Novel pseudo random key & cosine transformed chaotic maps based satellite image encryption

Image encryption converts the images into unrecognizable forms that seems like a white noise. Digital Chaos has also emerged as one of the important technique to design secure and efficient image encryption schemes. The chaos theory possesses various desirable properties required for the encrypting images like initial state sensitivity, unpredictability and behavioural complexity. However, image encryption schemes suffer from vulnerabilities like differential attack, statistical, known/chosen plaintext attack, and brute force attack. This paper proposes a novel approach to generate a pseudorandom key. This pseudo random key is combined with Logistic map (LM), cosine transformed Logistic map (CTLM) and cosine transformed Logistic-Sine Map (CTLSM), one by one, to implement three secure and efficient methods for satellite image encryption. The scheme uses the 384-bit of share key to perform encryption during the process. The proposed approaches are tested for parameters such as Entropy, Correlation Coefficient (CC), Number of Changing Pixel Rate (NPCR), Unified Averaged Changed Intensity (UACI), Avalanche effect, Bit Correct Ratio (BCR) and Peak Signal to Noise Ratio (PSNR). The work also analyses the various cryptanalytic attacks on the proposed chaos and novel Pseudo random key combinations. The results show that the proposed pseudo random key and CTLSM combination outperforms the other two combinations, and is more efficient in resisting all type of attacks as well.


Introduction
The exchange of digital images through various communicating networks creates the demand to secure the image through the communicating channels. Therefore, various methods have been proposed by various researchers to provide security to these images. Out of these encryption & watermarking are the two most frequent used techniques. In the watermarking [17], the image authenticates the identity of the sender, whereas in the encryption method [16,36], the original image is converted into the noise like image. Chaotic theory is also one of the important concept to encrypt the images. A chaos map provides a nonlinear and random behavior at a particular interval, and also, it is sensitive to the initial conditions [27,31,48].
The chaos based image encryption scheme comprises of two stages, i.e. permutation and diffusion [14,25,65]. The permutation operation consists of shuffling the positions of the pixels of the image whereas, the diffusion phase consists of modifying the pixel value of the pixels. However, the traditional permutation and diffusion algorithms consist of only secret key, and it uses the same keystream for various plain images i.e. the secret key for number of images remains same. Hence, these algorithms suffer from known or chosen plaintext attack [13,15,39,51,66] . To defend these attacks, there have been various approaches or techniques developed by the researchers such as self-adaptive approach [28,40,58,60], variable control parameters technique [38], dynamic state variable selection [9,12], orbit variation [11,49], and plaintext related techniques [44,59]. All these schemes do change the keystreams during the implementation of the encryption process [21,22,37,61].
The pseudo-random (PR) key performs essential role in the image encryption process. The benefit of pseudo-random key is, to provide the randomness to the shared key. Due to this, the different keystreams generate from the same share key during the encryption process, which results in different cipher images generation for the same plain-image. Hence, it becomes difficult for the adversary to compute the plain-image from the cipher image. There are various algorithms designed by the various researchers for the image encryption process. Section 1.1 describes the various existing schemes for the image encryption.

