# Secure Communication of Fractional Complex Chaotic Systems Based on Fractional Difference Function Synchronization

Authors: Jiaxun Liu; Zuoxun Wang; Minglei Shu; Fangfang Zhang; Sen Leng; Xiaohui Sun
Journal: Complexity
 (2019)

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## Abstract

Fractional complex chaotic systems have attracted great interest recently. However, most of scholars adopted integer real chaotic system and fractional real and integer complex chaotic systems to improve the security of communication. In this paper, the advantages of fractional complex chaotic synchronization (FCCS) in secure communication are firstly demonstrated. To begin with, we propose the definition of fractional difference function synchronization (FDFS) according to difference function synchronization (DFS) of integer complex chaotic systems.FDFS makes communication secure based onFCCS possible. Then we design corresponding controller and present a general communication scheme based onFDFS. Finally, we respectively accomplish simulations which transmit analog signal, digital signal, voice signal, and image signal. Especially for image signal, we give a novel image cryptosystem based onFDFS. The results demonstrate the superiority and good performances ofFDFS in secure communication.


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## Body

## 1. Introduction

With the booming of multimedia and wireless networks, secure communication of information is more and more significant in real applications. In particular, encryption of voice, images, and other real information has got a lot of attentions, such as recording of business meetings, military and architectural images, computer code, and so on. Many of the theoretical results make improvements in communication complexity as justification and a measure of success, and this seems logical as the communication complexity is the lower bound of the computation complexity [1].In the last three decades, chaotic synchronization of real variables has been a hotspot in nonlinear science since Caroll and Pecora [2] firstly achieved chaotic synchronization in electronic circuit. Chaos can exhibit a noise-like behavior such as randomicity, ergodicity, unpredictability, high sensitivity for the initial condition, and broadband nature. It is the great advantage in chaos communication and allows chaos to elegantly cover up the communication information. Moreover, chaos communication is usually easy to be realized by numerous simple circuits, which is more convenient than traditional cryptosystem.Since Fowler et al. [3] came up with the concept of complex chaos in 1982, many studies have been proposed in the field of new complex chaotic systems (CCS) and their applications [4–12]. Some actual physical models also were discovered asCCS, such as amplitudes of electromagnetic and detuned laser systems [13–17]. The number of the variables inCCS is twice as many as real chaotic systems (RCS), which is the major difference betweenRCS andCCS. In some sense, it means thatCCS have nature superiority in increasing transformation and channels. Moreover, complex variables are simpler to be accomplished by RLC circuit in actual applications than real variables. Therefore, the complicated dynamics and easy implementations ofCCS are born to secure communication. Some methods of integer complex chaotic synchronization (ICCS) used in secure communication were discussed in literatures. Reference [18] firstly put complex function projective synchronization (CFPS) into secure communication and got the superb result. However, when the signal of the master system is close to zero inCFPS, it affects the encryption since the denominator is close to zero. Therefore, [9] gave the definition of difference function synchronization (DFS) and solved this problem.DFS is proposed to the point of difference in two state variables and breaks the previous concept that scholars only study synchronization from the proportional relation between state variables. It is somewhat great innovation.DFS extends the difference between two state variables from zero to any desired functions. Therefore, it is the synchronization with much broader context. Complete synchronization and phase synchronization are its special cases. E.Mahmoud and Abo-Dahab studied another chaotic complex nonlinear framework and achieved communication of analog signal based on complex antisynchronization [19]. As far as we know, all above literatures of secure communication were just on the basis ofICCS.Fractional calculus has been recommended over three hundred years, but it had not received too much attention because of the unclear physical background of fractional calculus. Until the last decade, some scholars found that it is more accurate than integer order calculus in describing some actual physical models, for instance, viscoelastic system [20], viscoelastic material [21], finance system [22], nuclear spin generator system [23], industrial system [24], human immunodeficiency virus model [25], and other interdisciplinary fields. Therefore, fractional chaos also caused abundant interests. Besides, because of complicated geometric interpretation of nonlocal effects of fractional derivatives in time and space [26], fractional chaos exhibits more unpredictable and complex nonlinear dynamic behaviors than integer chaotic systems. These merits were caught hold of by a few researchers who focused on chaos communication. Reference [27] studied the modified generalized projective synchronization of fractional real hyperchaotic systems and applied it to secure communication. Sarah et al. [28] proposed a novel secure image transmission method based on fractional real discrete-time chaotic systems. Muthukumar et al. [29] presented a fractional sliding mode controller and studied its application for a cryptosystem. Li and Wu accomplished secure communication of fractional chaotic systems with teaching-learning-feedback based optimization [30]. Mohammadzadeh and Ghaemi researched a secure communication with uncertain fractional hyperchaotic synchronization [31]. These proposed literatures make significant contributions in secure communication of fractional chaotic systems. However, we find most papers were based on fractional real chaotic synchronization.Enlightened by the above discussions, we will put theFCCS withDFS into secure communication in this paper and gain higher security and better reliability than traditional methods and other chaos communication schemes. It will combine the advantages ofDFS in complex chaotic synchronization and fractional chaotic systems. Compared with secure communication based on integer real chaos synchronization,FCCS not only increase the number of potential channels, but also complicate the types of encryption keys. When it comes to secure communication based on fractional real chaos synchronization,FCCS have nature superiority in transmitting complex signal and choosing variable signal channels. As for secure communication based on integer complex chaos synchronization,FCCS could generate more unpredictable secret keys by means of different fractional orders. Due to the existence of the double variables and fractional derivative factors, it is difficult for the unauthorized third party to extract the useful information because of its complicated dynamic behavior.The main contributions of this paper are as follows: (1) Firstly, we extend theDFS toFDFS and investigate the general controller. (2) Four types of transmitted signals including analog signal, digital signal, voice signal, and image signal are accomplished to verify the high security and good performance of the communication scheme based onFDFS. (3) As for the image signal, we propose a novel cryptosystem withFDFS.The structure of the remaining paper is organized as follows: some basic mathematical theorems are given in Section2. Section 3 introduces theFDFS and general controllers. The application ofFCCS is finished by four types of signals and the method of image cryptosystem is proposed in Section 4. Finally, the conclusions are drawn in Section 5.


