, vertical, diagonal, and anti-diagonal directions. The corresponding scatterplots are depicted in Figure 18. The outcomes of the correlation analysis are presented in Table 1. The correlation coefficient between adjacent pixels can be represented as:(Equation 22) {rxy=cov(x,y)D(x)D(y)cov(x,y)=1N∑i=1N(xi−E(x))(yi−E(y))D(x)=1N∑i=1N(xi−E(x))2E(x)=1N∑i=1Nxi

where xi and yi constitute the i-th pair of neighboring pixels, N is the total number of neighboring pixels, cov(x,y) is the covariance between pixel values x and y, D(x) and D(y) are the pixel value x and pixel value y mean-square error, E(x) and E(y) are the expected values of pixel value x and pixel value y, respectively. And rxy is the correlation coefficient of pixel values x and y.Figure 18 Adjacent pixels’ correlation of plaintext image and ciphertext image

(A) R channel.

(B) G channel.

(C) B channel.

Table 1 The comparison results of the correlation coefficients of adjacent pixels

Component	Direction	Original Image	Proposed	
R channel	Horizontal	0.9903	−0.0080	
Vertical	0.9793	0.0088	
Diagonal	0.9692	−0.0025	
Anti-diagonal	0.9782	−0.0163	
G channel	Horizontal	0.9786	−0.0566	
Vertical	0.9837	0.0032	
Diagonal	0.9725	0.0131	
Anti-diagonal	0.9559	0.0044	
B channel	Horizontal	0.9602	−0.0162	
Vertical	0.9292	0.0426	
Diagonal	0.9183	−0.0140	
Anti-diagonal	0.9334	−0.0019	

The experiment shows that our encryption algorithm effectively reduces the correlation between pixels, resulting in almost imperceptible correlation in the ciphertext.

Information entropy analysis

Information entropy is an important indicator of the uncertainty of image information. When the uncertainty within an image increases, its information entropy also increases, which enhances confidentiality. Conversely, a decrease in information entropy indicates a reduction in the image’s uncertainty, leading to a decrease in confidentiality. The equation for calculating information entropy is as follows:(Equation 23) H(x)=−∑i=1LP(xi)log2P(xi)

where L is the total number of symbols xi∈x and P(xi) denotes the probability of the symbols.

Table 2 presents the information entropy results of this study, where HI represents the plaintext image information entropy, and HC represents the ciphertext image information entropy. The information entropy of the ciphertext approaches the ideal value of 8 after encryption, indicating excellent algorithm performance in terms of confidentiality. Table 3 compares the data from this article with other approaches.Table 2 Information entropy values of different images

