# A Novel Memcapacitor Model and Its Application for Image Encryption Algorithm

Authors: Nan Li; Junwei Sun; Yanfeng Wang
Journal: Journal of Electrical and Computer Engineering
 (2019)

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## Abstract

In this paper, a novel chaotic circuit with one memcapacitor is designed, which has abundant dynamic behaviors. As a potential application, combining the DNA binding mode and the characteristics of the chaotic system, a new image encryption scheme is proposed. Firstly, the original image is processed and generates a set of hash values by the SHA-3 algorithm. The generated hash value and original image are performed XOR operation. The hash values are processed to generate the initial value of the chaotic system by the Hamming distance. Furthermore, the elliptic curve with the hyperchaotic sequence is used to construct the Hill encryption matrix, and the plaintext image is permuted and encrypted. At last, the proposed chaotic system is used to scramble the image. The simulation results and theoretical analysis show that the chaotic system has abundant dynamic behaviors, and the proposed algorithm has a better anti-interference ability for image and information encryption.


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## Body

## 1. Introduction

With the rapid development of multimedia technology and Internet technology, digital images have become one of the most important information carriers for people to communicate. It is so easy to obtain digital images through network and further use, process, reproduce, and distribute them. However, the digital images are vulnerable to attacks in unauthorized network transmission, so people have to take more attention on security and confidentiality of multimedia information, and encryption technology must be used to ensure the security of image transmission. Some traditional encryption algorithms such as Data Encryption Standard (DES), Advanced Encryption Standard (AES), and International Data Encryption Algorithm (IDEA) do not take into account the strong correlation between the pixels in the digital image itself. Therefore, finding a new encryption method applied to image encryption will be a hot research topic at present.Generally, there are two major approaches which are used to protect digital images. One is information hiding [1] which includes watermarking, anonymity, and cover channel. The other is encryption which includes conventional encryption and others such as chaotic encryption. There have been a large number of image-encrypted documents [2–5], based on chaotic systems and DNA algorithms. The low-dimensional system has the advantages of simple form and short time of generating chaotic sequence. But the disadvantage is that the key space is too small to be easily attacked. Therefore, the chaotic image encryption algorithm develops along the direction of complex chaotic systems, such as low-dimensional to high-dimensional chaos and hyperchaos. However, most of the processing technology is based on the computer software platform, which is difficult to meet the growing processing quantity and fast and timely processing requirements. So, it is necessary to explore the hardware implementation of digital image processing, that is, the design and implementation of high speed and accurate image encryption circuit. In order to solve these problems, with the research of DNA computing, DNA cryptography is born as a new cryptographic field emerged, in which DNA is used as information carrier and the modern biological technology is used as implementation tool [6, 7]. Some researchers suggest that DNA computing [8, 9] is an effective way to improve the security of chaotic image encryption technology.As we all know, DNA has the feature with large-scale parallelism, ultra-low energy consumption, and high storage density inherent in molecules. DNA computation was first introduced by Adleman in the literature [10]. More specifically, DNA encryption computation is based on DNA as the information carrier, with modern biotechnology as the realization tool, to excavate the inherent high storage density and high parallelism of DNA and realize cryptographic work such as encryption and decryption. An image encryption algorithm based on chaos and DNA dynamic encoding is proposed in the literature [11], which contains dynamic DNA encoding rules. Hanis et al. [12] designed an image encryption scheme using an improved chaotic map and a butterfly-like structure. Xue et al. proposed a digital image encryption technique based on DNA sequence and multichaotic mapping, which uses DNA sequences, cubic radiation, and logistic maps to fuse the encrypted images in the literature [13]. The method makes the pixel correlation of the encrypted image become higher, but the implementation process is complex. Wong et al. [14] proposed an image encryption with the chaotic standard map. The algorithm has large key space and high sensitive, but their security is not high enough. A novel encryption scheme based on the DNA sequence is proposed by Guesmi et al. [15]. They used the natural DNA sequences to encoding the information and encrypted an image by using the hash algorithm SHA-2. Such experiments can only be done in a well-equipped lab using current technology, and it needs high cost. For these reasons, the researches of DNA cryptography are still much more theory than practicality. As a result of ongoing studies, some promising image encryption schemes and improvements to the previous algorithms have been proposed recently [16–18].On the other hand, more than 40 years ago, according to the principle of symmetry, Chua proposed the fourth fundamental circuit element memristor [19, 20]. But it was not confirmed until found that TiO2 has the characteristics of the memristor in the nanoscale in Hewlett Packard Labs [21]. Since then, memristors have attracted widespread attention and become a hot research area. Compared with the conventional memories technology, memristors bring new technological innovation by virtue of its characteristics, such as instance, high density, low power, and good memory characteristics, which are utilized to design memory device, construct chaotic circuits [22, 23], improve neural networks [24], and so on. Some examples are circuits with two HP memristors in antiparallel [25] and a practical implementation of memristor-based chaotic circuits in a programmable gate array [26]. Also, the memristor-based chaotic circuit for pseudorandom number generation has been analyzed in a cryptography application study [27].Compared with the development of memristor, memcapacitor and meminductor have received relatively little attention. Although actual solid-state memcapacitor has not been yet realized so far, it is important to design effective memcapacitor models and make prospective studies for its applications [28]. In 2009, Ventra and Chua [29] extended the concept of memristors to memcapacitors and meminductors. Memcapacitor has the characteristic of variable capacitance according to its accumulated state, e.g., charge and voltage [30]. It would be worthwhile to prospectively study effective memcapacitor models and its applications. Several memcapacitor models, such as piecewise linear, quadric, and cubic function models, memristor-based memcapacitor models, and memcapacitor emulators, were proposed in [31], and a mathematical memcapacitor model and a corresponding circuit model are established in [32].In this work, a new image encryption scheme is proposed by analyzing the characteristics of the memcapacitor system and DNA encoding. Referring to the previous researches [2, 14] on chaotic encryption, there are some advantages which make our research more attractive and meaningful. At first, the original image is processed and generates a set of hash values by the SHA-3 algorithm. The generated hash value and original image are performed XOR operation. The hash values are processed to generate the initial value of the memcapacitor chaotic system by the Hamming distance. Then, the combination of the memcapacitor system and elliptic curve is used to construct the Hill encryption matrix to permute and encrypt the image. Finally, the hyperchaotic sequence is used to scramble the image.The outline of this paper is organized as follows. In Section2, a new chaotic system base on memcapacitor is proposed and analyzed. In Section 3, the related preparatory work is introduced. Section 4 presents the chaotic encryption scheme, and Section 5 gives some simulation results and security analysis. Finally, some conclusions and suggestions for future work are given in Section 6.


