Color image encryption algorithm based on Mackey–Glass time-delay chaotic system and quantum random walk

To ensure the confidentiality and integrity of image data and prevent unauthorized data tampering and privacy leaks. This study proposes a new color image encryption scheme based on the Mackey–Glass time-delay chaotic system and quantum random walk. This approach fully leverages the unpredictability of quantum random walks to generate random values. It combines the differences in Hamming distance between the three RGB channels of color images to create a highly complex and random key. The overall image and the three independent RGB channels are arranged in ascending order using Logistic-tent chaotic mapping and the Mackey–Glass time-delay chaotic system to obfuscate the image data. The deformed fractional-order Lorenz chaotic system is introduced, integrated with DNA encoding and decoding technology, and XOR operations are performed to achieve encryption at the spatial and pixel levels, thereby increasing the complexity of decryption. Through extensive experimental research, this solution has demonstrated excellent results in tests such as adjacent pixel correlation, information entropy, and key sensitivity. It has an excellent ability to protect the privacy of images and provides a reliable guarantee for the security of image data.


Introduction
In the era of rapid digitalization, color images have become an essential part of our daily lives.However, due to the digitization of information and the widespread use of network transmission, color images are increasingly susceptible to unauthorized access and malicious tampering [1].Color image encryption is an important information security technology that offers an effective method for safeguarding the confidentiality of these crucial images [2][3][4].Unlike traditional image encryption algorithms [5,6], color image encryption involves complex data structures and high-dimensional information content [7,8], necessitating special technologies to address increasing security challenges and uphold privacy.
In the field of image encryption and image processing, quantum random walks have garnered significant attention from researchers and are extensively utilized [9][10][11].Compared to traditional encryption technology, quantum encryption technology stands out for its faster encryption speed and excellent encryption effectiveness, which has injected strong impetus into improving image security [12,13].In 1993, Aharonov innovatively proposed the quantum random walk algorithm.By applying the principles of classical random walk to the field of quantum mechanics, researchers achieved a significant milestone in quantum computing and established a strong foundation for further research [14].In 2021, Abd-El-Atty B and colleagues proposed an image encryption algorithm that utilizes quantum random walk and dual random phase encoding technology [15].Through simulation experiments on key sensitivity, histograms, and adjacent pixel correlation, research in this area has yielded remarkable results.In the same year, Ma et al developed a clever color image encryption scheme using quantum random walk and discrete cosine transform [16].This approach utilizes the control parameters of quantum random walks instead of random masks as the keys in the encryption process, which enhances key management and transmission efficiency.In 2023, Mou et al proposed a color image encryption algorithm that integrates quantum random walks and various chaotic systems [17].This algorithm not only features fast calculation but also has a wide key space, demonstrating strong resistance to attacks.Hao et al proposed a hybrid NEQR image encryption cryptosystem based on two-dimensional quantum random walk and quantum coding technology [18].Shi et al proposed a chaotic image encryption method based on boson sampling.This method fully utilizes the chaotic random characteristics of boson sampling probability distribution and cleverly combines chaotic random sequences with image processing technology to scramble the pixels of an image and change pixel values.This unique encryption scheme not only enhances data security but also broadens the application of boson sampling to solve practical problems [19].Wang et al creatively combined the enhanced Zigzag scrambling technology with random block-based DNA coding to develop a chaotic encryption scheme.Through numerous experimental simulation analyses, the scheme has demonstrated strong security and reliability [20].Hu et al proposed a non-equilibrium chaotic system with implicit attractors using fractional-order dynamics.However, upon analyzing its chaotic performance, researchers found that the chaotic phase diagram did not meet their expectations and needed further research and improvement [21].Chen et al proposed an asymmetric image encryption algorithm.Through a detailed analysis of experimental data, researchers found that its encryption effect is not ideal and is only suitable for scenarios that do not require high-quality image reconstruction [22].Rani et al developed a novel image encryption algorithm for magic squares, utilizing a combination of differential coding and chaos.This algorithm utilizes differential coding technology to enhance bit-level security while altering image pixel values.However, the introduction of magic squares and complex chaos leads to a decrease in the efficiency of algorithm encryption, making it difficult to meet the needs of practical applications [23].Kumar et al proposed an efficient image encryption algorithm based on hybrid chaos, which processes the red (R), green (G), and blue (B) channels of color images, respectively.However, upon evaluation, it was determined that this algorithm is not significantly different from typical image encryption algorithms.While it aims for efficiency, it overlooks security, leading to unsatisfactory evaluations across multiple algorithm indicators [24].Kang et al proposed a color image encryption algorithm based on DNA operations and spatiotemporal chaos systems.The original image was converted into three DNA matrices using DNA random encoding rules.Then, the scrambling matrix generated by the nonlinear coupling mapping system was used to perform a confusion operation.[25].Jian Zhang et al proposed an innovative image encryption algorithm that dynamically selects 8 DNA encoding rules and 8 DNA addition and subtraction rules.Although the algorithm is flexible and customizable, the encryption process is relatively complex [26].Compared to traditional chaotic systems, this scheme successfully overcomes the problem of periodic degradation, expands the key space, and enhances security.Based on the above discussion, the primary focus areas for color image encryption research are algorithm development and enhancement, security assessment and standardization, and addressing varied requirements in different application domains [27,28].
In this study, we have developed a novel color image encryption scheme based on the Mackey-Glass time-delay chaotic system and quantum random walk.This approach is unique and innovative in both theory and experimental verification, and it has shown remarkable results in improving the reliability and security of image encryption.The primary contributions of our research are as follows: a.By harnessing the unpredictability of quantum random walks to generate random values, and considering the difference in Hamming distance between the three RGB channels of color images, a more highly random and complex key is created, ensuring reliable encryption.b.Combine the Mackey-Glass time-delay chaotic system with trigonometric functions to enhance the complexity of the model, generate an output with randomly chaotic sequence values, and apply it to the pixel data of a color image.This process introduces richer diversity to image processing through sorting confusion.c.By cleverly integrating the randomly chaotic sequence values generated by the deformed fractional-order Lorenz chaotic system with the DNA encoding and decoding process, a unique dynamic DNA encoding mode is created, making the DNA encoding process more flexible and diverse.d.Combine the randomly generated chaotic sequence values from the deformed fractional-order Lorenz chaotic system in pairs to perform various DNA XOR operations.This process achieves a deep-level scrambling of image information, effectively enhancing the obfuscation and privacy protection of image data.
Later in the article, we will delve into the related research used in color image encryption schemes.Next, we will provide a more detailed explanation of the specific implementation steps and algorithm flow to offer greater clarity.Simultaneously, we will conduct thorough research and analysis of experimental data and results to comprehensively evaluate the feasibility of the solution.Finally, we also offer an outlook on the future development of color image encryption, providing valuable forward-looking insights to foster significant progress in this field.

