301 © 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License Using a 3D Chaotic Dynamic System as a Random Key Generator for Image Steganography Mohammed Abod Hussein 1* , Saad Al-Momen 2 1,2 Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq. *Corresponding Author. Received: 25 June 2023 Accepted: 1 October 2023 Published: 20 July 2025 doi.org/10.30526/38.3.3619 Abstract In today's digital era, the importance of securing information has reached critical levels. Steganography is one of the methods used for this purpose by hiding sensitive data within other files. This study introduces an approach utilizing a chaotic dynamic system as a random key generator, governing both the selection of hiding locations within an image and the amount of data concealed in each location. The security of the steganography approach is considerably improved by using this random procedure. A 3D dynamic system with nine parameters influencing its behavior was carefully chosen. For each parameter, suitable interval values were determined to guarantee the system's chaotic behavior. Analysis of chaotic performance is given using the Lyapunov exponents, fractal dimension, and bifurcation diagrams. Furthermore, an algorithm is suggested to generate a random binary key, serving as the controller for the embedding process. And the randomness of the generated key was checked. Moreover, this paper introduces a technique that utilizes the generated random key to govern both the embedding process in the spatial domain and the frequency domain. The results of this study are promising and its potential applications can be extended to various fields that require discreet communication and robust data protection. Keywords: Random Key Generator, Chaotic Dynamic System, Information Security, Spatial Domain, Frequency Domain, Chaos, Lyapunov Exponents, Fractal Dimension, Bifurcation. 1. Introduction With the advent of digital communication methods in the modern era, ensuring information security has become crucial for information and communication technology. Two significant techniques in this field are cryptography and steganography (1). Cryptography is employed to safeguard the confidentiality of communication, preserving the message's content and nature. Occasionally, it is necessary to exchange information with a hidden secret message that no one knows about its existence other than the sender and the recipient; this implementation technique is steganography which is refers to the practice of concealing sensitive information within another file known as a cover (2). In other words, Steganography is the art of concealing a confidential message (file, data, image, video, audio) inside another message (file, data, image, video, audio) (3). In this particular study, text was https://orcid.org/0009-0009-5567-9667 mailto:mohammed.hasan2103m@sc.uobaghdad.edu.iq https://orcid.org/0000-0002-9715-4529 mailto:saad.m@sc.uobaghdad.edu.iq https://orcid.org/0009-0009-5567-9667 mailto:mohammed.hasan2103m@sc.uobaghdad.edu.iq https://orcid.org/0000-0002-9715-4529 mailto:saad.m@sc.uobaghdad.edu.iq https://orcid.org/0009-0009-5567-9667 mailto:mohammed.hasan2103m@sc.uobaghdad.edu.iq https://orcid.org/0000-0002-9715-4529 mailto:saad.m@sc.uobaghdad.edu.iq https://orcid.org/0009-0009-5567-9667 mailto:mohammed.hasan2103m@sc.uobaghdad.edu.iq https://orcid.org/0000-0002-9715-4529 mailto:saad.m@sc.uobaghdad.edu.iq https://orcid.org/0009-0009-5567-9667 mailto:mohammed.hasan2103m@sc.uobaghdad.edu.iq https://orcid.org/0000-0002-9715-4529 mailto:saad.m@sc.uobaghdad.edu.iq https://orcid.org/0009-0009-5567-9667 mailto:mohammed.hasan2103m@sc.uobaghdad.edu.iq https://orcid.org/0000-0002-9715-4529 mailto:saad.m@sc.uobaghdad.edu.iq 302 employed as the secret message to be concealed, while an image served as the cover to hide this message within it (4). This paper presents a steganography technique based on the utilization of a chaotic method. In this approach, a chaotic system is employed to generate a series of random numbers, which are then utilized to modify the cover data. The resulting sequence of numbers, generated through the chaotic method, exhibits a high level of unpredictability, thereby making it challenging for an attacker to detect the presence of the hidden message (5). Therefore, the chaotic sequence is used as a key in the hiding process, enhancing the security of the steganographic method (6). 