## 2. A Novel Base on Memcapacitor Chaotic System

### 2.1. Memcapacitor Chaotic System

Memcapacitor has the characteristic of variable capacitance according to its accumulated state, e.g., charge and voltage. In this section, the schematic of proposed four-order passive circuit base on the voltage-controlled memcapacitor is proposed. The chaotic circuit is shown in Figure1, which contains two linear inductors, L1 and L2, a linear resistor, R, a linear capacitor, C, and a voltage-controlled memcapacitor, CM. Its charge-voltage relation is(1)q=CMq,v,t.Figure 1
Four-order passive circuit based on voltage-controlled memcapacitor.Considering a voltage-controlled memcapacitor whose capacitance is dependent on the applied voltage, the device’s capacitance changes linearly or nonlinearly from the minimum value to maximum value according to the voltage add-on. The nonlinear capacitance-voltage relationship can be designed as follows:(2)CMv=CMmax1−e−βv+CMmine−βv=CMmax−CMmax−CMmine−βv,where CMmax is the maximum capacitance of the memcapacitor while the applied voltage approaches to infinite and CMmin is the minimum capacitance when applied voltage comes to zero.The current flowing across a voltage-controlled capacitor is the derivation of charge to time, as(3)i=dqdt=Cvdvdt+vdCvdt,dCvdt=signvβCMmax−CMmine−βvdvdt.According to Kirchho’s law, the KCL and KVL equations of this circuit are given in equation (4), where A and w are the amplitude and angular speed of this voltage source, respectively:(4)dVCdt=Asinwt−VC/R−iL1C=Asinwt−VCRC−iL1C,diL1dt=VC−VCML1,dVCMdt=iL1−iL2CM,diL2dt=VCML2.Letx=Vc, y=iL1, z=VCM, and u=iL2, finally, we get the state equation as(5)x˙=Asinwt−VC/R−iL1C,y˙=VC−VCML1,z˙=iL1−iL2CM,u˙=VCML2.Numerical simulation using MATLAB ode45 integration was adopted to research the dynamic behaviors of this proposed voltage-controlled memcapacitor circuit. The circuit’s parameter conditions are set as follows:A = 20, w=30rad/s, R = 100 Ohm, C = 0.2 F, L1 = 0.1H, L2 = 0.1H, CMmax = 0.1 F, CMmin = 0.01 F, and β = 10. If the initial conditions are x = 0, y = 0, z = 0, and u = 0, then equation (5) can be written as(6)x˙=sin30t−x20−5y,y˙=10x−10y,z˙=y−u0.1+0.09e−10z,u˙=10z.The Lyapunov exponents of the spectrum are LE1 > 0, LE2 > 0, LE3 < 0, and LE4 < 0 in Figure 2, which imply that system (6) is a hyperchaotic system. The attractor of the chaotic system (6) with initial conditions x0=0, y0=0, z0=0, and u0=0 is shown in Figures 3(a)–3(d).Figure 2
Lyapunov exponents of chaotic system (6) vs.t.Figure 3
Phase portraits of system (6). Phase portraits of (a) system (6)x vs. y, (b) system (6) x vs. z, (c) system (6) y vs. z, and (d) system (6) x vs. z vs. z.
(a)
(b)
(c)
(d)


### 2.2. Waveform Drawing

The time domain waveform of state variablesVc, iL1, VcM, and iL2 is shown in Figures 4(a)–4(d).Figure 4
Waveform drawing of state variable: (a) Voltage of C; (b) current flow of inductorL1; (c) voltage of CM; (d) current flow of inductor L2.
(a)
(b)
(c)
(d)


### 2.3. Dissipativity and Equilibrium Point Set

It is well known that being dissipative is a necessary condition for nonlinear circuit to generate chaotic attractors. In order to evaluate the dissipativity of the circuit in Figure1, mathematical expression of the exponential constrain rate is deduced from (6) as(7)∇V=∂x˙∂x+∂y˙∂y+∂z˙∂z+∂u˙∂u=−120−10−0.9y−ue−10z0.1+0.09e−10z2.Considering that the exponential function is greater than or equal to zero, equation (7) can be simplified as(8)∇V=∂x˙∂x+∂y˙∂y+∂z˙∂z+∂u˙∂u≤−20120<0.Obviously, the dissipativity of the circuit in Figure1 is negative, implying that all orbits are ultimately confined to a specific subset of zero volume, and the asymptotic motion settles onto an attractor in the neighborhood of the unstable equilibrium points. Its dynamical behaviors can be qualitatively determined by evaluating the eigenvalues of the corresponding Jacobian matrix at each of the equilibrium points. The equilibrium points of system (6) are obtained by solving the following equations:(9)x˙=sin30t−x20−5y=0,y˙=10x−10y=0,z˙=y−u0.1+0.09e−10z=0,u˙=10z=0.Clearly, the equilibrium point set is(10)E=x,y,z,ux=y=u=20101sin30t,z=0.Their values are determined by the specified circuit parameters ofA,R,C,L1,andL2. This property is different from the conventional memristor-based chaotic circuits.


### 2.4. Dynamics Dependent on System Parameters

Similar to dynamical analysis of the general chaotic circuit, by utilizing the conventional dynamical analysis tools such as bifurcation diagram, Lyapunov exponent spectra, and so on, the dynamical behaviors of the chaotic system (6) shown in Figure1 are studied under the variation of system parameters. According to formula (5), we have the following assumptions: 1/RC=a,1/C=b,1/L1=c,and1/L2=d.


#### 2.4.1. Fixb = 5, c = 10, and d = 10 and Vary a

For the above circuit parameters, the initial conditions of system (6) arex0=y0=z0=u0=0. Figures 5 and 6 show the spectrum of Lyapunov exponents and bifurcation diagrams of system (6) with respect to parameter a (to clearly see positive Lyapunov exponents, the horizontal axis is limited in [17, 24]). Obviously, when a ∈ [17, 24], it can be observed from Figure 5 that the memcapacitor-based chaotic circuit has two positive Lyapunov exponents (LE1 > 0 and LE2 > 0), which implies that this system should be in hyperchaotic state.Figure 5
Lyapunov exponents of chaotic system (6) versus parametera.Figure 6
Bifurcation diagrams of chaotic system (6) versus parametera.Observed from Figures5 and 6, when the parameter a ∈ [17, 24], the Lyapunov exponents LE1 and LE2 are greater than 0 but they are closer to zero in Figure 5. System (6) can produce complex chaotic behaviors, which are transient hyperchaos phenomena in Figure 6. Also, some further examples of typical points are described in Figure 7. For example, when a equals to 18 and 23, the transient hyperchaos behaviors are observed in Figures 7(a) and 7(b), respectively. The orbits of the system start from different initial states and have different dynamics. In the numerical simulation analysis, the initial states (0.1, 1, 0, 0) and (0.2, 0, 0.1, 1) are implemented, and the trajectories are shown in Figure 7 by the blue curve and the red curve, respectively [19, 21].Figure 7
Phase portrait of system (6) with different circuit parameters (a)a = 18 and (b) a = 23.
(a)
(b)


#### 2.4.2. Fixa = 20, b = 5, and d = 10 and Vary c

When the system parameters and initial conditions of system (6) arex0=y0=z0=u0=0, the system parameter c is a varying parameter. When parameter c belongs to [0, 15] and increases gradually, the Lyapunov exponent spectra and the corresponding bifurcation diagram of the state are shown in Figures 8 and 9, respectively. It can be observed from Figure 8 that the memcapacitor-based chaotic circuit has a positive Lyapunov exponent (LE1 > 0), which implies that this system should be in a chaotic state, and the bifurcation behavior is produced in Figure 9. In addition, several periodic windows have been generated within the chaotic region of c ∈ [0, 15]. It is obvious that Figure 10(a) (c = 8) and Figure 10(b) (c = 12) are observed complicated state phenomena in the memcapacitor-based chaotic circuit, respectively. In the numerical simulation analysis, the initial states (1, 0, 1, −1) and (2, 1, 0.1, 1) are implemented, and the trajectories are shown in Figure 10 by the blue curve and the red curve, respectively [28, 32].Figure 8
Lyapunov exponents of chaotic system (6) versus parameterc.Figure 9
Bifurcation diagrams of chaotic system (6) versus parameterc.Figure 10
Phase portrait of system (6) with different circuit parameters (a)c = 8 and (b) c = 12.
(a)
(b)


