Fast synchronization of non-identical chaotic modulation-based secure systems using a modified sliding mode controller

Chaos, Solitons and Fractals 84 (2016) 49–57

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Fast synchronization of non-identical chaotic modulation-based secure systems using a modified sliding mode controller Amin Kajbaf a, Mohammad Ali Akhaee b,∗, Mansour Sheikhan a a b

Department of Electrical Engineering, Islamic Azad University, South Tehran Branch, Tehran, P.O. Box 11365-4435, Iran Department of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, P.O. Box 14588-89694, Iran

a r t i c l e

i n f o

Article history: Received 27 December 2014 Accepted 5 December 2015 Available online 23 January 2016 Keywords: Chaotic modulation Synchronization Sliding mode control Secure communication Non-linear input

a b s t r a c t In this paper, a secure communication scheme based on chaotic modulation is proposed using a reversible process and a robust controller with efficient cost and complexity to synchronize two different chaotic systems. In the controller design, a sliding mode control with an adaptive rule is used for non-linear inputs. The adaptive rule is applied to ensure the synchronization when uncertainties, non-modeled dynamics or external distortions are at work. The message signal is recovered at the receiver using a recursive process at the end. The effectiveness of the proposed algorithm is confirmed via the simulation results for the synchronization of the transmitted signal modulated by Chen chaotic system at the transmitter and Genesio chaotic system at the receiver, and those for the information recovery process. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Synchronization is one of critical issues for the validity control of the received signals in communication systems. Recently, chaotic systems have attracted growing interest for the synchronization of the communication systems. Yamada and Fujisaka in 1983 proposed a method for the synchronization of the chaotic systems [1]. However, the interest in this field increased when Pecora et al. discussed the synchronization of the chaotic systems in the telecommunication problems [2]. They stated that the chaotic systems with sub-Lyapanov exponents laid at the left halfplane could be synchronized. In the literature, there are many research projects studying secure communication systems based on chaos

∗

Corresponding author. Tel.: +982182089746; fax: +982188778690. E-mail addresses: [email protected] (A. Kajbaf), [email protected], [email protected] (M.A. Akhaee), [email protected] (M. Sheikhan). http://dx.doi.org/10.1016/j.chaos.2015.12.002 0960-0779/© 2015 Elsevier Ltd. All rights reserved.

theory [3–6]. Because of some specifications of chaotic systems such as non-linearity, there are some common aspects between secure communication and chaotic systems. On the other hand, the demolishing narrow-band effects such as frequency-selective fading and narrow-band distortion can be handled thanks to the spread spectrum property of the telecommunication signals and the noise-like wide band spectrum of the chaos. The number of system components can be decreased and the maximum gain of the carrier power can be achieved with the help of chaotic systems. In spite of their deterministic nature, the chaotic systems exhibit many features of a stochastic process very critical for the spread spectrum communication such as being bounded, aperiodic, uncorrelated output and the autocorrelation function with its peak in zero and rapidly decreasing tails [7,8]. Furthermore, an appropriate feature which can be added to this list is their significant dependence on the initial condition. This principal property of the chaotic systems means that even with the least modifications in the initial state, the intermediate states and