2. Related Works There is a considerable number of researchs that explore the fusion of chaos theory and steganography. Noteworthy investigations in this realm encompass the following studies. In (7) introduced a novel steganography technique that leverages a chaotic logistic map. This technique offers simplicity, speed, efficiency, and high imperceptibility. The chaotic logistic map is utilized for encoding and embedding using Discrete Cosine Transform (DCT), enhancing security and incomprehensibility. The sensitivity of the initial state in the logistic map generates diverse sequences with distinct initial values. In 2017 (8) discussed several alternative schemes for hiding the grayscale image in a color cover image. They found that the chaotic, one-dimensional maps took less time to use to select the pixels in which the secret message would be embedded. However, 2D chaotic maps have more parameters to adjust which makes them more robust against attacks if a secret message is suspected. As for (9) proposed a safe LSB technique for hiding images using the concept of a nonlinear dynamic system (chaos). Where image files hosted in the spatial domain were used to hide the presence of sensitive information regardless of its format. An analysis of the performance of the proposed technique after comparison with the 3-3-2 LSB technique was very encouraging. In 2019, (10) introduced a novel algorithm for grayscale image processing. The algorithm incorporated Mandelbrot's chaotic logistic map and eclipse clusters, along with Huffman coding compression. The results of this study demonstrated that the proposed algorithm outperforms existing methods in terms of both image quality (as measured by the perception deficiency quality index) and data hiding capabilities. Moreover, the algorithm enhances security measures. In 2021, (11) introduced a technique that employs a 3D chaotic cat map to conceal images. The resulting chaotic map is utilized to determine the pixel locations and color properties, thereby facilitating the embedding of a confidential message into the original image. This research paper introduces a significant advancement in the field of chaos-based steganography techniques. The proposed approach enables the hiding of a secret message within an image by utilizing a randomly generated key derived from a three-dimensional chaotic dynamic system. The system's sensitivity to initial conditions ensures a strong and secure fusion process. Furthermore, the proposed method used both spatial and transformation domains, making it more versatile and increasing its overall effectiveness. 3. Materials and Methods In order to obtain a random binary key for the purpose of randomly embedding data within images, the chaotic nature of a dynamic system can be leveraged. This involves identifying a function that converts the system's solution from continuous values to a binary string. 303 3.1. 3D Chaotic Dynamic System Maghool and Naji (12), in their investigation the impact of fear on a three-species food chain, they suggest the following 3D dynamic system: [ ( )] [ ( ) ] (1) [ ] System (1) is controlled by nine parameters, each of which plays a crucial role in determining its behavior. For Example, Starting from the initial point ; system (1) had a rich chaotic behavior with the following parameter’s values [ ] (2) Figure (1 a) shows the strange attractor for the 3D dynamic system, while Figure (1b-d) show the projection in the xy-, xz-, and yz-plane, respectively. (a) (b) (c) (d) Figure 1. The trajectories of system (1) using parameters values (2). 3.2. Investigating the Chaotic Nature of the System The calculation of the Lyapunov exponent provides a quantitative approach to measure the sensitive dependence on initial conditions (SDIC) in accordance with nonlinear dynamical theory. It represents the average rate of divergence or convergence between two neighboring trajectories. The three Lyapunov exponents with parameter’s values in (2) are: , , and 304 The positive value of the largest Lyapunov exponent demonstrates the presence of chaotic behavior for this system. One of the typical features of chaos is the Fractal dimension which determined through Lyapunov exponents by Kaplan-Yorke dimension, where can be form as (13): ∑ where is the largest integer for which ∑ In this study, for the suggested 3D dynamic system, the value of , hence ∑ The fractional Lyapunov dimension shows that the system exhibits a fractal nature. Due to this fractal nature, the proposed system displays non-periodic orbits, and the trajectories of neighboring points diverge. Thus, it can be concluded that the proposed system is in a state of chaos (14). The chaotic behavior of system (1) is apparent, not only at the specified values (2), but also within a range of values for each parameter. The vectors LB and UB represent the lower and upper bounds of this parameter range, respectively. By keeping the remaining parameter values constant as specified in (2), the system exhibits chaotic behavior when the parameters fall within the corresponding range. [ ] (3) [ ] (4) In Figures 2-10, the bifurcation diagrams depict the variation of Max(x), Max(y), and Max(z) with respect to the parameter values , where . The parameter values are constrained within the intervals [ ], as indicated. 