 Research
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Image encryption based on 2DNA encoding and chaotic 2D logistic map
Journal of Engineering and Applied Science volume 70, Article number: 60 (2023)
Abstract
DNA has recently been investigated as a possible medium concerning ultracompact information storage and ultrascale computation. The development of secure image encryption systems has recently received a certain effective and new direction from chaosbased cryptographic algorithms. This paper proposes a novel image encryption algorithm, 2DNALM, based on doubledynamic DNA sequence encryption and a chaotic 2D logistic map. The three phases regarding the suggested approach are as follows: the first phase involves permuting the positions of the pixels using a position keybased scrambling operation. The second phase involves double DNA encoding on scrambled images using various rules by DNA cryptography concept to produce an encoded image, and in the final step, an image which has been encoded is encrypted using XOR operation and chaotic keys created through a chaotic 2D logistic map. The entropy analysis and experimental findings show that the suggested scheme exhibits great encryption and withstands several common attacks.
Introduction
Recently, there has been an increased demand for secure communication systems for transferring sensitive information, including images, to prevent unauthorized access to important information. Encryption to change the transmitted data’s shape is one way to achieve secure communication. An encryption technique and a secret key are used during the encryption process for encrypting the data. Across a wide range of applications, image encryption uses various encryption techniques. As a result, image encryption has emerged as a popular and efficient method for guaranteeing the security of image transmission [1].
Before data is transmitted across an unsecured channel, encryption is defined as transforming it into a structure difficult for unauthorized individuals to decipher or understand. Decryption, opposite to encryption [2], transforms the data into a readable state. Chaos theory is currently referred to as the science of unpredictability. It handles nonlinear events whose behavior is challenging to control or predict. Since it accurately captures the system’s complexity, a crucial characteristic in various applications, chaos theory has garnered much interest. This attribute significantly affects cryptography and its three primary features (positive Lyapunov exponent, sensitive dependence on initial condition, ergodicity, and stochasticity) [3]. It can produce pseudorandom sequences that can be utilized as encryption keys. It takes two steps to create chaotic coding systems: scrambling and diffusion. Scrambling involves shifting the locations of image pixels according to the key—the process of distribution changes pixel values in an image. The two methods produce a new encoded image different from the original [4, 5].
The twodimensional logistic map used for image encryption is unlike the conventional onedimensional logistic map; the twodimensional logistic map has chaotic behaviors—two dimensions with both basins and attractors in evolution. Consequently, the pseudo number sequences generated from the twodimensional logistic map for image encryption are more randomlike and complicated [6].
In living things, genetic information is passed down from generation to generation via deoxyribonucleic acid (DNA). T (thymine), G (guanine), C (cytosine), and A (adenine) are the four major bases. Following its [0, 1] encoding, these are employed in the encryption procedure [7]. One of the coding methods used is DNA coding, which transforms the pixels of a plain image into DNA format utilizing DNA’s basic rules. Due to its large storage capacity, parallel processing ability, and other benefits, DNA coding technology can be defined as significant encryption technology. As a result, various researchers have worked to create a variety of cryptosystems depending on DNA technology. Furthermore, DNA coding technology generates an encoded image which differs from the plain image [8].
The benefits of both methods are combined in the case when DNA coding and chaotic coding are used. This mixing results in a robust coding system with high entropy, which is complicated for attackers to attack [9, 10].
This paper suggested that the image encryption algorithm for the first time is 2DNALM based on a combination of double dynamic DNA sequence encryption and chaotic 2D Logistic Map. In the beginning, scrambling for the image is achieved to boost security before using 2DNA coding by applying various rules from Table 1. The outcome is an encoded image encrypted through a chaotic 2D logistic map utilized to generate chaotic keys. The paper focuses on encrypting the image faster than other existing algorithms [11], with more complex chaotic behaviors found in 2D logistic maps than in 1D. This mix protects the suggested technique from differential, statistical, and bruteforce attacks. This study’s remaining sections are organized in the following way. Related work has been explained in Sect. 2 of the presented work. The chaotic 2D logistic map’s preliminaries are described in Sect. 3. Preliminaries of 2DNA encoding are illustrated in Sect. 4. The proposed approach’s implementation is described in Sect. 5. The results of an experiment are shown in Sect. 6. The findings of the security analysis have been illustrated in Sect. 7. The relevant findings are provided in Sect. 8.
