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

Abstract

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.

[ProQuest: [...] denotes non US-ASCII text; see PDF]

Academic Editor:Herve G. E. Kadji

Harbin Engineering University, Harbin, Heilongjiang 150001, China

Received 9 October 2015; Revised 8 December 2015; Accepted 24 February 2016; 26 April 2016

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

With the development of the mobile, PC, cloud computing, the Internet of things, and wearable devices, the data-intensive 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 fractional-order 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 [26-28]. 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 fractional-order 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 three-chaotic-image encryption algorithm is proposed.

2. The Lorenz-Like System with Varying Parameter

2.1. The New Chaotic System

The Lorenz-like system is given by [31] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are real constants and [figure omitted; refer to PDF] is a bifurcation parameter. Compared with the traditional Lorenz system, [figure omitted; refer to PDF] is not in the second equation. If we replace parameter [figure omitted; refer to PDF] with a function of [figure omitted; refer to PDF] , such as [figure omitted; refer to PDF]

then a new system is generated and can be written as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are real constants, [figure omitted; refer to PDF] is a state feedback control function, and [figure omitted; refer to PDF] is the threshold. From (2), we can know that [figure omitted; refer to PDF] switches between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] under the control of [figure omitted; refer to PDF] ; then the Lorenz-like system (3) shows the bifurcation under the control of state variable [figure omitted; refer to PDF] .

When choosing [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , we can get a bifurcation diagram (Figure 1) of chaotic system (3) with the change of [figure omitted; refer to PDF] . For convenience, the Lorenz-like system when [figure omitted; refer to PDF] is denoted as [figure omitted; refer to PDF] chaotic system and when [figure omitted; refer to PDF] as [figure omitted; refer to PDF] chaotic system. From Figure 1, we can get that when [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , the new Lorenz-like system is equivalent to the [figure omitted; refer to PDF] chaotic system, while when [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; it is equivalent to the [figure omitted; refer to PDF] chaotic system. When [figure omitted; refer to PDF] , [figure omitted; refer to PDF] switches between 85 and 55 under the control of the state variable [figure omitted; refer to PDF] ; in other words, the new Lorenz-like system automatically switches between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] chaotic systems (Figure 2). And when [figure omitted; refer to PDF] , system appears periodic oscillation obviously.

Figure 1: Bifurcation diagram of chaotic system (3) with the change of [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 2: [figure omitted; refer to PDF] plane phase diagram of chaotic system (3) (a) [figure omitted; refer to PDF] ; (b) [figure omitted; refer to PDF] ; (c) [figure omitted; refer to PDF] ; (d) [figure omitted; refer to PDF] ; (e) [figure omitted; refer to PDF] ; and (f) [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

(e) [figure omitted; refer to PDF]

(f) [figure omitted; refer to PDF]

As shown in Figure 2, the blue part denotes [figure omitted; refer to PDF] chaotic system and the red part [figure omitted; refer to PDF] chaotic system. With the change of parameter, the nonlinear dynamical behaviors change significantly. When [figure omitted; refer to PDF] or [figure omitted; refer to PDF] , a strange attractor appears in Figures 2(b) and 2(d). In Figure 3, the three-dimensional phase diagram of chaotic system (3) is given with [figure omitted; refer to PDF] , 20, 30, and 37.

Figure 3: Three-dimensional phase diagram of chaotic system (3): (a) [figure omitted; refer to PDF] ; (b) [figure omitted; refer to PDF] ; (c) [figure omitted; refer to PDF] ; and (d) [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

2.2. Chaotic Characters

2.2.1. Symmetry and Invariance

For system (3), let [figure omitted; refer to PDF] ; the system equation remains the same. Then the system is symmetrical about the [figure omitted; refer to PDF] -axis, and the symmetry is not associated with the system parameters. If we let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] be any value, the system equation can transform into [figure omitted; refer to PDF] ; that is, the system will move on [figure omitted; refer to PDF] -axis and will be stable at the origin.

2.2.2. Dissipation and the Existence of Attractor

For system (3), [figure omitted; refer to PDF]

So, we can conclude that the system is dissipative and converges by exponential [figure omitted; refer to PDF] ; that is, the volume element with the initial volume [figure omitted; refer to PDF] converges to [figure omitted; refer to PDF] at time [figure omitted; refer to PDF] . When [figure omitted; refer to PDF] , each small volume that contains the system trajectories converges to zero at an exponential rate of [figure omitted; refer to PDF] . All the trajectories of the system will eventually be limited to a subset of zero volume, and this limit subset is called attractor.

