A novel auto-encoder induced chaos based image encryption framework aiding DNA computing sequence

3.1.1 Training SAE

At the outset, the SAE is an edifice with input and an output layer and three numbers of hidden layers. The size of the input layer is represented as ‘Is’ indicates the total number of pixels in a grayscale image. In contrast, the second and fourth layers’ size is equivalent to the total number of images, which are encrypted and denoted by ‘Es.’ Consequently, the size of the middle layer is said to be (Es-1).

While training the network, the input data (IN) is encoded to generate the values of the first hidden layer, say L1 is then followed with the decoding of L1 to generate the output (IN’) with an error E, where E =(IN’- IN). Again, the network encodes L1 to generate output L2, whereas the error E here is (L1’- L1); during this process, the weights between L1 and L2 are well-tuned. The process continues until the output layer is reached, which is the same as that of the input layer.

During the fine-tuning process, a backpropagation algorithm is utilized for training the weights to achieve the optimum results. The initial values of the weights must be both unequal and small. Equality of the initial weights prevents the network from learning when the problem requires that the network learns by unequal weights. This is due to the proportionality between the backpropagated error through the network and the value of the weights. For instance, if all the weights are equal, the backpropagated errors will be similar, and consequently, all of the weights will be updated equally. Thus, to avoid this equilibrium problem, the initial weights of the network should be unequal. Moreover, the initial weights should be small. This is because the weights in the BP algorithm are updated proportionally to the derivative of the activation function. Reaching the output layer will tend to estimate the grade error using Equation (1): E g n = - ( d - α ( k n ) ) . α ′ ( k n )(1)

Furthermore, the grade for each hidden layers are calculated with the aid of Equation (2): E g l = ( ( w ( l ) ) s E g l + 1 ) . α ′ ( d l )(2)

Whereas the partial derivatives are as follows: ∇ w ( l ) J ( w , b ; x , y ) = E g ( l + 1 ) ( α ( d l ) ) s(3)

Where, d = ws * x + b; w represents the weights, x and b be the input and offset values, respectively. y is denoted as the expected output and describes the sigmoid function. Once after completing network training, the weights are calculated, say (W1 and W2 respectively), and fed to the generator to obtain the shuffling matrix.

The input images are said to have an ‘n’ number of pixel components of similar dimensions. Here, the obtained weights are converted into a single-dimensional array (S _a ), whereas the array values range between 0 and 1. While carrying out this process, the weights of the prior epochs are added with the S _a if Sa’s size is smaller than n. The shuffling matrix is represented by Equation (4). S a = mod ( roundoff ( w 2 ( j ) ) × 10 4 ) ; j = 1 , 2 , . . . , m * n(4)

After establishing the shuffling matrix, the plain images are allowed to shuffle. The pixel values of the plain image are replaced with the corresponding shuffling matrix value with the position the same as the pixel value of the picture. Thus the sum of the shuffling matrixes is finally determined to obtain the permuted image (Q₁).

It then succeeds in confusing the permuted image by the chaotic matrix to produce an encrypted image. Thus with this whole process, a better permutation is achieved with less time complexity and is followed by generating the chaotic matrix aiding DNA encoding and decoding.

3.2.1 Establishing chaotic maps

A cryptographic hash function SHA-256 is deployed to establish the chaotic maps and prevent intrusions and attacks, which generates the secured secret keys for the chaotic system. For that, initially, SHA-256 hash value H1 has to be calculated. H1 is nothing but the hash value of the original image (IN). H 1 = H 1 ( 1 ) , H 1 ( 2 ) , . . . , H 1 ( 32 )(5)

The secret keys that are generated initially are one-dimensional chaotic systems comprising of certain Equations (6 and 7): p n + 1 = ( γ p n ( 1 - p n ) + ( 4 - γ ) 4 cos ( x , arccos ( p n ) ) ) mod 1(6) p n + 1 = ( γ sin ( π p n ) + ( 4 - γ ) 4 cos ( x , arccos ( p n ) ) ) mod 1(7)

