A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter

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Communications in Nonlinear Science and Numerical Simulation 14 (2009) 863–879 www.elsevier.com/locate/cnsns

A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter Arman Kiani-B a,*, Kia Fallahi b, Naser Pariz a, Henry Leung b a

Advanced Control and Nonlinear Laboratory, Electrical Engineering Department, Ferdowsi University of Mashhad, Mashhad 91775-1111, Iran b Electrical and Computer Engineering Department, University of Calgary, Alberta, Canada T2N 1N4 Received 23 July 2007; received in revised form 16 November 2007; accepted 16 November 2007 Available online 28 November 2007

Abstract In recent years chaotic secure communication and chaos synchronization have received ever increasing attention. In this paper, for the first time, a fractional chaotic communication method using an extended fractional Kalman filter is presented. The chaotic synchronization is implemented by the EFKF design in the presence of channel additive noise and processing noise. Encoding chaotic communication achieves a satisfactory, typical secure communication scheme. In the proposed system, security is enhanced based on spreading the signal in frequency and encrypting it in time domain. In this paper, the main advantages of using fractional order systems, increasing nonlinearity and spreading the power spectrum are highlighted. To illustrate the effectiveness of the proposed scheme, a numerical example based on the fractional Lorenz dynamical system is presented and the results are compared to the integer Lorenz system. Ó 2007 Elsevier B.V. All rights reserved. PACS: 05.30.Pr; 05.45.a; 05.45.Gg; 05.45.Jn; 05.45.Pq; 05.45.Tp; 05.45.Vx; 05.45.Xt; 87.53.Vb; 87.64.Aa; 89.70.+c; 89.75.k Keywords: Fractional chaotic systems; Chaotic cryptosystems; Security; Chaotic masking; Encryption; Fractional extended Kalman filter; Chaos states; Synchronization

1. Introduction Although fractional derivatives have a long mathematical history, for many years they were not used in physics and engineering. One possible explanation of such unpopularity could be the multiple on equivalent definitions of fractional derivatives [1]. Another difficulty is that fractional derivatives have complex geometrical interpretation because of their non-local character [2]. Different approaches to geometric interpretation of fractional integration and fractional differentiation has been suggested [3–5], for example Podlubny proposed a simple geometric interpretation of fractional integrals as ‘‘changing shadows on the wall” and some pictures describing this changing process were given [3]. *

Corresponding author. Tel.: +98 9153255583. E-mail address: [email protected] (A. Kiani-B).