Related work
Due to the significant degree of pixel correlation in the images and the difficulty in designing an effective image encryption method, chaotic systems are frequently used to generate secret keys in image encryption. A new algorithm of image encryption depending on an established bitlevel image encryption method using chaos theory, DNA coding, and the random grid was presented in [2020]. Those security keys are used to dilute and confuse the input images to decrypt the photos. An input image’s pixel positions can be changed using permutation operation, and diffusion modifies the values of pixels in an input image. Guanine (G), adenine (A), cytosine (C), and thymine (T) coding sequences are used as information carriers in DNA computing, which has significant benefits in storage capacity, energy consumption, and parallelism. In conjunction with other DNA coding techniques, chaotic mapping transmits image data via DNA coding, producing an effective encryption mechanism [12].
The need for an effective and reliable RGB image encryption method has increased since D. Ibrahim et al. announced their color image encryption approach in 2022. It is one of the most significant fields receiving greater attention for image protection. RGB image encryption guarantees confidentiality regarding color images throughout transmission and storage by combining chaotic logistic functions with multiple DNA encoding rounds. This method produces extremely promising and effective results regarding randomness, nonlinearity, and resistance to common attacks. However, a reliable, effective, and secure method against many threats is still required [11].
Huda Rashid Shakir and colleagues suggested several methods to encrypt image data in [2022], the most effective and popular of which is chaoticbased encryption because it fits the essential characteristics of cryptography by being sensitive to the initial state, randomness, and nonlinear. Because of the DNA molecule’s many benefits, including its extremely low power consumption, strong parallel processing ability, and large data capacity, image encryption using DNA, and chaotic systems combined for achieving high degrees of security is effective for color images. Yet, image encryption consisting of chaotic systems and DNA encoding has specific disadvantages, such as independency regarding the key generation approach from the original image, the slow encryption speed, and limitations of DNA operations. The research revealed that unique DNA encoding and 4Dhyperchaotic systems were presented as solutions to the abovementioned issues. Initially, chaotic sequences permute the positions of the pixels. Second, according to DNA cryptography, the DNA encoding sequence is subjected to various operations, including DNA XOR, DNA addition, DNA subtraction, shift left, and shift right, to encrypt color images and produce effective encryption results [13].
The need for information security has become urgent due to the constantly changing nature of the Internet and wireless communications and the daily generation of enormous volumes of multimedia. Mohamed Gabr et al. suggested a 3stage image cryptosystem approach in 2022. A tan variation of the logistic map is utilized to carry out deoxyribonucleic acid (DNA) encoding in the first stage. For the second encryption stage, the numerical solution of the Lorenz differential equations and a linear descent algorithm are jointly employed to build a robust Sbox. The logistic map in its original form is utilized in the third stage. Diffusion is guaranteed through the first and third encryption stages, while confusion is certified through applying the Sbox in the second encryption stage. They carry out the confusion and diffusioninducing steps, resulting in encrypted images that are completely asymmetric to their original (plain) counterparts. Combining DNA cryptography with chaotic functions and Sboxes in image encryption algorithms provided many security measures that enhanced the amount of data confidentiality and achieved better performance [14].
Every image encryption algorithm aims to generate a noisy image’s topquality to keep information secret. Additionally, digital communication has become broader by the fast development of Internet technology. People can send digital images on the Internet anytime and anywhere. It has resulted in the development of digital image encryption. Recently, many scholars have combined DNA technology with chaos theory and applied them in the field of image encryption. Simiao Wang et al. The proposed combines a fourdimensional chaotic system with DNA technology to design a color image encryption algorithm [2022]. The algorithm converts each pixel value into two 4bit binary using the windmilllike scanning scrambling method (WSSM). The proposed 4D chaotic system generates four sequences to determine the DNA encoding, decoding, and calculation rules. Finally, the three R, G, and B matrices are divided into blocks and scrambled. And by this, the image encryption algorithm can effectively resist various attacks [15].
Digital images are widely used in numerous fields of society with everincreasing developments in computer network technology. On the other hand, the security of the digital image is threatened critically. We should take precautions to prevent illegal data and information distribution. Digital images are supposed to be built, transmitted, and received securely. Because of this, cryptography becomes an effective way to transmit images securely. For this purpose, Melih Yildirim proposed using deoxyribonucleic acid (DNA) encoding as an encryption method with Rössler in [2022]. The number of DNA encoding rules in the proposed study has excessively increased compared to previous studies on DNA encoding techniques. Rössler attractor is employed as a chaotic system consisting of three differential equations. Thus, the security of the encryption scheme is enhanced [16].
