Modified Projective Synchronization between Different Fractional-Order Systems Based on Open-Plus-Closed-Loop Control and Its Application in Image Encryption

A new general and systematic coupling scheme is developed to achieve the modified projective synchronization (MPS) of different fractional-order systems under parameter mismatch via the Open-Plus-Closed-Loop (OPCL) control. Based on the stability theorem of linear fractional-order systems, some sufficient conditions for MPS are proposed. Two groups of numerical simulations on the incommensurate fraction-order system and commensurate fraction-order system are presented to justify the theoretical analysis. Due to the unpredictability of the scale factors and the use of fractional-order systems, the chaotic data from the MPS is selected to encrypt a plain image to obtain higher security. Simulation results show that our method is efficient with a large key space, high sensitivity to encryption keys, resistance to attack of differential attacks, and statistical analysis.


Introduction
Fractional calculus, which is a mathematical topic with more than 300-year history, was not applied to physics and engineering until recent decades.A fractional-order system is characterized as a dynamical system described by fractional derivatives and integrals.It is demonstrated that some fractional-order differential systems behave chaotically or hyperchaotically, such as the fractional-order Lorenz system [1], fractional-order Lü system [2], fractional-order Rössler system [3], and fractional-order Arneodo system [4].Recently, the control and synchronization of the fractionalorder chaotic systems start to attract a great deal of attention due to their potential applications in secure communication and control processing.Some approaches have been proposed to achieve chaos synchronization between fractionalorder chaotic systems, such as adaptive control [5], a scalar transmitted signal method [6], sliding mode control [7], and fuzzy logic constant control [8].
Other than the above studies, the Open-Plus-Closed-Loop (OPCL) control method is a more general and physically realizable coupling scheme that can provide stable synchronization in identical and mismatched oscillators [9,10].The advantage of the OPCL coupling includes the following two aspects.First of all, OPCL coupling provides synchronization in all systems without restrictions on the symmetry class of a dynamical system.Secondly, in the synchronization regimes, the OPCL coupling can realize stable amplification or attenuation in identical and mismatched systems.Until now, many researchers have achieved their synchronization scenarios for integer-order or fractional-order systems through OPCL control [11][12][13].It should be noted that most of the existing works focus on synchronization between identical chaotic systems.However, in practice applications, most systems are nonidentical and parameter mismatches are inevitable because of noise or other uncertain factors.Our coupling strategies need to be formulated to ensure stable synchronization in the presence of mismatch.As a matter of fact, OPCL control can be utilized to achieve synchronization of fractional-order chaotic systems with different structure.
Specially, we will realize modified projective synchronization (MPS) of two different fractional-order systems with parameter mismatches.In MPS, the states of the drive and response systems synchronize up to a constant scaling matrix 2 Mathematical Problems in Engineering with the complete synchronization, antisynchronization, and projective synchronization as the special cases.Based on the OPCL control, a general coupling method is proposed for MPS of two nonidentical fractional-order systems.The proposed coupling scheme is theoretically proved based on stability theory of linear fractional differential equations and its effectiveness is verified by two groups of numerical simulations.Finally, based on the realized MPS, an image encryption scheme with diffusion and confusion is designed.Both the unpredictability of scaling matrix and the use of fractional-order systems will raise the security level of the encryption scheme.According to the analysis of simulations, really satisfactory results are obtained, with large key space, high sensitivity to initial conditions, and high security.

Theory Analysis.
There are several definitions of fractional derivatives.The Caputo derivative is more popular in the real applications, because the inhomogeneous initial conditions are allowed, if such conditions are necessary.The Caputo definition of the fractional derivative [15], which sometimes is called smooth fractional derivative, is defined as where  is the smallest integer larger than ,   denotes the Caputo definition of the fractional derivative,  () () is the -order derivative in the usual sense, and Γ stands for gamma function.
As to the fractional-order chaotic systems, we will briefly describe how to synchronize two different systems via the OPCL coupling method.Assume the fractional-order chaotic system in the drive part is as follows: where  ∈   ,  :   →   is a continuous vector function, and Δ() contains mismatch parameters.If the system parameters are not disturbed in the theory, we set zero to the value of Δ(). = ( 1 ,  2 , . . .,   )  for 0 <   < 1 ( = 1, 2, . . ., ) is the order of fractional-order system.If  1 =  2 = ⋅ ⋅ ⋅ =   , we call the system (2) a commensurate fractional-order system, otherwise an incommensurate fractional-order system [16].
Then, the controlled response system is constructed as where  ∈   ,  :   →   is a continuous vector function, and () is the controller to be designed.
Definition 1 (MPS).For the drive system (2) and controlled response system (3), it is said to be modified projective synchronization (MPS), if there exists a constant matrix  = diag( According to the OPCL control [9,10], we design the controller () as in the form of  () =    −  () + ( −  ()) ( − ) , (4) where  = /() is the Jacobian matrix of the dynamic system and  ∈ ( × ) is an arbitrary constant matrix.Then, () can be written, using the Taylor series expansion, by Keeping the first order terms in (5) and putting ( 5) and ( 4) into (3), the error dynamics between systems (2) and ( 3) is then obtained to be In order to research the synchronization stability of the two incommensurate or two commensurate fractional-order systems by OPCL coupling, we provide the following two theorems.
From the two theorems, we can easily obtain the following two corollaries.Corollary 6.When system (2) and system (3) are incommensurate fractional-order systems, set  as the lowest common multiple of the denominators   of   , where   = V  /  , gcd(  , V  ) = 1.The zero solution of the error system (6) is asymptotically stable if all roots  of the equation Corollary 7. When system (2) and system (3) are commensurate fractional-order systems, the error system (6) is asymptotically stable if and only if | arg()| > /2 is satisfied for all eigenvalues  of .

