Asymmetrical novel hyperchaotic system with two exponential functions and an application to image encryption

: In this article, asymmetrical novel system with two exponential functions, which can show hyperchaotic behavior, has been proposed. Although new system possesses only one unstable equilibrium. The dynamical behaviors of such system are discovered by computing the Lyapunov exponents and bifurcation diagram. Furthermore, the synchronization of the proposed system are also presented by an adaptive synchronization approach of two identical hyperchaotic systems. An application to image encryption has been obtained.


Introduction and formulation of the system
In the last two decades, a large number of chaotic systems have been studied with their application in weather forecasting [1], telecommunication [2], biological modeling [3], and so on.Chaotic systems can be classified into three categories by physical, dynamical, and algebraic features according to the number of dimensions, the number of wings, and number of equilibrium points, respectively.After Lorenz's dis-covery (1964 [4]), researchers have attempted to provide chaotic systems with unique peculiarities (physical and dynamical) such as biological model of Rössler that contain only one nonlinear term [5], electronic circuit of Chua that exhibit two scroll chaotic behavior [6], and simplification of the Lorenz system by the Chen and Lu system [7,8].On the other side, chaotic systems, with special algebraic structure were provided, such as Wei's system that has no equilibria [9] and Wang's system that has only one stable equilibrium [10].In the late 1970s, Rössler proposed a very interested choatic system with more then one positive Lypunouv exponent, later was called hyperchoatic [11].After that, hyperchaotic systems got the interest of researchers in different areas, and many of them have been introduced, especially, 4D hyperchaotic Lorenz-type system [12].Note that such kind of systems with unusual peculiarities can have neither heteroclinic orbit nor homoclinic orbit, and thus, the Shilnikov method [13] may not help to verify the chaos.Thus, hyperchaotic systems are more complicated.
From a computational point of view, a hidden attractor was observed in some kinds of chaotic and hyperchaotic systems such as systems with no equilibria.To have better understanding to hidden attractor, we refer to the studies by Kuznetsov [14] and Leonov and Kuznetsov [15].
In this article, a novel hyperchaotic system with two exponential functions is proposed.This article is organized as follows: Sections 2 introduces the theoretical model of the system and studies some of it is fundamental properties.In Section 3, the ultimate boundedness of the proposed system has been obtained.In Section 4, the possibility of the synchronization scheme of the proposed systems is studied.In Section 5, an application to image encryption was processed.
where α β c , , , and d are system parameters and xz xy , exp( ) are the nonlinear terms.System (1) shows complex dynamics when = =− = α β c 0.78, 0.5, 0.37, and = d 2.5.Besides, system (1) has two equilibria, and due to this fact, one can easily check that (1) is not topologically equivalent to the original Lorenz or any Lorenz like systems.In the study by Pham et al. [16], the following 4D hyperchaotic system was constructed: Pham et al. [16] reported that system (2) has no equilibrium and it can be classified as a chaotic system with the hidden strange attractor where a basin of attraction does not contain neighborhoods of the equilibrium points.
On the basis of the idea of system (2), we introduce asymmetrical four-dimensional hyperchaotic system with two exponential functions such as follows: where y is a sigmoid function; x y , , and z are state variables; and α β m d , , , , and r, are real constant para- meters of systems (3) to control.
In what follows, we test the most important peculiarities of complex dynamical systems such as: disspativity, stability, Lyapunov exponents (LEs), and bifurcation diagram.First, system (3) is disspative for < β 0 with the divergence For example, considering = α 1 and = − β 2, we have ∇ = − V 2. Thus, the considered systems are dissipative.The exponential contraction rate is calculated as follows: Therefore, each volume containing the system trajectory shrinks to zero as → ∞ t and system orbits are ultimately bounded.Thereby, the existence of attractor of system (3) is proved.
To set the stability, the following algebraic equations must be held ( ) From (4) follows that the system (3) have no equilibrium points and this similar to system (2) which means that system (3) is hidden strange attractor in contrast to system (2), one can easily observe that systems (3)  and are shown in Figure 1(a).The bifurcation diagrams show the local maxima of the variable z t ( ) when changing the value of r β , , and d, which is shown in Figure 1(b)-(d).The dynamics of (3) is shown in Figure 2.

Ultimate boundedness
Chaotic systems are bounded, and estimating the boundedness is one of the most important and difficult tasks.The bounds of chaotic systems play a very significant role in chaos control and chaos synchronization [17,18].Here, we discuss the boundedness property of system (3).Theorem 3.1.Suppose that the parameters α β m d , , , , and r of system (3) are positives.Then, all the system orbits, including chaotic orbits, are trapped in a bounded region.
Proof.Construct the following Lyapunov function: Along the orbits of (3), we have   , , ,  ( ) Consequently, we have the surface x y z w V , , , , , ,  {( ) | ( ) }is confined a trapping region of all solutions of system (3).