Related work
Chaos based image encryption is classified into two categories of the symmetric key cipher i.e. chaotic block cipher and chaotic stream cipher. Chaotic block ciphers are designed using the confusion and diffusion operations. There are various block ciphers that are designed for the image encryption using chaos theory [20,24,28,46,50,61]. However, some of the block ciphers are vulnerable to various types of attacks [2,9]. On the other hand, the stream ciphers encrypt individual bits of the image. The stream ciphers are generated using the XOR operation between the plain image and the keystream output of the key [4,30,52,54,57].
Liu et al. [41] in 2018, proposed a chaotic based scheme for medical image encryption that uses fourth order chaotic based system. This scheme consists of four initial conditions and six control parameters for encrypting the medical image. Bisht et al. [6] proposed an encryption scheme using the multiple chaotic map and fractional random transform. After that, Li et al. [37] in 2019, proposed image encryption approach using the dynamic state variable and orbit perturbation. This scheme uses the logistic adjusted sine map to perform the diffusion as well as permutation. Also, it depends upon the initial conditions and control parameters. The author claimed that their proposed scheme provides resistance from chosen plaintext attack.
Kovalchuk et al. [32][33][34][35] in 2019, proposed various works to improve the popular Rivest, Shamir, Adleman (RSA) method. RSA algorithm, in some cases, suffers from the weakness of leaving object outlines in an encrypted image, which in turn results in additional limitations while using it for image encryption [32]. Kovalchuk et al. [32], used topological image coverage concept with elements of RSA algorithm to overcome this limitation, and increase cryptostability of RSA apprach. Kovalchuk et al. [33], applied projective transformation with RSA algorithm to propose two-element and element-by-element image encryption-decryption for both grayscale and color images. The modified proposed RSA approach not only avoids the contours appearance effect in encrypted images, but also is resistant to unauthorized decryption.
One more mathematical approach named, Quaternary Fractional-Linear Operations, was used by Kovalchuk et al. [34], in addition with RSA approach to prevent contours in encrypted image and increase RSA algorithm's resistance from unauthorized decryption. In work of [35], Kovalchuk et al., combined RSA algorithm with bitwise binary operations to propose a more cryptostable approach and secure than basic RSA technique.
Researchers have also combined Sine and Cosine Functions with chaotic maps to improve dynamicity and complexity of the existing chaotic maps [3,26]. Alwaida et al. [3]combined cosine function with LM and Henon chaotic maps to propose digital chaotic cosine map for cryptographic applications. In 2019, Hua et al. [27] proposed cosine transformation based scheme for image encryption. This scheme consists of three phases, where, the first phase consists of scrambling the image such that the correlation between the original and encrypted image gets decreased. In the second phase, the scrambled image is rotated by 90 o in clockwise direction, and the last phase comprises of substitution of the image to perform the diffusion. The authors also designed three hybrid chaotic maps Logistic-Sine Cosine (LSC) map, Sine-Tent Cosine map (STC), and Tent-Logistic cosine (TLC) map. Ge et al. [23] in 2019, proposed a 3D cat map with Markov properties based scheme for image encryption. This scheme uses sum of all the binary bits as a secret key to defend against the known/chosen plaintext attack. The proposed approach analysed security in terms of key space analysis and differential attack.
Alawida et al. [2] in 2019, proposed a new chaos based encryption scheme based on the perturbation. This scheme contains combination of two chaotic map to perform the encryption. The authors claimed that their method provides resistance from differential attack, known plaintext and chosen plaintext attacks. Saljoughi et al. [50] in 2019, proposed a 3D Logistic map for the image encryption that consists of three random sequences for the permutations. The researcher used the chaotic based encryption for the secure data transfer from the satellite communication. Bentoutou et al. [5] proposed image encryption using the two-dimension Logistic Adjusted Sine (LAS) map. The authors have also combined optimization methods with different chaos maps to improve efficiency of the existing encryption schemes [8,18,19].
Motivated by these works, the proposed work in this paper creates a three new image encryption schemes by combining the novel pseudo random key with the three different chaotic sequences, individually. Initially, this scheme generates a novel pseudo random key by applying various operations like XOR, left circular shift (Left Rotate), modular. Then, it combines the generated pseudo random key with Logistic map (LM), cosine transformed LM (CTLM) and cosine transformed Logistic-Sine Map (CTLSM), individually, to propose convert three different encryption approaches. These proposed schemes use 384-bit initial shared key during the process of image encryption. All the proposed schemes provide the resistance from various security vulnerabilities like known/ chosen plaintext attack, differential attack, statistical attack, brute force attack. However, the CTLSM combination with the proposed novel pseudo random key outperforms the other two proposed combinations. This rest of the paper is organized as: Section 2 describes the proposed scheme; Security analysis of the proposed scheme is described in Section 3; Section 4 compares the proposed technique with existing works: Section 5 concludes the paper with discussion of future work.

