Chaos-based Color Image Steganography Method Using 3 D Cat Map

Steganography is a technique to hide a secret message within a different multimedia carrier so that the secret message cannot be identified. The goals of steganography techniques include improvements in imperceptibility, information hiding, capacity, security, and robustness. In spite of numerous secure methodologies that have been introduced, there are ongoing attempts to develop these techniques to make them more secure and robust. This paper introduces a color image steganographic method based on a secret map, namely 3-D cat. The proposed method aims to embed data using a secure structure of chaotic steganography, ensuring better security. Rather than using the complete image for data hiding, the selection of the image band and pixel coordination is adopted, using the 3D map that produces irregular outputs for embedding a secret message randomly in the least significant bit (LSB) of the cover image. This increases the complexity encountered by the attackers. The performance of the proposed method was evaluated and the results reveal that the proposed method provides a high level of security through defeating various attacks, such as statistical attacks, with no detectable distortion in the stego-image. Comparison results ensure that the proposed method surpasses other existing steganographic methods regarding the Mean Square Error (MSE) and Peak Signal-to-Noise Ratio(PSNR).


Introduction
Recently, the data security concern has attained significant attention, considering that millions of users are often sending and receiving data [1]. Improvements in computer security have demonstrated that steganography is a better technique for securing data than cryptography [2]. The output of cryptography is twisted, meaning it can draw the attention of a third party to encrypted messages. Whereas steganography deals with the ability of embedding data into a digital cover so that the secret message is unable to be recognized where the output is not visible [3]. The steganography aim is to transfer the secret message to another via concealing the secret message in a carrier object [4]. Typically, all kinds of files, like text, image, video, and audio can be used as carriers. The carrier that has a high rate of redundancy is the more suitable media for the steganography. Since text files do not include a large amount of redundant data, they are used rarely in steganography; also audio and video are complicated to apply. Therefore the image is the best cover for hiding information [5]. An efficient steganography method should provide invisibility or perceptual transparency, high hiding capacity, robustness (i.e. the capability of the method to keep the data embedded in the cover), tamper resistance (the ability to avoid change, removal, or hiding of a different message). Although some of these requirements are disproportionate to each other, one or two of them can be achieved by one method. It is difficult to satisfy all these requirements in one algorithm [6]. Steganography strategy consists of several components; plaintext is the secret message needing to be transferred to another party; cover is the media that is used as a container of the secret message; and finally, the stego is the resultant media after hiding the secret message in the cover [7]. On the other hand, steganalysis involves discovering the presence of steganography in addition to trying to retrieve hidden messages from the cover text [8]. In recent decades, the evolution of chaotic theory has led to its extensive use in secure communications. It produces some benefits, such as high security, speed and sound computational overheads. Furthermore, steganalysis of chaotic steganographic algorithms demonstrates extraordinary results compared to conventional algorithms [9]. The goals of steganography techniques are improvements in imperceptibility, hiding information capacity, security, and robustness. In spite of numerous secure methodologies that have been introduced, there is an attempt to develop these techniques to make them more secure and robust [10].

Related work
Many research papers are focused on the combination of chaos theory with steganography. Some of these researches are as follows. Anees et al. in 2014 proposed a steganographic method based on a logistic, tangent delay ellipse reflecting cavity map system (TD-ERCS) within a nonlinear chaotic algorithm (NCA) for determining the pixel coordinate. The cover image was divided into two parts. The sensitive data was converted into binary. The most significant bits (MSBs) and LSBs of sensitive data were embedded into the upper and lower parts of the cover image, respectively. The results showed that the steganographic image is similar to the cover image [11]. Tiwari in 2014 suggested a method for image steganography using chaotic maps for embedding messages in a color image. The results illustrated that the method is highly capable of keeping the message secret [12]. Ghebleh and Kanso in 2014 proposed a method for embedding a binary message randomly in selected detail coefficients of the discrete wavelet transform of a cover image using a 3D chaotic cat map. The results showed that the proposed method has good imperceptibility and high sensitivity to the secret key [13]. Martnez-Gonzlez et al. in 2015 proposed a method for hiding text in color images. First, the data was encrypted based on a chaotic map. Then the encrypted data was hidden in a random way in a color image Bernoulli chaotic map. The results revealed that the proposed algorithm produced an improvement in the peak signal to noise ratio compared with similar algorithms' results [14]. Krishnagopal et al. in 2015 proposed an image hiding and encryption method. First, the cover image was encrypted using the logistic map and cat map. Then the encrypted image was embedded using a Lorenz map that defines the location of the pixels to be secreted in the cover. The results showed that the proposed method provides efficient security [15]. Rajendran and Doraipandian in 2017 suggested an image hiding method based on a symmetric key. The random symmetric key was generated via the 1D logistic map that was used for selecting the pixel coordinate for hiding the secret image. The results showed that the method has efficient security [16]. Ogras in 2019 introduced an image hiding algorithm using a least significant bit (LSB), Logistic map and XOR operation that was used to decode the message. The results confirmed that the method satisfied high visual quality and good security [17]. ALabaichi1 et al. in 2020 introduced a method for hiding a secret message in the LSB of the cover image using 3D Chebyshev and 3D logistic maps. The results showed that the method provides efficient data hiding and good visual quality of the steganographic image [9]. The main contribution of this paper is to introduce a new, effective, LSB-based image steganographic method that inserts a secret message into a cover image randomly, depending on a secure pseudorandom sequence generated by adapting a 3D cat map which is sensitive to initial state and controlled by the secret key. The rest of the paper is organized as follows: The mathematical model of the 3D chaotic map is described in section 2. The proposed steganographic method is introduced in section 3. Section 4 clarifies the assessment of the proposed method and presents a comparison with other methods. Finally, section 5 presents the concluding explanations and future work.