Filename	Description	HI	HC	
4.1.01	Female (NTSC test image)	6.8981	7.9991	
4.1.02	Couple (NTSC test image)	6.2945	7.9990	
4.1.03	Female (from Bell Labs?)	5.9709	7.9991	
4.1.04	Female	7.4270	7.9992	
4.1.05	House	7.0686	7.9990	
4.1.06	Tree	7.5371	7.9989	
4.1.07	Jelly beans	6.5835	7.9990	
4.1.08	Jelly beans	6.8527	7.9989	
4.2.01	Splash	7.2428	7.9998	
4.2.03	Mandrill (a.k.a. Baboon)	7.7624	7.9998	
4.2.05	Airplane (F-16)	6.6639	7.9998	
4.2.06	Sailboat on lake	7.7622	7.9997	
4.2.07	Peppers	7.6698	7.9997	
2.1.01	San Diego (Miramar NAS)	7.4705	7.9998	
2.1.02	San Diego	7.3311	7.9998	
2.1.03	San Francisco (Golden Gate)	6.8709	7.9998	
2.1.04	Oakland	7.0728	7.9998	
2.1.05	San Diego (North Island NAS)	7.5787	7.9998	
2.1.06	Woodland Hills, Ca.	7.3983	7.9998	
2.1.07	Foster City, Ca.	7.3983	7.9998	
2.1.08	San Diego	6.3833	7.9997	
2.1.09	San Diego (Point Loma)	6.8026	7.9998	
2.1.10	San Diego (Shelter Island)	6.9127	7.9998	
2.1.11	Earth from space	7.3511	7.9998	
2.1.12	San Diego (Downtown)	6.5674	7.9998	
2.2.01	San Diego	7.6133	7.9999	
2.2.02	San Diego	6.5786	7.9999	
2.2.03	San Diego	6.3301	7.9999	
2.2.04	Richmond, Ca.	6.9188	7.9999	
2.2.05	San Diego (Miramar NAS)	7.4349	7.9999	
2.2.06	San Francisco (Bay Bridge)	6.4133	7.9999	
2.2.07	Oakland	6.4492	7.9999	
2.2.08	San Diego	7.6470	7.9999	
2.2.09	San Francisco	6.8672	7.9999	
2.2.10	Richmond and San Rafael	6.7117	7.9999	
2.2.11	Stockton	6.7947	7.9999	
2.2.12	San Francisco and Oakland	6.9619	7.9999	
2.2.13	Stockton	7.3534	7.9999	
2.2.14	Shreveport	7.0507	7.9999	
2.2.15	San Francisco	6.7776	7.9999	
2.2.16	San Francisco	7.1279	7.9999	
2.2.17	San Francisco	7.1177	7.9999	
2.2.18	Stockton	6.9236	7.9999	
2.2.19	Stockton	6.6772	7.9999	
2.2.20	Stockton	7.0353	7.9999	
2.2.21	San Francisco	6.9940	7.9999	
2.2.22	San Francisco and Oakland	6.7777	7.9999	
2.2.23	San Diego	6.3572	7.9999	
2.2.24	Stockton	7.3338	7.9999	

Table 3 Comparison of information entropy between different algorithms

Filename	Proposed	Zhang et al.41	Peng et al.42	Zhang and Hu43	Song et al.44	
Airplane	7.9998	7.9983	7.9994	7.9992	/	
Couple	7.9990	7.9987	/	/	7.9973	
House	7.9990	7.9988	7.9978	7.9994	7.9968	
Mandrill	7.9998	7.9986	/	7.9992	7.9992	
Peppers	7.9997	7.9992	7.9994	7.9989	7.9971	
San Diego	7.9998	7.9995	7.9998	/	/	
Tree	7.9989	7.9994	/	/	/	
Female	7.9992	/	7.9974	/	7.9971	
Oakland	7.9999	/	7.9998	/	/	
Stockton	7.9999	/	7.9998	/	/	

Image quality analysis

Peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) are frequently used to evaluate the quality of encryption in image processing. PSNR includes mean square error (MSE) as a component, which is defined as:(Equation 24) {MSE=1H×W∑i=1H∑j=1W(X(i,j)−Y(i,j))2PSNR=10×log10(Q2MSE)

where MSE represents the mean square error between the plaintext image X and the ciphertext image Y, with H and W representing the height and width of the image, respectively. Q denotes the pixel level of the image. SSIM is a metric that quantifies the similarity between two images, defined as:(Equation 25) SSIM(X,Y)=(2μXμY+(0.01L)2)(2μXY+(0.03L)2)(μX2+μY2+(0.01L)2)(μX2+μY2+(0.03L)2)

where the mean values of the images X and Y are denoted by μX and μY, respectively, while their respective standard deviations are represented as well. The dynamic range of pixel values is denoted as L.