Proposed architecture
This section discusses the steps involved in the implementation of the proposed method of image encryption. Figure 1 a,b show the encryption and decryption processes performed by using the proposed scheme. The encryption makes use of the proposed pseudo random key and chaotic function to encrypt the plain image, whereas the decryption process makes use of same pseudo random key and chaotic to decrypt the cipher image. The proposed method takes an initialization random variable X of 128-bit and share key of 384-bit to encrypt the image file. The share key is divided into three part for each red, blue, green components of the image, respectively. The decryption process is just reverse of the encryption process. The following sections discuss both in detail. Table 1 shows various notation used in this paper. Figure 2 describes the detailed architecture of the proposed image encryption approach. As shown by the figure, the pseudo random key generation uses Hamming weight (hw), XOR, modulus (mod), and left rotate (Rot) operations. These operations are used to enhanced the security of the proposed scheme. The LM uses mod and divide (div) operations to generate x0, the initial seed for the chaos map. The whole process uses operation Addition (Add) and mod to convert Plane Image (PI) to Cipher Image (CI).

Encryption
Function 1 gives the pseudo code and following steps describe the flow for converting plain image into cipher image. The subsequent sections discuss these steps in detail.
Step 1: Take variable X as 128-bit random number and generate the pseudo-random key.
Step 3: Generate the cipher image using Logistic map, pseudorandom key as well as plain image.
Step 4: Repeat the step 1 to 3 for i in size of image  and Key3 should not equal to zero. Figure 3 shows the structure of share key during the image encryption process. The final output of pseudo random key generation process gives us three pseudo-random keys for each component of the image, where each pseudo random key is generated using the initial random variable X and the 128-bit share key of the corresponding component. The variable X is computed using eq. (1).
The significance of this sequence is that a single change in bit by the adversary will change number of bits in the result. For Instance, let us consider two different numbers initial vector A

SRK← Key
where, the col + (cols * row) calculate the position of the plain image. The pseudo key generated for red, Green and blue component of the image are computed as: The proposed scheme uses Logistic map (LM), Cosine Transformed Logistic Sine map (CTLSM), Cosine Transformation of Logistic map (CTLM) in the encryption process. Generally, the Logistic map is defined in eq. 8.
where r ϵ [0, 4], x t is initial parameters for generating the logistic map and x t + 1 is next value generating for the logistic map. Here, x0 is the initial seed of the chaos function. The behaviour of the chaos function depends upon the parameter r. If r is kept between 0 to 3, the behaviour of the chaos function independent of the initial condition as the population quickly approaches the value r−1 r . Keeping r between 3 to 3.44949, 3.44949 to 3.54409, and beyond 3.54409 oscillates the population among 2, 4 and 8 values, respectively. The logistic function shows chaotic behaviour, when parameter r varies from approximately 3.56995 to approximately 3.82843, and beyond r = 4, almost all initial values eventually leave the interval [0,1] and diverge [47,53].
In the proposed scheme, the initial variable of logistic map (x0) is calculating using the two different hamming weights hw(X ⊕ Key[i]) and hw(X) in eq. 9.
LM is simple and easy to implement. However, it suffers from issues such as stable windows, blank windows and small key space [26]. To overcome these issues, researchers have either replaced LM with more complex and dynamic chaos maps, or combined it with some complex functions. The proposed work in this paper also uses two such strategies: the cosine transformation of LM (CTLM) and cosine transformation of Logistic-Sine (CTLSM). The CTLM is mathematically defined as [3]: where, r ϵ [0, 4] and k ϵ [10,24] are control parameters for the Cosine of Logistic map. The CTLSM is represented as [27]: where r ϵ [0, 1] control parameters for the cosine of Logistic-Sine map. The research works of [3,27] show that both CTLM and CTLSM are more secure, complex and dynamic than simple LM map.

Cipher image creation
After the creation of pseudo-random key and the Logistic map, the cipher image is generated at a i th component of the image. The add and mod operator applied to the Plain _ Image, pseudo _ Key and x0 * X[ham64 : ham64 + 32] to generates the i th component of the image.

Decryption
To decrypt the cipher image, the pseudo-random key is generated using the initial parameter and the random number. Then, the chaos map and the pseudo-random key produce the plain image from the cipher image. Function 2 defines the procedure of the decrypting cipher image into color image. The algorithm takes initial parameter, shared key, number of rows and columns in color image, r, and, Cipher _ Image for input and produce an output of plain image. There are various steps defined to decrypting color image.
Step 1: Compute the pseudorandom key for the decryption process Step 2: Generate the chaos function using LM or CTLM or CTLSM.
Step 3: Generate the decrypting image using chaos function, pseudorandom key and cipher image as: Step 4: Repeat the step 1 to 3 for i in size of image

Experimental setup and results analysis
The proposed scheme has been implemented using the Python language on Windows 10 Operating on Intel Core i3 2370M processor with 6 GB RAM, and 256 GB SSD drive.  [1] have been used for testing the purposed method. These images have been taken from the flickr website. These images were taken by the "NASA Goddard Space Flight Center" under the licence of CC-BY. The Fig. 4 shows these four images. The Section 3.1 describes the security analysis of our proposed scheme.

