Fast chaotic encryption scheme based on separable 1 moments and parallel computing 2

In this paper, we propose three novel image encryption algorithms. 7 Separable moments and parallel computing are combined in order to enhance 8 the security aspect and time performance. The three proposed algorithms 9 are based on TKM (Tchebichef-Krawtchouk moments), THM (Tchebichef- 10 Hahn moments) and KHM (Krawtchouk-Hahn moments) respectively. A novel 11 chaotic scheme is introduced, which allows for the encryption steps to run si- 12 multaneously. The proposed algorithms are tested under several criteria and 13 the experimental results show a remarkable resilience against all well-known 14 attacks. Furthermore, the novel parallel encryption scheme exhibits a drastic 15 improvement in the time performance. The proposed algorithms are compared 16 to the state-of-the-art methods and they stand out as a promising choice for 17 reliable use in real world applications. 18


Introduction
[10]. The latter algorithms instead of image pixels deal with the coefficients ob-48 tained in the transform domain, these algorithms were reported to have higher 49 efficiency, they are more robust against image processing operations and can 50 make lossless recovery of the original image [3] [16]. 51 A common problem in image encryption domain is the "speed vs. secu-52 rity" dilemma [29]. While the recently proposed algorithms tend to be more 53 and more secure, they come at a cost, which is computation speed [9] [42]. In 54 fact, chaos based algorithms -which are the predominant schemes in image 55 encryption area-are generally a "two stages based process", namely, confu-56 sion and diffusion. These two separated stages are repeated until a satisfactory 57 level of security is obtained [21]. The more these steps are repeated the more 58 secure the algorithm is, and the slower it gets [21] [12]. Several works have 59 been introduced in order to enhance the time performance of image encryp-60 tion algorithms while keeping a required level of security [32] [23] [9] [33]. While 61 these works present much enhancement toward resolving the dilemma men-62 tioned above, they are not suitable for real world applications, because they 63 either require some specific settings or are still limited in time performance 64 for real world scenarios. Thus, our work aims to tackle the above-mentioned problem by presenting a fast and secure encryption algorithm based on chaos 66 and parallel computing. In this work, we have opted for using separable mo-67 ments as the transform domain for encryption. This choice is motivated by the 68 remarkable results shown in different areas of image processing [36] [2] [35]. In 69 fact, the theory of moments have been introduced into the image encryption 70 domain and showed remarkable results [6] [18] [15]. In [11] we made a first at-71 tempt to explore the use of the transform domain of moments for encryption. 72 In this work, we used a logistic map to confuse and then diffuse the moments' 73 coefficients obtained using: Tchebichef, Krawtchouk, Hahn, Dual Hahn and 74 Racah moments. We argued that the moments' based encryption algorithms 75 outperform state-of-the-art methods [ In this paper, we propose three novel encryption algorithms based on sep-85 arable moments and chaos. We propose a novel scheme that allows several 86 steps to run simultaneously. We add an new step in the encryption process

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-Introducing SM moments into the domain of image encryption. 105 We have organized the rest of this paper in the following way: background 106 on separable moments and chaos theory is given in section 2. Section 3 presents  Our aim is to make this paper as self-contained as possible. In this regard, 112 the current section serves as a theoretical background for the concepts used 113 in this article. We briefly discuss the theory of image moments and show how 114 they are be computed then we present the logistic map and give some of its 115 important properties.
Where kernel nm is the moments kernel that consists of specific polynomials  The original image can be reconstructed from the moments' coefficients 141 using the formula: The properties of the separable moments used in this paper are summarized 143 in the table 1. Table 1 Moments' Kernel form Polynomial form x n is the state variable, n the number of iterations and µ is a parameter      the key is divided into 3 segments: K 1 , K 2 and K 3 . These segments are used as 193 the initial conditions for the logistic maps involved in the encryption process.

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Logistic maps take an initial value between 0 and 1, so in order to adapt 195 the segments K 1 , K 2 and K 3 as initial conditions we perform the following 196 mathematical operations: 197 We note each segment K i in its binary form: Then the initial values for 199 the logistic maps are computed as follows: Where X 0 , Y 0 and Z 0 are the initial values for the logistic maps X, Y and 204 Z respectively.

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Three logistic maps are used, X, Y and Z to generate random sequences.

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The generated sequences are used for block permutation, pixels' permutation 208 and pixels' diffusion respectively.

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Thread 2: change block position to L i .

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-Store the block at the position L i in the matrix B.

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Input: matrix B, S and T (random sequences generated in step 2).

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-Convert the image into an array A of size M × N .

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-For each pixel j:

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Thread 1: diffusion, change pixel's value according to A j XOR T j .

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Thread 2: confusion: Change pixel's position to S j .

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-Store calculated value at position S j in matrix E. Steps 1 and 2: The same encryption key generated in the step 1 and the 249 random sequences generated in the step 2 of the encryption algorithms are 250 used for the decryption process.

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Input: E encrypted image, S and T (random sequences generated in step 253 2).

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-The image is transformed to an array of size M × N notated W .

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Thread 1: change pixel value according to W i XOR T i .

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Thread 2: change pixel position according to S i .

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-Reverse the computed array to a matrix F of size M × N .

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Input: matrix F , L (random sequence generated in step 2).    Where W and H are the image width and height. D(i, j) is defined as, When the N P CR is around 99.6 and U ACI reaches approximately 33.4, 292 the encryption is secure against differential attacks [29] [24]. 293 We encrypt the test images listed in table 6 using the proposed encryption art algorithms, which makes them more secure against differential attacks. formula: Where x and y are pixels' positions of plain-text image and cipher-text image 327 respectively, Cov(x, y) is covariance, V AR(x) is variance at pixel position x, 328 σ x is standard deviation and N is the total number of pixels. 329 We compute the correlation coefficient between two vertically adjacent pix-   without compromising the security performance of the proposed algorithms.