A Robust Steganographic Algorithm based on Linear Fractional Transformation and Chaotic Maps Steganographic Algorithm based on S-Boxes

—The fundamental objectives of a steganographic technique are to achieve both robustness and high-capacity for the hidden information. This paper proposes a steganographic algorithm that satisfies both of these objectives, based on enhanced chaotic maps. The algorithm consists of two phases. In the first phase, a cryptographic substitution box is constructed using a novel fusion technique based on logistic and sine maps. This technique overcomes existing vulnerabilities of chaotic maps, such as frail chaos, finite precision effects, dynamical degradation, and limited control parameters. In the second phase, a frequency-domain-based embedding scheme is used to transform the secret information into ciphertext by employing the substitution boxes. The statistical strength of the algorithm is assessed through several tests, including measures of homogeneity, correlation, mean squared error, information entropy, contrast, peak signal-to-noise ratio, energy, as well as evaluations of the algorithm's performance under JPEG compression and image degradation. The results of these tests demonstrate the algorithm's robustness against various attacks and provide evidence of its high-capacity for securely embedding secret information with good visual quality.


I. INTRODUCTION
History has demonstrated that secret communication has always been an essential requirement in human society. As time progressed, more advanced techniques have been introduced. In the last two decades, the rapid development of digital communication systems has significantly increased the demand for secure data exchange through digital multimedia. Innovative strategies to protect confidential information from intruders have become the focus of recent research. In this regard, modern techniques of cryptography, watermarking, and steganography have gained unusual importance in the last few years [1][2][3][4][5]. Cryptography involves converting useful information into dummy data to protect it from unintended recipients, while watermarking is associated with protecting the data's copyright. Steganography, on the other hand, conceals confidential information into other information [6][7][8].
With the advent of modern computer technology, remarkable skills for surreptitious communication have been developed. Steganography involves embedding secret information in either the spatial or transform (frequency) domain. In spatial domain embedding, the LSB-substitution technique is most commonly used. However, in the transform domain, invertible transforms such as (DCT) and (DWT) are typically applied to transform the image into its frequency representation [9][10][11]. While both domains have advantages and disadvantages, frequency domain embedding is robust, while spatial domain offers increased capacity for hiding data. This motivates researchers to deploy a combination of both domains. The transforms DCT and DWT are frequently used in image compression applications due to their favorable features. DCT is the most widely used, requiring fewer computational resources; however, DWT is considered more efficient in quality. Many multimedia applications and algorithms in recent literature are based on the joint applications of DCT and DWT [12][13][14][15]. In our proposed method, we apply a hybrid of these transforms for improved and robust outcomes.
In the last decade, chaos has been widely used to enhance the security level of confidential communication. Chaotic systems possess prime features such as irregularity, butterfly effect and unpredictability which making them well-suited for multimedia security applications. Consequently, the study and analysis of chaos-based steganographic techniques have gained popularity in recent years. However, it has been observed that some chaos-based methods are vulnerable to statistical analysis because of the limited chaotic range of the used maps [16][17][18][19][20]. To overcome this issue, authors of [19] proposed a nonlinear combination of one-dimensional chaotic maps that enhances the chaotic range of the resulting system. Such systems are applied in image encryption applications, but to the best of our knowledge, they have not been applied in steganographic methods yet.
The fundamental objectives of a steganographic technique are to achieve both robustness and high-capacity for the hidden information. This paper proposes a steganographic algorithm that satisfies both of these objectives, based on enhanced chaotic maps. The algorithm consists of two phases. In the first phase, a cryptographic substitution box is constructed using a novel fusion technique based on logistic and sine maps. This technique overcomes existing vulnerabilities of chaotic maps, such as frail chaos, finite precision effects, dynamical degradation, and limited control parameters. In the second phase, a frequency-domain-based embedding scheme is used to transform the secret information into ciphertext by employing the substitution boxes. In addition, we exploit the combination of the spatial and www.ijacsa.thesai.org transform domains to achieve a significantly high capacity level for embedding secret data. Our technique uses a hybrid of DCT and DWT, and we observe that the combined effect of DCT and DWT increases robustness against several image processing attacks. The strength of the proposed method is evaluated through the most frequently used analysis techniques, and we prove that our technique produces coherent results. The use of enhanced chaotic systems in steganography has been demonstrated to be effective in various studies, highlighting the potential for further development in this field. The structure of the paper is organized as follows: Section II presents the chaotic map fusion technique. Section III describes the design of the S-box. Section IV presents a novel steganographic technique and Section V provides the security analysis and simulation results. Finally, Section VI presents the conclusions.