### 2.5. Realization of the Circuit

In this section, the basic complex dynamic characteristics of chaotic system (6) are realized by experimental circuit by Spice in Figure11. The circuit in Figure 11 can be used as a schematic diagram to study system (6)’s parameters and characteristics and has certain theoretical significance. The schematic diagram consists of resistors, capacitors, operational amplifiers, analog multipliers, and peripheral circuits. The resistors and capacitance values are chosen as R1 = R2 = R3 = R6 = R7 = R8 = R9 = R10 = R11 = R12 = R13 = R14 = R15 = R16 = R17 = R18 = R19 = R20 = R21 = R22 = R23 = 1 KΩ, R4 = 20 KΩ, R5 = 0.5 KΩ, C1 = 1 mF, C2 = C3 = C4 = 0.1 mF, V1 = 1 V, V2 = 0.1 V, V3 = 10 V, and V4 = 0.09 V.Figure 11
Realization of the circuit by Spice.Furthermore, the motion trajectories of the physical circuit are observed by oscilloscope in Figure12. The images of the oscilloscope are consistent with the simulation images of MATLAB, which proves the correctness of system (6).Figure 12
Debugging diagram of oscilloscope.
(a)
(b)
(c)


## 2.1. Memcapacitor Chaotic System

Memcapacitor has the characteristic of variable capacitance according to its accumulated state, e.g., charge and voltage. In this section, the schematic of proposed four-order passive circuit base on the voltage-controlled memcapacitor is proposed. The chaotic circuit is shown in Figure1, which contains two linear inductors, L1 and L2, a linear resistor, R, a linear capacitor, C, and a voltage-controlled memcapacitor, CM. Its charge-voltage relation is(1)q=CMq,v,t.Figure 1
Four-order passive circuit based on voltage-controlled memcapacitor.Considering a voltage-controlled memcapacitor whose capacitance is dependent on the applied voltage, the device’s capacitance changes linearly or nonlinearly from the minimum value to maximum value according to the voltage add-on. The nonlinear capacitance-voltage relationship can be designed as follows:(2)CMv=CMmax1−e−βv+CMmine−βv=CMmax−CMmax−CMmine−βv,where CMmax is the maximum capacitance of the memcapacitor while the applied voltage approaches to infinite and CMmin is the minimum capacitance when applied voltage comes to zero.The current flowing across a voltage-controlled capacitor is the derivation of charge to time, as(3)i=dqdt=Cvdvdt+vdCvdt,dCvdt=signvβCMmax−CMmine−βvdvdt.According to Kirchho’s law, the KCL and KVL equations of this circuit are given in equation (4), where A and w are the amplitude and angular speed of this voltage source, respectively:(4)dVCdt=Asinwt−VC/R−iL1C=Asinwt−VCRC−iL1C,diL1dt=VC−VCML1,dVCMdt=iL1−iL2CM,diL2dt=VCML2.Letx=Vc, y=iL1, z=VCM, and u=iL2, finally, we get the state equation as(5)x˙=Asinwt−VC/R−iL1C,y˙=VC−VCML1,z˙=iL1−iL2CM,u˙=VCML2.Numerical simulation using MATLAB ode45 integration was adopted to research the dynamic behaviors of this proposed voltage-controlled memcapacitor circuit. The circuit’s parameter conditions are set as follows:A = 20, w=30rad/s, R = 100 Ohm, C = 0.2 F, L1 = 0.1H, L2 = 0.1H, CMmax = 0.1 F, CMmin = 0.01 F, and β = 10. If the initial conditions are x = 0, y = 0, z = 0, and u = 0, then equation (5) can be written as(6)x˙=sin30t−x20−5y,y˙=10x−10y,z˙=y−u0.1+0.09e−10z,u˙=10z.The Lyapunov exponents of the spectrum are LE1 > 0, LE2 > 0, LE3 < 0, and LE4 < 0 in Figure 2, which imply that system (6) is a hyperchaotic system. The attractor of the chaotic system (6) with initial conditions x0=0, y0=0, z0=0, and u0=0 is shown in Figures 3(a)–3(d).Figure 2
Lyapunov exponents of chaotic system (6) vs.t.Figure 3
Phase portraits of system (6). Phase portraits of (a) system (6)x vs. y, (b) system (6) x vs. z, (c) system (6) y vs. z, and (d) system (6) x vs. z vs. z.
(a)
(b)
(c)
(d)


## 2.2. Waveform Drawing

The time domain waveform of state variablesVc, iL1, VcM, and iL2 is shown in Figures 4(a)–4(d).Figure 4
Waveform drawing of state variable: (a) Voltage of C; (b) current flow of inductorL1; (c) voltage of CM; (d) current flow of inductor L2.
(a)
(b)
(c)
(d)


## 2.3. Dissipativity and Equilibrium Point Set

It is well known that being dissipative is a necessary condition for nonlinear circuit to generate chaotic attractors. In order to evaluate the dissipativity of the circuit in Figure1, mathematical expression of the exponential constrain rate is deduced from (6) as(7)∇V=∂x˙∂x+∂y˙∂y+∂z˙∂z+∂u˙∂u=−120−10−0.9y−ue−10z0.1+0.09e−10z2.Considering that the exponential function is greater than or equal to zero, equation (7) can be simplified as(8)∇V=∂x˙∂x+∂y˙∂y+∂z˙∂z+∂u˙∂u≤−20120<0.Obviously, the dissipativity of the circuit in Figure1 is negative, implying that all orbits are ultimately confined to a specific subset of zero volume, and the asymptotic motion settles onto an attractor in the neighborhood of the unstable equilibrium points. Its dynamical behaviors can be qualitatively determined by evaluating the eigenvalues of the corresponding Jacobian matrix at each of the equilibrium points. The equilibrium points of system (6) are obtained by solving the following equations:(9)x˙=sin30t−x20−5y=0,y˙=10x−10y=0,z˙=y−u0.1+0.09e−10z=0,u˙=10z=0.Clearly, the equilibrium point set is(10)E=x,y,z,ux=y=u=20101sin30t,z=0.Their values are determined by the specified circuit parameters ofA,R,C,L1,andL2. This property is different from the conventional memristor-based chaotic circuits.


## 2.4. Dynamics Dependent on System Parameters

Similar to dynamical analysis of the general chaotic circuit, by utilizing the conventional dynamical analysis tools such as bifurcation diagram, Lyapunov exponent spectra, and so on, the dynamical behaviors of the chaotic system (6) shown in Figure1 are studied under the variation of system parameters. According to formula (5), we have the following assumptions: 1/RC=a,1/C=b,1/L1=c,and1/L2=d.


### 2.4.1. Fixb = 5, c = 10, and d = 10 and Vary a

For the above circuit parameters, the initial conditions of system (6) arex0=y0=z0=u0=0. Figures 5 and 6 show the spectrum of Lyapunov exponents and bifurcation diagrams of system (6) with respect to parameter a (to clearly see positive Lyapunov exponents, the horizontal axis is limited in [17, 24]). Obviously, when a ∈ [17, 24], it can be observed from Figure 5 that the memcapacitor-based chaotic circuit has two positive Lyapunov exponents (LE1 > 0 and LE2 > 0), which implies that this system should be in hyperchaotic state.Figure 5
Lyapunov exponents of chaotic system (6) versus parametera.Figure 6
Bifurcation diagrams of chaotic system (6) versus parametera.Observed from Figures5 and 6, when the parameter a ∈ [17, 24], the Lyapunov exponents LE1 and LE2 are greater than 0 but they are closer to zero in Figure 5. System (6) can produce complex chaotic behaviors, which are transient hyperchaos phenomena in Figure 6. Also, some further examples of typical points are described in Figure 7. For example, when a equals to 18 and 23, the transient hyperchaos behaviors are observed in Figures 7(a) and 7(b), respectively. The orbits of the system start from different initial states and have different dynamics. In the numerical simulation analysis, the initial states (0.1, 1, 0, 0) and (0.2, 0, 0.1, 1) are implemented, and the trajectories are shown in Figure 7 by the blue curve and the red curve, respectively [19, 21].Figure 7
Phase portrait of system (6) with different circuit parameters (a)a = 18 and (b) a = 23.
(a)
(b)