1007-5704/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2007.11.011

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However, during the last 10 years, fractional calculus has started to attract the attention of physicists and mathematicians much more. It was found that various, especially interdisciplinary applications, can be elegantly modeled with the help of the fractional derivatives. For example, the nonlinear oscillation of earthquakes can be modeled with fractional derivatives [6] while a fluid-dynamic traffic model with fractional derivatives [7] can eliminate the deficiency arising from the assumption of continuum traffic flow. Based on experimental data fractional partial differential equations for seepage flow in porous media are suggested in [8]. As the relation between memory and fractional mathematics have shown in [1,9,10], differential equations with fractional order have recently proved to be valuable tools to the modeling of many physical phenomena. In the past 20 years, on the other hand there has been a great deal of interest in the study of nonlinear dynamical systems. Deterministic dynamical systems are those whose states change with time in a deterministic way. The introduction of chaos into communication systems offers several opportunities for improvement. This is because of the random nature of chaotic systems. Since a chaotic dynamical system is a deterministic system, its random-like behavior can be very helpful in disguising modulation as noise [11]. A small perturbation eventually causes a large change in the state of the system. In the digital world nowadays, the security of the digital signal has become very important since the proliferation of wireless products [12]. Compared to conventional communication systems, there are several unique features of chaotic communication systems [13–15]. Potential benefits of chaotic communications include the efficient use of the bandwidth of a communication channel, utilization of the intrinsic nonlinearities in communication devices, large-signal modulation for efficient use of carrier power, reduced number of components in a system, and security of communication by chaotic encryption [16]. Chaotic dynamics, with its noise-like broadband power spectra, is a good candidate to fight narrow-band effects, such as frequency-selective fading or narrow-band disturbances in communication systems. Another attractive feature of the chaotic signal is its dependence on the initial condition, which makes it difficult to guess the structure of the generator and to predict the signal over a longer time interval. This feature is of interest in cryptography, where highly complex and hard-to-predict signals are employed. Chaotic signals are deterministic so there is no random component in differential equations. However, trajectories are noise-like. Also, since they are bounded and on successive generations of chaos, the states stay in a finite range. Moreover, they are aperiodic, in that the same state is never repeated twice. Chaotic output streams are completely uncorrelated, and the auto-correlation of a chaotic signal has a large peak at zero and decays rapidly. Thus a chaotic system shares many properties of a stochastic process, which is a basic requirement for spread spectrum communications. In a typical chaotic synchronization communication scheme the information to be transmitted is carried from the transmitter to the receiver by a chaotic signal through an analog channel. It is possible to implement a chaotic communication system either with chaos synchronization (coherent) or without chaos synchronization (noncoherent) [17–19]. Most of the research activities in chaotic communication have so far addressed systems based on synchronization of chaos between a transmitter and a receiver linked by a transmission channel. For such systems, chaos synchronization is mandatory, while the quality of communication, measured by the bit-error rate (BER) of a decoded message at the receiver, depends crucially on the accuracy and robustness of synchronization. Following these approaches, different methods have been developed in order to mask the contents of a message using chaotic signals [20,21]. However, it has been shown that most of these methods are not secure or have a low level of security because one can extract the encoded message signal from the transmitted chaotic signal by using different unmasking techniques [22,23]. Therefore, to overcome the problem of unmasking the information message from the chaotic carrier, different approaches for designing cryptosystems based on chaos have been recently introduced [19,24]. In these schemes, both conventional cryptographic methods and synchronization of chaotic systems are combined so that the level of security of transmitted chaotic signals is enhanced. Typically these approaches are based on the synchronization properties of simple chaotic systems. Thus, it is of considerable interest to achieve secure encoding of the digital information signal by considering the masking of chaotic encoding signals to be just as important as the masking of information signals. For this purpose, combining of the advantages of digital encryption techniques and chaos synchronization methods, the level of security of the transmitted signal can be potentially enhanced. In this paper, we proposed new scheme to improve security based on fractional chaotic systems. The main advantages of using these kinds of systems, in addition to those mentioned, are adding more complexity because fractional derivatives have complex geometrical interpretation because of their non-local character [3] and high nonlinearity.

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The extended Kalman filter has been widely used in the state estimation of nonlinear dynamic systems and is also as an important algorithm for the implementation of chaotic synchronization [19]. Much research work demonstrates that EKF can synchronize different chaotic maps for applications in secure communications. In order to use fractional chaotic systems, the fractional Kalman filter is needed to estimate the states. The identification of parameters or states in fractional order systems and especially the fractional orders of these systems are not as easy as in the case of integer order systems because of their high nonlinearity. In this work, the fractional Lorenz dynamic system is used to modulate sinusoid data via masking modulation. The modulated signal is transmitted through the additive white Gaussian noise (AWGN) channel. At the receiver, EFKF is employed to estimate states of the fractional chaotic systems. According to our previous work [25], this proposed approach is based on a simple masking scheme to illustrate the advantages of using fractional systems. In this article, the application of the extended fractional Kalman filter for state reconstruction of fractional nonlinear systems in a chaotic communication scheme is presented. The proposed scheme uses a fractional Lorenz system as chaos generators to encrypt data using masking modulation. The results are compared with integer Lorenz system. The proposed scheme is not restricted to the Lorenz system and, in fact, other kinds of chaotic systems can also be used. The receiver consists of an extended fractional Kalman filter for state reconstruction, and a chaos masking demodulator. Therefore, for enhancing the security of chaotic cryptosystem in this paper for the first time, we introduce the concept of using fractional order chaotic systems in cryptography. Following points make this new cryptosystem distinctive and advantageous compared to the integer order nonlinear chaotic schemes:  The power spectrum of fractional order chaotic systems fluctuates complexly and this cryptosystem enhance the security both in frequency and time domain.  The existence of the derivative order. One advantage of using fractional order chaotic systems is for computational complexity goal. Although the computational complexity is enhanced, the derivative orders can be used as secret keys as well.  High complexity due to high dimensionality and chaoticity. It is obvious that the attack complexity is determined by the size of the key space and the complexity of the verification of each key.  Using extended fractional Kalman filter in synchronization which is the extension of well-known EKF to estimate in fractional nonlinear dynamics. This paper is organized as follow: Section 2 illustrates chaotic cryptosystems and chaotic masking modulation. In Section 3, nonlinear fractional systems, numerical solution of fractional differential equations, and extended fractional Kalman filter are described. In Section 4, the proposed chaotic encryption scheme and numerical examples are presented. Section 5 describes simulation experiments for the proposed system. 2. Chaotic cryptosystems The basic idea of these cryptosystems is based on using a chaotic nonlinear oscillator as a broadband pseudo-random signal generator. This signal is combined with the message to produce an unintelligible signal, which is transmitted through the insecure communication channel. At the reception, the pseudo-random signal is regenerated, so that by combining it with the received signal through the inverse operation, the original message is recovered [21]. To fight against vulnerable points of simple masking, we have implemented fractional chaotic systems which will be described in Section 3. 2.1. Chaotic masking The chaotic signal is added to the information signal and, at the receiver, the masking is removed. In order for this scheme to properly work, the receiver must synchronize robustly enough so as to admit the small perturbation in the driving signal due to the addition of the message. The power level of the information signal must be much lower than that of the chaotic signal to effectively bury it [21]. One of the chaotic system states is masked with the message signal sðtÞ to encrypt and modulate the plaintext signal. The resultant modulated signal cðtÞ can be obtained as follows:

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Fig. 1. Block diagram of chaotic masking modulation.

cðtÞ ¼ yðtÞ þ sðtÞ

ð1Þ

in which, yðtÞ is the chaotic output signal and sðtÞ is the message signal. The block diagram of the proposed chaotic communication scheme is shown in Fig. 1. 3. Fractional systems and extended fractional Kalman filter Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary (non-integer) order. This subject, as old as the ordinary differential calculus, goes back to times when Leibniz and Newton invented differential calculus. The problem raised by Leibniz in a letter dated September 30, 1695 on a fractional derivative has been an ongoing topic for more than hundreds of years. Many famous mathematicians contributed to this theory over the years, including Liouville, Riemann, Weyl, Fourier, Abel, Lacroix, Leibniz, Grunwald and Letnikov. For more details, refer to the books by Oldham and Spanier [26]. Fractional derivatives have been extensively applied in many fields which have experienced an overwhelming growth in the last three decades. Examples abound which involve fractional derivatives: models admitting backgrounds of heat transfer, viscoelasticity, electrical circuits, electro-chemistry, economics, polymer physics, and even biology [7,8,10,27]. 3.1. Nonlinear fractional systems Actually, fractional derivative based approaches establish far superior models of engineering systems than ordinary derivative based approaches do in many applications. The concepts of fractional derivatives generalize the concepts of ordinary derivatives to some extent. Thus, as mentioned in [2], there is no field that has remained untouched by fractional derivatives. However, progress still needs to be made before ordinary derivatives can be truly interpreted as a subset of fractional derivatives. In particular, fractional differential equations, as an important research branch of fractional derivatives attract much attention. Theories on the local existence, uniqueness and structural stability of the solutions of specific fractional differential equations have been fully successfully established [9]. Also, varieties of schemes for numerical solutions of fractional differential equations are being proposed. It is even reported that specific fractional differential equations may exhibit complex dynamical evolution, even chaotic dynamics. However, that is not the case for ordinary differential equations due to the famous Poincare´–Bendixson theorem [29]. In addition, some topics of great importance and potential application, involving chaos control and synchronization for fractional differential equations, have recently been numerically investigated [28]. Fractional calculus is a generalization of integration and differentiation of the non-integer order fundamental operator a Dat , where a and t are the limits of the operation. The two definitions generally used for the fractional integral are the Grunwald–Letnikov (GL) definition for discrete systems and the Riemann–Liouville (RL) definition for continuous systems [8]. The RL integral definition is Z t 1 1a I ðaÞ f ðtÞ ¼ ðt  sÞ f ðsÞds; a; t  0: ð2Þ CðaÞ 0 With this definition of integral, the next two equations, the Riemann–Liouville and Caputo fractional derivatives, can be defined as