Adaptive control of the proposed chaotic system
Let consider the driver system defined as follows: Here, we consider the adaptive synchronization of identical novel chaotic systems with unknown parameters.The response system is presented as follows: where x x x , , 1 2 3 , and x 4 are the states, α β m d , , , , and r are unknown system parameters, and = U u u u u , , , T [ ] is the adaptive controller to be determined.
We consider the adaptive controller defined by where ε ε ε ε , , , α β m d , and ε r are estimates of α β m d , , , , and r, respectively, and By substituting ( 7) into (6), we obtain the closed-loop system: with parameter estimation errors defined as follows: Next, we reduce (8) to Theorem 4.1.If the controller are chosen as (7) and update laws of parameters are given by Then the synchronization between the driver system (5) and the response system ( 6) is achieved if k k k , , 1 2 3 , and k 4 are positive constants.Proof.We consider the Lyapunov function defined by

( ) (
) Taking time derivative of the aforementioned function along the trajectories of (11), we have which is a negative definite function for . Thus, due to the Lyapunov stability theory, we obtain that In other words, the synchronization occurred between the driver system (5) and the response system (6).
5 An application to image encryption

Encryption procedure
In this section, we study the exploit of the proposed system in theory of image cryptosystem.The scheme of the Asymmetrical novel hyperchaotic system with two exponential functions  5 presented image cryptosystem is shown in Figure 3(a).In particular, the following steps need to be done: 1. Calculate the following value: and then updates the value of X using the formula: where MN is the total number of pixels in the image, ∑ P is the sum of the pixel values, K is the system's dimensions, and = i K 2,…, .2. Solve the system (3) using the initial values from the previous step.

Sort the output of the equations the system
(3) and store the sorting indices.4. Rearrange the original image after convert it to vector R of pixel values according to the order of the sorted indices form the previous step.

Reshape the vector of pixel values into a matrix with M
rows and N columns and store it as matrix R, then extract × 2 2 block of elements from matrix R after that store it in matrix Cx.In particular, the two blocks are chosen based on their position in the image by iterating over every second row (for = i M 1 : 2 : ) and every second column (for = j N 1 : 2 : ) of the image, and after selecting a × 2 2 block of pixels centered at that position.

The matrix multiplication Cx A
* is performed, and the obtained matrix is stored in the corresponding block of , which represents the encrypted image, where A is an arbitrary × 2 2 matrix, which we called secret.

Decryption procedure
The decryption is usually a reverse of encryption process.The decryption scheme corresponding to the proposed encryption approach is displayed in Figure 3(b).
After convert, the encrypted image to a double-precision floating-point array P and obtaining the size of it M N * , ( ) then according to the number of rounds that is used in encryption and by going from the last round to the first round, we apply a linear transformation to each × 2 2 block of pixels in the encrypted image using the adjoint of matrix A that is used in encryption.Then we sort the elements of the transformed image using the indices in the key for that round, flatten the transformed image into a single vector, and sort the vector using the same indices.After that we reshape the sorted vector into a matrix with the same size as the original image and apply the modulo operation elementwise to ensure that all values are within the range 0-255.After all the rounds have been completed, the decrypted image converted to an unsigned 8-bit integer.For more information about the recent encryption, methods based on chaotic and as well hyperchaotic system, we refer the readers to the previous studies [21][22][23][24].
6 Performance and security analysis

NPCR and UACI tests
Its well known that in cryptosystem, minor modifications are possible to the plain image, and to test the sensitivity, there are two measures, shortly denoted by UACI and NPCR.The equations for computing the NPCR and UACI are as follows, respectively: where M and N are the rows and columns in the image, respectively.i j dist , ( ) is the difference between C 1 and C 2 , given by the following equation.