II. FUSION OF 1-DIMENSION CHAOTIC MAPS
The study of security protocols has demonstrated the extensive applications of one-dimensional chaotic maps due to their simple structure and computational convenience. In our research, we use the logistic map and the sine map to develop a stronger chaotic system for our problem. In the upcoming sections, we explored the essential features of these maps and their implementation in our research.

A. The Logistic Map
It is a mathematical function that has been used in various applications in information security due to its ability to generate chaotic and pseudorandom sequences. The logistic map is defined by a quadratic recurrence, which can be written as: Where ( is called the control parameter. It is also known as a catalyst for chaos as the behavior of the map varies when the value of parameter changes. It is clear from the bifurcation diagram (as shown in Fig. 1(a)) that produces the chaotic effect only when Although the Logistic map is widely used, it has been noted that its chaotic range is limited, which can be a drawback in some applications.
Another important characteristic of a dynamical system is Lyapunov Exponent, which is a quantitative measure of chaos. A system is chaotic if the value of the Lyapunov exponent is strictly positive, as then a minor disturbance in the initial conditions may cause exponential divergence. The larger the Lyapunov exponent, the better is chaotic performance.

B. The Sine Map
It is a type of chaotic system that has received considerable attention in the area of nonlinear dynamics. It is a straightforward mathematical model that is capable of displaying intricate and unpredictable patterns. Sine map equation is represented as: Where ( ) Like the logistic map, the sine map also exhibits chaotic behavior. However, it has been observed that the chaotic range of the sine map is also limited, as shown in the bifurcation diagram presented in Fig. 1(b). The variations of the Lyapunov exponent of the sine map are shown in Fig. 2(b) Study regarding the combinations of onedimensional chaotic maps shows that by introducing suitable combinations of such maps, the chaotic range can be enhanced [19]. In the next section, we discuss such combinations in detail.

C. Combinations of Chaotic Maps
Our approach involves creating new chaotic systems that have chaotic properties across their entire domain. We accomplish this by constructing nonlinear combinations of the underlying seed maps, which includes the LLS and SSS. The mathematical expression used for the nonlinear combination LLS is given below: Where ( ) , is called an adjustment function. The larger the value of , the better is the chaotic performance. For both LLS and SSS we choose = 14. Eq.
(3) can be rewritten as: The figures depicting the bifurcation diagram and the Lyapunov exponent of the LLS chaotic system can be observed in Fig. 3(a) and Fig. 4(a), respectively. A comparison of the bifurcation diagram of LLS with that of the www.ijacsa.thesai.org logistic map in Fig. 2(a) reveals that the former exhibits a significantly wider chaotic range than the latter. Additionally, the improved uniform distribution of the density function in LLS compared to its seed maps ensures better chaotic behavior of the newly generated system and its efficient application in information security. The other two chaotic systems discussed in the following subsections exhibit similar properties. The expression for the Sine-Sine system can be obtained by using Eq. (1).
The bifurcation diagram and the variations of Lyapunov exponent are exhibited in Fig. 3(b) and Fig. 4(b). Just like the previous case, one may observe the improved performance of the new chaotic system.