### 2.4.2. Fixa = 20, b = 5, and d = 10 and Vary c

When the system parameters and initial conditions of system (6) arex0=y0=z0=u0=0, the system parameter c is a varying parameter. When parameter c belongs to [0, 15] and increases gradually, the Lyapunov exponent spectra and the corresponding bifurcation diagram of the state are shown in Figures 8 and 9, respectively. It can be observed from Figure 8 that the memcapacitor-based chaotic circuit has a positive Lyapunov exponent (LE1 > 0), which implies that this system should be in a chaotic state, and the bifurcation behavior is produced in Figure 9. In addition, several periodic windows have been generated within the chaotic region of c ∈ [0, 15]. It is obvious that Figure 10(a) (c = 8) and Figure 10(b) (c = 12) are observed complicated state phenomena in the memcapacitor-based chaotic circuit, respectively. In the numerical simulation analysis, the initial states (1, 0, 1, −1) and (2, 1, 0.1, 1) are implemented, and the trajectories are shown in Figure 10 by the blue curve and the red curve, respectively [28, 32].Figure 8
Lyapunov exponents of chaotic system (6) versus parameterc.Figure 9
Bifurcation diagrams of chaotic system (6) versus parameterc.Figure 10
Phase portrait of system (6) with different circuit parameters (a)c = 8 and (b) c = 12.
(a)
(b)


## 2.4.1. Fixb = 5, c = 10, and d = 10 and Vary a

For the above circuit parameters, the initial conditions of system (6) arex0=y0=z0=u0=0. Figures 5 and 6 show the spectrum of Lyapunov exponents and bifurcation diagrams of system (6) with respect to parameter a (to clearly see positive Lyapunov exponents, the horizontal axis is limited in [17, 24]). Obviously, when a ∈ [17, 24], it can be observed from Figure 5 that the memcapacitor-based chaotic circuit has two positive Lyapunov exponents (LE1 > 0 and LE2 > 0), which implies that this system should be in hyperchaotic state.Figure 5
Lyapunov exponents of chaotic system (6) versus parametera.Figure 6
Bifurcation diagrams of chaotic system (6) versus parametera.Observed from Figures5 and 6, when the parameter a ∈ [17, 24], the Lyapunov exponents LE1 and LE2 are greater than 0 but they are closer to zero in Figure 5. System (6) can produce complex chaotic behaviors, which are transient hyperchaos phenomena in Figure 6. Also, some further examples of typical points are described in Figure 7. For example, when a equals to 18 and 23, the transient hyperchaos behaviors are observed in Figures 7(a) and 7(b), respectively. The orbits of the system start from different initial states and have different dynamics. In the numerical simulation analysis, the initial states (0.1, 1, 0, 0) and (0.2, 0, 0.1, 1) are implemented, and the trajectories are shown in Figure 7 by the blue curve and the red curve, respectively [19, 21].Figure 7
Phase portrait of system (6) with different circuit parameters (a)a = 18 and (b) a = 23.
(a)
(b)


## 2.4.2. Fixa = 20, b = 5, and d = 10 and Vary c

When the system parameters and initial conditions of system (6) arex0=y0=z0=u0=0, the system parameter c is a varying parameter. When parameter c belongs to [0, 15] and increases gradually, the Lyapunov exponent spectra and the corresponding bifurcation diagram of the state are shown in Figures 8 and 9, respectively. It can be observed from Figure 8 that the memcapacitor-based chaotic circuit has a positive Lyapunov exponent (LE1 > 0), which implies that this system should be in a chaotic state, and the bifurcation behavior is produced in Figure 9. In addition, several periodic windows have been generated within the chaotic region of c ∈ [0, 15]. It is obvious that Figure 10(a) (c = 8) and Figure 10(b) (c = 12) are observed complicated state phenomena in the memcapacitor-based chaotic circuit, respectively. In the numerical simulation analysis, the initial states (1, 0, 1, −1) and (2, 1, 0.1, 1) are implemented, and the trajectories are shown in Figure 10 by the blue curve and the red curve, respectively [28, 32].Figure 8
Lyapunov exponents of chaotic system (6) versus parameterc.Figure 9
Bifurcation diagrams of chaotic system (6) versus parameterc.Figure 10
Phase portrait of system (6) with different circuit parameters (a)c = 8 and (b) c = 12.
(a)
(b)


## 2.5. Realization of the Circuit

In this section, the basic complex dynamic characteristics of chaotic system (6) are realized by experimental circuit by Spice in Figure11. The circuit in Figure 11 can be used as a schematic diagram to study system (6)’s parameters and characteristics and has certain theoretical significance. The schematic diagram consists of resistors, capacitors, operational amplifiers, analog multipliers, and peripheral circuits. The resistors and capacitance values are chosen as R1 = R2 = R3 = R6 = R7 = R8 = R9 = R10 = R11 = R12 = R13 = R14 = R15 = R16 = R17 = R18 = R19 = R20 = R21 = R22 = R23 = 1 KΩ, R4 = 20 KΩ, R5 = 0.5 KΩ, C1 = 1 mF, C2 = C3 = C4 = 0.1 mF, V1 = 1 V, V2 = 0.1 V, V3 = 10 V, and V4 = 0.09 V.Figure 11
Realization of the circuit by Spice.Furthermore, the motion trajectories of the physical circuit are observed by oscilloscope in Figure12. The images of the oscilloscope are consistent with the simulation images of MATLAB, which proves the correctness of system (6).Figure 12
Debugging diagram of oscilloscope.
(a)
(b)
(c)


## 3. Preparatory Theory

### 3.1. DNA Sequence

A DNA sequence has four nucleic acids, which are adenine (A), cytosine (C), guanine (G), and thymine (T). A is complement to T, and C is complement to G. As we all know, 0 and 1 complement each other in binary numbers, so that 00 and 11, 01 and l0 are complementary, respectively. Therefore, one way to encode 00, 11, 01, and 10 using DNA bases is A, T, C, and G, respectively. Using this way, due to each pixel of an image can be represented by 8-bit binary digits, each pixel value can be expressed as a DNA sequence with a length of 4. For example, if the gray scale value of an image pixel is 210, its binary value 11010010 can be DNA encoded as TCAG, from which pixel value can be obtained back through DNA decoding by replacing DNA bases with their binary sequences. Obviously, there are 24 kinds of coding schemes. But there are only 8 kinds of coding schemes that satisfy the Watson–Crick complementary base pairing rule shown in Table1. Because the color image could be divided into three channels, which are red channel, green channel, and blue channel, the three channels can be represented by using the DNA codes. The following base operations and transformation rules are defined at the same time.Table 1
8 kinds of schemes encoding rule of DNA sequence.
Rule 1
Rule 2
Rule 3
Rule 4
Rule 5
Rule 6
Rule 7
Rule 8
00-A
00-A
00-C
00-C
00-T
00-T
00-G
00-G
01-C
01-C
01-A
01-A
01-G
01-G
01-C
01-C
10-G
10-T
10-G
10-T
10-A
10-C
10-A
10-T
11-T
11-G
11-T
11-G
11-C
11-A
11-T
11-A


