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Complexity, Dynamics, Control, and Applications of Nonlinear Systems with Multistability

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Research Article | Open Access

Volume 2020 |Article ID 8034196 | https://doi.org/10.1155/2020/8034196

Fei Yu, Li Liu, Shuai Qian, Lixiang Li, Yuanyuan Huang, Changqiong Shi, Shuo Cai, Xianming Wu, Sichun Du, Qiuzhen Wan, "Chaos-Based Application of a Novel Multistable 5D Memristive Hyperchaotic System with Coexisting Multiple Attractors", Complexity, vol. 2020, Article ID 8034196, 19 pages, 2020. https://doi.org/10.1155/2020/8034196

Chaos-Based Application of a Novel Multistable 5D Memristive Hyperchaotic System with Coexisting Multiple Attractors

Academic Editor: Sundarapandian Vaidyanathan
Received23 Nov 2019
Revised08 Feb 2020
Accepted17 Feb 2020
Published30 Mar 2020

Abstract

Novel memristive hyperchaotic system designs and their engineering applications have received considerable critical attention. In this paper, a novel multistable 5D memristive hyperchaotic system and its application are introduced. The interesting aspect of this chaotic system is that it has different types of coexisting attractors, chaos, hyperchaos, periods, and limit cycles. First, a novel 5D memristive hyperchaotic system is proposed by introducing a flux-controlled memristor with quadratic nonlinearity into an existing 4D four-wing chaotic system as a feedback term. Then, the phase portraits, Lyapunov exponential spectrum, bifurcation diagram, and spectral entropy are used to analyze the basic dynamics of the 5D memristive hyperchaotic system. For a specific set of parameters, we find an unusual metastability, which shows the transition from chaotic to periodic (period-2 and period-3) dynamics. Moreover, its circuit implementation is also proposed. By using the chaoticity of the novel hyperchaotic system, we have developed a random number generator (RNG) for practical image encryption applications. Furthermore, security analyses are carried out with the RNG and image encryption designs.

1. Introduction

In recent years, chaos systems have become the subject of many studies in the fields of science and engineering. A large number of new chaotic systems have been proposed one after another, and their application scopes are more and more extensive [18]. With the progress of science and technology, chaos has been applied not only to communication [912], image processing [1315], complex networks [1621], synchronization [2227], electronic circuits [2830], and optimization [3135] but also to encryption studies [3641]. This is because chaotic signal has good pseudorandom, initial-value sensitive, and long-term unpredictable characteristics, which enhances the confusion and diffusion of encrypted data.

Due to the more complex structure and dynamic behavior of the hyperchaotic system, in order to better meet the needs of secure communication and information hiding, people propose to construct hyperchaotic systems to improve the complexity of the systems. At present, hyperchaotic systems are usually constructed by loading feedback controller on 3D or 4D continuous chaotic systems [4246]. The feedback controllers are divided into linear and nonlinear, among which the nonlinear-feedback term will further increase the complexity and unpredictability of the system, which is more suitable for the construction of hyperchaos [4751].

Memristor is a kind of hardware implementation component of memory nonlinear electronic memristor chaotic circuit, which has research significance in chaotic secure communication, image encryption, neural networks, and other fields [5256]. It describes the relationship between magnetic flux and charge. The concept of the memristor was proposed by Chua in 1971 [57], and it was not until 2008 that HP laboratory realized the first real memristor [58]. Because of the nonlinear and memory characteristics of the memristor, as the feedback term of the hyperchaotic system, it can produce complex nonlinear dynamic phenomena, which provides a new development space for the design of the hyperchaotic system. At present, the main method is to use the memristor as the feedback term in typical chaotic systems to construct hyperchaotic systems. In [59], a novel 5D hyperchaotic four-wing memristive system (HFWMS) was proposed by introducing a flux-controlled memristor with quadratic nonlinearity into a 4D hyperchaotic system, the dynamic characteristics of the HFWMS were analyzed, and the FPGA realization of the 5D HFWMS was also reported. In [60], a new memristive system was presented by replacing the resistor in the circuit of modified Lü system with the flux-controlled memristor, respectively, which could exhibit a hyperchaotic multiwing attractor, and the values of two positive Lyapunov exponents were relatively large. The dynamical behaviors and the circuit implementation were also carried out.