Where, P _n be the initial value, γ be the control parameter, and x ε N is the degree of Chebyshev map. Here, Equations (6) and (7) represent the Logistic Chebyshev map and Sine Chebyshev map. With this proposed work, the initial conditions and the control parameters of the generated secret keys are updated with the aid of the SHA-256 hash function and are given by: p 0 ′ = 1 3 ( p 0 + 2 × 1 256 × bin 2 dec ( H 1 ( 9 ) ⊕ . . . ⊕ H 1 ( 16 ) ) )(8) p 1 ′ = 1 3 ( p 1 + 2 × 1 256 × bin 2 dec ( H 1 ( 1 ) ⊕ . . . ⊕ H 1 ( 32 ) ) )(9) γ 0 ′ = 1 3 ( γ 0 + 2 × 1 64 × bin 2 dec ( H 1 ( 1 ) ⊕ . . . ⊕ H 1 ( 8 ) ) )(10) γ 1 ′ = 1 3 ( γ 1 + 2 × 1 64 ( ∑ i = 1 32 H 1 ( i ) mod 256 ) )(11)

A function, which renovates the binary string into a decimal number and ⨀ is the symbol, denotes the bit-XOR operation. Thus the secret keys are updated with the aid of Equations (8–11) and are used in our proposed work to perform image encryption.

The permuted image (Q₁) obtained using the shuffling matrix is now divided into four blocks, say (C _i ) where i = 1, 2, 3, and 4. Moreover, Q _j , (j = 1, 2, . . k). It is denoted as the jth pixel in the block C _i , whereas k is assumed as the total number of pixels found at each block. Now, the blocks are transformed using the concept of S-box (S _Ci ), where i ranges between (1, 2, 3, and 4) as discussed in [26].

Consider, S _j  = S _Ci  (Q _j  + 1), where S _j is said to be the new pixels and S _Ci has replaced the Q _j to develop a new matrix say Q₂. The matrix Q₂ is then transformed into its binary form Q _b of dimension MX8N and is supposed to undergo DNA encoding, resulting in creating a MX4N DNA matrix say (Q₃).

After accomplishing DNA encoding, a complementary matrix has to be generated. A chaotic matrix Y ^MX4N has to be obtained by iterating the Equation (6) to hide the transient effects using the updated parameters and conditions established in Equations (8–12). This would result in the formation of a chaotic map Y given by Equation (12). Y ( i , j ) = { 1 , if Y ( i , j ) > 0.5 0 , if Y ( i , j ) ⩽ 0.5(12)

Now to calculate the complementary matrix of Q₃, Q 4 ( i , j ) = { Q 3 ( i , j ) Q 3 ( i , j ) if Y ( i , j ) = 1(13)

Finally, at the encryption side, Q₄ DNA decoded to obtain MX8N matrix say, (Q _de ). Iterate Equation (7) for (l≥500) times to avoid the transient effects with Equations (7). Here, Q _de is decimal converted to obtain a matrix Q₅ with MXN dimension.

Generate a chaotic matrix U with the continuous iteration of Equation (7) for MXN times as per the Equation (14): V = ( U × 10 15 ) mod 256(14)

In addition, a scrambled matrix Q₆ is to be generated in the bit-level diffusion layer to obtain the encrypted image (Z) by Equation (15). Q 6 = Q 5 ⊕ V(15)

Repeat the process using initial conditions and parameters in Equations (8–11) as secret keys to obtain the encrypted image Z. A flowchart representing the overall encryption process is as follows:

Image decryption is the process in which the encrypted image is decrypted to retrieve the original image by a similar procedure carried out during encryption but in reverse order. Here the process begins with the bitwise-OR operation for the between the encrypted image and the matrix obtained by Equation (15).

Convert Z to its binary form Z _b , (MX8N), followed by the RNA encode to obtain the chaotic matrix Z₂, (MX4N). Consequently, the complementary matrix for Z₂ is derived with the aid of Equation (16): Z 3 ( i , j ) = { Z 2 ( i , j ) Z 2 ( i , j ) if A ( i , j ) = 1(16)

A is the chaotic matrix generated in the encryption process with the aid of secret keys stated in Equation (8–11). Generate a decoded matrix Z₄, (MX8N), and convert it to its decimal form, say, Z d MXN . It is then encoded with inverse s-boxes, and the resultant matrix is denoted as Z₅.

Thus with the initial encryption process, the final decrypted image S has been obtained securely. Thereby, the proposed methodology ensures a secured network communication with less time complexity and better permutation performance.

In below encryption process flow is explained in step by step manner as follows:

Input the plain image I_N, where N be the dimension of the image matrix.

