The Design and Its Application in Secure Communication and Image Encryption of a New Lorenz-Like System with Varying Parameter

A new Lorenz-like chaotic system with varying parameter is proposed by adding a state feedback function. The structure of the new designed system is simple and has more complex dynamic behaviors. The chaos behavior of the new system is studied by theoretical analysis and numerical simulation. And the bifurcation diagram shows a chaos-cycle-chaos evolution when the new parameter changes. Then a new synchronization scheme by a single state variable drive is given based on the new system and a chaotic parameter modulation digital secure communication system is also constructed.The results of simulation demonstrate that the new proposed system could be well applied in secure communication. Otherwise, based on the new system, the encryption and decryption of image could be achieved also.


Introduction
With the development of the mobile, PC, cloud computing, the Internet of things, and wearable devices, the dataintensive science such as big data [1] has become the main topic of the technological reform.The most prominent features of the big data are enormous volume of data, wide variety of data types, lower value density, and faster processing.But the big data has both advantages and disadvantages.It brings great convenience to individuals and enterprises; at the same time the data security is an urgent problem to be solved.From the view of information security, the traditional cryptography and secure communication model can be cracked easily, so that great security risks exist in the information system in every country.Therefore it is very urgent to improve the information security technology for the country and the enterprise in which the big data is main stream.
Due to the characteristic of the long-term unpredictability and extreme sensitivity to initial values, the chaos system has been researched deeply in the secure communications and the cryptography.Since Pecora and Carroll [2] first proposed the master-slave synchronization method in 1990, many synchronization types were presented, such as complete synchronization [3], lag synchronization [4], generalized synchronization [5], modified projective synchronization [6], modified function projective synchronization [7], phase synchronization [8], and dislocation synchronization [9].Now more and more people paid their attention to chaotic secure communication, and the research mainly focuses on two aspects: one is to find a safer secure communication scheme, such as chaotic masking [10,11], chaotic modulation [12], chaos shift keying [13], and chaos spreading spectrum [14] and the other is to research chaotic systems with a better encryption performance, such as fractionalorder chaotic systems [15,16], time-delay chaotic systems [17], complex chaotic system [18], and multiscroll chaotic systems [19].Kiani-B et al. [20] applied the fractional-order Kalman filter in secure communications system.Mei [21] proposed a new secure communication scheme based on uncertain time-delay chaos system.Mahmoud et al. [22] researched the projective synchronization for complex hyperchaotic system and achieved secure communications with four-order complex Lorenz system.In addition, other secure communication schemes based on fractional-order [23], time-delay [24], and multiscroll [25] chaotic systems have been proposed also.In the field of cryptography, compared with the traditional password, the generated mechanisms for chaos passwords are different and have real-time, so it has a greater advantage in terms of image encryption and video and other multimedia data encryption, and therefore the research on chaos image encryption has attracted more and more people [26][27][28].The quality of the chaotic password is closely related to the chaos systems.For the low-dimensional chaotic system, because of its simple form, small key space, and low chaos sequence complexity, its security is not high enough.So many scholars focus on the hyperchaotic systems and fractionalorder chaotic systems.Zhu and Sun [29] analyzed the security of the hyperchaos image encryption (HIE) algorithm, improved hyperchaos image encryption (IHIE) algorithm, and proposed the enhanced hyperchaos image encryption algorithms.Zhao et al. [30] gave an image encryption scheme based on an improper fractional-order chaotic system.
So the more complicated structure of the chaotic systems, the better performance of the secure communications and cryptography.However, these complicated chaotic systems are usually not easy to design synchronous controller, which decreases the communication efficiency.Therefore it is necessary to seek a new chaotic system with simple structure and complex behaviors.
In this paper, we propose a new Lorenz-like system with varying parameter by adding a state feedback factor in Lorenz-like system [31].By theoretical analysis and numerical simulation, the structure of the new system is simple and easy to construct.At the same time, it has more complicated behaviors.This paper is divided into three parts as follows.Firstly, the new Lorenz-like chaotic system with varying parameter is designed based on the Lorenz-like system and analyzes its chaos characteristics theoretically.Secondly, a synchronization scheme driven by a single state variable is achieved based on the new proposed system, and the chaotic parameter modulation digital secure communications system is constructed.Finally, the designed variable parameter chaotic system is applied to image encryption and a threechaotic-image encryption algorithm is proposed.