### 3.2. Value Generation

In communication theory, the Hamming distance, named after Richard Hamming, is the number of positions in two strings of equal length for which the corresponding elements are different. Put another way, it measures the minimum number of substitutions required to change one into the other. The Hamming distance is used in telecommunication to count the number of flipped bits in a fixed-length binary word, an estimate of error, and so is sometimes called the signal distance. Hamming weight analysis of bits is used in several disciplines including information theory, coding theory, and cryptography. The Hamming distanceHx,y of sequence x = ｛x1, x2, …, xn｝ and y = ｛y1, y2, …, yn｝ can be defined as(11)hxi,yi=0,xi=yi,1,xi≠yi,Hx,y=∑i=1nhxi,yi.The first set of hash values produced by SHA-3 was used as key K, which was used to generate the initial value of the memcapacitor system. The hash values produced by SHA-3, even if the original image has one bit difference, the hash values and the key will be completely different. The antiattack is 2256, and the generated key has the advantage of randomness, periodicity, and long key space in the way. Combining the original image information with the key, the algorithm will effectively resist the known plaintext attacks. Divide K by byte, which can be expressed as b1, b2, b3, …, b32. The initial value of the memcapacitor chaotic system is computed by(12)x0=α+2∗HC1,C2−C24,y0=β+2∗HC1,C2−C34,z0=γ+2∗HC1,C2−C44,u0=δ+2∗HC1,C2−C14,where C1 = b1 ⊕ b2 ⊕ b3 ⊕ b4 ⊕ b5 ⊕ b6 ⊕ b7 ⊕ b8, C2 = b9 ⊕ b10 ⊕ b11 ⊕ b12 ⊕ b13 ⊕ b14 ⊕ b15 ⊕ b16, C3 = b17 ⊕ b18 ⊕ b19 ⊕ b20 ⊕ b21 ⊕ b22 ⊕ b23 ⊕ b24, and C4 = b25 ⊕ b26 ⊕ b27 ⊕ b28 ⊕ b29 ⊕ b30 ⊕ b31 ⊕ b32 and α,β,γ,andδ are given values.


### 3.3. Dynamic DNA Encoding

On the basis of the above theories, the DNA encoding is given. Reference to Table1, we know that there are eight different encoding rules in the DNA computation. Dynamic DNA coding technology is based on the position of the matrix to be encoded in the image matrix I. The plain image is m∗n grayscale image, where each pixel is represented using a byte. The pixel values in the plain image according to formula (13) are encoded:(13)ri,j=roundmodi−1∗n+j,8+1,where i ∈ {1, 2, …, m}, j ∈ {1,2, …, n}, and i and j represent the position of the pixel P(i, j) in the matrix, respectively. The encoding rules are selected according to the pixel position, which increases the diversity of coding rules.


### 3.4. Elliptic Curve

The application of elliptic curves to the field of cryptography has been relatively recent. Elliptic curve cryptography is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC needs smaller keys compared to non-ECC to provide equivalent security. Nowadays, elliptic curves are used in key agreement, digital signatures, pseudorandom generators, and other tasks.The elliptic curve refers to the plane curve determined by the Weierstrass equation:(14)y2+a1xy+a3y=x3+a2x2+a4x+a6.F is a domain, where ai ∈ F (i = 1, 2, 3, 4, and 6). The number (x, y) that satisfies the Weierstrass equation is called the point of the elliptic curve in the F domain. The F domain can be a rational domain or a complex domain. All points of the elliptic curve, plus a special infinite point O, are defined as the elliptic curve on the finite domain F.There are two kinds of elliptic curves that are often used: the elliptic curves with prime values on the prime number domainFp, where P is the odd prime number, and the elliptic curve on the finite field F2m with the eigenvalue 2. The elliptic curve used in this paper is the eigenvalue with prime number domain Fp. In Fp, an elliptic curve L with a large prime number factor is randomly searched, and the base point G0 = (xʹ0, yʹ0) of the prime number p on E. a and b are the parameters of the elliptic curve L. The elliptic curve L defined on the prime number domain Fp is(15)y2=x3+ax+bmodp,where a, b ∈ Fp and satisfy 4a2+27b3≠0 (mod p) and p is the order of the elliptic curve.Key generation: each user generates its own key; the sender Alin randomly selects an integerda as the private key in the M sequence and 1 ≤ da ≤ p − 1 and the receiver Bill randomly selects an integer db as the private key in the M sequence and 1 ≤ db ≤ p − 1. If the public key of the sender Alin is Pa = daG0 and the public key of the receiver Bill is Pb = dbG0, then da ∗ Pb = db ∗ Pa=(xʹ, yʹ). da and db are the private keys of the users Alin and Bill, respectively. Pa and Pb are the public keys of Alin and Bill, respectively.


### 3.5. Hill Encryption Matrix

In classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra. Invented by Lester S. Hill in 1929, it was the first polygraphic cipher in which it was practical to operate on more than three symbols at once. The Hill code is a replacement code. The key to the Hill code is the encryption matrix. If the encryption matrix is irreversible, the ciphertext will not be restored to the plaintext. In this paper, the elliptic curve andM sequence are used to construct the encryption matrix.The image to be encrypted is grouped by 8 pixels, and each group of pixels is converted into an 8× 1 matrix I8∗1. By constructing an 8 × 8 reversible matrix K, Hill encryption is performed for each group of images. The encryption formula is as follows:(16)E=K∗Imod256=k11k12⋯k17k18k21k22⋯k27k28⋮⋮⋮⋮k71k72⋯k77k78k81k82⋯k87k88∗I11I21I31I41I51I61I71I81mod256E11E21E31E41E51E61E71E81,where E is the result of Hill encryption, I11∼81 is a set of pixels to be encrypted, and k11∼88 is the Hill encryption matrix K. The ciphertext is decrypted by using the inverse matrix K−1 of K, I = (K−1 ∗ E) mod256= (K ∗ E) mod256. The encryption matrix K is divided into four parts:(17)K=K11K12K21K22,K11=k11k12k13k14k21k22k23k24k31k32k33k34k41k42k43k44.(1)
(xʹ, yʹ) = da ∗ Pb = db ∗ Pa, (k11, k12) = xʹG0, (k13, k14) = m1G0, (k21, k22) = m2G0, (k23, k24) = m3G0, (k31, k32) = m4G0, (k33, k34) = m5G0, (k41, k42) = m6G0, (k43, k44) = yʹG0, and the submatrix K11 is obtained. m1, m2, …, m6 are the first six elements in the M1 sequence.(2)
The submatrixK11 is used as the following calculation to generate the submatrix K12, K12 = n ∗ (I − K11).(3)
SubmatrixK22 = I − K12.(4)
SubmatrixK21 = (I + K12).(5)
The generated four submatrixesK11, K12, K21, and K22 are merged to obtain the reversible cipher matrix K.


## 3.1. DNA Sequence

A DNA sequence has four nucleic acids, which are adenine (A), cytosine (C), guanine (G), and thymine (T). A is complement to T, and C is complement to G. As we all know, 0 and 1 complement each other in binary numbers, so that 00 and 11, 01 and l0 are complementary, respectively. Therefore, one way to encode 00, 11, 01, and 10 using DNA bases is A, T, C, and G, respectively. Using this way, due to each pixel of an image can be represented by 8-bit binary digits, each pixel value can be expressed as a DNA sequence with a length of 4. For example, if the gray scale value of an image pixel is 210, its binary value 11010010 can be DNA encoded as TCAG, from which pixel value can be obtained back through DNA decoding by replacing DNA bases with their binary sequences. Obviously, there are 24 kinds of coding schemes. But there are only 8 kinds of coding schemes that satisfy the Watson–Crick complementary base pairing rule shown in Table1. Because the color image could be divided into three channels, which are red channel, green channel, and blue channel, the three channels can be represented by using the DNA codes. The following base operations and transformation rules are defined at the same time.Table 1
8 kinds of schemes encoding rule of DNA sequence.
Rule 1
Rule 2
Rule 3
Rule 4
Rule 5
Rule 6
Rule 7
Rule 8
00-A
00-A
00-C
00-C
00-T
00-T
00-G
00-G
01-C
01-C
01-A
01-A
01-G
01-G
01-C
01-C
10-G
10-T
10-G
10-T
10-A
10-C
10-A
10-T
11-T
11-G
11-T
11-G
11-C
11-A
11-T
11-A