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 dm  ðmaÞ I f ðtÞ ; m dt   m ðaÞ ðmaÞ d f ðtÞ ; D f ðtÞ ¼ I dtm

DðaÞ f ðtÞ ¼

867

ð3aÞ ð3bÞ

where m  1 < a 6 m; m 2 N. In this paper, we enhanced security by using the fractional order model of chaotic systems. The premise of the proposed model is the fact that fractional order models possess memory and exhibit more complex dynamic evolution than integer chaotic dynamics in the chaotic masking which is described in Section 2.1. 3.2. Numerical solution of fractional differential equations Unlike the numerical solving of an ordinary differential equation, the numerical simulation of a fractional differential equation is not unproblematic. In the literature of fractional chaos field, two approximation methods have been proposed to solve a fractional differential equation numerically. The first method, known as frequency domain approximation, is based on the approximation of the fractional order system behavior in the frequency domain. In [33], an algorithm has been proposed to calculate linear transfer function approximations of 1=sq where, the Laplace transform of the fractional derivative is Lf0 Dat f ðtÞg ¼ sa F ðsÞ 

m1 X

sk ½0 Dtak1 f ðtÞt¼0 ;

m  1 6 a < m; m 2 N

ð4Þ

k¼0

and in zero initial condition, the Laplace transform of fractional derivative is  a  d f ðtÞ L ¼ sa Lff ðtÞg: dta

ð5Þ

Also according to the Riemann–Liouville fractional integral definition, we can define this operator as follows: Z t 1 1 a1 t  f ðtÞ; I ðaÞ f ðtÞ ¼ ðt  sÞa1 f ðsÞds ¼ ð6Þ CðaÞ 0 CðaÞ where Lfta1 g ¼ CðaÞsa ;

ð7Þ

so, the Laplace transform of the fractional integral is LfI ðaÞ f ðtÞg ¼ sa F ðsÞ:

ð8Þ

In this method, the aim is to find zeros and poles of a transfer function that has a similar amplitude diagram as 1=sa in a given frequency range. 1=sa has a Bode diagram characterized by a slope of 20a dB=decade. Therefore, in this method the 20a dB=decade line is approximated by a number of zigzag lines connected together with alternate slops of 0 dB=decade and 20a dB=decade. According to this method, we can obtain a linear approximation of the fractional order integrator with any desired accuracy over any frequency band. The order of this linear approximation system depends on the desired bandwidth and accuracy. The second method is an improved version of the Adams–Bashforth–Moulton algorithm [29,30] and is proposed based on the predictor-correctors scheme for this system [31,32]. To explain the method, we consider the following differential equations: Dqt yðtÞ ¼ rðt; yðtÞÞ; ðkÞ

y ð0Þ ¼

ð0Þ y0 ;

0 6 t 6 T;

k ¼ 0; 1; . . . ; m  1:

This differential equation is equivalent to the Volterra integral equation [33]: Z t ½q1 X ðkÞ tk 1 q1 þ yðtÞ ¼ y0 ðt  sÞ rðs; yðsÞÞds: CðqÞ k! 0 k¼0

ð9Þ

ð10Þ

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Now, set h ¼ NT ; tn ¼ nh (n = 0, 1, . . . , N). Then (10) can be discredited as follows [35]: y h ðtnþ1 Þ ¼

½q1 X

ðkÞ

y0

k¼0

where aj;nþ1

n X tk hq hq þ rðtnþ1 ; y ph ðtnþ1 ÞÞ þ aj;nþ1 rðtj ; y h ðtj ÞÞ; k! Cðq þ 2Þ Cðq þ 2Þ j¼0

8 qþ1 q > < n  ðn  qÞðn þ 1Þ ; j ¼ 0; ¼ ðn  j þ 2Þqþ1 þ ðn  jÞqþ1  2ðn  j þ 1Þqþ1 ; > : 1; j ¼ n þ 1