Coexisting attractors depend on the symmetry of the systems and the initial condition of the systems [61]. Multistability refers to the phenomenon that the system shows different dynamic characteristics and different attractors coexist under same parameters [62]. In recent years, the study of multistability and coexistence attractors is a hot topic in nonlinear dynamics [6370]. Lai et al. [63] showed the coexistence behavior of different attractors under different initial conditions and parameter values, such as four limit cycles, and two double-scroll attractors with a limit cycle. In [65], a new 4D fractional order chaotic system was proposed by adding a variable to the 3D chaotic system. This new system had no equilibrium point, but it could also show rich and complex hidden dynamics. Zhang et al. [66] introduced a state variable into a 3D chaotic system and then analyzed the dynamic characteristics of the new system under different initial conditions, proving that the new system has extreme multistability. In fact, various systems exhibiting multistability have been proposed. However, a review of literature revealed that this remarkable behavior is rare in 5D memristive hyperchaotic system with coexisting multiple attractors. Such systems cannot be ignored. Because of their complexity, the generated signals are usually used for secure communication and random number generation.

With the development of communication technology and the coming of information age, people are more and more aware of the important role of information security [7176], and the research of various security protection has become the current research hotspot [7782]. As an important part of information security transmission, random number generator (RNG) has been paid more and more attention. The unpredictable and unrepeatable random number sequence which can be produced by RNG plays an important role in information encryption. Based on Shannon information theory, in order to ensure the absolute security of communication, the RNG with high speed, unpredictability, and good randomness has great research value [8389]. The chaotic system is a kind of complex nonlinear motion, which is highly sensitive to the initial conditions, and its orbit is unpredictable for a long time. Therefore, the chaotic system shows very good cryptography characteristics.

In recent years, people are committed to the research and design of chaos-based RNGs [9093]. Sometimes the key of generating random sequence by chaos is the choice of chaotic systems. However, most RNGs based on chaos have a typical disadvantage. That is to say, the limited precision of all processors may cause the chaotic system to degenerate into periodic function or fixed point [94]. In order to overcome this disadvantage, a generator based on hyperchaos was proposed in [94]. The self-shrinking generator was used to disturb the hyperchaotic sequence to reduce the period degradation and improve the sequence performance, which was superior to many other linear-feedback-shift register-based generators. Random numbers created in the chaotic systems are tested according to the randomness tests with the highest international standards such as AIS-31 and NIST 800 22 and then are ready to be used in encryption applications [95, 96]. In encrypted applications, it is not enough to encrypt data only. Encrypted data must also be equipped with the highest possible reliability. In order to prove the high level of reliability, some security analysis must be carried out according to the data type. Key space, sensitivity, floating frequency, histograms, correlation, and information entropy analysis are common security analysis in the literature [9799].

Motivated by undiscovered features of systems with coexisting multiple attractors, we introduce a novel multistable 5D memristive hyperchaotic system with a line of equilibrium and its practical chaos-based application in the present work. The rest of this work is organized as follows. Section 2 describes the mathematical model of the novel multistable 5D memristive hyperchaotic system. Dynamical properties and circuit realization of the system are investigated in Sections 3 and 4, respectively. Section 5 presents a random number generation (RNG) using the chaoticity of the multistable 5D memristive hyperchaotic system, while security analyses are also carried out with the RNG designed. To validate the performance of the RNG, the application of image encryption is employed in Section 6, we also employ standard security analysis whose outcome is compared alongside available state-of-the-art methods. Finally, we conclude in Section 7.

2. A Novel Multistable 5D Memristive Hyperchaotic System

Recently, Yu and Wang [100] proposed a 4D four-wing chaotic system, and its mathematical model iswhere x, y, z, and are the state variables and a, b, c, d, and e are the system parameters. When a = 10, b = 12, c = 60, d = 2, and e = 3, system (1) can display a fully four-wing chaotic attractor under the initial conditions .