3.3. The Proposed Algorithm to Generate a Random Binary Key The generation of a random binary key involves several steps. First, a keyword is selected and converted into its corresponding ASCII code. This ASCII code is then transformed into a binary sequence. From this sequence, the initial 8 bits are extracted and converted into a decimal number, referred to as , which should fall within the range [ ]. Using , the parameter to be modified is determined. The parameter number, denoted as , is calculated through the equation: (5) where . Additionally, the bits 9 to 12 of the binary sequence are converted into another decimal number, , which should be in the range [ ]. The range of parameter is divided into subintervals of length , where is calculated using the equation (6) The vectors UB and LB respectively contain the upper and lower bounds of all parameter values. Furthermore, bits 13 to 16 of the binary sequence are converted into a decimal number, , which falls within [ ]. To determine the serialization of the value within the specified range of the parameter, is calculated as (7) with . The value of parameter is then modified according to (8) Utilizing the vector that contains the initial specified values of all the nine parameters. 305 Figure 2. The bifurcation diagrams for the system when [ ] vs. (a) Max (x) (b) Max (y) and (c) Max (z). Figure 3. The bifurcation diagrams for the system when [ ] vs. (a) Max (x) (b) Max (y) and (c) Max(z). Figure 4. The bifurcation diagrams for the system when [ ] vs. (a) Max (x) (b) Max (y) and (c) Max(z). Figure 5. The bifurcation diagrams for the system when [ ] vs.(a) Max (x) (b) Max (y) and (c) Max(z). Figure 6. The bifurcation diagrams for the system when [ ] vs. (a) Max (x) (b) Max (y) and (c) Max(z). 306 Figure 7. The bifurcation diagrams for the system when [ ] vs. (a) Max (x) (b) Max (y) and (c) Max (z). Figure 8. The bifurcation diagrams for the system when [ ] vs. (a) Max(x) (b) Max (y) and (c) Max (z). Figure 9. The bifurcation diagrams for the system when [ ] vs.(a) Max (x) (b) Max (y) and (c) Max (z). Figure 10. The bifurcation diagrams for the system when [ ] vs. (a) Max (x) (b) Max (y) and (c) Max (z) The new parameter values are employed to solve the 3D chaotic dynamic system, resulting in the generation of three vectors: , , and . To address any discrepancies in these vectors, a normalization process takes place. Each vector ( , , and ) is normalized using the equations , , (9) where , , , , , and represent the minimum and maximum values of each respective vector. 307 After the process of normalization, the authentic numeric entities of , , and and are transformed into their corresponding binary values. This conversion is achieved using the equations , , (10) . Finally, a sequence is calculated by applying the equation (11) If the resulting sequence successfully passes at least 12 NIST tests, it is deemed a random binary key. However, if the sequence does not fulfill the criteria, an alternative keyword is chosen, and the entire procedure is restarted anew. Algorithm 1 details the methodology for generating an unpredictable binary key, whereas Figure (11). presents a graphical representation of the entire key formation procedure. Algorithm 1: Random Binary Key Generator Input: Keyword (string). Output: Random Binary Sequence Step 1: Convert the keyword to its corresponding ASCII code and store it as an integer value keyword_ascii. Step 2: Convert keyword_ascii to a binary sequence keyword_binary. Step 3: Extract the initial 8 bits from keyword_binary and convert them to a decimal number . Step 4: Calculate the parameter number using the equation: . Step 5: Extract bits 9 to 12 from keyword_binary and convert them to a decimal number . Step 6: Calculate the parameter range length using the equation: . Step 7: Extract bits 13 to 16 from keyword_binary and convert them to a decimal number . Step 8: Calculate the serialization index j using the equation: . Step 9: Modify the value of parameter using the equation: . Step 10: Solve the 3D chaotic dynamic system using the modified parameter values, resulting in three vectors: , , and . Step 11: Normalize each vector (X, Y, and Z) using the equations: Step 12: Convert the normalized values XN, YN, and ZN to binary values using the equations: Step 13: Calculate the sequence op using the equation: . Step 14: Perform at least 12 NIST tests on the op sequence.  