Statistical analysis
Here, we will check the following statistical indices: I. Correlation analysis.By N p , we denote the neighboring pixels.In the usual images, N p values are very close to each other, and this means that connected pixels are   highly correlated in the original images.For specialist, highly correlated feature can be used to break the cipher.In particular, in the cipher image, N p must be highly uncorrelated.The correlation coefficient between any two pixels is given in the following equation: where .
{ } indicates the expected values of the random variables, α i and β i be the grayscale values of N p and K t is the total number of pixels taken for the calculation.In Figure 4(a) and (b), we show the correlation between the original image and the encrypted/decrypted image, respectively.The correlation coefficient for the original vs encrypted image is −0.002197, while the correlation coefficient for the original vs decrypted image is 1.These results indicate that the process has successfully scrambled the pixels of the original image, making it difficult to infer any information about the original image from the encrypted image.In contrast, the decryption process has successfully restored the original image from the encrypted image.II.Histogram and Chi-square test.To visual description to the distribution of pixel intensities in the image, we need to consider the histogram of an image, that gives a graphic view.
Note that original images usually have non-uniform histograms because pixel intensities are limited within some range.Its well known that this property can be used by cryptanalysts to intercept the cipher using histogram-based attacks.Thus, a secure encryption should produce cipher images with uniform histograms.Figures 5 and 6 show two grayscale images, namely Airplane and Peppers, along with their histograms with depicts the encrypted analogs of the original images along with their histograms.It can be observed that the original images exhibit non-uniform histograms while In the experiment process, the distribution is considered to be uniform when the Chi-square test is found to be more than a significance level ∈ μ μ 0, 1 , ( [ ]) and this way, the null hypothesis is accepted.Table 2 shows the success rate of the Chi-square test of different encryption schemes.III.Entropy information.One of the most important measure in dynamical systems theory is entropy, and the statistical test for calculating randomness in a sequence is defined as follows: where X is he source, p x k ( ) is the probability of the ele- ment x k , and K c is the number of different elements generated by X.The ideal of entropy is to obtain when all pixel levels appear with an equal probability showing that all pixel uniformly distributed.In Table 3, we show different entropies considered in the literature.

Keyspace analysis
For a good encryption process, the keyspace must be large enough to withstand bruteforce assaults.The suggested algorithm's key consists of the starting values x y z w , , , as well as the parameters α β d m r , , , , and .Also, the opera- tions of the proposed encryption algorithm are governed by parameters α β d m , , , , r and matrix A. In particular, the algorithm iterates for a specified number of rounds to enhance security.The keys in this algorithm are not explicitly generated; rather, they are derived from the parameters α β d m r , , , , , and A, which means that these parameters play a crucial role in determining the behavior of the algorithm and, consequently, the resulting encryption.The choice of values for these parameters significantly impacts the resulting ciphertext.For instance, larger values may lead to more complex transformations, potentially increasing the size of the keyspace.In addition, the matrix A introduces another layer of complexity, as its elements influence the linear transformation applied to the image data.As a result, when an attacker attempts to brute-force the key, they must account for the influence of A along with the other parameters.This increases the complexity of the encryption process and enlarges the keyspace, making it even more challenging for an attacker to deduce the correct key through trial and error.

Occlusion attack
It is well known that cipher images will loss data when it transferred through a communication channel.Completely or partially, the lost data can affect the decryption process.
To measure the strength of the presented cryptosystem, the occlusion attack test is applied to the cipher images.Testing is conducted using 1/16, 1/8, ∕ 1 4, and 1/2 data losses.We can compare Figure 7(a)-(d), which show the encrypted images with above mentioned losses, and Figure 7(e)-(h), which show analogous ciphered images.
Visually, we can observe that most of the information is retained even after information of half of the encrypted image's information is lost.

Noise attack
Particularly, while images goes through media channels that are often subjected to the noise.Obviously, this noise can affect the quality of the decrypted image if the corresponding cipher image is subjected to it.By contaminating the cipher image with salt and pepper nose where densities 0.005, 0.05, 0.1, and 0.3.The analogous decrypted images are shown if Figure 8(a)-(d).
From the aforementioned figures, one can see that these images are noisy but readable.

Discussion
In this article, the complex dynamics of a four dimensional hyperchaotic system involved in two exponential terms was investigated.Ultimate bound of this system was obtained.Adaptive control of the new system was obtained.As an application to the proposed system, the proposed cryptosystem consists of several stages: first, some initial values are calculated by applying proposed equation in Section 5.1, where its parameters depending on the dimensions of the image to be encrypted, the initial values that were calculated previously were used to solve a system of differential equations in (3), and after that, the output indices of the differential equations will be the initial values that are used to rearrange the pixels of the input image.The resulting image is rebuilt in 2 matrices.Each matrix multiplies a secret matrix of dimension × 2 2. Also, the resulting matrix is converted into unit 8, which represents the encrypted image.At the end, throughout this study, we observed that the proposed system involved to sigmond function is making the system weaker.In particular, the other dynamics of the introduced system of this article are expected to be further studied.
large, so that for all x y z w

Figure 3 :
Figure 3: (a) The scheme of the encryption algorithm; Airplane as an example and (b) the scheme of the encryption algorithm; Airplane as an example.

Figure 4 :
Figure 4: Distribution of connected pair pixels in image of airplane.(a) vertically and (b) diagonally.

Table 3 :
Entropy information Figure 7: Occlusion attacks result.Asymmetrical novel hyperchaotic system with two exponential functions  9