Memristor is a passive two terminal device which describes the relationship between flux φ and charge q. In this paper, the memristor is controlled by flux, and the relationship between the current flowing through the two terminal device and the port voltage can be expressed as follows:where W(φ) is the memductance function of the flux-controlled memristor and defined as

Based on system (1), by introducing the memristor model in (3) to the third equation of system (1), a novel 5D memristive hyperchaotic system is presented as follows:where a, b, c, d, e, f, and are the system parameters. When the typical parameters are fixed as a = 10, b = 12, c = 30, d = 2, e = 4, m = 0.1, and n = 0.01 and the initial conditions are chosen as , the memristive system (4) exhibits a four-wing hyperchaotic attractor, as shown in Figure 1, from which it can be seen that the system has topologically more complex attractor structure than system (1) presented by [100]. The memristive chaotic system (4) has the same symmetry as the original 4D chaotic system (1) and remains unchanged under the coordinate transformation .

Equilibrium points of system (4) are obtained by setting its right-hand side to zero, that is,

According to equation (5), it is easy to see that system (4) has a line equilibrium point , where l is any real constant. The Jacobian matrix at the online equilibrium point O of system (4) is

According to (6), the characteristic equation can be obtained as

It is easy to get λ1 = 0, λ2 = −e, λ3 = −a, λ4 = −c, and λ5 = b because the values of system parameters a, b, c, and e are greater than zero, so λ2, λ3, and λ4 are negative, λ5 is positive, so system (4) has unstable saddle point. The dissipativity of memristive chaotic system (4) can be described as

Since − a + bce = −32 satisfies ∇V < 0, system (4) is dissipative.

3. Dynamic Analysis of the Novel 5D Memristive Chaotic System

In this section, we will use the tools of bifurcation diagram, Lyapunov exponent spectrum, time series, and phase diagram and use the fourth-order Runge–Kutta algorithm to study the complex dynamic behavior of system (4) through MATLAB. The proposed memristive chaotic system (4) has particularly complex dynamic characteristics, including coexistence attractors of the same type and different types, multistability, and transient transfer phenomena.

3.1. Lyapunov Exponent Spectrum and Bifurcation Diagram

It is very interesting that there are different dynamic behaviors (such as periodic phenomena, quasi-periodic, chaotic attractors, and hyperchaotic attractors), according to different differential equations of parameter values. The system parameters are set as b = 12, c = 30, d = 2, e = 4, m = 0.1, and n = 0.01, the initial conditions are chosen as , and the parameter a is the bifurcation parameter of the system. Figure 2(a) is the corresponding Lyapunov exponent spectrum (in order to make the graph display clear, the fifth Lyapunov index is omitted here), and Figure 2(b) is the bifurcation diagram when the parameter a of the system changes from 0 to 20 with the state variable x. It can be seen from Figure 2(b) that as the parameter a gradually increases in the range, the system leads from periodic state to chaos and then to period, with some quasi-periodic windows and transient transfer phenomena in the middle. Table 1 lists the dynamic behavior of parameter a in different ranges and its Lyapunov exponent. Therefore, it can be shown that system (4) has a very rich and complex dynamic behavior:(i)When 0 ≤ a ≤ 1.6, the maximum Lyapunov exponent of system (4) is zero (λ1 = 0, λ2,3,4,5 < 0), so the system is in a multiperiod state.(ii)When 1.6 ≤ a < 2.2, 5.8 < a < 11.5, and 12.4 < a < 13.1, the system has a positive Lyapunov exponent (λ1>0, λ2 = 0, λ3,4,5 < 0) and is in a chaotic state.(iii)When 3.1 ≤ a ≤ 14.8, system (4) has two positive Lyapunov exponents (λ1,2 > 0, λ3 = 0, λ4,5 < 0), so the system is hyperchaotic.(iv)When 14.8 < a ≤ 17.9, the Lyapunov exponent of the system has two zeros (λ1,2 = 0, λ3,4,5 < 0), and the system is quasi-periodic.(v)When 17.9 < a ≤ 20, the maximum Lyapunov exponent of system (4) is zero (λ1 = 0, λ2,3,4,5 < 0), which is different from that of the system in the multiperiod state (0 ≤ a ≤ 1.6), but the parameter a is only in the limit cycle state in this range.(vi)When 2.2 ≤ a ≤ 5.8 and 11.5 ≤ a ≤ 12.4, the most interesting and also very important is the existence of transient chaos and steady-state periodic phenomena. Firstly, the system has a positive Lyapunov exponent, but when it reaches a certain time range, the maximum Lyapunov exponent becomes zero.