The total number of keys that can be used in the process of encryption and decryption is called the keyspace of the algorithm [30]. The keyspace is calculated using how many keys are used in permutation and diffusion processes. The keyspace of an algorithm should be more significant than 2¹²⁸ to resist brute-force attack efficiently. Keys used in this algorithm are,

Original keys ‘xm’, ‘xx 1’,’ ym,’ & ‘yx1’ of PWLCM systems.

Table 1 presents the total keyspace of the proposed algorithm. In this cryptosystem, a keyspace of 10¹⁵ is used for each of the individual keys. This is because the algorithm uses a 64-bit floating-point standard, and the IEEE suggested a keyspace of 10¹⁵ for the 64-bit floating-point standard [30]. Also, the proposed algorithm uses a keyspace of 2¹²⁸ for the “Secure Hash Algorithm SHA-256” for resisting the best attack in that algorithm. Therefore, the total keyspace is 10128, which is more significant than 2128 to resist the brute-force attack strongly.

4.3.1 Information entropy

It has been determined by using the following Equation (17): G ( l ) = ∑ i = 0 2 n - 1 q ( l i ) log 1 q ( l i )(17)

Where q(l_i) be the probability of l_i, the entropy of a random image having 256 gray levels must be eight unless there arises a chance of predictability if the cipher image entropy is less than 8.

Evaluating the plain well’s local randomness as the cipher image Equation (18) has been used. H k , T B ¯ ( m ) = ∑ i = 1 k H ( S i ) k(18)

It analyses the plain and the cipher images horizontally, vertically, and diagonally with the aid of the following Equations (19–21): r x , y = F ( ( a - F ( a ) ) ( b - F ( b ) ) ) E ( a ) E ( b )(19) F ( a ) = 1 N ∑ i = 1 N a i(20) E ( a ) = 1 N ∑ i = 1 N ( a i - F ( a ) ) 2(21)

N is the sum of pixels selected from image a, and b denotes the two adjacent pixels’ grey level values. Are the expectation and variance of variable a, correspondingly.

Unified average changing intensity (UACI) and Number of pixel change rate (NPCR) metrics are used to analyze the resisting performance and are calculated using Equations (22, 23). NPCR = ∑ i , j E ( i , j ) X × I × 100 %(22) UACI = 1 X × I [ ∑ i , j | D 1 ( i , - j ) - D 2 ( i , j ) | 255 ] × 100 %(23)

Where X and I represents the width and height of the image correspondingly and D₂ represents the cipher images, the original image is amended after and before one pixel.

It is analyzed to examine the pixel distribution of the image; whereas, if the probability acquired for each gray level of a cipher image are equal, then the encryption technique is said to be highly resistant to various statistical attacks. To carry out this, Y ² test and variances are evaluated using Equations (24 and 25): Y 2 = ∑ i = 0 255 ( a i - a / 256 ) 2 a / 256(24) Var ( G ) = 1 a 2 ∑ i = 1 a ∑ j = 1 a ( g i - g j ) 2(25)

Where a _i is the incidence frequency of gray level i and a/256 be the expected incidence frequency of each gray level. Here, Equation (25) G = (g₁, g₂, … . . , g₂₅₆) denotes the histogram vector values; g _i and g _j represents the numbers of pixels; whereas, the gray values are equivalent to i and j, correspondingly.

It is carried out with the aid of the performance metrics such as Uniform Histogram Deviation (UHD) and Irregular Deviation (ID) using Equations (26 and 27). ID = ∑ l = 0 255 H Dl(26) UHD = ∑ l = 0 255 ( | H cs - H c | ) Z(27)

It calculates the histogram of total discrepancy amid plaintext image P and ciphertext image C. Where H _c is the histogram of the ciphertext image, and H _cs is the frequency of incidence at index l.

Moreover, the contrast of the image is calculated using the Equation (28): C a = ∑ ( | i - j | 2 ) × Z ( i , j )(28)

Where Z (i, j) determines the number of gray-level co-incidence matrices.

Thus to analyze the performance of the proposed work, all the above metrics are to be calculated for this work as follows:

At first, the information entropy test for the four different images is carried out, and the outcomes are scheduled in Table 2.

Let us consider Table 2 having a theoretical value close to 8; this, in turn, indicates that the proposed work has effectively prevented the information leakage and data loss in a wide range. Thus it concludes that this work has exhibits improved performance.

Here in our proposed framework, let us consider the number of test images L as 256, where k is 32 and TB is 1938. The results obtained in Table 3 shows that generated cipher images exhibit greater values, say, 7.90, which represents better local randomness.