Equations 24 and 25 are used to calculate the values of MSE, PSNR, and SSIM. Additionally, to ensure generality, several images were selected to test the encryption module. The encrypted images should have a PSNR of less than 10 dB and an SSIM value close to 0. Table 4 details the results of the three tests. The experimental results demonstrate the excellent encryption performance of our algorithm.Table 4 MSE, PSNR, and SSIM values of different images

Filename	Size	MSE	PSNR	SSIM	
4.1.01	256	36379.0000	2.5223	0.0082	
4.1.02	256	46247.0000	1.4800	0.0052	
4.1.03	256	19769.0000	5.1709	0.0099	
4.1.04	256	25468.0000	4.0709	0.0093	
4.1.05	256	25058.0000	4.1413	0.0111	
4.1.06	256	29732.0000	3.3986	0.0132	
4.1.07	256	27010.0000	3.8155	0.0106	
4.1.08	256	26639.0000	3.8757	0.0103	
4.2.01	512	33629.0000	2.8637	0.0097	
4.2.03	512	25797.0000	4.0151	0.0114	
4.2.05	512	31054.0000	3.2097	0.0096	
4.2.06	512	30328.0000	3.3124	0.0100	
4.2.07	512	30302.0000	3.3160	0.0096	
2.1.01	512	22328.0000	4.6423	0.0109	
2.1.02	512	25417.0000	4.0796	0.0100	
2.1.03	512	28461.0000	3.5883	0.0058	
2.1.04	512	22496.0000	4.6098	0.0084	
2.1.05	512	24661.0000	4.2107	0.0093	
2.1.06	512	23237.0000	4.4689	0.0109	
2.1.07	512	23237.0000	4.4689	0.0109	
2.1.08	512	27995.0000	3.6600	0.0099	
2.1.09	512	26393.0000	3.9160	0.0108	
2.1.10	512	26627.0000	3.8776	0.0109	
2.1.11	512	22605.0000	4.5888	0.0105	
2.1.12	512	27889.0000	3.6765	0.0108	
2.2.01	1024	27657.0000	3.7127	0.0080	
2.2.02	1024	26238.0000	3.9416	0.0106	
2.2.03	1024	28353.0000	3.6048	0.0104	
2.2.04	1024	27052.0000	3.8088	0.0109	
2.2.05	1024	22284.0000	4.6508	0.0106	
2.2.06	1024	25786.0000	4.0170	0.0112	
2.2.07	1024	27591.0000	3.7231	0.0107	
2.2.08	1024	27254.0000	3.7766	0.0094	
2.2.09	1024	26243.0000	3.9406	0.0112	
2.2.10	1024	26277.0000	3.9351	0.0105	
2.2.11	1024	23685.0000	4.3861	0.0106	
2.2.12	1024	22190.0000	4.6693	0.0108	
2.2.13	1024	23396.0000	4.4394	0.0098	
2.2.14	1024	21637.0000	4.7788	0.0106	
2.2.15	1024	23083.0000	4.4978	0.0101	
2.2.16	1024	21364.0000	4.8339	0.0110	
2.2.17	1024	22622.0000	4.5855	0.0108	
2.2.18	1024	23532.0000	4.4143	0.0108	
2.2.19	1024	24447.0000	4.2486	0.0108	
2.2.20	1024	22628.0000	4.5844	0.0106	
2.2.21	1024	26925.0000	3.8293	0.0099	
2.2.22	1024	26607.0000	3.8808	0.0101	
2.2.23	1024	26033.0000	3.9756	0.0109	
2.2.24	1024	23049.0000	4.5043	0.0088	

Differential statistical analysis

The number-of-pixels change rate (NPCR) serves as a widely used metric for comparing the similarity of two images, defined as the count of changed pixels between them. It is frequently employed to evaluate the performance of encryption algorithms against differential attacks, which are among the most effective methods for block cipher attacks. In a typical scenario, the attacker seeks to extract the key by analyzing the impact of specific plaintext differences on corresponding ciphertext differences, thereby attempting to compromise the algorithm’s security. To assess the proposed algorithm’s resilience against differential attacks, we perform numerical calculations and compare NPCR using Equation 26. Due to the large amount of data, the images are grouped by size, with each group size being 256, 512 and 1024, corresponding to Tables 5, 6, and 7, respectively. Meanwhile, Table 8 compares the data from this article with other approaches. The NPCR equation is expressed as follows:(Equation 26) NPCR=1M×N×∑i=1M∑i=1ND(i,j)×100%

where M×N is the size of the image. D can be defined by the following equation:(Equation 27) D(i,j)={0,v1(i,j)=v2(i,j)1,v0(i,j)≠v2(i,j)