aBehavior of dynamics

0 ≤ a ≤ 1.6Multiperiod
1.6 ≤ a < 2.2Chaotic attractor
2.2 ≤ a ≤ 5.8Transient chaos, stable state period-2
5.8 < a < 11.5Chaotic attractor
11.5 ≤ a ≤ 12.4Transient chaos, stable state period-3
12.4 < a < 13.1Chaotic attractor
13.1 ≤ a ≤ 14.8Hyperchaotic attractor
14.8 < a ≤ 17.9Quasi-periodic state
17.9 < a ≤ 20Limit cycle

3.2. Multistability in the 5D Memristive Chaotic System

In order to study the coexistence attractors and other characteristics of the system better, it is necessary to give some disturbance to the initial conditions under the condition of keeping the system parameters constant. Figure 3 shows the dynamic behavior with coexistence bifurcation, in which the initial conditions of blue trajectory and red trajectory are and , respectively. It can be seen from Figure 3 that, under these two initial conditions, the bifurcation mode of the system is almost the same, so the system has exactly the same coexistence attractor under these two conditions. Table 2 is a summary of the dynamic characteristics of different parameter values a. Figure 4 shows coexisting multiple attractors of system (4) for different parameter values a. Figure 4(a) shows that the system has the coexisting two-wing period-1 attractors for a = 1; Figure 4(b) shows that the system has two-wing chaotic attractors coexisting when a = 2; Figure 4(c) shows that the phenomenon is very rare, the system has transient chaos, and then transfers to stable state of period-2 for a = 3.2. When a = 8, Figure 4(d) is very similar to the two-wing chaotic attractors, as shown in Figure 4(b); The system has four-wing chaotic attractors coexisting for a = 10.1 (see Figure 4(e)). It is very similar to the phenomenon in Figure 4(c), but it is different that Figure 4(f) has the coexistence of stable state of period-3 for a = 11.7. It is different from the previous two kinds of two-wing chaotic attractors; when a = 14.6, the system has the coexisting two-wing hyperchaotic attractors, as shown in Figure 4(g). Figure 4(h) shows that when a = 17, the system has coexistence quasi-periodic phenomenon. Figure 4(i) shows that when a = 18.2, the system has coexistence limit cycle with period-1 under two different initial conditions.


aDynamicsFigure

1.0Limit cycle with period-1Figure 4(a)
2.0Two-wing chaotic attractorFigure 4(b)
3.2Stable state period-2Figure 4(c)
8.0Two-wing chaotic attractorFigure 4(d)
10.1Four-wing chaotic attractorFigure 4(e)
11.7Stable state period-3Figure 4(f)
14.6Hyperchaotic attractorFigure 4(g)
17.0Quasi-periodicFigure 4(h)
18.2Limit cycle with period-1Figure 4(i)

If a chaotic system has different states of coexistence attractors under different initial conditions, the system has better randomness and is more suitable for random number generation, image encryption, secure communication, and other fields. As shown in Figure 5, system (4) has coexistence of various types of attractors under the initial conditions and , such as two-wing multiperiod and two-wing period-5 coexist (Figure 5(a)), different two-wing chaotic attractors coexist (Figure 5(b)), periodic-2 and two-wing chaotic attractors coexist (Figures 5(c) and 5(d)), two-wing chaotic attractors coexist with quasi-period (Figure 5(e)), and two-wing chaotic attractors coexist with four-wing chaotic attractors (Figure 5(f)).

3.3. Transient Chaos

Due to the appearance of nonattractive saddle point in phase space, chaos appears in the system in a limited period of time. After a period of time, the system finally becomes a nonchaotic state, which is called transient chaos. In practice, transient chaos is more common than permanent chaos. A close observation of Figure 2 shows that, in the interval ranges of system parameter a, a periodic window appears in Figure 2(b), but Figure 2(a) does indicate that the system is in a chaotic state in this range. This dynamic behavior with two different characteristics is called transient transfer behavior. With the evolution of time, system (4) changes from chaotic behavior to periodic behavior.