## 2. Mathematical Background

There are three main types of definition of fractional derivative, such as Caputo definition, Riemann Liouville definition, and Grunwald Letnikobv definition. In this paper, we use the Caputo definition as it includes the conventional initial conditions and Caputo derivative of the constant is zero.Definition 1 (see [32]).
The Caputo derivative definition is as follows:(1)Dmft=1Γn-a∫att-τ-a+n-1fnτdτ,where n=[m]+1,[m] is integer part of m and Γ(∗) is the gamma function. t and a are the upper and lower bounds, and Dm is called the m order Caputo differential operator. The gamma function is(2)Γw=∫0∞e-ttw-1dt,Γw+1=wΓw.Lemma 2 (see [33]).
There is an autonomous fractional system(3)Dqxt=Qxt,x0=x0,where x∈Rc,Q∈Rc×c and 0<q<1. The system is asymptotically stable if and only if(4)arg⁡eigQ>qπ2.And the component of the state decays toward 0 like t-q.


## 3. Fractional Difference Function Synchronization

### 3.1. The Definition of FDFS

Recently, a new type of chaotic synchronization was proposed, which is calledDFS in [9]. It is the expanding form of complete synchronization (CS) and phase synchronization (PHS). Particularly, for some signals near zero, it is effective forDFS in secure communication. In this part, we expand theDFS intoFDFS and give the general synchronization controller, aiming to lay the foundation for secure communication in the following part.Delighted by the concept ofDFS, we firstly present theFDFS as follows:Definition 3.
For twor-dimensional and q-order general form of fractional chaotic systems Dq1X(t) and Dq2Y(t), we call the difference Dq3G(t) between Dq1X(t) and Dq2Y(t) as the fractional difference function vector,(5)Dq3Gt=Dq1Yt-Dq2Xt,where Gt=g1(t),g2(t),⋯,gn(t)T, Xt=x1(t),x2(t),⋯,xn(t)T, Yt=y1(t),y2(t),⋯,yn(t)T, and 0<q1≤1,0<q2≤1,0<q3≤1.Definition 4.
For two arbitrary fractional chaotic state vectorxn(t),yn(t) and difference vector gn(t) in (5), they are said to beFDFS if there exists(6)limt→+∞⁡ent=ynt-xnt-gnt=0,where · is the matrix norm.Remark 5.
Whenq1=q2=q3=1, theFDFS would beDFS.Remark 6.
CS indicates the difference vectors with different or same initial value converge to zero when Dq3G(t)=0 and q1=q2=1. TheCS is a special case of theFDFS.Remark 7.
Whenq1=1,q2=1,q3=0, the difference function vector Dq3G(t) will be a constant value, soFDFS is also the extension ofPHS.