Security analysis
This section describes the analysis of the various metrics used to evaluate the security of the proposed scheme. The various metrics used are: Brute force attack, Key sensitivity, Histogram analysis, Differential attack, and Known or Chosen plaintext attack.

Brute force attack
The adversary tries to compute all the possible combinations to decrypt the cipher image into original image. This scheme uses the 384-bit key use for the encryption process. Therefore, the adversary will be required to perform 2 384 steps to decrypt the share key, which is impractical.
To compute the cipher image, the adversary will chooses a key i and variable X, which will require 2 128 each. Hence, the total complexity to guessing the pseudo key is 2 128 * 2 128 . The complexity to compute the initial value for the chaos function is 192 and the chaos map is multiplied by X[ham64 : ham64 + 32]. The value of ham64 lies between 0 to 64, and the X[ham64 : ham64 + 32] selects the 32-bit values from the 0 to 96 bit of the variable X. Hence, the total complexity to generate the cipher image from the chaos function is 2 96 * 192. Therefore, the total complexity is 2 128 * 2 128 + 2 96 * 192.
Similarly, while doing decryption using Function 2, the adversary knows about the cipher image (CI). Hence, the adversary will try to guess the variable map, X[ham64 : ham64 + 32] and pseudo _ Key. The complexity of guessing the map, X[ham64 : ham64 + 32], and pseudo _ Key is 192, 2 32 , and 2 32 respectively. Hence, the total complexity for guessing one byte of the cipher image is 192 * 2 32 * 2 32 . Therefore, the complexity to guessing the overall each component of image is 192 * 2 32 * 2 32 * 2 rows * cols . Hence, this scheme provides good resistance from the brute force attack.

Histogram analysis
The histogram shows the distribution of the digital image according to the pixel values. Ideally, a cipher image should have flat distribution for the image pixels. Table 2 shows the histograms of the cipher image generated for the different components of the plain image Cosmic_Fair_Lights using all the three proposed combinations. The results clearly show that the histograms are uniform for all the proposed approaches.

Differential attack
In differential attack, the attacker produces two cipher images. The first image is the original image converted into the cipher image. The second one is the cipher image obtained by making trivial change in the original image. Then, the encrypted image and the original image are compared using the two parameters i.e. UACI and NPCR.
NPCR Number of changing pixel rate (NPCR) [7] determines, how the change in one-pixel value of the original image produces the change in the encrypted image. It determines the percentage of change of pixel value in the original and encrypted image. Hence, NPCR should have higher value for the better image encryption. The NPCR is calculated as: where size refers to the size of the image, and C 1 , C 2 are the two different cipher images. UACI United average change intensity (UACI) determines the difference between the average intensity of the two images. Generally, UACI is higher for the encrypted image for better encryption.
where size is the size of the image, and C 1 , C 2 are the two cipher images. Table 3 results show the UACI and NPCR values of the different images corresponding to their color components i.e. Red(R), Green(G), and Blue(B). The results clearly show that the CTLM combination with novel pseudo random key outperforms other two combinations.