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subsequently the output of such systems undergo considerable changes. This property makes them very unpredictable and appropriate for the application in the secure communications [7]. Based on specifications of chaotic systems, several techniques have been proposed for chaos based telecommunication systems such as chaotic masking [9,10], chaotic shift keying or chaotic switching [11], and chaotic modulation [12]. Extensive supplementary research has been conducted on each of the above-mentioned methods. In chaotic masking systems which can be considered as conventional methods, the information signal is added to one of the varying states of the chaotic system in the transmitter, and is recovered at the receiver synchronized with the transmitter by applying the subtraction to the pseudonoise signal received after passing the public channel [13]. The information signal is switched between two or N chaotic systems in chaotic switching. This method is very effective against noise but the security level is lowered in the case of non-identical chaos systems in the transmitter and receiver [13]. In the chaotic modulation systems, the task of modulation of the information is addressed by the parameters or the states of the chaotic systems using a reversible process. This approach elevates the security level compared with the chaotic masking method [13]. In the conventional chaos based telecommunications, two chaotic systems known as drive or master, and response or slave are connected by a public communication channel. Due to the high sensitivity of the chaotic systems to the initial condition and the importance of bit error rate (BER), the synchronization of two systems is critical to achieve a precise long-term prediction. In order to achieve the appropriate synchronization, a controller is required. Since the chaotic systems are intrinsically non-linear, the application of the controller with the non-linear input and low cost and complexity is inevitable. Several methods are proposed for the synchronization of the chaotic systems, such as feedback control [14,15], adaptive control [16–20], optimal control [21,22], intermittent control [23,24], digital redesign control [25], back-stepping control [26,27], linear state feedback by linear matrix inequalities (LMI) technique [28], H-inf based control [29], passive control [30], active control [31], fuzzy control [32–34], neural-based control [35–37], and sliding mode control [38,39]. Since the unknown parameters, uncertainties, and external distortions may cause severe problems or failure in the synchronization process, their effect must be carefully considered in the design of chaos based synchronization. Up to now, studies have been conducted on sliding mode control by considering the adaptive version. This adaptive rule helps ensure the synchronization at the presence of the unknown bounded parameters, external distortions and bounded uncertainties. Moreover, a reliable secure modulation plan can improve the communication security. Previous works, to the best of our knowledge have not considered the chaotic secure modulation in the presence of all the aforementioned destructive factors and are mainly focused on simple masking and modulations with a few or none of these factors. For example, the study on the adaptive synchronization of Rossler and Chen chaotic systems was performed considering unknown time-varying

parameters in [40]. As another instance, an adaptive sliding mode controller for the synchronization of the master– slave chaotic systems at the presence of the uncertainties was proposed in [38]. Moreover, a secure communication scheme using hyper-chaos synchronization and chaotic masking at the presence of uncertainties and distortions was suggested in [9]. Another important point that is less considered in such studies is that the chaotic nature which is the basis of the desired security must be preserved. This study presents a chaotic secure communication scheme with the aid of two different chaotic systems, highly secure modulation, fast and accurate synchronization of the two chaotic systems by an improved sliding mode controller with nonlinear input. In this scheme, the message will be hidden at the heart of chaotic system dynamics using an innovative way in two steps: chaos modulation as the first step and mapping as the second. Besides contributing to the establishment of the system structure, this mapping also increases the security level. After transmission over the channel and considering the uncertainty problem in the receiver, the resultant signal with a high level of security is synchronized with the transmitter by the improved sliding mode controller. Eventually, using a specific algorithm, the message signal will be extracted in the receiver. We also employed an adaptive law in the design of the controller. This guarantees the synchronization of two different chaotic systems in the presence of bounded uncertainties, external disturbances, and bounded unknown parameters. The high speed and accuracy of controller in synchronizing the transmitter and receiver systems made the accurate and low error message extraction possible. This means a lower bit error rate (BER) and, accordingly, a higher string length. Totally, in the suggested scheme, it is attempted to develop a method for chaotic secure modulation and also a controller design that leads to a secure communication system which, in addition to simplicity and low complexity, is highly secure and entails a low cost in practical implementation. Therefore, given the structure of the proposed system, it can be implemented using simple processors which enable us to develop rapid and economic systems compared to conventional systems. Meanwhile, exploiting two different chaotic systems results in secure and easy implementation of a secure communication system in comparison with two similar chaotic systems. The rest of the paper is organized as follows. The problem formulation and the proposed method are given in Section 2. Simulation results of the proposed scheme are presented in Section 3. Section 4 discusses the advantages and disadvantages of the proposed method, while Section 5 concludes the paper.

2. Proposed method In this section, the generic design of the proposed method is introduced. Then, the problem definition including chaotic modulation equations for the input signal, design of the controller and the synchronization of the received signal are discussed. Finally, the demodulation

A. Kajbaf et al. / Chaos, Solitons and Fractals 84 (2016) 49–57

process leading to the extraction of the original signal from the received signal is explained.