### 3.2. The Control Laws for FDFS

In order to increase generality, we consider a general form of coupled fractional chaotic system as follows:(7)Dwx=Ax+F1x,u=F2x,y,Dwy=By+F3y+u,where w is the fractional operator and 0<w≤1, F1,F3 are nonlinear continuous vector functions, and A,B are the Jacobian matrices of systems Dwx,Dwy. u is the controller part of the coupled system and F2 is the combination between nonlinear and linear function.As for the state vectorsx,y, one has the following.Case 1.
When they are real vectors, thenx=x1,x2,⋯,xnT, and y=y1,y2,⋯,ynT, u=u1,u2,⋯,unT, where n=1,2,3,….Case 2.
When they are complex vectors,x=xr+jxi,y=yr+jyi,u=ur+jui, where xr,yr,ur are the real parts and xi,yi,ui are the imaginary parts. j is the imaginary unit and j2=-1. xr=x1r,x2r,⋯,xnrT, xi=x1i,y1i,⋯,yniT, yr=y1r,y2r,⋯,ynrT, yi=y1i,y2i,⋯,yniT, ur=u1r,u2r,⋯,unrT, and ui=u1i,u2i,⋯,uniT.
We assume the difference function ispn(t), then P=Dwpn(t) where P=P1,P2,⋯,Pn. According to (7) and the definition ofFDFS, we have the error system as follows:(8)Dwe=Dwy-Dwx-P=By-Ax+F3y-F1x+F2x,y-P+u,where e=e1,e2,⋯,enT.According to the fractional stable theory and Lemma2, we will get the following theorem.Theorem 8.
For the coupled fractional chaotic system (7) with the difference function pn(t) and the initial value x(0),y(0), it could accomplishFDFS with the following controller:(9)u=Ax-By+F1x-F3y+P+ke,where parameter matrix k satisfies arg⁡(eig(k))>wπ/2.Proof.
In order to verify the effect of the control law, we put (9) into (8) and get (10)Dwe=By-Ax+F3y-F1x-P+Ax-By+F1x-F3y+P+ke=ke.According to Lemma 2, since arg⁡eigk>wπ/2 in the coupled fractional chaotic system, the system could arriveFDFS asymptotically with the controller (9) and the error function Dew could tend to zero.


## 3.1. The Definition of FDFS

Recently, a new type of chaotic synchronization was proposed, which is calledDFS in [9]. It is the expanding form of complete synchronization (CS) and phase synchronization (PHS). Particularly, for some signals near zero, it is effective forDFS in secure communication. In this part, we expand theDFS intoFDFS and give the general synchronization controller, aiming to lay the foundation for secure communication in the following part.Delighted by the concept ofDFS, we firstly present theFDFS as follows:Definition 3.
For twor-dimensional and q-order general form of fractional chaotic systems Dq1X(t) and Dq2Y(t), we call the difference Dq3G(t) between Dq1X(t) and Dq2Y(t) as the fractional difference function vector,(5)Dq3Gt=Dq1Yt-Dq2Xt,where Gt=g1(t),g2(t),⋯,gn(t)T, Xt=x1(t),x2(t),⋯,xn(t)T, Yt=y1(t),y2(t),⋯,yn(t)T, and 0<q1≤1,0<q2≤1,0<q3≤1.Definition 4.
For two arbitrary fractional chaotic state vectorxn(t),yn(t) and difference vector gn(t) in (5), they are said to beFDFS if there exists(6)limt→+∞⁡ent=ynt-xnt-gnt=0,where · is the matrix norm.Remark 5.
Whenq1=q2=q3=1, theFDFS would beDFS.Remark 6.
CS indicates the difference vectors with different or same initial value converge to zero when Dq3G(t)=0 and q1=q2=1. TheCS is a special case of theFDFS.Remark 7.
Whenq1=1,q2=1,q3=0, the difference function vector Dq3G(t) will be a constant value, soFDFS is also the extension ofPHS.