## 3.2. Value Generation

In communication theory, the Hamming distance, named after Richard Hamming, is the number of positions in two strings of equal length for which the corresponding elements are different. Put another way, it measures the minimum number of substitutions required to change one into the other. The Hamming distance is used in telecommunication to count the number of flipped bits in a fixed-length binary word, an estimate of error, and so is sometimes called the signal distance. Hamming weight analysis of bits is used in several disciplines including information theory, coding theory, and cryptography. The Hamming distanceHx,y of sequence x = ｛x1, x2, …, xn｝ and y = ｛y1, y2, …, yn｝ can be defined as(11)hxi,yi=0,xi=yi,1,xi≠yi,Hx,y=∑i=1nhxi,yi.The first set of hash values produced by SHA-3 was used as key K, which was used to generate the initial value of the memcapacitor system. The hash values produced by SHA-3, even if the original image has one bit difference, the hash values and the key will be completely different. The antiattack is 2256, and the generated key has the advantage of randomness, periodicity, and long key space in the way. Combining the original image information with the key, the algorithm will effectively resist the known plaintext attacks. Divide K by byte, which can be expressed as b1, b2, b3, …, b32. The initial value of the memcapacitor chaotic system is computed by(12)x0=α+2∗HC1,C2−C24,y0=β+2∗HC1,C2−C34,z0=γ+2∗HC1,C2−C44,u0=δ+2∗HC1,C2−C14,where C1 = b1 ⊕ b2 ⊕ b3 ⊕ b4 ⊕ b5 ⊕ b6 ⊕ b7 ⊕ b8, C2 = b9 ⊕ b10 ⊕ b11 ⊕ b12 ⊕ b13 ⊕ b14 ⊕ b15 ⊕ b16, C3 = b17 ⊕ b18 ⊕ b19 ⊕ b20 ⊕ b21 ⊕ b22 ⊕ b23 ⊕ b24, and C4 = b25 ⊕ b26 ⊕ b27 ⊕ b28 ⊕ b29 ⊕ b30 ⊕ b31 ⊕ b32 and α,β,γ,andδ are given values.


## 3.3. Dynamic DNA Encoding

On the basis of the above theories, the DNA encoding is given. Reference to Table1, we know that there are eight different encoding rules in the DNA computation. Dynamic DNA coding technology is based on the position of the matrix to be encoded in the image matrix I. The plain image is m∗n grayscale image, where each pixel is represented using a byte. The pixel values in the plain image according to formula (13) are encoded:(13)ri,j=roundmodi−1∗n+j,8+1,where i ∈ {1, 2, …, m}, j ∈ {1,2, …, n}, and i and j represent the position of the pixel P(i, j) in the matrix, respectively. The encoding rules are selected according to the pixel position, which increases the diversity of coding rules.


## 3.4. Elliptic Curve

The application of elliptic curves to the field of cryptography has been relatively recent. Elliptic curve cryptography is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC needs smaller keys compared to non-ECC to provide equivalent security. Nowadays, elliptic curves are used in key agreement, digital signatures, pseudorandom generators, and other tasks.The elliptic curve refers to the plane curve determined by the Weierstrass equation:(14)y2+a1xy+a3y=x3+a2x2+a4x+a6.F is a domain, where ai ∈ F (i = 1, 2, 3, 4, and 6). The number (x, y) that satisfies the Weierstrass equation is called the point of the elliptic curve in the F domain. The F domain can be a rational domain or a complex domain. All points of the elliptic curve, plus a special infinite point O, are defined as the elliptic curve on the finite domain F.There are two kinds of elliptic curves that are often used: the elliptic curves with prime values on the prime number domainFp, where P is the odd prime number, and the elliptic curve on the finite field F2m with the eigenvalue 2. The elliptic curve used in this paper is the eigenvalue with prime number domain Fp. In Fp, an elliptic curve L with a large prime number factor is randomly searched, and the base point G0 = (xʹ0, yʹ0) of the prime number p on E. a and b are the parameters of the elliptic curve L. The elliptic curve L defined on the prime number domain Fp is(15)y2=x3+ax+bmodp,where a, b ∈ Fp and satisfy 4a2+27b3≠0 (mod p) and p is the order of the elliptic curve.Key generation: each user generates its own key; the sender Alin randomly selects an integerda as the private key in the M sequence and 1 ≤ da ≤ p − 1 and the receiver Bill randomly selects an integer db as the private key in the M sequence and 1 ≤ db ≤ p − 1. If the public key of the sender Alin is Pa = daG0 and the public key of the receiver Bill is Pb = dbG0, then da ∗ Pb = db ∗ Pa=(xʹ, yʹ). da and db are the private keys of the users Alin and Bill, respectively. Pa and Pb are the public keys of Alin and Bill, respectively.


## 3.5. Hill Encryption Matrix

In classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra. Invented by Lester S. Hill in 1929, it was the first polygraphic cipher in which it was practical to operate on more than three symbols at once. The Hill code is a replacement code. The key to the Hill code is the encryption matrix. If the encryption matrix is irreversible, the ciphertext will not be restored to the plaintext. In this paper, the elliptic curve andM sequence are used to construct the encryption matrix.The image to be encrypted is grouped by 8 pixels, and each group of pixels is converted into an 8× 1 matrix I8∗1. By constructing an 8 × 8 reversible matrix K, Hill encryption is performed for each group of images. The encryption formula is as follows:(16)E=K∗Imod256=k11k12⋯k17k18k21k22⋯k27k28⋮⋮⋮⋮k71k72⋯k77k78k81k82⋯k87k88∗I11I21I31I41I51I61I71I81mod256E11E21E31E41E51E61E71E81,where E is the result of Hill encryption, I11∼81 is a set of pixels to be encrypted, and k11∼88 is the Hill encryption matrix K. The ciphertext is decrypted by using the inverse matrix K−1 of K, I = (K−1 ∗ E) mod256= (K ∗ E) mod256. The encryption matrix K is divided into four parts:(17)K=K11K12K21K22,K11=k11k12k13k14k21k22k23k24k31k32k33k34k41k42k43k44.(1)
(xʹ, yʹ) = da ∗ Pb = db ∗ Pa, (k11, k12) = xʹG0, (k13, k14) = m1G0, (k21, k22) = m2G0, (k23, k24) = m3G0, (k31, k32) = m4G0, (k33, k34) = m5G0, (k41, k42) = m6G0, (k43, k44) = yʹG0, and the submatrix K11 is obtained. m1, m2, …, m6 are the first six elements in the M1 sequence.(2)
The submatrixK11 is used as the following calculation to generate the submatrix K12, K12 = n ∗ (I − K11).(3)
SubmatrixK22 = I − K12.(4)
SubmatrixK21 = (I + K12).(5)
The generated four submatrixesK11, K12, K21, and K22 are merged to obtain the reversible cipher matrix K.