Now, applying the mapping of (3)–(2) results in:

x˙ 1 (t ) = x2 (t ) x˙ 2 (t ) = x3 (t )

2.1. Overall design

51





x˙ 3 (t ) = F (x(t )) = P Q −1 (z(t )) = H (z(t ))

The generic structure of the proposed secure communication system based on the chaotic synchronization is shown in Fig. 1. The proposed system includes the transmitter, communication channel and the receiver. At the transmitter, the message signal m(t) is modulated at the output of the master chaotic system according to the method that is explained in the following section. Output signal of this modulation scheme is received at the receiver after being passed through the channel. The controller designed at the receiver performs the task of the reliable and quick synchronization of the transmitted and the received signal in the presence of uncertainties, distortions, and unknown parameters. The message signal m(t) is extracted using an invertible process at the end. In our proposed scheme, m(t) is embedded into the chaotic system dynamics that provides high security level and unpredictability of the information signal as discussed later. The received signal is compared with the output of the slave chaotic system and the resulting error signal e(t) serves as the input for a sliding mode adaptive controller. The output of the sliding mode controller is then used as the feedback to the input of the slave chaotic system to provide the synchronization between the received signal and the output of the chaotic system. After synchronization, the information signal will be reconstructed based on the available parameters and equations.

= −ca2 (z1 (t ) −z2 (t )) + (ac−a2 )(c−a )z1 (t ) + c (ac−a2 )z2 (t ) − (ac − a2 )z1 (t )z3 (t ) + a2 z3 (t )(z1 (t ) −z2 (t )) −az12 (t )z2 (t ) + abm(t )z1 (t )z3 (t ) = f (x, t ).

(4)

The modulated signal will be transmitted to the receiver through the channel. Slave system equations are those of Genesio chaotic system with three dimensions described as below:

y˙ 1 (t ) = y2 (t ) y˙ 2 (t ) = y3 (t )

y˙ 3 (t ) = −a y1 (t ) − b y2 (t ) − c y3 (t ) + y21 (t ) + g(y, t ) + d (t ) + u(t )

(5)

where y1 , y2 and y3 are the states of the Genesio chaotic system and a , b and c are constant non-negative parameters. Also, d(t), g(y, t) and u(t) denote the external distortion, uncertainty parameter representing the varying structures or non-modeled dynamics, and the control input of the slave system, respectively. In addition, we have:

g(y, t ) = −a y1 (t ) − b y2 (t ) − c y3 (t ) + y21 (t )

(6)

Dynamic error equation that is used to achieve the synchronization is calculated through subtracting the differential equations in (4) and (5):

e˙ 1 (t ) = e2 (t ) e˙ 2 (t ) = e3 (t )

2.2. Chaotic modulation and the controller design

e˙ 3 (t ) = f (x, t ) −g(y, t ) − g(y, t ) −d (t ) − u(t ).

A 3D Chen chaotic system is assumed at the transmitter, of which the equations can be stated as follows:

z˙ 1 (t ) = −a(z1 (t ) − z2 (t ))

ei (t ) = xi (t ) − yi (t ),

z˙ 2 (t ) = (c − a )z1 (t ) + cz2 (t ) − z1 (t )z3 (t ) z˙ 3 (t ) = z1 (t )z2 (t ) − bz3 (t )

(1)

where z1 , z2 and z3 are chaotic system states and a, b and c are the constant non-negative parameters. Input signal m(t) is combined with the Chen chaotic system as below:

z˙ 1 (t ) = −a(z1 (t ) − z2 (t ))

e˙ i (t ) = ei+1 (t ),

z˙ 3 (t ) = z1 (t )z2 (t ) − bm(t )z3 (t ).

Defining e3 (t) as

Exploiting the following mapping, Eq. (2) turns into this new form:



E (t ) = Q (z(t )) =

⎡ =⎣

x1 (t ) x2 (t ) x3 (t )



z1 t

(8)

e3 (t ) = −

2

i = 1, 2.

hi ei (t ),

(9)

(10)

i=1

a standard form of the error stated in (9) is achieved where hi is a positive parameter which determines the stability of (9). With this definition, the above-mentioned switching level can be stated in this way:

⎤

⎦. 2 acz1 (t ) + (ac − a )z2 (t ) − az1 (t )z3 (t ) −a(z1 (t ) − z2 (t ))

i = 1, 2, 3.