ð11Þ

1 6 j 6 n;

and bj;nþ1 ¼

hq q q ððn þ 1  jÞ  ðn  jÞ Þ q

and y ph ðtnþ1Þ ¼

½q1 X

ðkÞ

y0

k¼0

n tk 1 X þ bj;nþ1 rðtj ; y h ðtj ÞÞ: k! CðqÞ j¼0

The error of this approximation is described as follows: maxj¼0;1;...;N jyðtj Þ  y h ðtj Þj ¼ Oðhp Þ, where p ¼ minð2; q þ 1Þ. The numerical solution of a fractional order system can be determined by applying the mentioned method. Consider the following fractional order system: 8 q1 d x > > > > dtq1 ¼ f1 ðx; y; zÞ; > > < q2 d x ð12Þ ¼ f2 ðx; y; zÞ; q2 > > > dtq3 > > > : d x ¼ f3 ðx; y; zÞ dtq3 with initial condition ðx0 ; y 0 ; z0 Þ. The above system can be discretized as follows: 8 n P q1 > xnþ1 ¼ x0 þ Cðqh þ2Þ ½f1 ðxpnþ1 ; y pnþ1 ; zpnþ1 Þ þ a1;j;nþ1 f1 ðxj ; y j ; zj Þ; > > 1 > j¼0 > > > < n P q2 p p p y nþ1 ¼ y 0 þ Cðqh þ2Þ ½f2 ðxnþ1 ; y nþ1 ; znþ1 Þ þ a2;j;nþ1 f2 ðxj ; y j ; zj Þ; 2 > j¼0 > > > n > P q3 > p p p > : znþ1 ¼ z0 þ Cðqh þ2Þ ½f3 ðxnþ1 ; y nþ1 ; znþ1 Þ þ a3;j;nþ1 f3 ðxj ; y j ; zj Þ 3

j¼0

and 8 n P > xpnþ1 ¼ x0 þ Cðq1 Þ b1;j;nþ1 f1 ðxj ; y j ; zj Þ; > > 1 > j¼0 > > > < n P y pnþ1 ¼ y 0 þ Cðq1 Þ b2;j;nþ1 f2 ðxj ; y j ; zj Þ; 2 > j¼0 > > > n > P > > : zpnþ1 ¼ z0 þ Cðq1 Þ b3;j;nþ1 f3 ðxj ; y j ; zj Þ; 3

where ai;j;nþ1

j¼0

8 q þ1 i  ðn  qi Þðn þ 1Þqi ; j ¼ 0; > : 1; j ¼ n þ 1

1 6 j 6 n;

ð13Þ

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and hqi ððn þ 1  jÞqi  ðn  jÞqi Þ: qi Due to specificity of the error estimation bound in the proposed Adams–Bashforth–Moulton algorithm, simulation results by this method are more reliable than simulation results of the first method. Therefore, this method was selected for our simulations. The next section describes the extended fractional Kalman filter, which is used in the estimation of fractional system states. bi;j;nþ1 ¼

3.3. Extended fractional Kalman filter The extended fractional Kalman filter is an important and fascinating algorithm in nonlinear theory which is extended to fractional systems. In order to present a picture of the designing process, a brief discussion of stochastic state estimation design will follow [19]. The EKF based synchronization approach employs extended Kalman filter at the receiver section and generates state estimations based on the noisy output measurement. Meanwhile, this method can render the processing noise as well. In this research, we implement the fractional chaotic system. As a result, the extended fractional Kalman filter (EFKF) is used to estimate the states of a fractional chaotic system; the process within the filter synchronizes to the transmitter as the estimations converge. In addition, decomposition is not required. This synchronization is insensitive to additive noise. The fractional order nonlinear state-space system model is obtained analogically to the integer order one and is defined as follows [34]: Definition 1 [34]. The fractional order Gru¨nwald–Letnikov difference is given by the following equation:   k 1 X j n n D xk ¼ n ð1Þ xkj ; ð14Þ h j¼0 j where n 2 R is a fractional order, R is the set of real numbers, and h isa sampling time later equal to 1, k is the  n number of samples for which the derivative is calculated. The factor can be obtained from the relation: j   ( 1; j ¼ 0 n ¼ nðn1Þðnjþ1Þ ð15Þ ; j > 0: j j! According to this definition, it is possible to obtain a discrete equivalent of the derivative (when n is positive), a discrete equivalent of integration (when n in negative). More properties of the definition can be found in [36]. Definition 2 [34]. The nonlinear stochastic discrete fractional order state-space system is given by the following set of equations: 8  D xkþ1 ¼ f ðxk ; uk Þ þ wk ; > > > < kP þ1 j ð16Þ xkþ1 ¼ D xkþ1  ð1Þ  j xkþ1j ; > j¼0 > > : y k ¼ hðxk Þ þ vk ; where     n1 n1  k ¼ diag  ; k k 2 n1 3 D x1;kþ1 6 7 .. 7; D xkþ1 ¼ 6 . 4 5 nN D xN ;kþ1