Statistical attack
The adversary can break the encryption technique using the Correlation Coefficient (CC) and Entropy of the image. To defend statistical attack, it is essential the correlation between the adjacent pixels of the encrypted image should be close to zero. Also, the entropy of encrypted image should be close to ideal value i.e. 8.
Information entropy Entropy reflects the unpredictability and randomness of the digital images. The entropy H (S) of an image is defined as: where, P (S i ) is the probability of the random variable S i. Ideally, the information entropy of the 8bit image is 8. The Table 3 shows that the information entropy values of various encrypted images for the corresponding test dataset. Here also, the PR-CTLSM combination dominates the results. Therefore, it is very difficult for the adversary to perform statistical attack on this scheme.
Correlation coefficient (CC) The Correlation Coefficient determines the relation between the adjacent pixels of the image [45]. The Correlation Coefficient is nearly equal to zero for the good quality image encryption. It is mathematically expressed as: where L is defined as the number of the pixel of the image. The variable x and y represent as plain image and cipher image, respectively. Figure 6 shows the correlation of the different component of the original and cipher images obtained using proposed schemes. Table 4 shows the values of the CC obtained for the horizontal (CCH), vertical (CCV) and diagonal (CCD) adjacent pixels of the original and encrypted images using the proposed schemes. These values have been generated using the randomly pairs of the pixel values, and these values are close to zero for the encrypted images, as shown in Table 4. Hence, all the proposed schemes provide resistance from the statistical attack.

Bit correct ratio (BCR)
The Bit correct ratio (BCR) is defined as the total number of bit errors between the original image and encrypted image. Mathematically, bit correct ratio is defined as: Where O (x, y) represent the pixel value at position x and y of original image, and R(x, y) represent the x, y pixel position of the received image to the receiver and BCR belongs to [0, 1]. The ideal value of the BCR is 1. Figure 7 shows decrypted different images at different percentage of salt and pepper noise at 1%, 1.5%, 2%, 2.5%, and 3%. Table 5 gives the values of average BCR at different noise levels, obtained using the proposed schemes.

PSNR
Peak signal to noise ratio (PSNR) define as the ratio between the maximum size and the noise of the image. PSNR calculates as follows: PSNR ¼ 10*log 10 255*255 Where, MSE define as the error occurred between the original image and received image during the transmission of the image. It is calculated as: where O(x, y) represent the pixel value at position x and y of original image, and R (x, y) represent the x, y pixel position of the received image to the receiver. Table 5 shows that the PSNR values of the received images at the different percentage of 'salt and pepper' noise.

Time analysis
Encryption and decryption time plays an important role in judging efficiency of an image encryption algorithm i.e. a good image encryption algorithm should take very less time encryption and decryption time. The start of section 3 described the system and image specifications that are used to implement the proposed image encryption approach. The proposed scheme takes approximately 8 s to encrypt or decrypt an image of size 512*512. Undoubtedly, the system with higher specifications will do the same job in lesser time. The time analysis has been carried out by using time () function of Python.

Comparison with earlier research works
This section of the paper discusses the comparison of the proposed work in this paper with some of the earlier proposed research works. The Table 6 describes the comparative analysis of the different existing schemes with the proposed schemes in terms of Correlation Coefficient, NPCR, UACI and Entropy. The color image "Lena" of size "512*512" has been used to perform this performance analysis. It can clearly be observed that combining the novel pseudo random key with LM, cosine transformed LM, cosine transformed Logistic Sine Map improves the efficiency of these maps to resist different types of attack. In this digital age, protecting information from unauthorized access is the biggest challenge. As researchers work to propose new algorithms to protect the vital information from attackers, the attackers also work on developing new threats/attacks to steal the digital information. Hence, developing a fast, more secure and efficient algorithm is need of the hour. In this paper, a novel pseudo random key has been combine with three different chaotic sequences to propose three image encryption techniques for encrypting satellite images. The pseudo-random key has been generated by applying various operations like XOR, left circular shift (Left Rotate), modular. Then, this key has been combined with Logistic map (LM), cosine transformed LM (CTLM) and cosine transformed Logistic-Sine Map (CTLSM) one by one to implement three different image encryption methods. All the proposed schemes use a 384-bit initial shared key during the process of image encryption. The comparative analysis has been done for all three methods by using standard encryption and decryption parameters. Also, analysis has also been performed for various security vulnerabilities like known/chosen-plaintext attack, differential attack, statistical attack, and brute force attack. It can be observed from the results that the  63 and 0.00007, respectively. It also achieves decryption parameter results very close to the values that are ideal for a secure and efficient image encryption algorithm. As shown in time analysis, the whole process of encryption or decryption can be carried out in just 8 s. However, like any other research work, the proposed scheme also has scope of future improvements. The work can further be extended by using multidimensional, faster and more complex chaos function with the proposed novel pseudo random key to achieve less image and decryption times.