Encryption algorithms are one of the methods to protect data during its transmission through an unsafe transmission medium. But encryption methods need a lot of time during encryption and decryption, so it is necessary to find encryption algorithms that consume little time while preserving the security of the data. Rawia Abdulla Mohammed and colleagues suggested a method to encrypt images in 2023; this method combined more than one algorithm to obtain high security with a short implementation time. A chaotic system, DNA computing, and Salsa20 were integrated. A proposed 5D chaos system was used to generate more robust keys in a Salsa algorithm and DNA computing. Also, the confusion is performed using a new SBox. The proposed chaos system achieves three positive Lyapunov values. So results demonstrate that the proposed scheme has a sufficient peak signaltonoise ratio, a low correlation, and a large key space. These factors make it more efficient than its classical counterpart and can resist statistical and differential attacks [17].
The limitations of related works are the following:
Year  authors  The limitations 

2020  Babu M, et al  The techniques utilize a single encryption algorithm for encrypting the entire data, posing a security threat. Therefore, this paper has adapted multiple image encryption algorithms for each segmented image to enhance security. Due to this, the complexity of the proposed algorithm is increased, thereby making the proposed framework vulnerable to attacks and taking more time to encrypt the image 
2022  Dina Ibrahim et al  The running time of the proposed technique on a core i5 processor with 8 GB of RAM, the proposed approach takes 7–9 s on average to encrypt a 256 × 256 RGB image. However, the proposed algorithm is 16 rounds, which is a lot of work. A smaller number of rounds, such as 4 rounds of DES, can be used to reduce time constraints, and it is still robust and secure 
2022  Huda Rashid Shakir et al  Because employing only DNA to encode images is insecure, recent research combined DNA coding with chaotic systems to improve image encryption techniques. However, image encryption made up of DNA encoding and chaotic systems has certain drawbacks, like the independency of the key generation method from the original image, the limitations of DNA operations, and the slow encryption speed 
2022  Mohamed Gabr et al  The proposed Sbox design did not score optimal values in all metrics. Due to not achieving better values in all evaluation metrics, the Lorenz system constructs the Sbox; the Lorenz system is continuous, with its computation heavily depending on the numerical method adopted in solving it. A better choice would be any discrete chaotic function 
2022  Simiao Wang et al  The chaotic sequences are generated by a SineLogistic map to obtain better results. We can design more complex composite discrete hyper chaotic dynamical systems with different simple chaotic systems, such as a 2dimensional logisticmodulatedsinecouplinglogistic chaotic map (LSMCL). We use the logistic map to modulate the Sine map and couple the result of the modulation and Sine map together. Compared with other existing chaotic maps, we can observe that LSMCL has better chaotic performance in terms of chaotic trajectory, Lyapunov exponent, and Kolmogorov entropy 
2022  Melih Yildirim  In the proposed study, the number of DNA encoding rules has excessively increased compared to previous studies on DNA encoding techniques, which leads to slow encryption speed 
2023  Rawia Abdulla Mohammed et al  Even though the proposed scheme has useful applications in image transmission, it still requires profound improvement in implementing the highintelligence scheme and verifying its feasibility on devices with the Internet of Things (IoT) enabled 
Preliminaries of chaotic 2D logistic map
In the investigation of dynamic nonlinear systems, chaotic maps frequently appear. Mathematical equations govern its behavior, and even a small alteration in the initial position could result in a noticeably different result. Although they appear disorganized and random, they follow certain patterns. A cryptosystem’s diffusion and confusion operations involve chaotic output signals that exhibit random statistical features. The control system used to generate chaotic maps could be complex or simple; thus, the chaotic maps show characteristics of chaotic systems. Several wellknown maps include logistic, quadratic, tent, and others. There are two fundamental problems with chaosbased cryptography: (a) 1D chaotic maps that do not meet the characteristic of unpredictability and (b) highdimensional (HD) chaotic maps that, despite having chaotic and complex behavior, demand more processing complexity [18]. More complex chaotic behaviors can be found on a 2D logistic map compared to 1D. Equation (1), in which (xi; yi) is the pairwise point at ith iteration, and r represents the system parameter, could be used to define such a 2D logistic map mathematically [19] discretely.