Related works 2.1. Quantum random walk
Quantum random walks have led to numerous significant applications and theoretical advancements in the fields of quantum information science and quantum computing [29,30].This method utilizes the superposition state and entanglement effect of qubits to generate random sequences, ensuring that the key generation process is highly random and unpredictable.A walker can simultaneously move left and right through the state of a quantum coin, reflecting the peculiar properties of quantum mechanics.After T time steps, the probability distribution of the quantum random walk is depicted in figure 1.
The action space of a quantum random walk is defined as Among them, H wC is the coin space and H wP is the walking space.Assume that the coin operator is always In special cases of θ, the values of C are shown in table 1 in detail.
The transfer operator R determines how the walker walks based on the coin status, and is defined as follows Among them, |0⟩ and |1⟩ mean walking in two opposite directions.
During the quantum random walk, H wP is formed by the position state {|x, y⟩, x, y ∈ Z} tensor, and the unitary operator U is expressed as: If the position of the walker at the initial moment is at (x, y) = (0, 0), then |φ 0 ⟩ = |00⟩ ⊗ (cos θ|0⟩ + sin θ|1⟩) can be expressed as the initial state of the quantum.After walking N step steps, the final quantum state of the entire system is φ N step = U Nstep |φ 0 ⟩, and the probability of the walker being found at position (x, y) is: When the coin state is |0⟩ or |1⟩, R y acts on the walker, making it walk up or down along the Y-axis, and R x acts on the walker, making it walk left or right along the X-axis.The general process of quantum random walk is shown in figure 2.

Hamming distance
The Hamming distance is a method used to measure the difference in sequence values between two sequences of the same length at corresponding positions [31].It is defined as follows Where ϕ = (ϕ 1 , ϕ 2 , . . ., ϕ n ) and φ = (φ 1 , φ 2 , . . ., φ n ) are two sequences of equal length.

Chaotic system 2.3.1. Logistic-tent chaotic mapping
Logistic chaotic mapping and Tent chaotic mapping have their own unique dynamic characteristics, and combining them can enhance the complexity and randomness of chaotic sequences [32].Its mathematical formula is defined as follows In the formula, r is the control parameter, g(i) is the system variable, and their value range is 3 which appears messy and difficult to predict and analyze.