If the sequence passes all tests, return op as the random binary key.  If the sequence fails any test, go back to step 1 with a new keyword. 308 Figure 11. Block diagram of random binary key generator. 4.Experimental Results and Analysis This section demonstrates the performance of the proposed random binary key generation algorithm using system (1) with the initial point (0.8, 0.7, 0.6) and the parameter values specified in Equation (12): [ ] (12) Considering the keyword "Student," its corresponding ASCII code is [ ] The binary representation for the keyword is: Based on this information, the values of , , and are determined as 167, 13, and 3, respectively. These values lead to being equal to 6, being equal to 0.0323, and being equal to 3. Consequently, the value of is defined as 1.6769, resulting in the following values for the nine parameters replace the values in (12) and the new set of parameter’s values be: [ ] (13) Figure (12 a) illustrates the strange attractor for the 3D dynamic system, while Figure (12 b- d) show the projection in the xy-, xz-, andyz-plane, respectively. An illustration of the conversion process from real values of the chaotic system outputs to binary values is presented in Table (1). Table 1. Example of converting the real values to binary values 0.3496 1.1743 0.1159 0.3450 0.8185 0.1004 0 1 0 1 0.2474 1.2481 0.1487 0.2423 0.8700 0.1289 1 0 1 0 0.1577 1.2650 0.1898 0.1524 0.8817 0.1645 0 1 1 0 0.1005 1.2040 0.2429 0.0947 0.8392 0.2105 1 0 1 0 0.0714 1.0775 0.3143 0.0655 0.7510 0.2725 1 0 1 0 0.0611 0.9012 0.4133 0.0551 0.6281 0.3583 1 1 1 1 0.0658 0.6809 0.5496 0.0599 0.4746 0.4765 1 0 1 0 0.0920 0.4317 0.7153 0.0861 0.3008 0.6201 1 0 1 0 0.1604 0.2192 0.8464 0.1549 0.1528 0.7337 1 0 1 0 0.3035 0.0988 0.8858 0.2987 0.0688 0.7679 1 0 1 0 309 (a) (b) (c) (d) Figure 12. The trajectories of system (1) using parameters values (13). To verify the statistical characteristics of the generated key, the SP800-22 test package, developed by the National Institute of Standards and Technology (NIST), is employed for detecting random performance (15). In this evaluation, a keystream sequence of 1,000,000 bits generated by the proposed keystream generator is subjected to the tests. The results of these tests are presented in Table (2), which includes the test name, the corresponding P- value, and the test result. Table 2. Evaluating key randomness Test P-value Result Frequency (Monobit) 0.9792 Success Block Frequency 0.9792 Success Runs 0.1822 Success Longest Run of Ones 0.8138 Success Binary Matrix Rank 0.5780 Success DFT 0.6202 Success Non Over Lapping Templates 0.4901 Success Over Lapping Template 0.2873 Success Universal Statistical 0.0803 Success Serial Test 0.4194 Success Approximate Entropy 0.4801 Success Cumulative Sums (Forward) 0.9144 Success Random Excursions Test 0.5954 Success Random Excursions Variant 0.4553 Success Linear Complexity 0.6541 Success 310 It is noteworthy that no deviation from a truly random sequence is observed, as indicated by all P-values exceeding the significance value . Therefore, the results in the table demonstrate the high quality of randomness achieved by the proposed keystream generator. Given that the binary sequence generated successfully passed all the randomness tests mentioned earlier, it becomes a viable option for employing as a key in various encryption techniques or steganography, as demonstrated in this study. In order to evaluate the outcomes of employing this key to conceal data within an image using random techniques, we utilized the two algorithms proposed in our previous research (16). Where, the first algorithm operates within the Spatial domain, while the second algorithm conducts the hiding process in the Frequency domain. Performance evaluation and image quality assurance involve utilizing several commonly used metrics. Key assessments include the signal-to-noise ratio (PSNR), mean squared error (MSE), and normalized cross- correlation (NCC), which are considered crucial in this context (17,18). Figure (13) showcases a collection of standard images, demonstrating their appearance before and after the inclusion of a hidden binary message. Additionally, the histograms for each image are presented. The final column of the figure showcases the randomization map, where red points represent the