A scatter plot of 30,000 points from the trajectory is shown in Fig. 1. The 2D logistic map with initial value (x_{0}, y_{0}) at (0:8909; 0:3342) and the parameter r = 1.19.
For r = 1.19, Fig. 2 depicts the phase portrait regarding the 2D logistic map. Such phase portrait agrees with the 2D logistic map’s mathematical representation for r = 1.19. An (x; y) trajectory about chaotic behavior is randomappearing yet is entirely predictable in the case where r and (x_{0}, y_{0}) are both known; hence, it could be utilized as the pseudonumber generator for cryptograph.
Preliminaries of 2DNA encoding
Knowledge of DNA sequences has become crucial for basic biological research and many practical fields, including diagnostics, forensics, biotechnology, and biological systematic. In a DNA sequence, there are four distinct nucleic acids: T (thymine), A (adenine), G. (guanine), and C (cytosine). Purine adenine (A) and pyrimidine thymine (T), and pyrimidine cytosine (C) always couple with each other according to the rules of base pairing (G). A basic DNA structure is shown in Fig. 3. One can conclude that C and G are also complementary to T and A [20].
Those relations are called Watson–Crick base pairing rules after the two scientists discovered their structural basis. As we all know, 0 and 1 are complementary in the binary system; hence, it may be said that 01 and 10 are complementary, as well as numbers 00 and 11. Table 1 introduces the decoding and coding map rules for the DNA sequence utilized in this study. Table 2 displays the XOR operation for DNA sequences for satisfying Watson–Crick base pairing rules [21].
Method
This work aims to create an image encryption/decryption scheme that can be used to transfer a secure image in an untrusted channel. This scheme is inspired by a combination of some functions from double DNA encoding and chaotic 2D logistic map (2DNALM) algorithms for the first time. This scheme breaks the link between the pixels and mostly concentrates on concepts such as the scrambling algorithm. This scheme was developed to achieve high encryption efficiency, resistance to various assaults with an appropriate execution time, memory savings, and complexity reduction. Figure 4 presents the structure of the proposed scheme.
To achieve the aim of the proposed methodology used decryption and encryption stages on different images as samples are Lena [22], House [23], Baboon [24], and Peppers [25], as explained in the following paragraphs:
Encryption stage
There are various steps in this stage. The general steps of the encryption are shown in Fig. 5.
Image scrambling operation
Image scrambling operations are carried out to improve the encryption technique’s security by breaking the correlation of adjacent pixels. This procedure involves scrambling image I with the dimensions (M × N), where M stands for the image’s height and N for its width. The pixel value of any coordinate (i,j) is expressed by A (i,j) where i = 0,1,2, …. M1 and j = 0,1,2,…N1. To create a scrambled image that scrambles the elements of I, as can be seen in algorithm1, the image is moved using a chaotic key (k), representing a random matrix. The permutation of the image’s pixels is carried out by the scrambling operations utilizing a symmetric key, with the key value as a straight swap. The random matrix elements act as new coordinates of the pixels in the scrambled image I'.
Algorithm 1. The scrambling pixels algorithm.
Image double DNA encoding
The 2DNA coding was applied to the permutation result from the preceding step. An RGB scrambling image is disintegrated into three components R, G, and B. Then, each component is converted separately to binary numbers, with every two bits converted to one of DNA symbols A, C, G, and T based on Rule 1 in Table 1 to produce three encoding components Rc, Gc, and Bc. Next, each component is encoded for the second time based on different rules in Table 1. The Rc component is encoding based on Rule 3, The Gc component is encoding based on Rule 5, and the Bc component is encoding based on Rule 8 to produce three encoding components: Rcc, Gcc, and Bcc, that combined into one image that represents the encoding image as shown in the algorithm 2.
Algorithm 2. The double DNA encoding algorithm.
Image encryption process
The proposed method uses Eq. 1 to build two chaotic sequences regarding real numbers, which are then transformed into two vectors as keys (key1, key 2), depending on the chaotic 2D logistic map produced by entering parameters and initial conditions. Each vector has a size equal to the (h*w) original image dimensions. The encryption operation is displayed in algorithm 3. At this stage, the image encoded with key1 is put into an even array after the DNAXOR operation in Table 2 has been done to the pixels of even rows. The pixels regarding the image’s odd rows encoded using key2 are saved in an odd array. The encrypted image is created by combining the two 2D arrays created after the encoded arrays from the two XOR operations.