Table 5 NPCR test values for images of size 256

Filename	Description	Size	Channel	NPCR	
4.1.01	Female (NTSC test image)	256	Red	99.5636	
4.1.01	Female (NTSC test image)	256	Green	99.6048	
4.1.01	Female (NTSC test image)	256	Blue	99.6353	
4.1.02	Couple (NTSC test image)	256	Red	99.6002	
4.1.02	Couple (NTSC test image)	256	Green	99.6201	
4.1.02	Couple (NTSC test image)	256	Blue	99.6002	
4.1.03	Female (from Bell Labs?)	256	Red	99.6017	
4.1.03	Female (from Bell Labs?)	256	Green	99.5865	
4.1.03	Female (from Bell Labs?)	256	Blue	99.5956	
4.1.04	Female	256	Red	99.5743	
4.1.04	Female	256	Green	99.6017	
4.1.04	Female	256	Blue	99.5987	
4.1.05	House	256	Red	99.6490	
4.1.05	House	256	Green	99.6017	
4.1.05	House	256	Blue	99.6277	
4.1.06	Tree	256	Red	99.5621	
4.1.06	Tree	256	Green	99.6704	
4.1.06	Tree	256	Blue	99.6338	
4.1.07	Jelly beans	256	Red	99.5895	
4.1.07	Jelly beans	256	Green	99.5911	
4.1.07	Jelly beans	256	Blue	99.6231	
4.1.08	Jelly beans	256	Red	99.5789	
4.1.08	Jelly beans	256	Green	99.6216	
4.1.08	Jelly beans	256	Blue	99.5712	

Table 6 NPCR test values for images of size 512

Filename	Description	Size	Channel	NPCR	
4.2.01	Splash	512	Red	99.6037	
4.2.01	Splash	512	Green	99.6105	
4.2.01	Splash	512	Blue	99.5972	
4.2.03	Mandrill (a.k.a. Baboon)	512	Red	99.5911	
4.2.03	Mandrill (a.k.a. Baboon)	512	Green	99.5995	
4.2.03	Mandrill (a.k.a. Baboon)	512	Blue	99.6181	
4.2.05	Airplane (F-16)	512	Red	99.6002	
4.2.05	Airplane (F-16)	512	Green	99.6269	
4.2.05	Airplane (F-16)	512	Blue	99.6098	
4.2.06	Sailboat on lake	512	Red	99.5998	
4.2.06	Sailboat on lake	512	Green	99.5979	
4.2.06	Sailboat on lake	512	Blue	99.6063	
4.2.07	Peppers	512	Red	99.6071	
4.2.07	Peppers	512	Green	99.6162	
4.2.07	Peppers	512	Blue	99.6216	
2.1.01	San Diego (Miramar NAS)	512	Red	99.6387	
2.1.01	San Diego (Miramar NAS)	512	Green	99.6002	
2.1.01	San Diego (Miramar NAS)	512	Blue	99.6269	
2.1.02	San Diego	512	Red	99.6101	
2.1.02	San Diego	512	Green	99.5899	
2.1.02	San Diego	512	Blue	99.6243	
2.1.03	San Francisco (Golden Gate)	512	Red	99.6101	
2.1.03	San Francisco (Golden Gate)	512	Green	99.6014	
2.1.03	San Francisco (Golden Gate)	512	Blue	99.6193	
2.1.04	Oakland	512	Red	99.6239	
2.1.04	Oakland	512	Green	99.6086	
2.1.04	Oakland	512	Blue	99.6105	
2.1.05	San Diego (North Island NAS)	512	Red	99.6056	
2.1.05	San Diego (North Island NAS)	512	Green	99.6014	
2.1.05	San Diego (North Island NAS)	512	Blue	99.6078	
2.1.06	Woodland Hills, Ca.	512	Red	99.6155	
2.1.06	Woodland Hills, Ca.	512	Green	99.6132	
2.1.06	Woodland Hills, Ca.	512	Blue	99.6311	
2.1.07	Foster City, Ca.	512	Red	99.6155	
2.1.07	Foster City, Ca.	512	Green	99.6132	
2.1.07	Foster City, Ca.	512	Blue	99.6311	
2.1.08	San Diego	512	Red	99.6101	
2.1.08	San Diego	512	Green	99.6136	
2.1.08	San Diego	512	Blue	99.6128	
2.1.09	San Diego (Point Loma)	512	Red	99.6185	
2.1.09	San Diego (Point Loma)	512	Green	99.6273	
2.1.09	San Diego (Point Loma)	512	Blue	99.6174	
2.1.10	San Diego (Shelter Island)	512	Red	99.5983	
2.1.10	San Diego (Shelter Island)	512	Green	99.5834	
2.1.10	San Diego (Shelter Island)	512	Blue	99.6048	
2.1.11	Earth from space	512	Red	99.5914	
2.1.11	Earth from space	512	Green	99.6136	
2.1.11	Earth from space	512	Blue	99.6078	
2.1.12	San Diego (Downtown)	512	Red	99.6162	
2.1.12	San Diego (Downtown)	512	Green	99.6105	
2.1.12	San Diego (Downtown)	512	Blue	99.6029	