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n1 ; . . . ; nN are the orders of system equations and N is the number of these equations. The nonlinear function f ðÞ and hðÞ can be linearized according to the Taylor series expansion: Theorem 1 [34]. For the nonlinear fractional order stochastic discrete state-space system given by Definition 2, the extended fractional Kalman filter (EFKF) is given by the following equations: D ~xkþ1 ¼ f ð^xk ; uk Þ; ~xkþ1 ¼ D ~xkþ1 

ð17Þ

kþ1 X

ð1Þj  j~xkþ1j ;

ð18Þ

j¼0

where ~xkþ1 is the state prediction for the system given by Definition 2 [34]. T Pe k ¼ ðF k1 þ  1 ÞP k1 ðF k1 þ  1 Þ þ Qk1 þ

k X

 j P kj  Tj :

ð19Þ

j¼2

Pe k is a prediction of an estimation error covariance matrix. As it is shown, the prediction of covariance error matrix depends on the value of covariance matrixes in previous: ^xk ¼ ~xk þ K k ðy k  hð~xk ÞÞ:

ð20Þ

^xk is a state vector estimation at time instant k P k ¼ ðI  K k H k Þ Pe k :

ð21Þ

P k is an estimation error covariance matrix

1 K k ¼ Pe k H Tk H Tk Pe k H Tk þ Rk :

ð22Þ

K k is called the Kalman filter gain vector. With initial conditions: xð0Þ ¼ x0 and h i T P 0 ¼ E ð~x0  x0 Þð~x0  x0 Þ ;

ð23Þ

where



of ðx; uk1 Þ ox  ohðxÞ Hk ¼ : ox x¼^xk



F k1 ¼

; x¼^xk1

Noise is white Gaussian and has the following characteristics: Efvg ¼ 0;

Efwg ¼ 0;

EfvvT g ¼ V ;

EfwwT g ¼ W :

ð24Þ

As mentioned above, extended fractional Kalman filter estimates the states of fractional order system and in the next section we will describe how to use extended fractional Kalman filter in receiver to recover original data from encrypted data. 4. Proposed chaotic encryption scheme and numerical example We have enhanced the security of the transmitted signal by using fractional order chaotic systems as instead of integer order chaotic systems. The synchronization is achieved by the extended fractional Kalman filter (EFKF) acting as the state estimator in the presence of noise. The block diagram of the proposed scheme is shown in Fig. 2. The proposed scheme does not need to know the initial condition of the chaotic signals between the receiver and the transmitter. The system consists of a transmitter module, a communication channel, and a receiver module. The transmitter module consists of a fractional chaotic system and an encryption mechanism (masking modulator). In this system the chaotic signal is generated by using the fractional Lorenz system. The fractional Lorenz system is described by the following differential equations:

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Fig. 2. Block diagram of the proposed communication scheme.