## 3.2. The Control Laws for FDFS

In order to increase generality, we consider a general form of coupled fractional chaotic system as follows:(7)Dwx=Ax+F1x,u=F2x,y,Dwy=By+F3y+u,where w is the fractional operator and 0<w≤1, F1,F3 are nonlinear continuous vector functions, and A,B are the Jacobian matrices of systems Dwx,Dwy. u is the controller part of the coupled system and F2 is the combination between nonlinear and linear function.As for the state vectorsx,y, one has the following.Case 1.
When they are real vectors, thenx=x1,x2,⋯,xnT, and y=y1,y2,⋯,ynT, u=u1,u2,⋯,unT, where n=1,2,3,….Case 2.
When they are complex vectors,x=xr+jxi,y=yr+jyi,u=ur+jui, where xr,yr,ur are the real parts and xi,yi,ui are the imaginary parts. j is the imaginary unit and j2=-1. xr=x1r,x2r,⋯,xnrT, xi=x1i,y1i,⋯,yniT, yr=y1r,y2r,⋯,ynrT, yi=y1i,y2i,⋯,yniT, ur=u1r,u2r,⋯,unrT, and ui=u1i,u2i,⋯,uniT.
We assume the difference function ispn(t), then P=Dwpn(t) where P=P1,P2,⋯,Pn. According to (7) and the definition ofFDFS, we have the error system as follows:(8)Dwe=Dwy-Dwx-P=By-Ax+F3y-F1x+F2x,y-P+u,where e=e1,e2,⋯,enT.According to the fractional stable theory and Lemma2, we will get the following theorem.Theorem 8.
For the coupled fractional chaotic system (7) with the difference function pn(t) and the initial value x(0),y(0), it could accomplishFDFS with the following controller:(9)u=Ax-By+F1x-F3y+P+ke,where parameter matrix k satisfies arg⁡(eig(k))>wπ/2.Proof.
In order to verify the effect of the control law, we put (9) into (8) and get (10)Dwe=By-Ax+F3y-F1x-P+Ax-By+F1x-F3y+P+ke=ke.According to Lemma 2, since arg⁡eigk>wπ/2 in the coupled fractional chaotic system, the system could arriveFDFS asymptotically with the controller (9) and the error function Dew could tend to zero.


## 4. Secure Communication of FCCS

In this section, we focus on secure communication ofFDFS with complex variables. Due to the broadband characters of chaotic systems, we could effectively cover up the signal by the chaos carrier. The block diagram of information transmission based onFDFS is illustrated in Figure 1. From Figure 1, we can know that the overall structure is composed of two parts approximately. One part S is the sending end, which provides the carrier of chaotic masking. Message modulation between chaotic driver system and information signal occurs here. The other main part R is receiving end, which aims at information demodulation. Response system and received signal have demodulated with the controller in part R. P(t) is the information signal and d(t) is the potential disturbance signal in process of transmitting. Ts(t) is the transmission signal, where Ts(t)=Dwy(t)+P(t)+d(t). Others(t) are the other alternative transmission channels. Fliter(t) is used for filter interference and we can get accurate recovered signals.Figure 1
The block diagram of secure communication based onFDFS.In order to verify the security of the proposed cryptosystem, we will simulate the transmission of four kinds of information signals in this part. In 2013, the coupled complex fractional Lorenz system was presented in [8], which is described as follows:(11)S:Dwy1=b1y2-y1+u1,Dwy2=b2y1-y2-y1y3+u2,Dwy3=12y¯1y2+y1y¯2-b3y3+u3,and(12)R:Dwx1=a1x2-x1,Dwx2=a2x1-x2-x1x3,Dwx3=12x¯1x2+x1x¯2-a3x3,where x1=x1r+jx1i,x2=x2r+jx2i,y1=y1r+jy1i,y2=y2r+jy2i and the overbar of x¯1,x¯2,y¯1,y¯2 means the complex conjugate of x1,x2,y1,y2,a1=b1=10,a2=b2=28,a3=b3=8/3. u1=u1r+ju1i,u2=u2r+ju2i,u3 are correlation controllers.We choose (9) as the controller in secure communication and the controller is(13)u1r=-a1e2r+k1e1r+p1r,u1i=-a1e2i+k2e1i+p1i,u2r=-a2e1+y1ry3-x1rx3+k3e2r+p2r,u2i=-a2e1i+y1iy3-x1ix3+k4e2i+p2i,u3=-y1ry2r+x1rx2r-y1iy2i+x1ix2i+k5e3+p3,where k1,k2,…,k5 are the scale parameters of controller and p1,p2,p3 are the difference factors ofFDFS which also represent information signal (p(t)={p1,p2,p3}T) in secure communication.For the disturbance signal, we choose the stochastic Gaussian noise as thed(t), which is described as follows:(14)nd=12πσexp⁡-d-d022σ2s,where the mean value d0=0 and variance σ=1 and s is the scale parameter of stochastic Gaussian noise.