The Lorenz-Like System with Varying Parameter
2.1.The New Chaotic System.The Lorenz-like system is given by [31] ẋ =  ( − ) , where  and  are real constants and  is a bifurcation parameter.Compared with the traditional Lorenz system,  is not in the second equation.If we replace parameter  with a function of , such as then a new system is generated and can be written as where , ,  1 ,  2 , and  are real constants,  is a state feedback control function, and  is the threshold.From (2), we can know that  switches between  1 +  2 and  1 −  2 under the control of ; then the Lorenz-like system (3) shows the bifurcation under the control of state variable .
When choosing  = 20,  = 8,  1 = 70, and  2 = 15, we can get a bifurcation diagram (Figure 1) of chaotic system (3) with the change of .For convenience, the Lorenz-like system when  = 85 is denoted as  chaotic system and when  = 55 as  chaotic system.From Figure 1, we can get that when  < 0,  = 85, the new Lorenz-like system is equivalent to the  chaotic system, while when  > 41.2,  = 55; it is equivalent to the  chaotic system.When 0 <  < 41.2,  switches between 85 and 55 under the control of the state variable ; in other words, the new Lorenz-like system automatically switches between  and  chaotic systems (Figure 2).And when 16.5 <  < 23.9, system appears periodic oscillation obviously.
As shown in Figure 2, the blue part denotes  chaotic system and the red part  chaotic system.With the change of parameter, the nonlinear dynamical behaviors change significantly.When  = 10 or  = 30, a strange attractor appears in Figures 2(b) and 2(d).In Figure 3, the threedimensional phase diagram of chaotic system (3) is given with  = 10, 20, 30, and 37.

Chaotic Characters
2.2.1.Symmetry and Invariance.For system (3), let (, , ) → (−, −, ); the system equation remains the same.Then the system is symmetrical about the axis, and the symmetry is not associated with the system parameters.If we let (0) = 0, (0) = 0, and (0) be any value, the system equation can transform into ż = −; that is, the system will move on -axis and will be stable at the origin.

Dissipation and the Existence of Attractor. For system (3),
So, we can conclude that the system is dissipative and converges by exponential / =  −(+) ; that is, the volume element with the initial volume (0) converges to (0) −(+) at time .When  → ∞, each small volume that contains the system trajectories converges to zero at an exponential rate of −( + ).All the trajectories of the system will eventually be limited to a subset of zero volume, and this limit subset is called attractor.It is easy to know that  is the shared equilibrium point for both  system and  system.When choosing  = 20,  = 8,  ).The distribution of equilibrium points in the phase space can be seen in Table 1, in which  −1 ,  0 , and  +1 denote three areas separated by  as follows:

The Existence and Stability of Equilibrium
For  chaotic system, the corresponding eigenvalues for each equilibrium point can be calculated as follows: Table 1: The distribution of equilibrium points in the phase space.
Obviously,  is a saddle point, and  +  ,  −  are saddlefocus equilibrium points.And all these three equilibrium points are unstable, which leads the orbits of system stretch in phase space.Under the interactive stretching and contractions, the chaotic motion is generated.In the same way, we also can draw a similar conclusion that  chaotic system also has three unstable equilibrium points and the chaotic condition is satisfied.
From Table 1, it is easy to know that the system has three equilibrium points when  ≤ 0 and it is equal to  chaotic  system.The system also can be treated approximately such that it has three equilibrium points when  ≫ 26.0768 and the system is equal to  chaotic system.But in addition to these two cases, the system has five equilibrium points.Of all the five points,  influences the trajectory in the whole region, while  +  and  −  influence the trajectory in  0 , and  +  and  −  influence the trajectory in  −1 and  +1 .With the increase of ,  0 gradually extends and  ±1 is reduced; that is, the influence of  +  and  −  gradually increases while  +  and  −  are reduced until they nearly disappear.So the system shows a complex dynamic chaos-cycle-chaos when the new parameter  changes.

Spectrum.
In Figure 4, we can see that spectrum of the system is continuous, which shows that the new designed system has the chaotic characteristics.