Algorithm 3. The image encryption algorithm.
Decryption stage
The decryption process is the inverse operation of the encryption process, as shown in algorithm 4. Figure 6 shows the proposed decryption process.
Algorithm 4. The image decryption algorithm.
Experiment
Some experimental setup is presented in this section. The test image is used for the experimental analysis picture of “Lena” (356*242 pixel) to evaluate the suggested image encryption’s performance algorithms. The simulation is done by Python 3.10 in a computer of Core i7, 8th Gen, CPU 2.20 GHz and memory 8.00 GB, to implement the entire encryption/decryption process, as shown in Fig. 7.
Figure 8 shows the cipher and original images. The findings demonstrate that all cipher images are completely distorted, demonstrating the effectiveness of the suggested approach for encryption.
Results and discussion
Several tests are applied to the suggested approach to assess the effectiveness of the provided encryption algorithm, key space, entropy, differential attack analysis, histogram analysis, image quality, correlation analysis, and time analysis. For evaluation and testing of results, a set of colored images are utilized. The following the illustration of these tests:
Key space and sensitivity analysis
It is regarded as a crucial metric for assessing encryption systems. It evaluates how sensitive an encryption system is to even the smallest change to the secret key used for decryption and encryption. In the suggested system, the secret key value that was produced by 2D logistic map encryption with the use of parameter r = 1.19 and the initial value (x_{0}; y_{0}) at (0:8909; 0:3342) that is explained in Eq. 1 was utilized to encrypt (Lena) image, as can be seen in Fig. 9a, b, and decrypted in Fig. 9c. After that change, the parameter was r = 1.12000001 and applied for decrypted the image, as shown in Fig. 9d; the image we obtained after decoding by slightly altering the secret key was entirely different from the original image. The suggested approach is, therefore, extremely sensitive to even the slightest modification in the secret key [26].
Entropy
Entropy, a measure of how much information is present in a dataset, shows how grayscale values are distributed in the image. Entropy is a constant positive value; the more it suggests that a variable carries much information, the more it indicates this. Equation 13 could be used to compute the entropy [27].
P(s_{i}) stands for probability of (s_{i}), s_{i} stands for gray scale, and n represents the total number of the gray scale. The results of the entropy test for encryption images and counterpart algorithms from the literature are displayed in Table 3. The findings show that the encryption image’s information entropy is close to 8, indicating that the system can resist attacks [28].
Differential attack analysis
The normalized mean change intensity (UACI) and pixel change rate (NPCR) detect differential attack factors and indicate important image information. NPCR stands for the percentage of pixels in the encrypted image that changes after a pixel’s value is arbitrarily changed in the original image. The pixel value change in the encrypted image after the original image’s pixel value has been modified at random is represented by UACI. The encryption algorithm could efficiently resist differential attacks when the pixel change regarding the original plaintext image is minor, and the encrypted image varies significantly [18]. The equation is as follows:
H and W represent the columns and rows for the image, respectively, and D is estimated depending on the following formula:
C_{1}(i, j) and C_{2}(i, j) represent pixel values of the cipher images, L represents the number of gray levels, and H and W are the image’s column and row, respectively. Table 4 shows the computed UCAI values, and Table 5 shows the computed NPCR values for various input image examples, showcasing how these values stand compared to other encryption techniques in the literature and demonstrating comparable results. Figure 10 displays each value of UCAI and NPCR after a series of images was tested using the suggested system [30].
Reporting UCAI and NPCR values when analyzing image encryption algorithms is common practice, and this is because the computation of NPCR is based on the value of UCAI. Table 4 displays columns of N/A under the headings of [31], not mentioning the UCAI values.
Histogram analysis
The encrypted image’s histogram should be smooth for an encryption algorithm to be effective. Figure 11 displays the color histogram of four images before and after encryption, as implemented by Matlab in 2013. Compared to plaintext images, the encrypted color image histogram’s distribution is equally distributed, and the interval’s distribution probability is roughly equal. As a result, it is challenging for attackers to predict the original image using statistical analysis. The findings demonstrate that the algorithm could successfully prevent statistical attacks [34].