8 ðq1 Þ > > > > x1 ðtÞ ¼ rðx2 ðtÞ  x1 ðtÞÞ; > < ðq2 Þ

x2 ðtÞ ¼ x1 ðtÞx3 ðtÞ þ rx1 ðtÞ  x2 ðtÞ; > > > > > : ðq3 Þ x3 ðtÞ ¼ x1 ðtÞx2 ðtÞ  bx3 ðtÞ;

ð25Þ

when r ¼ 10, r ¼ 28, b ¼ 1:25 and q1 ¼ 0:96; q2 ¼ 0:98; q3 ¼ 1:1 system (25) behaves chaotically [29]. Process noise is considered in chaos states. The encryption process of the algorithm can be described as follows. The information signal sðtÞ is added with the second state according to the masking modulation. Then the encrypted signal cðtÞis passed through an AWGN channel. The first state of chaos, x1 ðtÞ, is also passed to the receiver module to improve synchronization. The major part of the receiver section consists of an extended fractional Kalman filter for state reconstruction and chaos masking demodulator. The fractional Lorenz states are estimated by the EFKF. It should be noted that the first state of the fractional Lorenz is used for chaotic synchronization. In the receiver, the first state goes to the EFKF and other states are estimated. Obviously, the fractional Lorenz model can also be presented as 3 2 ðq1 Þ 2 32 3 6 x1 ðtÞ 7 r r 0 x1 ðtÞ 7 6 6 ðq2 Þ 7 6 76 7 ð26Þ 6 x ðtÞ 7  4 r 1 x1 ðtÞ 54 x2 ðtÞ 5 ¼ AðxðtÞÞxðtÞ: 6 2 7 4 ðq Þ 5 0 x1 ðtÞ x3 ðtÞ b 3 x3 ðtÞ Matrix AðxðtÞÞ in our proposed scheme is 2 3 r r 0 6 7 AðxðtÞÞ ¼ 4 r 1 x1 ðtÞ 5; 0

x1 ðtÞ

ð27Þ

b

where r ¼ 10, r ¼ 28, b ¼ 1:25 and q1 ¼ 0:96; q2 ¼ 0:98; q3 ¼ 1:1. The sum of the squared errors (SSE) in state estimation is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 SSE ¼ R3i1 ðxi ðtÞ  ^xi ðtÞÞ : The first chaotic state is employed for synchronization. Therefore, the output measurement matrix is

ð28Þ

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C ¼ ½1

0

0 :

The initial conditions for the fractional chaotic system and the EFKF are xð0Þ ¼ ½ 1

10 1 T ;

^xð0Þ ¼ ½ 40

17

17 T :

The variances of the process and channel noise used in the EFKF are V ¼ 0:002;

W ¼ 0:00262:

The employed information signal for evaluating the performance of the proposed system is sðtÞ ¼ sinð2pftÞ;

f ¼ 2 Hz:

Fig. 3. Attractor of fractional Lorenz dynamical system.

Fig. 4. First state of fractional Lorenz system.

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Remark. The proposed chaotic communication scheme is totally different from the traditional cryptosystem where a fractional chaotic system enhances both the time and frequency domain characteristics of the encrypted signal. It should be pointed out that, in this approach, regarding the different employed chaotic states for the objectives of synchronization, masking modulation and meanwhile, noting the fact that all chaotic states of a same chaos attractor are inherently absolutely different from each other, there is no

Fig. 5. Second state of fractional Lorenz system.

Table 1 Convergence time in state estimation and data recovery Synchronization time

State 1 (s)

State 2 (s)

Maximum (s)

Recovered data (s)

Fractional Lorenz Integer Lorenz

0.72 0.13

0.4 0.4

0.72 0.4

1.611 0.43

Fig. 6. SSE in state estimation (fractional Lorenz).

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requirement for purely transmission of the chaotic key and so extended fractional Kalman filter can synchronize and recover original data. This in turn, heightens the security level of the system.

5. Simulation experiments In this section, the performance of the proposed scheme will be studied. As mentioned in Section 3.2, we have implemented the improved Adams–Bashforth–Moulton algorithm for numerical simulation in MATLAB. In Fig. 3, the attractor of the fractional Lorenz system can be seen. Figs. 4 and 5 show the first two states of the fractional Lorenz system and their estimations in the time interval of [0, 5]. After a while, the EFKF will synchronize the estimations to the original states. The convergence times of the first two states of the fractional Lorenz system are presented in Table 1. The maximum of the two values is considered as the convergence time

Fig. 7. Masked data with fractional Lorenz system.

Fig. 8. Original data and recovered data (fractional Lorenz).