Numerical Method for Solving Fractional-Order Systems.
An efficient method for solving fractional-order differential equations is the improved predictor-corrector algorithm [19], which will be used in numerical simulation section.The algorithm can be interpreted as a fractional variant of the classical second-order Adams-Bashforth-Moulton method.

Numerical Examples.
In this section, to demonstrate the effectiveness of the proposed OPCL based MPS scheme for different fractional-order systems, we provide two groups of numerical examples.Firstly, fractional-order Arneodo system and fractional-order Lü system are used to verify the incommensurate synchronization.Secondly, fractionalorder Lorenz system and fractional-order financial system are introduced to validate the commensurate case.
From system (13), we can obtain the Jacobian matrix: The constant matrix  for response Lü system is selected as On the basis of Definition 1, the error vector of MPS can be expressed by Consequently, define (12) as the drive system and the response system controlled by OPCL coupling is obtained as The time evolutions of states for coupled system (12) and system (17).
Figure 2 displays the error state trajectories of the two systems.And the error state trajectories asymptotically converge to zero, which implies that the MPS between the incommensurate system (12) and system (17) is realized.

MPS between Fractional-Order Lorenz System and
Fractional-Order Financial System.The fractional-order Lorenz system with parameter perturbation is expressed as where Δ, Δ, and Δ are the mismatches in parameters.When (, , ) = (10, 28, 8/3) and  ≥ 0.993, the Lorenz system exhibits chaotic behavior.
The simulation results of the two examples demonstrate that the nonidentical fractional-order chaotic systems with mismatches can achieve the MPS under the OPCL coupling.

A Novel Image Encryption Scheme
Based on MPS 3.1.Scheme Description.Based on the MPS between fractional-order Arneodo system and fractional-order Lü system, an image encryption scheme is designed for the sake of higher security.
The typical image encryption framework is used to encrypt plain image, which is illustrated in Figure 4.
The image cryptosystem in Figure 4 includes two stages, chaotic confusion and pixel diffusion, where the former process permutes a plain image and the latter process changes the value of each pixel one by one.As shown in Figure 4, the confusion and diffusion processes are both repeated several times to enhance the security of this cryptosystem.Suppose that the size of image is  ×  and the detailed encryption algorithm is described as follows.
(2) In the confusion process,  utilizes the discrete data of system (17) to permute the position of pixel; set   = abs(fix( 3 ( 1 ))) and   = abs(fix( 3 ( 1 +  2 ))), where fix (⋅) is the function to obtain the integer part,  1 >  0 , and  2 is the time interval of the two parameters; the position of pixel is permuted as follows: where (  ,   ) and ( +1 ,  +1 ) are considered as the positions of image pixel before and after permutation.
(3) In the diffusion stage, the pixel value of image is substituted with its position information by ; according to the chaotic stream , we can obtain two substitution parameters: where round( ) is rounding function and  is a positive integer; the biggest value of the parameter  relates to the precision of the computer; therefore, the range of parameter  is from 1 to 14 in current experiment, which can be used as secret key; the substitution of pixel value is in the form of where  and V are the pixel values of image before and after substitution and  is the grey level of pixel.The decryption procedure is similar to that of encryption process with reverse operational sequences to those described above.When  receives the cipher image, it uses the chaotic stream  1 = ( 1 (),  2 (),  3 ()),  >  0 , generated by the system (12) and the initial condition of system (12) and scaling matrix  to generate  2 = ( 1 (),  2 (),  3 ()),  >  0 , by  1 () =  1  1 (),  2 () =  2  2 (), and  3 () =  3  3 ().Firstly, substitute the grey values in cipher image back to original ones, namely, for every position (  ,   ) and corresponding grey value V of cipher image; compute original grey value as follows: where substitution parameters  and  can be computed by (26).After all pixels return to original grey values, then, the pixel in position ( +1 ,  +1 ) should be moved back to the original position (  ,   ) by following inverse operation: where the values of   and   are the same as they are in (25).After the two steps are followed, the plain image can be resumed and the process of decipher is over.