Image quality
A standard requirement for image encryption methods is that the cipher image deviates greatly from the original image. Two criteria, the MSE and PSNR, could be used to compare the encrypted and original images, and they may be calculated with the use of (17) and (18) [35, 36].
where H and W represent the height and width of the image, respectively, p(i, j) and C(i, j) are the pixel values of the plaintext image and the cipher image at position (i, j), respectively.
As Eq. (17) shows, the PSNR is inversely proportional to MSE. Such mathematical representation steers the preference of the PSNR to be the inverse of the preference of MSE; hence, a minimal value is ideal. Table 6 shows the computed MSE values, and Table 7 shows the computed PSNR values for various input image examples, showcasing how these values stand compared to other encryption techniques in the literature and demonstrating comparable results. Figure 12 displays each value after a series of images was tested using the suggested system.
Correlation coefficient
The correlation coefficient, or c, measures the linear group regarding two random variables’ range and trend. The correlation coefficient is near a value of 1 when two variables are closely connected, and the two variables are unrelated if the coefficient is near 0. Equation (19) could be used for calculating the coefficient c [41].
When the correlation coefficient’s value is close to 0, the correlation coefficient’s value is close to 1. Since it uses a 2Dchaotic method for encrypting the image and 2DNA encoding, this system has a high correlation coefficient. As indicated in Table 8 and Fig. 13, the range of the value correlation coefficient for the four original tests was between 0.9550 and 0.9064, and the range for the encryption image was between − 0.0210 and − 0.0432. The correlation coefficient between the encryption and the original images is different, as shown in Fig. 14. This status showed an extremely robust system [42].
Time analysis
Since our technique is based on the direct manipulation of each pixel’s bits, the cryptographic speed’s complexity is determined by the pixel count in the image. That makes the proposed technique very efficient concerning the time consumed for encryption and memory allocation. The proposed approach takes 6–8 s on average to encrypt a 256 × 256 RGB image used to evaluate the running time of the proposed method on a core i7 processor with 8 GB of RAM; the proposed approach takes 6–8 s on average to encrypt a 256 × 256 RGB image. However, a GPU processor can be used to reduce time constraints and is still robust and secure. In the future, we intend to present a parallel version of the algorithm to make it more efficient. Moreover, a customized parallel version of the algorithm can divide the image into blocks. Encrypting these blocks in parallel as if each block was an independent image can reduce the time greatly.
As presented in Sect. 7, it is obvious that the performance of the combined DNA coding and chaotic coding technique gives good results within a short time and could successfully resist different threats.
Conclusions and future works
A reliable decryption and encryption approach was suggested in this study and used for the first time image encryption algorithm is 2DNALM based on a combination of two dynamic DNA sequence encryption and a chaotic 2D Logistic Map. Before performing 2DNAencoding, the image is scrambled using various methods to increase the security layer. The encryption process is completed later, resulting in an encrypted image. The inverse operation from encryption is used for decryption. An encryption approach was suggested for the simulation experiment—entropy, differential attack analysis, histogram analysis, sensitivity analysis, key space, and image quality. The results of the experiments demonstrate that the algorithm has good safety performance and can successfully resist different threats. According to the findings of the experiments, our proposed solution is competitive with previous chaotic image encryption algorithms and meets the criteria for image security. Future works could (a) attempt used other types of data encryption, like audio and video encryption, and (b) boost security; a higherorder 5D logistic chaotic map can produce chaotic keys instead of a chaotic 2D logistic map.
Availability of data and materials
All data generated or analyzed during this study are included in the article (and in its supplementary materials).
Abbreviations
 DNA:

Deoxyribonucleic acid
 2DNALM:

2 DNA logistic map
 RGB:

Red, green, and blue
 D:

Dimension
 T:

Thymine
 G:

Guanine
 C:

Cytosine
 A:

Adenine
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Each author has made substantial contributions to the conception and design of the work. AH has analyzed and designed the work and created a new method presented in the work. MAA has performed the data curation formal analysis and interpretation of the data, has utilized the software, and has attained manuscript review and editing. AT has substantively revised it. The authors read and approved the final manuscript.
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Alrubaie, A.H., Khodher, M.A.A. & Abdulameer, A.T. Image encryption based on 2DNA encoding and chaotic 2D logistic map. J. Eng. Appl. Sci. 70, 60 (2023). https://doi.org/10.1186/s44147023002282
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DOI: https://doi.org/10.1186/s44147023002282