### 4.1. Communication of Analog Signals

In this section, we put the analog signal into transmission system. In order to get the clear result of simulations, we firstly choose the information signalp(t)=10sin⁡0.1πt+j15cos⁡(0.1πt); then we have(15)Pt=Prt=10sin⁡0.1πt+0.5πw,Pit=15cos⁡0.1πt+0.5πw,where P(t) is the w-order derivative of p(t). Pr(t),Pi(t) are the real and imaginary parts of P(t), respectively.As for the choice of communication channels, we selecty1r,y1i as the sending end to transmit the real and imaginary part of analog signal and x1r,x1i as the receiving end to compute the real and imaginary part of encrypted analog signal. To further enhance the confidentiality of the transmission system and simulate the potential disturbance signal, we add stochastic Gaussian noise into transmitting process. Set s=10 in (14). In the receiving end, we could use some easy filters in real engineering and it allows assuring the accuracy of recovered information.Figure2 shows the encrypted transmission signal, which is complicated and irregular. It is hard for unauthorized third party to find the information signal. The recovered signal, original signal, and their error are shown in Figure 3. With the controller (13) in the receiving end, information signal has been recovered correctly in Figure 3. Figure 4 shows the error of receiving end and sending end. According to the definition ofFDFS, as the choice of communication channel is complex variable y1 and x1, their error is the information signal. Other errors of complex variables y2 and x2 and real variables y3 and x3 are zero asymptotically.Figure 2
Picture of transmitted signalTs(t), where s=10,k=-1,w=0.995, the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8), and all iterations are 500.Figure 3
Picture of original signal and recover signal, wheres=10,k=-1,w=0.995, the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8), and all iterations are 300.Figure 4
Diagram of error of receiving end and sending end, wheres=10,k=-1,w=0.995, the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8), and all iterations are 300.


### 4.2. Communication of Digital Signal

Firstly, the original digital signal produced by the random function is shown in Figure5. In order to achieve a fast transmission without increasing the system complexity, the O binary bits are transformed into one corresponding fractional difference functions by 2O-ary. In this part, we set O=4 and transform digital signal to the fractional difference function by 2O=16-ary. The signal duration is 200 iterations. We choose the y1i of the drive system as the sending terminal and x1i as the receiving terminal. Figure 5 shows the transmitted process of digital signals without noise, where the error between original digital signal and the recovered signal is zero and the transmitted signal Ts(t) is an analog signal which completely covers up the binary digital sequence.Figure 5
Diagram of digital signals transmission without noise, wheres=0,k=-1,w=0.995, the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8), and all iterations are 3000.Considering that there is potential disturbance in transmitting process of digital signals, we get the transmitting process shown in Figure6. The transmitted signal Ts(t) is almost noise-like and someone like espionage hardly extracts the information signal without authorization. In order to reduce the effect of noise, we firstly compute the average value of the fractional difference function during each signal duration and then round off to the nearest integer that is bounded between 0 and 2O-1. Therefore, this scheme guarantees the accuracy and low bit error rate in process of recovery.Figure 6
Diagram of digital signals transmission with noise, wheres=10,k=-1,w=0.995, the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8), and all iterations are 3000.