2.2.3. The Existence and Stability of Equilibrium Point

For system (3), the equilibrium points are [figure omitted; refer to PDF]

It is easy to know that [figure omitted; refer to PDF] is the shared equilibrium point for both [figure omitted; refer to PDF] system and [figure omitted; refer to PDF] system. When choosing [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , the other two equilibrium points of [figure omitted; refer to PDF] system are [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , and the other two equilibrium points of [figure omitted; refer to PDF] system are [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The distribution of equilibrium points in the phase space can be seen in Table 1, in which [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] denote three areas separated by [figure omitted; refer to PDF] as follows: [figure omitted; refer to PDF]

Table 1: The distribution of equilibrium points in the phase space.

The scope of [figure omitted; refer to PDF]	Region
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF] [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF] [figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]		[figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]

Note : [dagger] represents [figure omitted; refer to PDF] and [double dagger] represents [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

For [figure omitted; refer to PDF] chaotic system, the corresponding eigenvalues for each equilibrium point can be calculated as follows: [figure omitted; refer to PDF]

Obviously, [figure omitted; refer to PDF] is a saddle point, and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are saddle-focus 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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] and it is equal to [figure omitted; refer to PDF] chaotic system. The system also can be treated approximately such that it has three equilibrium points when [figure omitted; refer to PDF] and the system is equal to [figure omitted; refer to PDF] chaotic system. But in addition to these two cases, the system has five equilibrium points. Of all the five points, [figure omitted; refer to PDF] influences the trajectory in the whole region, while [figure omitted; refer to PDF] and [figure omitted; refer to PDF] influence the trajectory in [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] influence the trajectory in [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . With the increase of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] gradually extends and [figure omitted; refer to PDF] is reduced; that is, the influence of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] gradually increases while [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are reduced until they nearly disappear. So the system shows a complex dynamic chaos-cycle-chaos when the new parameter [figure omitted; refer to PDF] changes.

2.2.4. 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.

Figure 4: The spectrum of Lorenz-like system with varying parameter when [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

2.2.5. 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 [figure omitted; refer to PDF] . All the Lyapunov exponents are listed in Table 2, and the curves with the change of [figure omitted; refer to PDF] are also given as in Figure 5. Obviously [figure omitted; refer to PDF] of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] systems is close to zero, while [figure omitted; refer to PDF] (when [figure omitted; refer to PDF] ) is a negative number for the reason of [figure omitted; refer to PDF] ; this implies that a new chaotic attractor occurred in the new system.

Table 2: Lyapunov exponents and Lyapunov dimension.

[figure omitted; refer to PDF]	Lyapunov exponents	Lyapunov dimension ( [figure omitted; refer to PDF] )
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF] ( [figure omitted; refer to PDF] system)	2.5110	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	2.0824
[figure omitted; refer to PDF]	2.2269	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	2.0204
[figure omitted; refer to PDF] ( [figure omitted; refer to PDF] system)	2.0533	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	2.0684

Figure 5: Lyapunov exponents curves of chaotic system (3) with the change of [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

For a [figure omitted; refer to PDF] -order system, the Lyapunov dimension can be calculated as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] while [figure omitted; refer to PDF] .

From the results in Table 2, we can get that all the Lyapunov dimensions are fractions and [figure omitted; refer to PDF] . 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.

2.2.6. A Brief Summary

This section shows a new Lorenz-like system (3) with varying parameter; several conclusions can be gotten as follows: (i) [figure omitted; refer to PDF] is a constant in Lorenz-like system (1) while it switches between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] or [figure omitted; refer to PDF] ; (ii) new chaotic behaviors occur when [figure omitted; refer to PDF] change (see Figures 1, 2, and 3); (iii) the equilibrium points are not fixed for the reason of [figure omitted; refer to PDF] as Table 1 shows; and (iv) the Lyapunov exponent [figure omitted; refer to PDF] 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).

3. 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 [figure omitted; refer to PDF] in place of [figure omitted; refer to PDF] in (3); then the master system is [figure omitted; refer to PDF]

And the slave system is [figure omitted; refer to PDF]

So the controller is designed as follows: [figure omitted; refer to PDF]

The designed controller only contains one state variable [figure omitted; refer to PDF] of the master system; thus it has simple structure and is driven by single variable. So it is easy to be achieved.

Let the error system be [figure omitted; refer to PDF]

Then, the error dynamics equation is [figure omitted; refer to PDF]

Select the Lyapunov function as [figure omitted; refer to PDF] ; then take the derivative of [figure omitted; refer to PDF] , so [figure omitted; refer to PDF]

When [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and when [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . According to the Lyapunov stability theorem, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] when [figure omitted; refer to PDF] ; that is, the synchronization between master and slave system has been achieved.