Table 7 NPCR test values for images of size 1024

Filename	Description	Size	Channel	NPCR	
2.2.01	San Diego	1024	Red	99.6017	
2.2.01	San Diego	1024	Green	99.6174	
2.2.01	San Diego	1024	Blue	99.6077	
2.2.02	San Diego	1024	Red	99.6071	
2.2.02	San Diego	1024	Green	99.6172	
2.2.02	San Diego	1024	Blue	99.6122	
2.2.03	San Diego	1024	Red	99.6010	
2.2.03	San Diego	1024	Green	99.6131	
2.2.03	San Diego	1024	Blue	99.6095	
2.2.04	Richmond, Ca.	1024	Red	99.6145	
2.2.04	Richmond, Ca.	1024	Green	99.6162	
2.2.04	Richmond, Ca.	1024	Blue	99.6024	
2.2.05	San Diego (Miramar NAS)	1024	Red	99.5968	
2.2.05	San Diego (Miramar NAS)	1024	Green	99.6024	
2.2.05	San Diego (Miramar NAS)	1024	Blue	99.6305	
2.2.06	San Francisco (Bay Bridge)	1024	Red	99.6078	
2.2.06	San Francisco (Bay Bridge)	1024	Green	99.6099	
2.2.06	San Francisco (Bay Bridge)	1024	Blue	99.6036	
2.2.07	Oakland	1024	Red	99.6194	
2.2.07	Oakland	1024	Green	99.6093	
2.2.07	Oakland	1024	Blue	99.6163	
2.2.08	San Diego	1024	Red	99.6169	
2.2.08	San Diego	1024	Green	99.6091	
2.2.08	San Diego	1024	Blue	99.6153	
2.2.09	San Francisco	1024	Red	99.6119	
2.2.09	San Francisco	1024	Green	99.6153	
2.2.09	San Francisco	1024	Blue	99.6025	
2.2.10	Richmond and San Rafael	1024	Red	99.6116	
2.2.10	Richmond and San Rafael	1024	Green	99.6074	
2.2.10	Richmond and San Rafael	1024	Blue	99.6157	
2.2.11	Stockton	1024	Red	99.6050	
2.2.11	Stockton	1024	Green	99.6008	
2.2.11	Stockton	1024	Blue	99.6136	
2.2.12	San Francisco and Oakland	1024	Red	99.6028	
2.2.12	San Francisco and Oakland	1024	Green	99.6106	
2.2.12	San Francisco and Oakland	1024	Blue	99.6032	
2.2.13	Stockton	1024	Red	99.6188	
2.2.13	Stockton	1024	Green	99.6126	
2.2.13	Stockton	1024	Blue	99.6035	
2.2.14	Shreveport	1024	Red	99.6032	
2.2.14	Shreveport	1024	Green	99.6158	
2.2.14	Shreveport	1024	Blue	99.6060	
2.2.15	San Francisco	1024	Red	99.6090	
2.2.15	San Francisco	1024	Green	99.6082	
2.2.15	San Francisco	1024	Blue	99.6139	
2.2.16	San Francisco	1024	Red	99.6090	
2.2.16	San Francisco	1024	Green	99.6163	
2.2.16	San Francisco	1024	Blue	99.6175	
2.2.17	San Francisco	1024	Red	99.6041	
2.2.17	San Francisco	1024	Green	99.6109	
2.2.17	San Francisco	1024	Blue	99.6103	
2.2.18	Stockton	1024	Red	99.5985	
2.2.18	Stockton	1024	Green	99.6150	