Mackey-Glass time-delay chaotic system
The Mackey-Glass time-delay chaotic system is a nonlinear dynamic model commonly utilized for simulating and studying the nonlinear behavior of complex systems [33] x Among them, x(t) is the state value of the chaotic system at time t , τ is the time constant, a and b respectively control the strength of the attenuation term and the strength of the nonlinear term.
The Mackey-Glass time-delay chaotic system is highly sensitive to the selection of initial conditions.The advantage of using trigonometric functions as the initial function is to increase the complexity of the model and clearly exhibit chaotic behavior.The waveforms change over time, and the difference between the waveforms as well as the phase diagrams of different planes are shown in figure 4.
The introduction of trigonometric functions diversifies the states and evolution paths of the waveform diagram, making the phase diagram more colorful

Deformed fractional Lorenz chaotic system
Compared to the traditional Lorenz chaotic system, the deformed fractional Lorenz chaotic system utilizes the concept of fractional calculus to enhance the system's characteristics and provide more diverse and complex dynamics [34] It is defined as follows When the parameters are selected as a = 15, b = 0.5, c = 27, d = 3, the three-dimensional evolution trajectory and waveform diagram of the deformed fractional Lorenz chaotic system are shown in figure 5.

DNA encoding technology
DNA encoding technology leverages the unique properties of DNA molecules in the field of biology, offering innovative possibilities for information security and data storage.It has emerged as a cutting-edge method for information storage and encryption.This field integrates molecular biology and computer science to utilize the unique characteristics of DNA molecules for hiding, encrypting, and storing information [35,36].The rules for DNA coding are presented in table 2.
DNA encoding technology applies operations such as XOR, addition, and subtraction in the field of image encryption.It not only offers superior security, reducing the potential risk of data leakage, but also effectively safeguards the confidentiality of image data.The operation rules are fully explained in figure 6.

Key generation
Quantum random walk operates and evolves on the quantum state, making full use of the superposition property of the quantum state, generating the corresponding random probability distribution matrix, and converting it into a highly random one-dimensional sequence k = {k 1 , k 2 , . . ., k n }.
Subsequently, we used three independent RGB channels of the color image to calculate the Hamming distance of H(R, G), H(R, B), H(G, B).Perform modulo operation with the sequence generated by quantum random walk to generate a new set of pseudo-random sequences for key generation, and use the key as the initial condition of the subsequent chaotic system (12)

Sort
Through the Logistic-tent chaotic mapping iteration 3wh times, a new chaotic sequence G is generated, where w and h represent the width and height of the color image I respectively.Combine the RGB channels of I into an array of size 3wh, and then use the chaotic sequence G to reorder it from small to large.After sorting, split it into three one-dimensional arrays of size wh, and give RGB three independent channels Fusion of the Mackey-Glass time-delay chaotic system and trigonometric functions creates three pseudo-random chaotic sequences X 0 , X 1 , X 2 of length wh.These sequences are used to obfuscate the pixel data in the three independent channels of the RGB.Finally, the element values in the three independent channels of RGB are arranged in ascending order according to the index value idxX (15)

DNA encoding
After 3wh iterations of the deformed fractional-order Lorenz chaotic system with the initial key u 0 , v 0 , w 0 , three pseudo-random chaotic sequences U, V, W are generated.These sequences are decomposed into three one-dimensional arrays, taking sequence V as an example, and combined with DNA encoding rules.Details can be found in table 3 Next, the XOR operation is performed.The XOR rules have been defined in detail in table 4. Three pseudo-random chaotic sequences U, V, W are combined and added separately, and modulo 8 is taken, corresponding to 8 different XOR rules.Then the RGB three channels of the color image are independently encoded, and combined with the XOR rules of the printing, XOR operation is performed on each pixel data Table 3.DNA encoding process combined with deformed fractional-order Lorenz chaotic system.
No. Control parameter conditions Encoding No.
Control parameter conditions Decoding

DNA decoding
The DNA decoding process is similar to the DNA encoding process.Details can be found in table 5.

Zigzag pixel scrambling
Zigzag pixel scrambling significantly alters the spatial structure of the image by rearranging the order of pixels, thereby increasing the complexity of image encryption and making decryption extremely difficult [37].This technology effectively enhances the security and robustness of color image encryption, making it resistant to various forms of attacks.The evolving trend of zigzag pixel scrambling is depicted in figure 7.