When a = 3, the time-domain waveform in the time interval [0,200] is shown in Figure 6(a), and Figures 6(b)6(e) are the phase portraits of the system in x-z plane in different time intervals. It is clear from Figure 6(a) that the system is chaotic in and periodic in . From Figures 6(b)6(e), it is verified that the system evolves from chaos to period gradually with time. Figure 7 also proves that the system does have transient chaos. Different from Figure 6, with the evolution of time, Figure 6 finally becomes a stable state period-2, while Figure 7 tends to a stable state period-3. The abovementioned two cases show that the nonlinear phenomenon from transient chaos to stable state period is not a sudden phenomenon, and it needs a process like chaos bifurcation. For example, when in Figure 6 is at a chaotic state but it is not just a stable state periodic burning, the chaotic phase portraits will change from Figures 6(b)6(e), which needs the same time interval (about [0, 100]) to completely change from chaos to period. Figures 6(b) and 6(c) are transient chaotic attractors, and Figures 6(d) and 6(e) are steady-state periodic states. Figure 6(a) is the time-domain waveform of state variable x, which is different from the time series generated by the general chaotic system. Before t = 40, the system is in chaotic state, and then it will slowly convert to periodic state.

4. Electronic Circuit Design

Using hardware circuit to realize the chaos mathematical model is a hot issue in practical application. The circuit design diagram of the 5D memristive hyperchaotic system (4) is shown in Figure 8. In the circuit design, LF347 is used as the operational amplifier, AD633JN is used as the multiplier chip, and the multiplication factor is 0.1/V. The operating voltage of the operational amplifier is ±E = ±15 V, and the actual saturation voltage measured by the operational amplifier and multiplier is. Since the variables in the phase portraits shown in Figure 1 are beyond the linear dynamic range, we must scale the system, and the relevant circuit equations are as follows:where R1 = R/a, R3 = R/b, R6 = R/c, R9 = R/e, . According to the parameters given in system (4), b = 12, c = 30, d = 2, e = 4, m = 0.1, and n = 0.01, we set Cx = Cy = Cz = = Cu = C = 10 nF, R = 100 kΩ, R2 = R4 = R5 = R8 = 10 kΩ, R3 = 8.25 kΩ, R6 = 3.32 kΩ, R9 = 25 kΩ, R11 = 500 kΩ, and R12 = 16.5 kΩ. Figure 9 shows the phase portraits which are obtained by Multisim simulator. Compared with the MATLAB simulation Figure 4, it can be clearly seen that the phase portraits of Figure 9 and system (4) in initial condition are exactly the same, which confirm the correctness of the proposed 5D memristive hyperchaotic system (4).

5. RNG Design with the Novel Multistable 5D Memristive Hyperchaotic System

5.1. The Design of RNG

Random numbers are widely used in image encryption, information security, computer, and other fields, so the research on RNGs is particularly important. Because the chaotic system has high sensitivity and strong complexity to parameters and initial conditions, random numbers generated by using the chaotic system as an entropy source of RNG have strong randomness. Algorithm 1 is a pseudocode for designing a RNG. As shown in Algorithm 1, (1) the initial conditions of the chaotic system, step value Δh, and sampling interval are given; (2) the fourth-order Runge–Kutta algorithm (RK4) is used to solve the differential equation of the chaotic system to obtain the 32 bit output of the chaotic system, in which 0–21 bit are used for the design of the RNG; (3) XOR the output 22 bit x, y, z, and , respectively, to improve the randomness; (4) the abovementioned two steps to obtain the test bit stream are combined

(1)start
(2)Given the initial condition, parameter value, step value Δh and sampling interval of chaotic system (4);
(3)while (least 100 M. Bit data) do
(4)Using RK4 algorithm to solve chaotic system (4), 32 bit x, y, z, , u has obtained;
(5)Select the last 22 bit number of 32 bit x, y, z, and ;
(6)Obtain the bit stream of the chaotic system (4) by XOR x and y, z, and ;
(7)Get test bit stream according to 5 and 6;
(8)end while
(9)End

In order to better evaluate the performance of generating random numbers of chaotic systems, NIST 800.22 with international high standard is used for random test. NIST 800.22 includes 15 test methods: frequency test, run test, overlapping templates test, linear complexity test, etc. The 22 bit sequence generated from the chaotic system must be large enough for RNG test. If the valueT of NIST 800.22 is more than 0.0001, it shows that the valueT is uniformly distributed and the sequence is random. NIST test is carried out with 130 sample sequences of 1M bit length generated by the chaotic random number generator. The test results are shown in Table 3. All valueT are greater than the threshold value of 0.0001, so RNG passed the test. The lowest pass rate for each statistical test is about 0.975.