2.2.18	Stockton	1024	Blue	99.6093	
2.2.19	Stockton	1024	Red	99.6112	
2.2.19	Stockton	1024	Green	99.6027	
2.2.19	Stockton	1024	Blue	99.6062	
2.2.20	Stockton	1024	Red	99.6054	
2.2.20	Stockton	1024	Green	99.6195	
2.2.20	Stockton	1024	Blue	99.6143	
2.2.21	San Francisco	1024	Red	99.6137	
2.2.21	San Francisco	1024	Green	99.6019	
2.2.21	San Francisco	1024	Blue	99.6056	
2.2.22	San Francisco and Oakland	1024	Red	99.6110	
2.2.22	San Francisco and Oakland	1024	Green	99.6089	
2.2.22	San Francisco and Oakland	1024	Blue	99.6134	
2.2.23	San Diego	1024	Red	99.6141	
2.2.23	San Diego	1024	Green	99.6037	
2.2.23	San Diego	1024	Blue	99.6011	
2.2.24	Stockton	1024	Red	99.6273	
2.2.24	Stockton	1024	Green	99.6122	
2.2.24	Stockton	1024	Blue	99.6112	

Table 8 Comparison of NPCR values between different algorithms

Filename	Proposed	Zhang et al.41	Peng et al.42	Zhang and Hu43	Song et al.44	
Airplane	99.6098	99.6283	99.6330	99.6092	/	
Couple	99.6002	99.5845	/	/	99.6130	
House	99.6017	99.6296	99.6399	99.6128	99.6110	
Mandrill	99.6181	99.6296	/	99.6131	99.6110	
Peppers	99.6071	99.6236	99.6174	99.6071	/	
San Diego	99.6101	99.6291	99.6172	/	/	
Tree	99.6338	99.6074	/	/	/	
Female	99.6017	/	99.5880	/	/	
Oakland	99.6093	/	99.6147	/	/	
Stockton	99.6050	/	99.6066	/	/	

Analysis of sensitivity to the key

Key sensitivity is a crucial aspect of encryption technology, referring to the significant differences observed in ciphertexts when encrypting the same image with slightly varying keys. In this section, we use the original key (key) and a key with a small perturbation (key+10−15) to encrypt the same plaintext. The generated ciphertexts are compared, and the NPCR and UACI are calculated using Equations 26 and 28, respectively. Table 9 presents the experimental results. The results indicate significant differences between the two ciphertext images obtained with the perturbed key. The NPCR and UACI values are close to the ideal values of 99.6094 and 33.4635, respectively, indicating the encryption scheme’s high sensitivity to the key. The UACI equation is:(Equation 28) UACI=1M×N×∑i=1M∑j=1N|v1(i,j)−v2(i,j)|255×100%

where v1(i,j), v2(i,j) are the ciphertext images before and after changing one pixel of the plaintext image, respectively.Table 9 Key sensitivity test