Lyapunov Exponents and Lyapunov Dimension.
Lyapunov exponent measures the exponential rates of divergence or convergence of nearby trajectories in phase space.A three-order nonlinear system has three Lyapunov exponents ( 1 ,  2 ,  3 ).All the Lyapunov exponents are listed in Table 2, and the curves with the change of  are also given as in Figure 5. Obviously  2 of  and  systems is close to zero, while  2 (when  = 4) is a negative number for the reason of ; this implies that a new chaotic attractor occurred in the new system.
For a -order system, the Lyapunov dimension can be calculated as follows: where From the results in Table 2, we can get that all the Lyapunov dimensions are fractions and 2 <   < 3. Thus, it is another evidence of chaos.In addition, both Lyapunov exponents' curves and bifurcation diagram can show the effect of parameter, so the same conclusion can be obtained from Figure 5 as Figure 1.

A Brief Summary.
This section shows a new Lorenzlike system (3) with varying parameter; several conclusions can be gotten as follows: (i)  is a constant in Lorenz-like system (1) while it switches between  1 +  2 and  1 −  2 in system (3), so the new system's structure has a slight difference with the Lorenz-like system (1) and it equals system (1) when  ≤ 0 or  ≫ 26.0678; (ii) new chaotic behaviors occur when  change (see Figures 1, 2, and 3); (iii) the equilibrium points are not fixed for the reason of  as Table 1 shows; and (iv) the Lyapunov exponent  2 is apparently different (see Table 2 and Figure 5).All the conclusions imply that the new proposed system has more complicated behaviors with respect to the Lorenz-like system (1).

The Application in Secure Communication
for the New Lorenz-Like System 3.1.Synchronization Design for Single Variable Drive.For a better description of the synchronization scheme, here we use notation ( 1 ,  2 ,  3 ) in place of (, , ) in (3); then the master system is And the slave system is So the controller is designed as follows: The designed controller only contains one state variable  1 of the master system; thus it has simple structure and is driven by single variable.So it is easy to be achieved.

Chaos Parameter Modulation Digital Secure Communications System.
Based on the synchronization scheme designed in the previous section, a chaotic parameter modulation digital secure communication system is given in Figure 7.
The transmitter system is The receiver system is where () is a digital signal to be transmitted.In order to encrypt (), we select  = 4 to represent "0" and  = 6 to represent "1"; that is, In this case, the topology of the phase diagram is similar, so it is difficult to crack encrypted signal and extract useful information by phase space reconstruction.
Based on the theory above, we simulate the digital secure communication system as in Figure 8.The digital signal () "0101110111001011101" will be transmitted and sent per symbol interval, that is, 10 seconds, where  1 is not only the encrypted signal but also the driven signal.The synchronization between the master system and slave system only can be reached when  = 4, and there is a large error between the response system and the drive system when  = 6 just as  1 in Figure 8. Finally the decrypted signal () can be gotten from  1 after detection, and compared with (), there is a nearly 10-second delay.

Image Encryption Algorithm Based on
Lorenz-Like System with Varying Parameter 4.1.A Three-Order Image Encryption Algorithm.Based on system (3), a three-order image encryption algorithm is given and the diagram of the image encryption and decryption is shown in Figure 9.
The chaotic image encryption is to disrupt the original image (plaintext) by chaotic sequence.The process of the algorithm is as follows: first, set the initial value of the system as the key; then iterate system (3) for 5000 times to make it fully chaotic.Later, continue to iterate system (3) to obtain (), (), and ().Then the chaotic sequences key(), key(), and key() can be gotten as below: where "floor" is the MATLAB function and floor() rounds the element  to the nearest integers less than or equal to .
Through the above process in (18), we get the chaotic sequences key(), key(), and key() which range between 0 and 255.Original image matrix (plaintext) is  and the ciphertext matrices are  1 ,  2 , and  3 , which are obtained after three-order encryption, respectively, where  3 is the final ciphertext image matrix.The encryption formula is as follows: "⊕" in (19) means "XOR", and the same is in (20).Just as Figure 9, the decryption process is the opposite of the encryption process.First, we should set the correct key; then for system (3), the decryption process is the same with the encryption to obtain the same chaotic sequences to decrypt correctly.Decryption formula is as follows:

Simulation and Analysis.
In this section, an image encryption experiment was given and a 512 × 512 color image "Lena" is chosen as the plaintext.In simulation, the step is selected as 0.01, and  = 20,  = 8,  1 = 70,  2 = 15, and  = 4.The encryption key is initial value ((0), (0), (0)) = (0.2, 0.7, 1.6).Based on the above algorithm, the image encryption system is designed to achieve the Lena encryption.The simulation results shown in Figure 10, and several tests have been carried out to demonstrate the effectiveness and efficiency of the proposed encryption algorithm.