III. CONSTRUCTION OF CRYPTOGRAPHIC S-BOX
Block ciphers are cryptographic algorithms that convert plaintext into ciphertext by breaking it up into fixed-length blocks and applying a series of substitution and permutation operations. They are widely used to secure data in various applications, such as electronic payments, online transactions, and communication systems. S-boxes, or substitution boxes, are a key component of block ciphers, as they determine the nonlinear substitution of input bits with output bits in the encryption process. The quality of the S-box plays a vital role in the overall security of the cryptosystem, as any weakness or vulnerability in the S-box can be exploited by attackers to break the encryption. To achieve resistance against differential and linear cryptanalysis in an encryption scheme, the selection of a suitable S-box is crucial. Differential and linear cryptanalysis are two common techniques used by attackers to break ciphers, and a well-designed S-box can help to mitigate these attacks and provide stronger security. To ensure the highest level of security, cryptographers must carefully design and test S-boxes to ensure they are resistant to attacks and produce high-quality encryption output. The selection and design of S-boxes is a complex and ongoing research area, as the security requirements and threats to cryptosystems are constantly evolving. Despite the challenges, the use of strong and secure S-boxes is crucial for the development of robust and reliable cryptosystems.
(IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 14, No. 3, 2023 456 | P a g e www.ijacsa.thesai.org 1) The detailed steganographic strategy is explained through the flowchart (Fig. 5). We explain the whole process in the following steps.
2) Take the host image (sized ) in the spatial domain and shape it into a vector of length .
3) Take the secret image (to be embedded), that is ⁄ of the host image size and substitute it using the S-box shown in table 1.
4) Break the substituted image into two parts and in the ratio 70% and 30% respectively. 5) First, is embedded at random positions of the host image, using the chaotic system LLS. This gives the partial stego image in the spatial domain.
6) Reshape this partial stego image into a matrix form and convert into the frequency domain by applying the combination of DWT and DCT.

7)
Revamp the obtained image into a vector of length m and pick the largest frequency components (30% of the whole).
8) Use the chaotic system SSS to embed at the random positions of the selected largest values. This produces the frequency domain version of the stego image. 9) At this stage, apply the inverse of DCT and DWT to reach the final version of the stego image.
10) The original embedded image can be extracted from the stego image by applying the reverse of the aboveexplained method.
The simulation results are expressed in Fig. 6, Fig. 7. We apply the steganographic algorithm on two images of baboon and Lena.

V. STATISTICAL SECURITY ANALYSIS
In this section, we measure the cryptographic strength of the proposed scheme with the help of some useful analysis such as entropy, contrast, correlation, homogeneity, peak signal to noise ratio (PSNR) and mean squared error (MSE).
We analyzed two benchmark images, baboon and Lena, using the proposed scheme. Notably, the scheme exhibits a remarkable feature of producing steganographic images that have high resemblance to their original counterparts. Moving on, let's discuss these security parameters one by one.

A. Contrast
The analysis of contrast is a method employed to evaluate the degree of sensitivity of image textures to variations in intensity. In simpler terms, it measures how much the texture of the image changes when the intensity of the image is changed. The contrast of an image is directly related to the texture of the image, and is an important factor in determining the quality of an image. Images with high contrast are usually considered to be of higher quality, since they have a greater range of intensity and a more distinct texture. The mathematical expression for the contrast is given as: Where Table II and III present the values of contrast for each image, which are computed from the (i,j) th-element of GLCM represented by P(i,j).

B. Information Entropy
Entropy analysis is used to measure the randomness or the degree of disorder of a system. Entropy is a critical security parameter used to evaluate the strength of a cryptographic scheme. It is a measure of randomness or uncertainty in the data, indicating the degree of unpredictability in the scheme. In the context of steganography, entropy measures the level of randomness in the distribution of pixels in the steganographic image. A higher entropy value implies that the distribution of pixel values in the steganographic image is more random and unpredictable, making it harder for an attacker to identify the hidden data. In general, higher entropy values are desirable as they indicate a higher level of security. A scheme with a low entropy value is more predictable, and hence less secure, as it makes it easier for an attacker to detect the presence of hidden data. Therefore, a higher entropy value indicates a higher degree of randomness in the scheme and, consequently, a more secure scheme. However, achieving a high entropy value may not always be practical, as it can negatively impact the quality of the steganographic image. A scheme that produces high entropy values but produces a visually distinguishable steganographic image may not be useful in practice. Thus, a balance between high entropy values and image quality needs to be struck to ensure that the scheme is both secure and practical. The mathematical formula for information entropy is given by: Tables II and III present the numerical outcomes for both the original and steganographic images. The algorithm we employed demonstrated favorable outcomes for entropy.