### 4.3. Communication of Voice Signal

In this section, a novel audio cryptosystem is presented for transmitting voice signal. In the sending end, we choose the mellifluous song “traveling light” as the information signal, which is shown in Figure7.Figure 7
Diagram of information signal and recovered voice signal, wheres=0,k=-1,w=0.995, the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8,w=0.995), and all iterations are 3000.There are five alternative communication channels to transmit the voice signal. We choosey1r,x1r to encrypt and decrypt the information signal.Because the voice signal has a great number of samples, we extract the sample from30000th to 33000th, which is enough to get an excellent simulation. From Figure 8, the error between the original voice and recovered voice approaches to zero quickly and the transmitted signal of encryption voice completely covers the information signal. Moreover, due to the existence of five alternative communication channels, there is less possibility for the eavesdropper to extract the original voice.Figure 8
Diagram of voice encryption transmitted signal and the error between original and recovered signal, wheres=10,k=-1,w=0.995, the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8), and all iterations are 3000.


### 4.4. Communication Image Signal

In this section, a new key cryptosystem is presented for sharing image messages to increase the security and anticrack ability of secure communication. Generally speaking, the purpose of key cryptography is to allow two different organizations to communicate the confidential message, even though they have never met and communicated with each other or they are supervised by an adversary. The proposed cryptosystem consists of three parts: key generation, encryption, and decryption.Key generationSending end and receiving end both agree on this:(1) The fractional orderw.(2) Complex variablex1,x2,y1,y2 and the real variable x3,y3 of couple system.(3) Initial value of fractional order complex chaotic system.(4) The parameters of couple system.(5) Scaling parametersb1,b2,b3,b4,b5.Encryption(1) The pixel matrix of original picture isMM(c×d).(2) There are also five zeros matrices,M1(c×d),M2(c×d),M3(c×d),M4(c×d),M5(c×d). As the coupled chaotic system has five alternative encryption sending terminals, y1r,y1i,y2r,y2i,y3, we could generate five arrays s1,s2,s3,s4,s5 with the initial value of coupled system, where the number of every array is more than 1/5c×d and c,d are the matrix size parameters.(3) LetM1(1:c×d/5)=s1(301:c×d/5+300) and M2(c×d/5+1:2c×d/5)=s2(301:c×d/5+300),M 3 ( 2 c × d / 5 + 1 : 3 c × d / 5 ) = s 3 ( 301 : c × d / 5 + 300 ) and M4(3c×d/5+1:4c×d/5)=s4(301:c×d/5+300),M 5 ( 4 c × d / 5 + 1 : 5 c × d / 5 ) = s 5 ( 301 : c × d / 5 + 300 ); if c×d/5 has remainder, we could adjust the number of M5,s5 correspondingly.(4)M(c×d)=b1M1+b2M2+b3M3+b4M4+b5M5+MM.(5) The sending end sentM to the receiving end.Decryption(1) On condition that we received theM and agree on the keys, we must form corresponding five decryption matrices N1(c×d), N2(c×d), N3(c×d), N4(c×d), and N5(c×d) in the decryption receiving terminal, x1r,x1i,x2r,x2i,x3 with the controller (9). The method of generation of these five decryption matrices is similar to encryption.(2)N(c×d)=b1N1+b2N2+b3N3+b4N4+b5N5.The recovered image pixel matrixRI(c×d)=M-N.(3) We can get the recovered image by the pixel matrixRI.Figures9 and 10 are the diagrams of original information image and recovered image, respectively. The picture of encrypted matrix is shown in Figure 11, where the encryption image completely covers the information picture. Due to the complexity of the secret key and the process of encryption, the unauthorized organization cannot recover the original image absolutely without the whole information of the secret key.Figure 9
Diagram of original picture, wheres=0,k=-1,w=0.995,b1=2000,b2=20000,b3=2000,b4=20000,b5=35, and the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8).Figure 10
Diagram of recovered picture, wheres=0,k=-1,w=0.995,b1=2000,b2=20000,b3=2000,b4=20000,b5=35, and the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8).Figure 11
Diagram of encrypted matrix, wheres=0,k=-1,w=0.995,b1=2000,b2=20000,b3=2000,b4=20000,b5=35, and the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8).