Key Space Analysis
. Key space size is the total number of different keys which can be used in an encryption process; it should be large enough to preclude the eavesdropping by brute-force attack.A single precision floating point format number has 2 32 kinds of possibilities; then the key space  in this paper will be up to 2 96 .Otherwise, if the system parameters are chosen as keys, it will get much larger key space, which greatly increases the security of the system.

Histogram Analysis.
From the histogram of a digital image, the distribution of the pixel values can be gotten; if the encrypted image is well encrypted, the histogram will be uniform, so the histogram attack can be prevented effectively.Figure 11 gives the histograms of both original image and encrypted image; it is obvious that the histograms of red, green, and blue for original image are steep and not flat enough; the histograms for encrypted image are all uniform and quite different from that of the original image when using the proposed algorithm.

Key Sensitivity Analysis.
A good cryptosystem must be highly sensitive at small changes in secret key in encryption and decryption process, so a full test contains two aspects: (i) slightly different keys to encrypt the same image are used and the difference between the corresponding encrypted images is computed; (ii) for an encrypted image only one correct key can decrypt it, so decrypt the encrypted image by an incorrect key which is similar to the correct one and observe whether it can be correctly decrypted.Table 3 gives some special cases to evaluate the sensitivity in encryption process, and the encrypted images were also shown in Figure 12; the difference ratio is really high, which means a good key sensitivity in encryption process.The test result in decryption process also can be seen in Figure 12; Figure 12(f) is the correct decrypted image; Figure 12(g) is the incorrect one with only a slight change 10 −14 for the key's first value.

Correlation Coefficients and Efficiency Analysis.
A good image encryption algorithm should have two characteristics: (i) high security, which is partly analyzed in key space and key sensitivity, will be analyzed by correlation coefficients complementally in this section; (ii) high efficiency, which means low time consumption in encryption and decryption process, will be analyzed in this section also.The correlation coefficient of an image can be measured as follows: where  is the number of pair of pixels and  and  are values of two adjacent pixels in grey scale.The correlation coefficients are calculated out based on 3000 random pixels, and all correlation coefficients of plaintext are greater than 0.96 while those of ciphertext are all near "zero," which implies a good information hiding for plaintext.Recently many chaotic systems with complex structure were found or applied to image encryption, such as systems in [29,30].Here we realize image encryption with different chaotic systems under the same simulation environment; then comparisons with the proposed algorithm were done and the results of correlation coefficients and cost are listed in Table 4. Compared with algorithms based on other chaotic systems in [29,30], we can know that the correlation coefficients of ciphertext are nearly, the time consumption is less for the proposed system.This comparison demonstrates that the proposed image encryption algorithm based on the new Lorenz-like system shows a good performance as well as algorithms based on other systems, and the efficiency is high for the reason of its simple structure.

Conclusion
A new Lorenz-like system with varying parameter is proposed by adding a state feedback function in this paper.Firstly, we analyze the influence of the threshold  to the chaotic behavior of the new system and found that the system shows a chaos-cycle-chaos evolution when  changes.Then we analyze the new system's chaotic characteristics.After that a new synchronization scheme using a single state variable drive based on the new system is designed.Finally, a chaotic parameter modulation digital secure communication system and image encryption based on the new system proposed is designed.The simulation results show that the new system has a good performance in application.Otherwise, according to the new system designed, we can modify many other systems to get more chaotic systems which have simple structure and complex dynamics.This will enrich the amount of chaotic signal sources, simplify designing, and improve the security of communication and image encryption.

Figure 1 :
Figure 1: Bifurcation diagram of chaotic system (3) with the change of .

Figure 4 :
Figure 4: The spectrum of Lorenz-like system with varying parameter when  = 4.

Figure 5 :
Figure 5: Lyapunov exponents curves of chaotic system (3) with the change of .

3 Figure 6 :
Figure 6: Synchronization error curve between master and slave systems when  = 4.

Figure 7 :
Figure 7: The schematic of digital secure communications.

Figure 9 :Figure 10 :
Figure 9: The diagram of image encryption and decryption.

Figure 11 :
Figure 11: Histogram of original image and encrypted image.

Table 4 :
Correlation coefficients and cost comparisons.