Figure 6 gives the synchronization error curves between the master system and slave system with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ; the initial value [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . From Figure 6, the master system traces the slave system to achieve synchronization quickly. The advantage of this chaotic synchronization system is that the controller is simple and only one signal is to be transmitted to complete the synchronization between the drive system and response system, which improves communication efficiency and conserves resources.

Figure 6: Synchronization error curve between master and slave systems when [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

3.2. 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.

Figure 7: The schematic of digital secure communications.

[figure omitted; refer to PDF]

The transmitter system is [figure omitted; refer to PDF]

The receiver system is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a digital signal to be transmitted. In order to encrypt [figure omitted; refer to PDF] , we select [figure omitted; refer to PDF] to represent "0" and [figure omitted; refer to PDF] to represent "1"; that is, [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] "0101110111001011101" will be transmitted and sent per symbol interval, that is, 10 seconds, where [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] , and there is a large error between the response system and the drive system when [figure omitted; refer to PDF] just as [figure omitted; refer to PDF] in Figure 8. Finally the decrypted signal [figure omitted; refer to PDF] can be gotten from [figure omitted; refer to PDF] after detection, and compared with [figure omitted; refer to PDF] , there is a nearly 10-second delay.

Figure 8: Chaotic parameter modulation digital secure communication.

[figure omitted; refer to PDF]

4. 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.

Figure 9: The diagram of image encryption and decryption.

[figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Then the chaotic sequences [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] can be gotten as below: [figure omitted; refer to PDF] where "floor" is the MATLAB function and floor( [figure omitted; refer to PDF] ) rounds the element [figure omitted; refer to PDF] to the nearest integers less than or equal to [figure omitted; refer to PDF] .

Through the above process in (18), we get the chaotic sequences [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] which range between 0 and 255. Original image matrix (plaintext) is [figure omitted; refer to PDF] and the ciphertext matrices are [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , which are obtained after three-order encryption, respectively, where [figure omitted; refer to PDF] is the final ciphertext image matrix. The encryption formula is as follows: [figure omitted; refer to PDF] " [figure omitted; refer to PDF] " 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: [figure omitted; refer to PDF]

4.2. Simulation and Analysis

In this section, an image encryption experiment was given and a [figure omitted; refer to PDF] color image "Lena" is chosen as the plaintext. In simulation, the step is selected as [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The encryption key is initial value [figure omitted; refer to PDF] . 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.

Figure 10: The simulation results of image encryption and decryption. (a) Original image (plaintext); (b) encrypted image (ciphertext); (c) decrypted image (plaintext); and (d) illegal decrypted image.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

4.2.1. 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 2³² kinds of possibilities; then the key space in this paper will be up to [figure omitted; refer to PDF] . Otherwise, if the system parameters are chosen as keys, it will get much larger key space, which greatly increases the security of the system.

4.2.2. 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.

Figure 11: Histogram of original image and encrypted image.

[figure omitted; refer to PDF]

4.2.3. 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.

Table 3: Differences between encrypted images produced by slightly different keys.

[...]Encryption keys (I) Figure 12(b)	Encryption keys (II)	Difference ratio between (I) and (II) (%)
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
			[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	9.3	Figure 12(c)	99.61
2.5	[figure omitted; refer to PDF]	9.3	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	9.3	Figure 12(d)	99.60
			[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	Figure 12(e)	99.60

Figure 12: Key sensitivity test. (a) Original image (plaintext); (b), (c), (d), (e) the encrypted images with different keys as Table 3; (f) decrypt (b) with key [figure omitted; refer to PDF] ; and (g) decrypt (b) with key (2.5- [figure omitted; refer to PDF] , - [figure omitted; refer to PDF] ).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

(e) [figure omitted; refer to PDF]

(f) [figure omitted; refer to PDF]

(g) [figure omitted; refer to PDF]

4.2.4. 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: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the number of pair of pixels and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] 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, but 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.

Table 4: Correlation coefficients and cost comparisons.

Chaotic systems used	Correlation coefficients	Cost (s)
Original image (plaintext)	Encrypted image (ciphertext)
Horizontal	Vertical	Diagonal	Horizontal	Vertical	Diagonal
Proposed system	0.9788	0.9677	0.9752	0.0086	0.0307	[figure omitted; refer to PDF]	10.9689
Reference [29]	0.9788	0.9677	0.9752	0.0167	[figure omitted; refer to PDF]	0.0371	13.9933
Reference [30]	0.9788	0.9677	0.9752	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0.0036	14.0401

5. 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 [figure omitted; refer to PDF] to the chaotic behavior of the new system and found that the system shows a chaos-cycle-chaos evolution when [figure omitted; refer to PDF] 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.

Acknowledgments

This research is funded by the National Natural Science Foundation of China (nos. 61203004 and 61306142) and the Natural Science Foundation of Heilongjiang Province (Grant no. F201220).

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