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of the system and it is 0.72 s in this case. Fig. 6 shows the sum squared error in the state estimation of three states at each point and obviously, it is very low and satisfactory. In Fig. 7, the data that is encrypted by masking can be seen. When looking at this signal, we cannot understand the content of the message. Fig. 8 presents the original sinusoid data and the recovered data in the same coordinate. We can see that the recovered data is nearly the same as the original data. As indicated in Table 1, after 1.611 s, the data is recovered and converged nearly to the original data. By using our proposed secure chaotic communication scheme in the presence of channel noise and processing noise, the data can be precisely recovered. The noise performance of this system is related to the use of the EFKF for state synchronization. To compare with the integer schemes, we simulated this strategy with both fractional and integer chaotic dynamical systems. The results of simulations indicate that the proposed method is much more secure than the integer Lorenz system because of increasing nonlinearity and spreading power spectrum. In Figs. 9 and 10, the

Fig. 9. First state of integer Lorenz system.

Fig. 10. Second state of integer Lorenz system.

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Lorenz chaotic states and their estimations are evident in the time interval of [0, 5] s. The convergence time of the first two states of Lorenz system is presented in Table 1. The maximum of the two values is 0.4 s in this case. Fig. 11 shows the sum squared error in the state estimation of three states at each point and it is as low as the case of the fractional Lorenz system. Fig. 12 presents the sinusoid data that is encrypted by masking modulation. By looking at this signal, as in the previous case, we could not understand the message. Fig. 13 shows the original sinusoid data and also the recovered data in the case of integer Lorenz system. As indicated in Table 1, after 0.43 s the data is recovered and converged nearly to the original data. From observing Fig. 13, it is obvious that the data recovery error is just as low as before. In this case, the data is converged more quickly than when using the fractional Lorenz. This is mainly because of increasing nonlinearity and complexity.

Fig. 11. SSE in state estimation (integer Lorenz).

Fig. 12. Masked with integer Lorenz system.

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Finally, the power spectrum of both fractional and integer chaotic systems can be found in Figs. 14 and 15. From Fig. 14, the power spectrum of the fractional Lorenz fluctuates and becomes more complex and acceptable than the power spectrum of the integer Lorenz in Fig. 15. The masking modulation that uses integer chaotic systems is less secure. This is because of the simple power spectrum and the fact that the masked data is obvious in the transmitted signal. We can conclude that one of the most significant advantages of using fractional chaotic masking is spreading the power spectrum and so increasing security. By increasing the complexity of power spectrum, the traditional cracking algorithms of chaotic masking [22,23] will be unusable, thus enhancing the security.

Fig. 13. Original data and recovered data (integer Lorenz).

Fig. 14. FFT of masked data (fractional Lorenz).

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Fig. 15. FFT of masked data (integer Lorenz).

6. Conclusion In this paper, the novel approach to enhance the security of data transmission and also improve the vulnerable points of chaotic masking is presented. For the first time, we implement fractional chaotic systems in a simple chaotic masking method to illustrate the heightening of security in communication. However, even though there are numerous challenging advantages in chaotic systems, several methods exist to extract masked data and reduce security. Many of these are based on the power spectrum. In this research, by using fractional systems, nonlinearity is increased and also the complexity of the power spectrum is enhanced in comparison to the integer order chaotic systems. Another advantage of using fractional dynamical systems is adding a free parameter which is the order of the derivative. This free parameter increases security and also disguises the portrait of a dynamic system making it very difficult for an eavesdropper to understand system type. The stochastic extended fractional Kalman filter is used for state reconstruction in a noisy environment. The proposed chaotic communication scheme is totally different from the traditional cryptosystems, due to employing different chaos states for synchronization and encryption. To inspect the performance of the proposed system, both integer order and also fractional order chaotic systems have been implemented. From the simulation results, the performance of the proposed systems seems to be satisfactory for secure communication applications. Also, the results are sufficiently acceptable for digital data and as a future research; we are going to check this schema for other fractional chaotic systems. Acknowledgements The authors would like to thank the kindest help and support of M. Rajabzadeh and M. Yarmohamadi from Electrical department of Ferdowsi University of Mashhad for discussion and helping us in hardware implementation of the proposed method. We also would like to thanks referees for their useful comments. References [1] [2] [3] [4] [5]

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