NIST statistical test valueTProportionResult

Frequency (monobit) test0.0371570.975Successful
Block frequency test0.7061490.983Successful
Cumulative sums test0.287306/0.2040760.983/0.983Successful
Runs test0.6024580.983Successful
Longest-run test0.0741771Successful
Binary matrix rank test0.4220340.983Successful
Discrete fourier transform test0.3924561Successful
Nonoverlapping templates test0.6058080.9875Successful
Overlapping templates test0.8043371Successful
Maurer’s universal statistical test0.6024580.975Successful
Approximate entropy test0.1951630.9917Successful
Random excursions test0.4075300.9844Successful
Random excursions variant0.4550040.9861Successful
Serial test 10.5510260.975Successful
Serial test 20.6371191Successful
Linear complexity test0.9850351Successful

5.2. Security Analyses
5.2.1. Key Space Analysis

The main purpose of designing a random number sequence generator is encryption, and the size of key space determines the ability to withstand exhaustive attack. The larger the key space, the better the encryption effect. In order to ensure the security of encryption, the key space should be greater than 2128. In this paper, the proposed multistable 5D memristive hyperchaotic system is used to construct a RNG, which can effectively increase the size of the key space. Five 16 bit keys are used to set the initial conditions of the hyperchaotic system, and seven 16 bit keys are used to set the parameters a, b, c, d, e, f, and of the hyperchaotic system. There are 192 bit keys in total, so the key space of this paper is 2192 > 2128, so the method used in this paper can effectively resist exhaustive attack.

5.2.2. Key Sensitivity Analysis

The chaos system is very sensitive to the initial value, so the random numbers generated by the chaotic system have good randomness. Generally, we make small changes to the initial value, and then judge the initial value sensitivity of the RNG by the bit change rate of two sequences. The closer the bit change rate is to 50%, the more sensitive it is to the initial value. Given x(0) = 2, x(0)′ = 2.00000001, a = 10, and a′ = 10.00000001 and the length of random number sequence is 10120000 bits, the change rate of bit with initial value is shown in Table 4. It can be seen that when the random sequence changes only 10−8, the system’s bit change rate is close to 50%, so the random sequence generator is very sensitive to the initial value of the 5D hyperchaotic system. Figure 10 is a time-domain waveform obtained by 50 iterations of the abovementioned two initial values. Figures 10(a) and 10(b) are time-domain oscillograms when the parameter value a and initial condition x change, respectively. The blue line represents the sequence generated when the system parameter value remains unchanged, and the red line represents the sequence generated by iteration when the initial value changes. As shown in Figure 10, when , the sequence curves of two different initial values coincide completely. After t = 8, the sequence curves of different initial values begin to separate, and the difference is more obvious with the increase of time. All the above show that the RNG is very sensitive to the initial value and small initial value changes will have a great impact on the sequence.


Initial valueAmount of changeChanged number of bits (%)

x(0)10–8506028350.0028
a10–8505955349.9956

5.2.3. Correlation Analysis

Correlation is another important measure of randomness. For an ideal random number sequence, the autocorrelation function is δ. The crosscorrelation function is 0. Figure 11 is the correlation graph of two random sequences generated by the RNG, given the initial conditions x(0) = 2 and x(0)′ = 2.00000001. Figure 11(a) is the autocorrelation graph of the sequence, and Figure 11(b) is the crosscorrelation graph of the sequence. From these two figures, it can be seen that the random sequence generated by the RNG based on the 5D hyperchaotic system has strong randomness. In order to further verify the key sensitivity of the generated random number, two similar equal length sequences are generated by the RNG through small changes in the initial value of the system, and the correlation coefficient is used for testing. Correlation coefficient can measure the statistical relationship between sequences. If the correlation coefficient is zero, then there is no correlation between the two sequences. If it is ±1, then there is a strong correlation between the two sequences. In the experiment, one initial condition of the 5D chaotic system (4) changes 10−8, all system parameters remain unchanged, and two groups of random sequences with a length of 4048000 bits are generated. The correlation value is calculated by MATLAB, and Table 5 is obtained. It can be noted that the correlation values obtained by changing the five initial conditions are very close to zero, so there is almost no correlation between the two sequences. This shows that the random number produced in this paper is very sensitive to the initial value.