Experimental Results and Security Analysis.
To demonstrate the validity and efficiency of our scheme, a group of experiments for gray Lena image (256 × 256) is carried out with results shown in Figure 5. Here, the key set is selected the same as Section 2.2. Figure 5(b) is the cipher image for original image in Figure 5(a).The histograms of plain image and cipher image illustrated in Figures 5(c) and 5(d) demonstrate that although the grey distribution of original images is not uniform, the grey values of cipher images become uniformly distributed and their statistical property is absolutely changed.A good encryption should be able to resist all kinds of known attacks and some security analyses have been performed on the proposed image encryption scheme.

Key Space.
The key space of a good image encryption algorithm should be sufficiently large to make brute-force attack infeasible.The key space of the proposed method is much larger than those of previous methods because system parameters, fractional derivative, and initial conditions of drive system (12) and diagonal elements of scaling matrix  are all cipher key ones; moreover, the mismatch parameters Δ, Δ, and Δ of drive system (12), time point  1 , time interval  2 , and positive integer  are all also secret keys.So this is enough to resist all kinds of brute-force attacks.

Key Sensitivity.
A good encryption scheme should be sensitive to cipher keys in process of both enciphering and deciphering.Namely, when an image is encrypted, tiny change of keys should receive two completely different cipher images and, when an image is decrypted, tiny change of keys can cause the failure of deciphering.(1) Key Sensitivity in Encryption.The following key sensitivity tests in encryption have been performed based on the 256 × 256 gray Lena image.
Test 1.One of the initial conditions of the drive system ( 12) is changed a bit; here, we let the first initial condition of system (12) be changed, using  1 (0) =  1 (0) + 10 −4 .
Test 2. One of the system parameters of the drive system (12) is changed slightly; here, we alter the second parameter, using  =  + 10 −4 .
Test 3. One of the fractional derivatives of the drive system (12) is changed, using  1 =  1 + 0.01.
Test 4. One element of the scaling matrix is altered, using  1 =  1 + 1.The differences of the two cipher images for the four tests are given in Table 1.From the table, it can be concluded that the proposed method is very sensitive to the key; a small change of the key will generate a different decryption result and one cannot get the correct plain image.
(2) Key Sensitivity in Decryption.In the encryption scheme, small changes to key can lead to completely incorrect image.For the image of gray Lena shown in Figure 5(a), the decryption result with right key is shown in Figure 6(a) and the incorrect decrypted image is shown in Figure 6(b) when the where  1 and  2 are the pixel value matrices of two different cipher images, respectively;  is the change of the corresponding pixel value, which is defined as  (, ) = { 0  1 (, ) =  2 (, ) 1  1 (, ) ̸ =  2 (, ) .
Next, two plain images are considered: one is the original image shown in Figure 5(a); the other is a changed image that adds 1 to the pixel value in the lower right corner of original image.When we encrypt the two plain images with the same encryption key, we can obtain two different cipher images  1 and  2 .Several comparisons of NPCR and UACI between our method and literature [14] with different values of  and  are given in Table 2. Compared with the results of literature [14], we can achieve a much more better performance NPCR > 0.996 and UACI > 0.334 with  =  = 4, which can be obtained with  = 6 in literature [14].

Statistical Analysis.
To test the correlation between two adjacent pixels, the following procedures are carried out.The correction coefficients   of two horizontally, vertically, and diagonally adjacent pixels in the plain image and the cipher image are calculated according to the following formulas: Here, we use the 256 × 256 gray Lena image, encrypted image with our method, encrypted image in literature [14], and random image for simulation.The results are given in Table 3.
Meanwhile, we randomly select 2000 pairs of two horizontally adjacent pixels from the Lena image.The correlation distribution of the pixels in the plain image and the cipher image is illustrated in Figure 7.Both the correlation coefficients and the figures justify that neighboring pixels of the plain image can be decorrelated by the proposed cryptosystem effectively.Therefore, the proposed algorithm has high security against statistical attacks.

Conclusions
In this paper, for the first time, an OPCL coupling scheme is utilized to achieve the MPS between two different fractionalorder dynamical systems in the presence of mismatch.Based on the stability theory of fractional-order system, the MPS

Figure 4 :
Figure 4: Block diagram of the image cryptosystem.

Figure 5 :
Figure 5: The encrypted results for Lena image: (a) plain Lena image; (b) histogram of Lena image; (c) cipher image; (d) histogram of cipher image.
where  and  are pixel values of two adjacent pixels in the image, () is the mean value of , and () is the variance of ,  =  × /2.

Table 1 :
Percentage difference between cipher images.

Table 3 :
The comparison of correlation coefficients between two adjacent pixels.