Image encryption process
Step 1: Use a quantum random walk to generate the corresponding random probability distribution matrix, and then convert it into a one-dimensional sequence k = {k 1 , k 2 , . . ., k n } with a high degree of randomness.
Step 2: Use the information of the three independent RGB channels of the color image to calculate the Hamming distance between each other.Step 3: Perform a modulo operation on the value generated by the Hamming distance and the random sequence, which is used to generate the key and serve as the initial condition of the subsequent chaotic system, Step 4: Perform 3wh iterations of the Logistic-tent chaotic map with the initial value g 0 to generate the chaotic sequence G.After arranging the sequence G in ascending order, split it and assign it to three independent channels of RGB.
Step 5: Perform 3wh iterations on the Mackey-Glass time-delay chaotic system with the initial value y 0 , z 0 , q 0 , generate the chaotic sequence X 0 , X 1 , X 2 , and re-sort the three independent channels RGB in ascending order.
Step 6: Perform 3wh iterations on the deformed fractional-order Lorenz chaotic system with the initial value u 0 , u 0 , w 0 to obtain the chaotic sequence U, V, W.
Step 7: Combine the chaotic sequence U, V, W with DNA encoding rules and decoding rules, and then perform an XOR operation to confuse the data in the three RGB channels to generate brand new image information.
Step 8: Create a new encrypted image by performing a Zigzag pixel scrambling operation on the new image information.
The process of color image encryption is shown in detail in figure 8.

Image decryption process
The specific steps of decryption may vary depending on the encryption method and key settings, but typically involve decryption key generation, data decryption, and image reconstruction.The decryption process in this study involves reversing the encryption steps using the same key and algorithm, with the goal of restoring the encrypted image to its original plaintext form.As a result, the specific steps for color image decryption will not be discussed in depth here.

Experimental results and analysis
This section conducts experimental simulations of the image encryption scheme and analyzes the results of the experiments.All experiments were conducted using Python 3.9.The hardware environment for the experiments was an AMD Ryzen 7 5800H with a CPU running at 3.20 GHz and 16.0 GB of RAM.This article uses test images from the USC-SIPI image database (https://sipi.usc.edu/database/) to ensure the generalizability of the study.This database contains 44 digitized images and is used to support image processing, analysis, and computer vision research.In order to demonstrate the practical application of the research, we selected six color images with a resolution of 512 × 512 as experimental objects.Such a selection not only represents the diversity of the database but also ensures the validity and robustness of the study.The specific experimental results are presented in figure 9.The research results show that this scheme can maintain the high definition and integrity of the image after decryption, with almost no information leakage or perceived distortion.

Key space analysis
In this study, the color image encryption scheme we proposed uses three different forms of chaotic systems, covering seven different initial parameter combinations, and the accuracy of each parameter is 10 10 .Specifically, the key space size of the Logistic-tent chaotic map is S LT = S g0 = 10 10 ∼ = 2 33.22 , the key space size of Mackey-Glass time-delay chaotic system is S MG = S x0 × S x1 × S x2 = 10 30 ∼ = 2 99.65 , and the key space size of the deformed fractional Lorenz chaotic system is S LOR = S u0 × S ν0 × S w0 = 10 30 ∼ = 2 99.65 .To sum up, the total key space of this encryption algorithm reaches S Total = S LT × S MG × S NOR = 10 70 ∼ = 2 232.52 , Significantly exceeds the minimum requirements required to resist brute force cracking in the field of image encryption.This effectively ensures the security, integrity and privacy of color image data.

Neighboring pixel correlation analysis
Neighboring pixel correlation analysis is a widely adopted method for assessing the security, efficiency, and performance of image encryption algorithms [38].A lower correlation typically indicates excellent privacy protection and data leakage prevention, while also demonstrating extremely high unpredictability, ensuring the reliability and stability of encrypted images [39].The calculation formula is as follows.
where R xy is the correlation between adjacent pixels, cov(x, y) is the covariance between x and y pixels, D(x) is the standard deviation, D(x) is the variance, and E(x) is the mean.In table 6, the correlation coefficients between adjacent pixels of all original color images and encrypted images are presented.Experimental data shows a strong correlation between adjacent pixels in the original image, indicating a trend close to 1.In contrast, the correlation in encrypted images is closer to 0. The encryption algorithm has been proven to successfully destroy the original pixel correlation, providing strong evidence of the unpredictability and security of the image.
The encrypted image underwent adjacent coefficient correlation analysis, and 6000 samples were selected.The specific results are shown in figure 10.At the same time, we compared the correlation of    are significantly lower, approaching 0. Our algorithm has been proven to have a superior scrambling effect in terms of adjacent pixel correlation, demonstrating higher robustness and security.