## 4. Encryption Algorithm

The image encryption algorithm is based on the proposed memcapacitor chaotic system and DNA encoding. The encryption algorithm is given as follows. At first, the original image is processed and generates a set of hash values by the SHA-3 algorithm. The generated hash value and original image are performed XOR operation. The hash values are processed to generate the initial value of the chaotic system by the Hamming distance. Then, the image pixels are dynamically encoded and converted to DNA sequences. Furthermore, the elliptic curve with the hyperchaotic sequence is used to construct the Hill encryption matrix, and the plaintext image is permuted and encrypted. At last, the proposed chaotic system is used to scramble the image again. The cryptographic flowchart is shown in Figure13. The specific encryption steps are as follows:(1)
Input the grayscale image matrixV(m ∗ n)(2)
The original image is splitted into unit pixels and encrypted according to the DNA encoding rules(3)
Hash function is used to calculate the hash value of the image matrixV, and the hash values generated and original image are performed XOR operation in Table 2, and the XOR results are encrypted according to the left cyclic shift 3 bits(4)
The first set of hash values produced by SHA-3 was used as keyK, and then divide K by byte, and the four sequences C1–C4 are obtained(5)
For the encrypted imageV, divide every 8 pixels into a block, according to formulas (11) and (12), the initial values of the chaotic system are obtained(6)
The image pixels are dynamically encoded and converted to DNA sequences according to formula (13)(7)
The elliptic curve with the hyperchaotic sequence is used to construct the Hill encryption matrix by Sections3.4. and 3.5, and the matrix value obtained is shifted(8)
The proposed chaotic system is used to scramble the image again, and the results are encoded by the DNA encoding ruleFigure 13
Encrypted flowchart.Table 2
XOR operation for DNA sequences.
XOR
A
T
C
G
A
A
T
C
G
T
T
A
G
C
C
C
G
A
T
G
G
C
T
A


## 5. Experiments and Analysis

In this section, simulation results indicate the effectiveness and feasibility of the above algorithm. An indexed image Lena of size 256 × 256 is used as a original image in Figure14(a), we use MATLAB 2017a to simulate the experiment, and in our experiment, we set α=0, β=0, γ=0, and δ=0. The encrypted image is shown in Figure 14(b).Figure 14
Encrypted Lena image. (a) Original Lena image. (b) Encrypted Lena image.
(a)
(b)


### 5.1. Secret Key Space Analysis

In this section, the performance and security of the proposed encryption scheme will be discussed and analyzed through several aspects. A good encryption algorithm should have a large key space to make it resist exhaustively and attack effectively.In this algorithm, it consists of four types of key parameters, which areα,β,γ,andδ and 256 bytes of the SHA-3 function. However, the calculation accuracy of α,β,γ,andδ is 1014, respectively. The key space of the memcapacitor system is 1014 × 1014 × 1014 × 1014, and the key space of SHA-3 (256 bytes) is 2128. The total key space is 1056 × 2128 ≈ 3.4 × 1094, which shows the algorithm has a sufficiently large key space to resist the brute force attack.


### 5.2. Sensitivity Analysis

An ideal cryptosystem should be of high sensitivity, including key sensitivity and plaintext sensitivity. Key sensitivity means that a tiny change of the secret key should produce a completely different encrypted image or the decryption will fail even via using a slight change key. In order to evaluate the key sensitivity of the proposed algorithm, the following key sensitivity tests have been performed. Figures15(a)–15(d) show the recovering images when only one of the initial values from four chaotic systems add 0.00001, respectively. Figure 16 shows the decrypted image under the wrong DNA encoding rule. Only when the decryption key is consistent with the encryption key can the image be decrypted properly. The different results show that our algorithm has excellent sensitivity.Figure 15
The recovering image under the worry secret key. (a)α=0.00001, other parameters are unchanged. (b) β=0.00001, other parameters are unchanged. (c) γ=0.00001, other parameters are unchanged. (d) δ=0.00001, other parameters are unchanged.
(a)
(b)
(c)
(d)Figure 16
The recovering image under the worry DNA encoding rule.


### 5.3. Differential Attack

Plaintext sensitive means that a tiny disturbance of the plain image will lead to dramatic changes in the cipher image. In order to measure the difference between the resulting cipher images, NPCR (number of pixel change rate) and UACI (unified average change intensity) are usually used to detect the ability of the image encryption scheme. NPCR and UACI are defined by(18)NPCR=∑i=1m∑j=1nCi,jm×n×100%,UACI=∑i=1m∑j=1nP1i,j−P2i,j255×m×n×100%,(19)ci,j=0,ifP1i,j=P2i,j,1,ifP1i,j≠P2i,j,where m and n represent the length and width of the image, respectively. P1(i, j) and P2(i, j) represent the corresponding ciphertext pixel values before and after the change of plaintext. For Lena images as shown in Table 3, the NPCR and UACI values of the algorithm are 99.62% and 33.38%, respectively, which verifies that the image encryption scheme has the ability to resist differential attack.Table 3
Comparison of NPCR and UACI in this paper and other references.
NPCR (%)
UACI (%)
Ours
99.62
33.38
Ref. [2]
99.02
32.83
Ref. [15]
99.59
33.12


### 5.4. Histogram Analysis

The image histogram is a method to evaluate the intensity distribution of pixels within one channel in color image, whose distribution shows a good permutation in an encryption algorithm and a high security against the potential statistical attack. The histograms of plain image and encrypted image are shown in Figure17. Figure 17(a) with a disorder distribution represents the histogram of plain image. The rests with a distribution shows the result after encryption in Figure 17(b). Obviously, no useful information can be extracted from the encrypted image, and high security can be guaranteed to resist statistical attacks.Figure 17
Histogram of (a) plain image and (b) encrypted image.
(a)
(b)


### 5.5. Correlation Coefficient Analysis

Correlation coefficient is an index to assess the image randomness. In this paper, it is employed to test the correlation between two adjacent pixels in plain image and ciphered image. In the original image, the correlation between two adjacent pixels is very high. In order to resist statistical attacks, the correlation of encrypted images must be reduced. 2500 pairs of two adjacent pixels are randomly selected from the plain and encryption image in horizontal, vertical, and diagonal directions to calculate the correlation between pixels. The function to calculate the correlation coefficient is shown as follows:(20)Ex=1N∑i=1Nxi,Dx=1N∑i=1Nxi−Ex2,COVx,y=1N∑i=1Nxi−Exyi−Ex,rxy=COVx,yDx×Dy,where x and y are values of two adjacent pixels selected in three directions randomly. N is the total number of duplets (x, y) obtained from the image. Ex and Dx are the expectation and the variance of x, respectively.The calculated correlation coefficients of plaintext images of Lena and its corresponding ciphered images in the proposed algorithm are listed in Table4. The correlation coefficient of adjacent pixels of the encrypted image is −0.000534. Therefore, the image encryption algorithm has a strong ability to resist statistical attacks. Table 4 and Figure 18 show the correlation comparison between the original image and the adjacent pixel of the encrypted image.Table 4
The correlation of the adjacent pixels for original and encrypted image.
Correlation coefficient
Horizontal
Vertical
Diagonal
Original image
0.9689
0.9486
0.9228
Encrypted image
−0.0029
−0.0031
−0.0037
Ref. [2]
0.0085
0.0021
0.0012
Ref. [15]
0.0265
0.0792
0.0625Figure 18
Correlation analysis of image: (a) plain image in horizontal; (b) cipher image in horizontal; (c) plain image in vertical; (d) cipher image in vertical; (e) plain image in diagonal; (f) cipher image in diagonal.
(a)
(b)
(c)
(d)
(e)
(f)


### 5.6. Information Entropy

The information entropy can test uncertainty. Entropy reflects whether grayscale values distribution is random or equality. The minimum and maximum values of entropy are zero and eight, respectively. The more uniform the distribution of the image gray value, the greater the entropy of the image. Therefore, the entropy value of the encrypted image should be as high as possible. Letm be the information source, and the equation for calculating information entropy is(21)Hm=−∑t=0tPmilog2Pmi,where P(mi) represents the probability that the information mi appears. Assume that there are 28 states of the information source and they appear with the same probability, according to formula (21), we can get the ideal H(m) = 8, which shows that the information is random. Hence, the information entropy of the encrypted image should be close to 8. The value of information entropy of the ciphered image in the proposed scheme is 7.98928, which indicate that the ciphered image obtained by the proposed algorithm could hardly divulge information.