## 4.1. Communication of Analog Signals

In this section, we put the analog signal into transmission system. In order to get the clear result of simulations, we firstly choose the information signalp(t)=10sin⁡0.1πt+j15cos⁡(0.1πt); then we have(15)Pt=Prt=10sin⁡0.1πt+0.5πw,Pit=15cos⁡0.1πt+0.5πw,where P(t) is the w-order derivative of p(t). Pr(t),Pi(t) are the real and imaginary parts of P(t), respectively.As for the choice of communication channels, we selecty1r,y1i as the sending end to transmit the real and imaginary part of analog signal and x1r,x1i as the receiving end to compute the real and imaginary part of encrypted analog signal. To further enhance the confidentiality of the transmission system and simulate the potential disturbance signal, we add stochastic Gaussian noise into transmitting process. Set s=10 in (14). In the receiving end, we could use some easy filters in real engineering and it allows assuring the accuracy of recovered information.Figure2 shows the encrypted transmission signal, which is complicated and irregular. It is hard for unauthorized third party to find the information signal. The recovered signal, original signal, and their error are shown in Figure 3. With the controller (13) in the receiving end, information signal has been recovered correctly in Figure 3. Figure 4 shows the error of receiving end and sending end. According to the definition ofFDFS, as the choice of communication channel is complex variable y1 and x1, their error is the information signal. Other errors of complex variables y2 and x2 and real variables y3 and x3 are zero asymptotically.Figure 2
Picture of transmitted signalTs(t), where s=10,k=-1,w=0.995, the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8), and all iterations are 500.Figure 3
Picture of original signal and recover signal, wheres=10,k=-1,w=0.995, the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8), and all iterations are 300.Figure 4
Diagram of error of receiving end and sending end, wheres=10,k=-1,w=0.995, the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8), and all iterations are 300.


## 4.2. Communication of Digital Signal

Firstly, the original digital signal produced by the random function is shown in Figure5. In order to achieve a fast transmission without increasing the system complexity, the O binary bits are transformed into one corresponding fractional difference functions by 2O-ary. In this part, we set O=4 and transform digital signal to the fractional difference function by 2O=16-ary. The signal duration is 200 iterations. We choose the y1i of the drive system as the sending terminal and x1i as the receiving terminal. Figure 5 shows the transmitted process of digital signals without noise, where the error between original digital signal and the recovered signal is zero and the transmitted signal Ts(t) is an analog signal which completely covers up the binary digital sequence.Figure 5
Diagram of digital signals transmission without noise, wheres=0,k=-1,w=0.995, the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8), and all iterations are 3000.Considering that there is potential disturbance in transmitting process of digital signals, we get the transmitting process shown in Figure6. The transmitted signal Ts(t) is almost noise-like and someone like espionage hardly extracts the information signal without authorization. In order to reduce the effect of noise, we firstly compute the average value of the fractional difference function during each signal duration and then round off to the nearest integer that is bounded between 0 and 2O-1. Therefore, this scheme guarantees the accuracy and low bit error rate in process of recovery.Figure 6
Diagram of digital signals transmission with noise, wheres=10,k=-1,w=0.995, the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8), and all iterations are 3000.


## 4.3. Communication of Voice Signal

In this section, a novel audio cryptosystem is presented for transmitting voice signal. In the sending end, we choose the mellifluous song “traveling light” as the information signal, which is shown in Figure7.Figure 7
Diagram of information signal and recovered voice signal, wheres=0,k=-1,w=0.995, the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8,w=0.995), and all iterations are 3000.There are five alternative communication channels to transmit the voice signal. We choosey1r,x1r to encrypt and decrypt the information signal.Because the voice signal has a great number of samples, we extract the sample from30000th to 33000th, which is enough to get an excellent simulation. From Figure 8, the error between the original voice and recovered voice approaches to zero quickly and the transmitted signal of encryption voice completely covers the information signal. Moreover, due to the existence of five alternative communication channels, there is less possibility for the eavesdropper to extract the original voice.Figure 8
Diagram of voice encryption transmitted signal and the error between original and recovered signal, wheres=10,k=-1,w=0.995, the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8), and all iterations are 3000.