The 3 D cat map
The cat map is a type of cut-out transformation that was introduced by Arnold [18]. The 2D Arnold cat map form is presented in Equation 1 [18]: where and are control parameters, and and [ ) The 3D cat map is introduced by six control parameters, namely as in Equation 2 [18] where [ ]

The proposed steganography method
The proposed method aims to embed data using a secure structure of chaotic steganography, thereby ensuring better security. Rather than using the complete image for data hiding, a selection of the image band and pixel coordination are exploited, using the 3D cat chaotic map for embedding the sensitive message in the cover image. This increases the complexity encountered by the attackers. The cover image is in RGB color, in which each pixel is 24 bits (i.e. 8-bit for red, 8-bit for green and 8-bit for blue). The LSB embedding method was adopted and the data was embedded in eight bits out of 24 bits (3 significant bits form red, 3 significant bits from green and 2 significant bits from blue components). Suppose that is a color cover image with width and height in which the secret message represented by is to be embedded. A stego-image denoted by can be generated by function that takes as presented in Equation 3. ) In the proposed method, is the 3D cat map with an extension that generates a random sequence for picking the pixel location for data concealing. The domain of the 3D cat map in Equation 3 is the interval (0, 1). However when the 3D cat map is used for selecting pixel coordinates and the component of the cover image for data hiding, the following Equations 4, 5, and 6 are adopted. In this way, the pixels will regulate distinctive chaotic positions and, accordingly, the secret message will be inserted randomly. Decompose of size into three components and 2.
Set the secret parameters of the 3D cat map to produce secret keys and using Equation 2 with , and 3.

4.
Convert to ASCII code and then to the binary representation,

5.
Determine the pixel coordinate ) and the component ) that is used to hide by calculating equation 4, 5 and 6.

6.
If then Convert ) to binary representation and insert 3 bits from into the 3 LSBs of . Transform the binary values to decimal values and store the result in ). elseif 7.
If then Convert ) to binary representation and insert 3 bits from into the 3LSBs of Transform the binary values to decimal values and store the result in ). elseif 8.
If then Convert ) to binary representation and insert 3 bits from into the 2 LSBs of . Transform the binary values to decimal values and store the result in ). Endif 9. Endfor 10. Combine to get the stego-image The inverse of function should be applied as presented in Equation 7 by the receiving party to retrieve the secret message from the stego-image. The receiving party performs the reverse steps of the embedding process and should be acquainted with the initial values of the 3D cat map to get the secret keys and for determining the pixel coordinate ) and the component ) that is employed to extract the message . ) (7)

RESULTS AND DISCUSSION
The proposed method was coded in MATLAB (R2014a). The experiments were conducted on a Dell computer with Intel(R) Core (TM) i3-3217U, CPU@ 1.80GHz, a Memory of 4.00 GB RAM and 64-bit system type. The six control parameters of the 3D cat map were set experimentally as and the initial values were set as , and for all the experiments. The performance of the proposed method was evaluated using MSE, PSNR [19], entropy [20], contrast, autocorrelation, energy, and homogeneity, over three standard images: Lena, Baboon and Peppers, of size . Table-1 quantifies the  and for the R, G and B components to show the quality of the stego-image acquired by the proposed method. It can be observed from the results given in the table that the proposed method can withstand statistical attacks and the distortion cannot be distinguished by human eyes considering the small values of and large values of Table-2    Comparisons were performed with the works of [6] and [9] to justify the performance of the proposed method, as shown in Table-3. Performance comparison results clarify that the proposed method revealed significantly improved performance in the quality of the stego-image over that reported by [6] and [9] regarding and .

Conclusions
In this paper, a 3D chaotic cat map was adopted for designing an image-steganographic method. The 3D cat map outputs were employed to determine the pixel coordinate and color component that control the process of embedding a secret message in the cover image. It is shown that there is no detectable distortion in the stego-image (i.e. high visual quality) and the proposed method provides a high level of security by defeating various attacks, such as statistical attacks. Furthermore, it is concluded that the proposed method outperforms other methods regarding visual quality and security. Future work should include an investigation of the impact of different chaotic maps on the steganography.