Table 4 shows the resultant values of the correlation coefficient obtained for the retrieved cipher images analyzed vertically, horizontally, and diagonally. Here the correlation coefficients obtained are below 0.03, which is significantly less when compared with the original images. Thus the table ensures that our proposed work has achieved better performance in terms of its efficiency in withstanding against various statistical attacks.

For the plain input images of size (512×512), a distribution diagram has been drawn for both the plain and the cipher images horizontally, vertically, and diagonally, which is shown in Fig. 7. The figure shows a strong correlation in the plain image amid adjacent pixels; the correlation coefficients are close to 1, while the pixels of the cipher image are strewn around the entire plane, and the strong correlations of adjacent pixels in the original image are highly lessened through the proposed encryption methodology.

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Fig. 7

Correlation coefficients of the input image.

Here, for the four different test images, NPCR and UACI are calculated and are shown in Table 5,

The obtained values for a 256-Gy scale image are 99.72% and 33.41% correspondingly, with highly closed results and robust against differential attack.

The histogram analysis of our proposed framework is illustrated in Fig. 8. When the considerable level is 0.05, the subsequent Y2 (0.045, 255) is 292.55. If the variance is lower, then the image uniformity will be higher. Initially, Y2 is executed, and the outcome is less than 292.55, which indicates that the proposed work ensures high security. The formulation of variance follows it. In contrast, the variance of the plain image is more significant than the cipher images, which specifies that the cipher images are uniformly distributed. It reveals the efficacy of the proposed work against the security threads.

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Fig. 8

Histogram analysis of various input images.

In terms of computational complexity, the proposed encryption scheme was calculated to require O(MNlog(MN)) iterations to encrypt an image with a width of 516 and a height of M and N, respectively. Table 6 compares the performance data of the proposed algorithm with other encryption schemes.

4.4.1 Occlusion attack analysis

During transmission of data from sender to receiver, there is a possibility that the encrypted data may get distorted due to an insecure channel. This distorted data is referred to as occluded data. For occlusion analysis, we have taken occluded parts of the image, and the results in Fig. 7 show that the decrypted concatenated images contain distortion. However, still, the images are detectable by appearance. Hence, it is inferred that our Hybrid Entangled Chaotic DNA Amalgam cryptosystem can survive up to 78% of occlusion attack.

4.4.2 Known-plaintext and chosen-plaintext attack analysis

Most of the image encryption algorithms have been broken by the known-plaintext attack and chosen plaintext attack. In this proposed cryptosystem, the keys (initial values and systems parameters) are generated from the given initial values and system parameters of PWLCM systems and the 256-bit hash value.0 of plain image. When any one of the plain images changes, the key values change, different chaotic sequences are produced, and different cipher images are obtained. So attackers cannot obtain cipher images by selecting some known plain images. This concludes that the proposed algorithm strongly resists known-plaintext attack and chosen-plaintext attack.

4.4.3 Noise attack analysis

To verify the robustness of the proposed image encryption algorithm against noise attacks, the plain image of ‘City’ is encrypted using the proposed image encryption algorithm. Then, salt and pepper noise with different intensities (0.01, 0.05, and 0.1) are added to the plain image. Finally, the noisy cipher images are decrypted to obtain the original plain image. The experiment is repeated using two perceptual hash algorithms: the average hash and the SHA-256, with hamming-distance threshold = 20. Figure 9 shows the cipher images of plain ‘City’ images for each intensity and the corresponding decrypted images. The similarity between the plain-image and decrypted images can be easily observed. This similarity can be verified with the high value of PSNR values shown in Table 7, which demonstrates the resistance of the proposed image encryption algorithm against noise attacks.

Table 8 proves the efficiency of the proposed work in terms of secured data transmission with encryption quality assessment. Here, the divergence is less, and the contrast values range between 10.534, 10.532, and 10.132 for colour, gray and binary images, respectively.

Hence the percentage difference of the two ciphertexts shown in Table 9 is calculated. It can be seen from Table 9 that the percentage differences for binary images are greater than 98% and for colour and gray images are even more significant than 99%. Thus key sensitivity test confirms that the proposed image encryption scheme is very sensitive to smaller changes in key.

Table 10 presents the keyspace comparison results between the proposed scheme and the existing algorithm. Using the proposed scheme, it shows the high resistivity of brute-force attack.