embedding of 1 bit per pixel, green points represent 2 bits per pixel, and blue points indicate 3 bits per pixel. Meanwhile, black points inwdicate skipped pixels. Furthermore, Table (3) presents a comprehensive overview of the numerical values associated with the three performance metrics mentioned earlier: PSNR, MSE, and NCC. Table 3. Performance metric in spatial domain Cover Image Image size Payload (bits) PSNR MSE NCC Lena 512×512 345216 43.5242 2.8884 0.9994 Barbara 512×512 345216 43.5302 2.8844 0.9994 Baboon 512×512 345216 43.5200 2.8912 0.9992 Peppers 512×512 345216 43.5210 2.8906 0.9997 Goldhill 512×512 345216 43.5364 2.8803 0.9994 Cameraman 512×512 345216 43.5658 2.8609 0.9996 Average 512×512 345215 43.5392 2.8826 0.9996 Figure (14) depicts the representation of embedding in the frequency domain. Table (4) presents the numerical values for the three metrics. It is important to highlight that the quantization step employed the modified quantization table proposed by Li and Wang (19). Table 4. Performance metric in frequency domain Cover Image Image size Payload (bits) PSNR MSE NCC Lena 512×512 159744 42.3139 3.8167 0.9992 Barbara 512×512 159744 35.3070 19.1594 0.9958 Baboon 512×512 159744 30.3099 60.5462 0.9843 Peppers 512×512 159744 42.2903 3.8375 0.9994 Goldhill 512×512 159744 40.8418 5.3567 0.9989 Cameraman 512×512 159744 40.1989 6.2114 0.9992 Average 512×512 159744 38.5436 16.4880 0.9961 311 N a m e Cover image Cover image histogram Stego image Stego image histogram Randomization map L en a B ar b ar a B ab o o n P ep p er s G o ld h il l C am er am an Figure 13.Images pre- and post-embedding of secret message in spatial domain 312 N a m e Cover image Cover image histogram Stego image Stego image histogram L en a B ar b ar a B ab o o n P ep p er s G o ld h il l C am er am an Figure 14. Images pre- and post-embedding of secret message in the frequency domain In addition, the frequency domain embedding results were compared with two other methods: the well-known Jsteg method and the method proposed by Senthooran and Ranathunga (20). To ensure a fair comparison, the same images and payloads as those used in (20) were employed. The images used for this comparison are shown in Figure (15). 313 Airplane Man Barbara Lena Roar Figure 15. The collection of images utilized for conducting the comparison The average results showcased in Table (5) demonstrate the superior performance of the proposed method compared to the other two methods. It achieves the lowest error and highest PSNR. Moreover, a visual depiction of the comparison, focusing on MSE and PSNR, is provided in Figures (16 and 17), respectively. Table 5. Comparative results for (20), and (16) Cover Image Payload MSE PSNR Jsteg (20) (16) Jsteg (20) (16) Airplane 105536 31.4586 26.1262 2.6343 33.1534 37.2589 43.9242 Man 121714 50.345 34.801 1.6070 31.1112 36.7316 46.0708 Barbara 122952 65.5876 55.8191 18.7003 29.9626 33.0522 35.4123 Lena 95936 19.8841 20.0296 3.0215 35.1457 33.0567 43.3286 Roar 59624 9.8284 12.8471 0.7914 38.206 37.4658 49.1468 Average 101152 35.4207 29.9246 5.3509 35.5158 35.5130 43.5765 Figure 16. Comparison of MSE values Figure 17. Comparison of PSNR values 5. Conclusion This paper presents an approach to hide data within images using a 3D chaotic dynamic system as a random key generator. Lyapunov exponents, fractal dimensions, and bifurcation diagrams are used here to check the chaotic behavior of the selected system. This analysis insure suitability of this system to generate random keys. Rigorous randomness tests confirm the effectiveness of the generated binary key. By employing this key to govern the data embedding process in both spatial and frequency domains, the steganography method achieves heightened security. The study's outcomes demonstrate the promising potential of utilizing chaotic dynamical systems as random key generators for data hiding, as the generated keys exhibit a remarkable level of unpredictability, thwarting attackers' efforts to detect hidden messages. Furthermore, by incorporating confusion during the data hiding process, discrete communication and robust data protection can be attained without compromising the appearance of the cover image. The proposed method offers simplicity, speed, efficiency, and 314 a high level of incomprehensibility while strengthening security. These findings pave the way for further exploration and application of chaotic dynamic systems in information security and various domains that demand confidential communication and robust data protection. Acknowledgment None. 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