C. Correlation
Correlation is another important security parameter used to evaluate the strength of a steganographic scheme. It measures the relationship between adjacent pixels in an image, indicating how much the pixel values are dependent on one another. A high correlation value implies that the adjacent pixel values are similar, and hence predictable, whereas a low correlation value indicates that adjacent pixels are less dependent on one another and more unpredictable. In steganography, a lower correlation value is desirable as it indicates a less predictable steganographic image, making it harder for an attacker to detect the hidden data. This is because the insertion of hidden data modifies the pixel values, disrupting the natural correlation between adjacent pixels. Therefore, a low correlation value indicates that the scheme is successful in hiding the data in the image, and thus more secure. However, it is important to note that achieving a low correlation value may not always be feasible, as it may come at the cost of image quality. If the scheme introduces too much noise or distortion into the image to reduce the correlation value, the resulting steganographic image may be of poor quality, making it distinguishable from the original image. Hence, a trade-off between low correlation and image quality needs to be achieved to ensure that the scheme is both secure and practical. Mathematical expression for correlation is given by: Where represent the expected values and standard deviations of and .

D. Homogeneity
Homogeneity is another security parameter that measures the uniformity of the intensity distribution in an image. It is a measure of the smoothness of the image, indicating how closely the pixel values are clustered around the average value. A higher homogeneity value indicates a more uniform distribution of pixel intensities in the steganographic image, making it more difficult for an attacker to detect the presence of hidden data. In the context of steganography, a high homogeneity value implies that the steganographic image closely resembles the original image, with minimal variation in pixel intensities. This is desirable as it indicates that the hidden data has been successfully embedded into the image without significantly altering its appearance. Thus, a high homogeneity value suggests a more secure scheme. However, as with other security parameters, achieving high homogeneity values while maintaining acceptable image quality can be challenging. A scheme that produces high homogeneity values but results in poor quality steganographic images may not be practical. Hence, a balance between high homogeneity and image quality needs to be achieved to ensure that the scheme is both secure and practical. The numerical results of the homogeneity analysis of the steganographic images are presented in Table II, providing a quantitative measure of the scheme's homogeneity. Its mathematical formula is:

E. Mean Squared Error
It is a security parameter used to evaluate the quality of a steganographic image. It measures the average squared difference between the pixel values of the original and steganographic images. A lower MSE value indicates a smaller difference between the original and steganographic images, and hence a higher quality steganographic image. In the context of steganography, a lower MSE value implies that the scheme has successfully embedded the hidden data into the image without significantly altering its appearance. This is because the scheme has introduced minimal distortion or noise into the image, resulting in a steganographic image that is visually similar to the original image. Thus, a low MSE value suggests a more secure scheme. However, achieving a low MSE value while maintaining high security can be challenging. A scheme that produces low MSE values but results in a visually distinguishable steganographic image may not be practical. Hence, a balance between low MSE values and image quality needs to be achieved to ensure that the scheme is both secure and practical. The numerical results of the MSE analysis of the steganographic images are presented in Table IV, providing a quantitative measure of the scheme's quality.
Where ( ) and ( ) represent the reference image and the steganographic image and represents the pixel's position in an image.

F. Peak Signal to Noise Ratio
It is a security parameter that measures the quality of a steganographic image by comparing the maximum possible signal value with the level of noise or distortion introduced by the embedding of hidden data. A higher PSNR value indicates a higher quality steganographic image with less noise or distortion. In steganography, a higher PSNR value indicates that the scheme has successfully hidden the data in the image without significantly degrading its quality. This is because a high PSNR value implies that the steganographic image is visually similar to the original image, with minimal distortion or noise. Thus, a high PSNR value suggests a more secure scheme. However, achieving high PSNR values while maintaining acceptable security can be challenging. A scheme that produces high PSNR values but results in a visually distinguishable steganographic image may not be practical. Hence, a balance between high PSNR values and security needs to be achieved to ensure that the scheme is both secure and practical. The numerical results of the PSNR analysis of www.ijacsa.thesai.org the steganographic images are presented in Table II, providing a quantitative measure of the scheme's quality. Where, a maximum pixel value in the image is given by . PSNR is defined by:

G. Robustness Analysis
To assess the resilience of the suggested steganographic technique, we perform several tests such as applying JPEG compression, adding noise, and cropping effect on the steganographic images. We determine the resemblance between the extracted steganographic image and the original one. Our algorithm's robustness is demonstrated by the high correlation between these two images. Mathematically, the estimation of similarity can be expressed as: Where and represent the corresponding elements. In the subsequent subsections, we will examine the impact of individual image processing operations one by one.