Initial conditionsAmount of changeChanged valueCorrelation value

x(0) = 210–8x(0)′ = 2.000000010.00066917
y(0) = 110–8x(0)′ = 1.00000001−0.00047467
z(0) = 110–8x(0)′ = 1.000000010.0017
= 210–8x(0)′ = 2.000000010.00071957
u(0) = 210–8x(0)′ = 2.00000001−0.00069381

6. Image Encryption

With the rapid development of computer technology, image information acquisition, processing, transmission, and other related technologies have been rapidly developed and applied and have been widely studied by scholars [101110]. Among them, image encryption plays an increasingly important role in the fields of information security, military, medicine, and meteorology and has become a hot issue of social concern. Chaotic systems show good randomness because of their strong initial value and parameter sensitivity, and they are widely used in the field of image encryption [111120]. In this section, as a typical application, we will use the random number generated by the proposed RNG for image encryption.

Suppose the size of the original image is m × n, where m and n are the number of rows and columns of the image pixel matrix, respectively, and the pixel gray value is an integer between 0 and 255. The specific operation steps of encrypting image with random number are as follows:Step 1: using the proposed multistable 5D memristive hyperchaotic system, the random sequence is generated iteratively according to the given system parameters and initial conditions.Step 2: transform the pixels in the image into a one-dimensional sequence I with a length of m × n in the order of traversal hierarchy.Step 3: ensure the randomness of the sequence and discard the previous n iterations. Continue the iteration to generate the binary sequence of m × n × 8 bits. Then, we convert every 8 bits of binary sequence into an integer, ranging from 0 to 255. Finally, we get an integer sequence of length m × n: i = 1, 2, …, M × N.Step 4: use the random sequence generated by the system to scramble all the pixel values in one-dimensional sequence I to get the scrambled sequence I′.Step 5: store the generated image as the final encrypted image.

Decryption is the reverse of encryption.

6.1. Simulation Results

In this paper, the Lena image with the size of 256 × 256 is used as the encrypted plain image (note that the same photo is used in all subsequent safety analysis comparisons with other references), and the keys are a = 10, b = 12, c = 30, d = 2, e = 4, m = 0.1, and n = 0.01 and . The results of encryption and decryption of Lena images are shown in Figure 12, where Figure 12(a) is the original plain image, Figure 12(b) is the encrypted image, and Figure 12(c) is the decrypted image successfully decrypted using the key. It can be seen that the encrypted image does not have the characteristics of the original plain image, and the decrypted image is exactly the same as the original plain image.

6.2. Security Analyses
6.2.1. Histogram Analysis

Histogram is used to display the distribution characteristics of pixels. In the encryption algorithm, changing the distribution characteristics is very important. If the probability of all intensity pixels generated is equal in the histogram of the encrypted image, the encryption has a high degree of symmetry and good uniformity. Figures 13(a) and 13(b), respectively, represent the histogram of the plain image and the encrypted image. It can be seen that the original plain image has obvious statistical characteristics, while the probability of each gray value of the encrypted image is almost equal. Therefore, encrypted images can effectively resist statistical analysis attacks.

6.2.2. Correlation Analysis

There is usually a strong correlation between adjacent pixels in an image, so a good encryption algorithm should be able to produce cipher images with low correlation, so as to hide image information and resist statistical attacks. The correlation of adjacent pixels is determined by the following formula:wherewhere and represent the expectation and variance of the variable x, and rx,y is the correlation coefficient of adjacent pixels x and y. Figure 14 shows the phase diagrams of Lena plain text image and cipher text image with adjacent pixel points in all directions upward (where (a) and (b) are horizontal directions, (c) and (d) are vertical directions, and (e) and (f) are diagonal directions). It can be seen from these figures that the adjacent pixel values of the plain image are located near the line with slope 1, indicating that the two adjacent pixels are highly correlated. The pixel values of the cipher image are scattered throughout the region, indicating a low correlation between the adjacent pixels. Table 6 shows the test values of correlation in three directions: horizontal, vertical, and diagonal. It can be seen that the adjacent pixels of the plain image have high correlation (rx,y ⟶ 1), and the adjacent pixels of the cipher image have low correlation (rx,y ⟶ 0). At the same time, compared with the corresponding results of References [111114], it shows that the proposed encryption algorithm has lower correlation between adjacent pixels and can more effectively resist statistical attacks.