## 4.4. Communication Image Signal

In this section, a new key cryptosystem is presented for sharing image messages to increase the security and anticrack ability of secure communication. Generally speaking, the purpose of key cryptography is to allow two different organizations to communicate the confidential message, even though they have never met and communicated with each other or they are supervised by an adversary. The proposed cryptosystem consists of three parts: key generation, encryption, and decryption.Key generationSending end and receiving end both agree on this:(1) The fractional orderw.(2) Complex variablex1,x2,y1,y2 and the real variable x3,y3 of couple system.(3) Initial value of fractional order complex chaotic system.(4) The parameters of couple system.(5) Scaling parametersb1,b2,b3,b4,b5.Encryption(1) The pixel matrix of original picture isMM(c×d).(2) There are also five zeros matrices,M1(c×d),M2(c×d),M3(c×d),M4(c×d),M5(c×d). As the coupled chaotic system has five alternative encryption sending terminals, y1r,y1i,y2r,y2i,y3, we could generate five arrays s1,s2,s3,s4,s5 with the initial value of coupled system, where the number of every array is more than 1/5c×d and c,d are the matrix size parameters.(3) LetM1(1:c×d/5)=s1(301:c×d/5+300) and M2(c×d/5+1:2c×d/5)=s2(301:c×d/5+300),M 3 ( 2 c × d / 5 + 1 : 3 c × d / 5 ) = s 3 ( 301 : c × d / 5 + 300 ) and M4(3c×d/5+1:4c×d/5)=s4(301:c×d/5+300),M 5 ( 4 c × d / 5 + 1 : 5 c × d / 5 ) = s 5 ( 301 : c × d / 5 + 300 ); if c×d/5 has remainder, we could adjust the number of M5,s5 correspondingly.(4)M(c×d)=b1M1+b2M2+b3M3+b4M4+b5M5+MM.(5) The sending end sentM to the receiving end.Decryption(1) On condition that we received theM and agree on the keys, we must form corresponding five decryption matrices N1(c×d), N2(c×d), N3(c×d), N4(c×d), and N5(c×d) in the decryption receiving terminal, x1r,x1i,x2r,x2i,x3 with the controller (9). The method of generation of these five decryption matrices is similar to encryption.(2)N(c×d)=b1N1+b2N2+b3N3+b4N4+b5N5.The recovered image pixel matrixRI(c×d)=M-N.(3) We can get the recovered image by the pixel matrixRI.Figures9 and 10 are the diagrams of original information image and recovered image, respectively. The picture of encrypted matrix is shown in Figure 11, where the encryption image completely covers the information picture. Due to the complexity of the secret key and the process of encryption, the unauthorized organization cannot recover the original image absolutely without the whole information of the secret key.Figure 9
Diagram of original picture, wheres=0,k=-1,w=0.995,b1=2000,b2=20000,b3=2000,b4=20000,b5=35, and the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8).Figure 10
Diagram of recovered picture, wheres=0,k=-1,w=0.995,b1=2000,b2=20000,b3=2000,b4=20000,b5=35, and the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8).Figure 11
Diagram of encrypted matrix, wheres=0,k=-1,w=0.995,b1=2000,b2=20000,b3=2000,b4=20000,b5=35, and the initial value is (y1r(0)=7,y1i(0)=8,y2r(0)=5,y2i(0)=6,y3(0)=12,x1r(0)=6.5,x1i(0)=8.3,x2r(0)=5.1,x2i(0)=6.6,x3(0)=12.8).


## 5. Conclusions

In the last two decades, chaos communication has been a hotspot and got astonishing progress. In this paper, we propose a novel secure communication scheme of fractional complex chaotic systems based onFDFS. We firstly extend theDFS from integer complex chaotic systems to fractional complex chaotic systems and design corresponding controller.FDFS is one ofFCCS in essence. In order to verify the effectiveness and advantages ofFCCS, we present novel secure communication schemes based onFDFS and transmit analog signal, digital signal, voice signal, and image. Moreover, we design an image cryptosystem with high security. The numerical simulations demonstrate the great effect of encryption, transmission, and decryption. Particularly, the results exhibit the advantages ofFDFS integrating with fractional complex chaotic system.Secure communication ofFCCS is a completely new field. We hope that more and more researchers will extend some traditional synchronization toFCCS and increase the diversity ofFCCS in secure communication, which will deeply develop chaos communication.



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*Source: 7242791-2019-08-18.xml*