H. JPEG Compression
In steganography, it is important to evaluate the robustness of the proposed scheme under different scenarios, including image compression. JPEG compression is a widely used lossy compression algorithm that can introduce significant distortion in the image, which may result in loss of hidden data. Therefore, it is crucial to test the proposed scheme's robustness under JPEG compression.
To evaluate the effect of JPEG compression on the steganographic images, we subjected the steganographic images of baboon and Lena to JPEG compression and studied the resultant images. The results of the study are presented in Fig. 8(a) and Fig. 9(a), which show the steganographic images before and after JPEG compression. Additionally, Table V presents the numerical results of the study, which prove the robustness of our proposed scheme. The results of the study demonstrate that the proposed scheme is robust to JPEG compression, as the hidden data remains intact even after compression. This is evidenced by the high values of security parameters such as entropy, correlation, homogeneity, PSNR, and low values of MSE for the compressed steganographic images. Thus, the proposed scheme is suitable for secure data transmission over channels that may introduce compression, such as the internet, where image compression is commonly used to reduce transmission time and bandwidth requirements.

I. Noise Addition
In addition to evaluating the robustness of the proposed scheme under JPEG compression, it is important to assess its performance under other forms of image degradation, such as noise. Salt and pepper noise is a common form of noise that can affect the steganographic images during transmission, and it is important to ensure that the proposed scheme is robust against such attacks.
To evaluate the effect of salt and pepper noise on the steganographic images, we subjected the steganographic images of baboon and Lena to varying levels of noise and studied the resultant images. The results of the study are presented in Fig. 8(b) and Fig. 9(b), which show the steganographic images before and after applying salt and pepper noise. Additionally, Table V presents the numerical results of the study, which demonstrate the proposed scheme's robustness against noise attacks. The results of the study indicate that the proposed scheme is quite robust against salt and pepper noise attacks, as the hidden data remains intact even after the introduction of noise. The high values of security parameters such as entropy, correlation, homogeneity, and PSNR, and the low values of MSE for the steganographic images with noise, suggest that the scheme can effectively protect the hidden data from being compromised.

J. Cropping Effect
The cropping effect is an important consideration in evaluating the effectiveness of a steganographic scheme. The cropping of an image refers to the removal of a portion of the image, which can be used to hide the secret data. Therefore, it is essential to assess the robustness of the steganographic scheme against cropping attacks. To evaluate the cropping effect on the proposed scheme, we cropped the steganographic images of baboon and Lena left side. Cropped image is shown in Fig. 8(c) and Fig. 9(c) respectively. Table V presents the numerical results of the study, which demonstrate the robustness of the proposed scheme against cropping attacks. The results of the study indicate that the proposed scheme is quite robust against cropping attacks, as the hidden data remains intact even after the removal of a significant portion of the image. The high values of security parameters such as entropy, correlation, homogeneity, and PSNR, and the low values of MSE for the steganographic images with different cropping percentages, suggest that the scheme can effectively protect the hidden data from being compromised.

VI. CONCLUSION
Based on the proposed steganographic algorithm using enhanced chaotic maps, the results of the statistical security analysis demonstrate its robustness and high capacity for securely embedding secret information with good visual quality. The use of a novel fusion technique based on logistic and sine maps in constructing the cryptographic substitution box overcomes existing vulnerabilities of chaotic maps such as frail chaos, finite precision effects, dynamical degradation, and limited control parameters. The algorithm's statistical strength was assessed through several tests, including measures of information entropy, correlation, contrast, energy, homogeneity, peak signal-to-noise ratio, mean squared error, as well as evaluations of the algorithm's performance under JPEG compression and image degradation. These tests demonstrate the algorithm's ability to resist various attacks while maintaining good visual quality. In summary, the proposed steganographic algorithm satisfies the fundamental objectives of achieving both robustness and high-capacity for hidden information, and it offers a secure and effective means of embedding secret information in digital images. The algorithm's strengths in terms of security and visual quality make it a promising tool for applications where the protection of sensitive information is critical.