Histogram analysis
Histogram analysis is utilized to record and display the distribution of pixel values in an image.When the frequency distribution of pixel values is more uniform, the degree of information leakage will be reduced.As a result, the image will appear irregular and difficult to analyze, which will enhance the security of image encryption [44].In figure 11, we present the histogram analysis illustrating the effect of the encryption scheme proposed in this article.The results indicate that the scheme is capable of effectively concealing sensitive information and demonstrates excellent resistance to attacks.

Information entropy analysis
Information entropy analysis is a method used to measure the unpredictability of information, and it is widely utilized in information theory and statistics.In the field of image processing and encryption, information entropy analysis is utilized to assess the scrambling effect and encryption strength of images.It specifically measures the randomness and complexity indicators of the image.The calculation formula is as follows.
where x i is the gray value of the image, and P(xi) is the probability of x i .We selected several images for analysis of information entropy in our experiments.At the same time, we compared the information entropy with other algorithms [32,[45][46][47], and the results are summarized in table 8.The experimental results clearly demonstrate that the information entropy value of the color image encryption algorithm proposed in this article is closer to the ideal value of 8, which is significantly superior to that of other algorithms.It has a greater capacity to withstand statistical attacks and offers a robust guarantee of reliability in practical applications support.
In addition, we took an additional step and randomly intercepted all encrypted images to generate partial sub-images of sizes 256×256, 128×128, and 64×64.Subsequently, we conducted a comprehensive local information entropy analysis on these subgraphs and presented the complete experimental results in table 9.The experimental data clearly show that even the local information of the encrypted image exhibits excellent randomness and a high level of security.

Key sensitivity test
Key sensitivity testing aims to assess the sensitivity of an encryption algorithm to small changes in the key in order to measure its robustness and security [48].In this process, key evaluation metrics include the pixel number change rate (NPCR) and unified average change intensity (UACI), which are used to measure the impact of significant changes on the differences between encrypted images.NPCR is used to quantify the  variation in pixel values between the encrypted image and the original image resulting from alterations in the encryption key when encrypting the same input image.Ideally, the NPCR value should be approximately 99.6094%.The formula for calculating NPCR is as follows: UACI is used to calculate the average variation in pixel values between the encrypted image and the original image when the key is altered.Ideally, the UACI value should be approximately 33.4635%.The mathematical equation for UACI is expressed as: D(i, j) is defined as: Where C 1 (i, j) is the original image pixel information of the encrypted image at (i, j), and C 2 (i, j) is the pixel information after the key has been changed.
First, the image 'Baboon' is encrypted as normal, then a small change is applied to the key Key[0], increasing 10 −10 by a negligible value, and encrypted again.Key sensitivity tests were conducted on these two images, and the experimental results are presented in table 10.Compared to other image encryption schemes [49][50][51][52], the algorithm in this paper is more responsive to subtle changes in the key.
To better illustrate the exceptional performance of this algorithm in terms of key sensitivity, we will make slight changes under different key conditions and then carry out decryption operations.Even if the key changes only slightly, the sensitive information in the image cannot be cracked and restored.The experimental results are presented in figure 12 which clearly show that the algorithm in this paper exhibits excellent randomness and significantly enhances the confidentiality and security of the image.

Fourier spectrum analysis
Fourier spectrum analysis provides detailed and comprehensive information, offering deep insights into the frequency structure of images [53].This analysis demonstrates the components of the image in various frequency domains by converting the image from the spatial domain to the frequency domain.This process helps us understand if the image has a periodic structure or texture, and also reveals hidden features and patterns within the image.We conducted Fourier spectrum analysis on several images, and the experimental findings are fully illustrated in figure 13 to better showcase the detailed variations in image frequency information.
Experimental results show that the original images display unique characteristics.In contrast, the encrypted image exhibits highly uniform features, suggesting that the encryption process reconstructs the pixel information and blurs the original image features.This further confirms that our proposed scheme successfully converts image data into ciphertext that is difficult to understand externally, effectively ensuring the confidentiality of image privacy, and also highlighting its excellent performance in maintaining the security of sensitive information.