Description	Size	Channel	a+10−15 NPCR	UACI	b+10−15 NPCR	UACI	c+10−15 NPCR	UACI	d+10−15 NPCR	UACI	
House	256	Red	99.6307	33.5170	99.5743	33.5130	99.5499	33.4093	99.6155	33.3815	
House	256	Green	99.5865	33.3260	99.6567	33.3445	99.5758	33.3546	99.6262	33.3243	
House	256	Blue	99.6017	33.5700	99.5926	33.3201	99.6185	33.5894	99.6063	33.4627	
Airplane (F-16)	512	Red	99.6174	33.4355	99.5720	33.3455	99.6021	33.3960	99.6120	33.4653	
Airplane (F-16)	512	Green	99.6025	33.4237	99.6044	33.3569	99.6143	33.3931	99.6101	33.4920	
Airplane (F-16)	512	Blue	99.5892	33.4288	99.6101	33.4242	99.5991	33.4436	99.6109	33.4780	
San Diego	1024	Red	99.6053	33.4061	99.6053	33.4301	99.6065	33.4460	99.6078	33.4842	
San Diego	1024	Green	99.5965	33.4564	99.6023	33.4271	99.5928	33.4474	99.6162	33.4703	
San Diego	1024	Blue	99.6129	33.4188	99.6026	33.4285	99.6117	33.4385	99.6124	33.4538	

Key space

The key space represents the range of the encryption key, and having a sufficiently large key space is crucial for effective protection against exhaustive attacks. The algorithm specifically chooses eight key parameters with an accuracy of 10−15. Therefore, the key space of the algorithm can be estimated to be about 1015×8=10120, which exceeds the theoretical requirement of 2100. At the same time, we compared the key spaces, as shown in Table 10.Table 10 Key space comparison

Proposed	Gan45	Zhang et al46	Wen et al47	Rehman et al48	Ponnaian and Chandranbabu49	
10120	1079	2299	2287	2200	1056	

Conclusion

This article presents a secure communication enhancement scheme based on face detection and chaos-driven three-dimensional partition permutation. The scheme provides an effective approach for dealing with privacy breaches. The face detection technology of edge recognition is used to identify facial information. Then, the hash feature values of the plaintext image are extracted as the key to the chaotic system to generate the required chaotic sequence for encryption. The ciphertext image is obtained through subsequent partition permutation, irregular diffusion, and bidirectional parallel confusion. In the experimental section, we conducted a thorough security assessment of the proposed algorithm and achieved good results through the analysis of various experimental data. The proposed algorithm demonstrates superior encryption performance and resistance to typical attacks, effectively safeguarding image privacy while meeting modern communication needs. Future plans include expanding our work to maintain high encryption security.

Resource availability

Lead contact

Further information for resources and materials should be directed to and will be fulfilled by the lead contact, Dr. H.W. (wenheping@uestc.edu.cn).

Materials availability

This study did not generate new unique reagents.

Data and code availability

• All experimental data is clearly explained in this article.

• This article does not report original code.

• Any additional information required to reanalyze the data reported in this article is available from the lead contact upon request.

Acknowledgments

This work was supported in part by 10.13039/501100021171 Guangdong Basic and Applied Basic Research Foundation under Grant 2023A1515011717 , Special Projects for Key Fields of the Education Department of Guangdong Province under Grant 2023ZDZX1041 , the Key Laboratory of Guangdong Higher Education Institutes under Grant 2023KSYS011 , and Special Funds for the Cultivation of Guangdong College Students’ Scientific and Technological Innovation (“Climbing Program” Special Funds) under Grant pdjh2024a428 and pdjh2023b0600 .