ImagePlain imageCipher image
HorizontalVerticalDiagonalHorizontalVerticalDiagonal

Ours0.945050.966530.919170.00068299−0.0007768−0.0036362
Reference [111]0.9642270.9824300.965609−0.038118−0.0291420.002736
Reference [112]0.8126880.8379590.7820530.001251−0.0035430.001449
Reference [113]0.918480.829210.807310.0118990.0180620.036784
Reference [114]0.971650.987300.954400.00312−0.00317−0.00310

6.2.3. Information Entropy

Information entropy is an important index to reflect the randomness of information. The more uniform the distribution of pixel gray value, the greater the information entropy, the greater the randomness, and the higher the security. The calculation formula is as follows:where pi is the probability of occurrence of pixel points with a pixel value of i. For grayscale images, the ideal value of information entropy is 8. As listed in Table 7, by comparing the information entropy of cipher and the cipher images in References [115118], it can be concluded that the information entropy value of the encrypted images in the algorithm in this paper is closer to the ideal value 8, and the encrypted images are closer to the random signal source, which can effectively resist the entropy attack.


ImageOursReference [115]Reference [116]Reference [117]Reference [118]

Results7.99747.99717.82327.99637.9951

6.2.4. Differential Attack

Pixels change rate (Number of Pixels Change Rate, NPCR) and normalized pixels flat change strong degree (Unified Average Changing Intensity, UACI) can be used to measure to express the sensitivity of the encryption algorithm, which is an important indicator of measuring algorithm ability to resist differential attack. NPCR and UACI, respectively, represent the proportion and degree of change in the pixel value of the corresponding position. The larger the proportion and the higher the degree of change, the stronger the antiattack capability of the algorithm. The calculation formulas are as follows:where M × N is the size of the image, and , respectively, represent the pixel values of the positions corresponding to the plain and cipher. When the NPCR and UACI of the image are close to the ideal values of 99.6094070% and 33.4635070%, the algorithm has good safety [112, 121]. As listed in Table 8, the algorithm in this paper is more sensitive to the plain than the NPCR and UACI values in References [111120] can meet the security requirements and have a good ability to resist differential attacks.


ImageOursReference [111]Reference [118]Reference [119]Reference [120]

NPCR99.595699.611499.551199.5799.58
UACI33.453533.452333.346133.3133.43

7. Conclusion

In this study, a novel multistable 5D memristive hyperchaotic system with line equilibrium is first introduced. Dynamical analysis is performed in terms of phase portraits, Lyapunov exponential spectrum, bifurcation diagram, and spectral entropy. Several interesting properties such as multistability and transient chaos have been revealed by using classical nonlinear analysis tools. Then, an electronic circuit is designed, and its accuracy is verified by Multisim simulation. As the engineering application, a new chaos-based RNG is designed and internationally accepted NIST 800.22 random tests are run. Security analyses are carried out and they have proved that the design can be used in cryptography applications. Finally, a chaotic image encryption is proposed based on the random number sequences; security analyses show that the algorithm has good security and can resist common attacks.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61504013, 61702052, 61772087, 61741104, 61674054, and 61901169, Natural Science Foundation of Hunan Province under Grants 2019JJ50648, 2016jj2005, 2017JJ2049, and 2019JJ40190, Scientific Research Fund of Hunan Provincial Education Department under Grant 18A137, National Key Research and Development Project under Grant 2018YFE0111200, Guizhou Provincial Science and Technology Foundation under Grant [2018]1115, and Guizhou Province Science and Technology Plan Project under Grant [2018]5769.

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Copyright © 2020 Fei Yu et al. 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.


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