Chi-square test analysis
Histograms allow us to visually perceive the impact of an algorithm on encryption, providing an intuitive way to assess the uniformity of pixel values.In order to analyze the values more accurately and quantitatively, we introduce the chi-square test as a measure.The chi-square test can statistically represent the uniform distribution of pixels across the gray scale values.The calculation formula is as follows: Among them, v i represents the observed value of the frequency distribution, and v represents the theoretical value of the frequency distribution.Under the condition of 255 degrees of freedom, with a critical value of 5% probability, the expected value of the chi-square test is ideally 293.2479 [54].We conducted a chi-square test analysis on all channels of the six images and have presented the detailed experimental results in table 11.The experimental results clearly show that the chi-square test values for all channels are lower than the ideal value, which fully confirms the reliability of our proposed scheme in safeguarding image privacy and data integrity.

Randomness test
The primary objective of robustness analysis is to assess whether the encryption algorithm can consistently maintain data integrity and provide effective privacy protection when facing different attacks or noise interference.

Evaluation indicators
There are several evaluation indicators for robustness analysis.We primarily use indicators such as mean square error (MSE), peak signal-to-noise ratio (PSNR), and bit error rate (BER) to comprehensively evaluate the performance of the encryption algorithm [55,56].The calculation formula is as follows.
where I(i, j) and I ′ (i, j) are the pixel information of the original image and the encrypted image at (i, j) respectively.

PSNR analysis
We conducted a PSNR test analysis on the three independent RGB channels of all encrypted images, and the detailed data are presented in table 12.The experimental results clearly indicate a significant disparity between the encrypted image and the original image, leading to noticeable image distortion.This further confirms the high reliability of the scheme proposed in this paper for safeguarding image complexity and security.

Anti-noise attack analysis
Noise attack resistance analysis is crucial for evaluating, improving, and verifying the security and reliability of image encryption algorithms.We apply various levels of noise attacks to the encrypted image and then conduct the decryption operation to observe the effects, as shown in figure 14.The detailed experimental data is presented in table 13.The research results clearly demonstrate that the proposed scheme in this article can effectively maintain stability in the presence of noise interference and attacks, thereby significantly enhancing the anti-interference and reliability of image encryption.

Analysis of anti-clipping attacks
Resistance to cropping attacks is essential for maintaining the confidentiality and integrity of images, ensuring their privacy and integrity.By randomly cropping the encrypted image in various sizes and then performing a decryption operation, we verify that the integrity of the image is effectively maintained and sensitive information remains secure.The experimental results are vividly displayed in figure 15 and the complete experimental data are presented in table 14.The article verifies the excellent performance of the proposed scheme in protecting image privacy and data integrity.

Summary and outlook
The rapid advancement of information technology has made the protection and security of image data increasingly crucial.In the field of information security, color image encryption plays a crucial role, and its primary challenge is to ensure that unauthorized access and tampering are effectively prevented during image transmission and storage.
This study presents a new color image encryption scheme based on the Mackey-Glass time-delay chaotic system and quantum random walk.By leveraging the characteristics of quantum random walk superposition states and calculating the Hamming distance, the randomness and complexity of the key are enhanced.The scheme employs multi-level encryption, including multiple chaotic systems, DNA encoding and decoding technology, and zigzag scrambling.While ensuring image quality, it also increases the difficulty of cracking.
The rich experimental research and analysis results fully demonstrate that the color image encryption scheme proposed in this article exhibits excellent security and robustness.It expands the range of choices and development possibilities for the field of image encryption, and offers a reliable guarantee for the transmission and protection of image information.Future research can explore more innovative methods and technologies, particularly applications that integrate artificial intelligence and deep learning, to address increasing security threats and enhance the secure transmission and storage of color images.

Figure 8 .
Figure 8.The general process of color image encryption.

Figure 12 .
Figure 12.Decryption effect of wrong key.

Figure 15 .
Figure 15.Resistance to clipping attack analysis results.

Table 1 .
Special values situations of C.

Table 5 .
DNA decoding process combined with deformed fractional-order Lorenz chaotic system.

Table 6 .
Adjacent pixel correlation test results.
[40][41][42][43]with other algorithms[40][41][42][43], and the results are summarized in table 7. The experimental results clearly demonstrate that the algorithm proposed in this paper exhibits a high level of randomness in its correlation coefficients.Whether in the horizontal, vertical, or diagonal direction, the correlation values

Table 7 .
Adjacent pixel correlation test results of encrypted images.

Table 10 .
Key sensitivity test results.

Table 11 .
Chi-square test analysis results.

Table 12 .
PSNR test analysis results.
Figure 14.Anti-noise attack analysis effect.

Table 13 .
Anti-noise attack analysis data.