Author contributions

Z.X.: Conceptualization, Methodology, Software, Visualization, Data Curation, Writing - Original Draft, Writing - Review and Editing. Y.L.: Conceptualization, Methodology, Project Administration, Resources, Writing - Original Draft, Writing - Review and Editing. T.L.: Visualization, Formal Analysis. H.W.: Funding acquisition.

Declaration of interests

The authors declare no competing interests.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER	
Software and algorithms	
	
MATLAB R2023b	MathWorks.Inc	https://ww2.mathworks.cn/products/matlab.html	
The USC-SIPI Image Database	University of Southern California	https://sipi.usc.edu/database/database.php	
The Pexels Image Database	Canva Germany GmbH	https://www.pexels.com	

Method details

The experiment used a computer running the Windows 11 operating system and equipped with the MATLAB R2023b experimental software. The host is equipped with an 11th Gen Intel Core i7-11800H @ 2.30 GHz eight-core processor and 32 GB of memory. All images used in this paper are from the USC-SIPI and Pexels database. All the software and data involved can be publicly available in the key resources table.

Quantification and statistical analysis

Adjacent pixel correlation analysis

The correlation coefficients are calculated as follows:rxy=∑i=1M(xi−1M∑j=1Mxj)(yi−1M∑j=1Myj)∑i=1M(xi−1M∑j=1Mxj)2∑i=1M(yi−1M∑j=1Myj)2

where xi and yi form the first pair of horizontal/vertical/diagonal/anti-diagonal adjacent pixels and M is the total number of horizontal/vertical/diagonal/anti-diagonal adjacent pixels.

Differential statistical analysis

Two standards are typically used to measure the dissimilarity between the source image and its encrypted version: the number of pixel change rate (NPCR) and the uniform average change intensity (UACI). In standard disparate attacks, attackers tend to make subtle modifications to the source image and then encrypt the original image using the proposed algorithm. This approach allows them to reveal the underlying relationship between the original and encrypted images. NPCR and UACI criteria are commonly used to evaluate the resistance of an encryption scheme to disparate attacks. The equations for calculating NPCR and UACI are as follows:{NPCR=1H×W×∑i=1H∑i=1WD(i,j)×100%UACI=1H×W×∑i=1H∑j=1W|v1(i,j)−v2(i,j)|255×100%

where H×W represents the size of the image, v1,v2 is the ciphertext image before and after changing one pixel of the plaintext image. D can be defined:D={0,ifv1(i,j)=v2(i,j)1,ifv0(i,j)≠v2(i,j)

Information entropy analysis

The degree of randomness in a system is typically assessed by using the entropy of the information as a standard metric. For an information source m, the information entropy H(m) is given by:H(m)=−∑i=0Lp(mi)log2p(mi)

where L represents the total number of pixels. The probability of mi is denoted by ip(m).

Image quality analysis

In the realm of image processing, Peak Signal to Noise Ratio (PSNR) and Structural Similarity (SSIM) are served as standard metrics for assessing the quality of encryption. The Mean Square Error (MSE) is part of the PSNR, defined as:{MSE=1H×W∑Hi=1∑Wj=1(X(i,j)−Y(i,j))2PSNR=10×log10(Q2MSE)

where MSE represents the mean square error between the plaintext image X and the ciphertext image Y. The vertical extent of the image is represented by H, the horizontal dimension of the image is denoted by W. The pixel level of the image is denoted by Q. SSIM is a measure of the similarity between two images, explained as:SSIM(X,Y)=(2μXμY+(0.01L)2)(2μXY+(0.03L)2)(μX2+μY2+(0.01L)2)(μX2+μY2+(0.03L)2)

where the mean values of the images X and Y are denoted by μX and μY, respectively, while their respective standard deviations are represented as well. The dynamic range of pixel values is denoted as L.
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