## 5.1. Secret Key Space Analysis

In this section, the performance and security of the proposed encryption scheme will be discussed and analyzed through several aspects. A good encryption algorithm should have a large key space to make it resist exhaustively and attack effectively.In this algorithm, it consists of four types of key parameters, which areα,β,γ,andδ and 256 bytes of the SHA-3 function. However, the calculation accuracy of α,β,γ,andδ is 1014, respectively. The key space of the memcapacitor system is 1014 × 1014 × 1014 × 1014, and the key space of SHA-3 (256 bytes) is 2128. The total key space is 1056 × 2128 ≈ 3.4 × 1094, which shows the algorithm has a sufficiently large key space to resist the brute force attack.


## 5.2. Sensitivity Analysis

An ideal cryptosystem should be of high sensitivity, including key sensitivity and plaintext sensitivity. Key sensitivity means that a tiny change of the secret key should produce a completely different encrypted image or the decryption will fail even via using a slight change key. In order to evaluate the key sensitivity of the proposed algorithm, the following key sensitivity tests have been performed. Figures15(a)–15(d) show the recovering images when only one of the initial values from four chaotic systems add 0.00001, respectively. Figure 16 shows the decrypted image under the wrong DNA encoding rule. Only when the decryption key is consistent with the encryption key can the image be decrypted properly. The different results show that our algorithm has excellent sensitivity.Figure 15
The recovering image under the worry secret key. (a)α=0.00001, other parameters are unchanged. (b) β=0.00001, other parameters are unchanged. (c) γ=0.00001, other parameters are unchanged. (d) δ=0.00001, other parameters are unchanged.
(a)
(b)
(c)
(d)Figure 16
The recovering image under the worry DNA encoding rule.


## 5.3. Differential Attack

Plaintext sensitive means that a tiny disturbance of the plain image will lead to dramatic changes in the cipher image. In order to measure the difference between the resulting cipher images, NPCR (number of pixel change rate) and UACI (unified average change intensity) are usually used to detect the ability of the image encryption scheme. NPCR and UACI are defined by(18)NPCR=∑i=1m∑j=1nCi,jm×n×100%,UACI=∑i=1m∑j=1nP1i,j−P2i,j255×m×n×100%,(19)ci,j=0,ifP1i,j=P2i,j,1,ifP1i,j≠P2i,j,where m and n represent the length and width of the image, respectively. P1(i, j) and P2(i, j) represent the corresponding ciphertext pixel values before and after the change of plaintext. For Lena images as shown in Table 3, the NPCR and UACI values of the algorithm are 99.62% and 33.38%, respectively, which verifies that the image encryption scheme has the ability to resist differential attack.Table 3
Comparison of NPCR and UACI in this paper and other references.
NPCR (%)
UACI (%)
Ours
99.62
33.38
Ref. [2]
99.02
32.83
Ref. [15]
99.59
33.12


## 5.4. Histogram Analysis

The image histogram is a method to evaluate the intensity distribution of pixels within one channel in color image, whose distribution shows a good permutation in an encryption algorithm and a high security against the potential statistical attack. The histograms of plain image and encrypted image are shown in Figure17. Figure 17(a) with a disorder distribution represents the histogram of plain image. The rests with a distribution shows the result after encryption in Figure 17(b). Obviously, no useful information can be extracted from the encrypted image, and high security can be guaranteed to resist statistical attacks.Figure 17
Histogram of (a) plain image and (b) encrypted image.
(a)
(b)


## 5.5. Correlation Coefficient Analysis

Correlation coefficient is an index to assess the image randomness. In this paper, it is employed to test the correlation between two adjacent pixels in plain image and ciphered image. In the original image, the correlation between two adjacent pixels is very high. In order to resist statistical attacks, the correlation of encrypted images must be reduced. 2500 pairs of two adjacent pixels are randomly selected from the plain and encryption image in horizontal, vertical, and diagonal directions to calculate the correlation between pixels. The function to calculate the correlation coefficient is shown as follows:(20)Ex=1N∑i=1Nxi,Dx=1N∑i=1Nxi−Ex2,COVx,y=1N∑i=1Nxi−Exyi−Ex,rxy=COVx,yDx×Dy,where x and y are values of two adjacent pixels selected in three directions randomly. N is the total number of duplets (x, y) obtained from the image. Ex and Dx are the expectation and the variance of x, respectively.The calculated correlation coefficients of plaintext images of Lena and its corresponding ciphered images in the proposed algorithm are listed in Table4. The correlation coefficient of adjacent pixels of the encrypted image is −0.000534. Therefore, the image encryption algorithm has a strong ability to resist statistical attacks. Table 4 and Figure 18 show the correlation comparison between the original image and the adjacent pixel of the encrypted image.Table 4
The correlation of the adjacent pixels for original and encrypted image.
Correlation coefficient
Horizontal
Vertical
Diagonal
Original image
0.9689
0.9486
0.9228
Encrypted image
−0.0029
−0.0031
−0.0037
Ref. [2]
0.0085
0.0021
0.0012
Ref. [15]
0.0265
0.0792
0.0625Figure 18
Correlation analysis of image: (a) plain image in horizontal; (b) cipher image in horizontal; (c) plain image in vertical; (d) cipher image in vertical; (e) plain image in diagonal; (f) cipher image in diagonal.
(a)
(b)
(c)
(d)
(e)
(f)


## 5.6. Information Entropy

The information entropy can test uncertainty. Entropy reflects whether grayscale values distribution is random or equality. The minimum and maximum values of entropy are zero and eight, respectively. The more uniform the distribution of the image gray value, the greater the entropy of the image. Therefore, the entropy value of the encrypted image should be as high as possible. Letm be the information source, and the equation for calculating information entropy is(21)Hm=−∑t=0tPmilog2Pmi,where P(mi) represents the probability that the information mi appears. Assume that there are 28 states of the information source and they appear with the same probability, according to formula (21), we can get the ideal H(m) = 8, which shows that the information is random. Hence, the information entropy of the encrypted image should be close to 8. The value of information entropy of the ciphered image in the proposed scheme is 7.98928, which indicate that the ciphered image obtained by the proposed algorithm could hardly divulge information.


## 6. Conclusion and Future Work

In this paper, a novel base on the memcapacitor chaotic system is proposed, and the waveform of the state variable is given. Moreover, the dynamic characteristics of the system are analyzed in detail, which indicate that the system has abundant dynamic behaviors. The corresponding implementation circuit is constructed by Spice. On this basis, a novel image encryption algorithm based on DNA sequence encoding operation and chaotic system is proposed. The Hill encryption matrix is constructed by the combination of the elliptic curve and hyperchaotic system, which avoids the complexity of elliptic curve encryption and reduces the correlation between encryption matrix elements. So, the encryption matrix has much more randomicity, and the encryption algorithm is more complex. Combined with the dynamic DNA encoding rules, the security of the encryption algorithm is increased, and the correlation between pixels is reduced, which makes the ciphertext difficult to crack. Experimental results show that this algorithm has better encryption effect, larger key space, and higher sensitivity to key.Compared with the previous encryption algorithm [2, 15], the computational time is reduced by 30%, and it has better rapidity and convenience. In addition, the algorithm can resist exhaustive attack, statistical attack, and differential attack. All these features show that our algorithm is very suitable for digital image encryption. In the future, we intend to carry encryption computation for images if necessary. The specific application and algorithm exploration are our future work.



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