Now the proper u(t) that satisfies limt→∞ e(t ) = 0 must be found. In this case, y tends to x when t → ∞. To find such a proper u(t), defining a proper switching level is required for the stability of the synchronization error dynamics and achieving the synchronization in the presence of all the factors of non-modeled dynamics, external distortions and non-linear control input. This is performed by a sliding mode controller (SMC) for which we have:

z˙ 2 (t ) = (c − a )z1 (t ) + cz2 (t ) − z1 (t )z3 (t ) (2)

(7)

Error vector between the transmitter and receiver systems is found according to (7) giving:

(3)

s(t ) = e3 (t ) +

2

i=1

hi ei (t ).

(11)

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Fig. 1. Generic block diagram of the proposed system.

 | f (x, t ) − g(y, t )| + γ + k

Theorem 1. According to (7) and the following explanation for (12), the control input of the slave system is found as:



u(t )=

2

≤ |s(t )|



hi ei+1 (t ) + | f (x, t ) −g(y, t )| +k+ μγˆ (t )

+|

i=1

Proof. Defining the Lyapanov 1 2 1 2 ˜ (t ) we have: 2 s (t ) + 2 γ

V˙ (t ) = s˙ (t )s(t ) + γˆ˙ γ˜ (t )



= s(t ) e˙ 3 (t ) +

2

function

from

+|

|s(t )| ˙ + γˆ γ˜ (t ) s(t )

2

 hi ei+1 (t )|

− η|s(t )| + γˆ˙ γ˜ (t )

i=1

 | f (x, t ) − g(y, t )| + γ + k

+|

+ γˆ˙ γ˜ (t ).

(14)

(15)

Now, exploiting γˆ˙ = |s(t )| results in:

(13)

i=1

 dV (t ) ≤ |s(t )| | f (x, t ) −g(y, t )| + | − g(y, t )| + | − d (t )|  2

+| hi ei+1 (t )| − ηs(t )sgn(s(t )) + γˆ˙ γ˜ (t )

hi ei+1 (t )| − η

 dV (t ) ≤ |s(t )| γ − μγˆ (t ) + γˆ˙ (t )(γˆ (t ) − γ ). dt

≤ |s(t )| | f (x, t ) − g(y, t )| + | − g(y, t )| + | − d (t )|

Inserting u(t) in the above equation results in:



Applying η from (12) we have:

i=1

 hi ei+1 (t )| − s(t )u(t ) + γˆ˙ γ˜ (t ).

2

i=1



i=1

− ηs(t )

 ≤ |s(t )| | f (x, t ) − g(y, t )| + γ + k

+ γˆ˙ γ˜ (t )

hi ei+1 (t ) + γˆ˙ γ˜ (t )

2

 hi ei+1 (t )|

i=1

hi e˙ i (t )

2

≤ |s(t )|

= s(t ) f (x, t ) − g(y, t ) − g(y, t ) − d (t ) − u(t ) 2

+|

V (t ) =

i=1

dt

− ηs(t )sgn(s(t )) + γˆ˙ γ˜ (t )

 ≤ |s(t )| | f (x, t ) − g(y, t )| + γ + k

(12)

where the external distortion is bounded between k1 and k2 , such that d(t) < k and non-modeled dynamics are limited to the constant and sufficiently large positive parameter γ , such that g(y, t) < γ . In (12), γˆ (t ) is an estimation of γ (t). Hence, the estimation error equals γ˜ (t ) = γˆ (t ) − γ . It is noteworthy that γˆ˙ = |s(t )| states the adaptive rule, thus, choosing this value for the control input results in the area below s(t) curve tending to zero.

+|

 hi ei+1 (t )|

i=1

×sgn(s(t )) = ηsgn(s(t ))

+

2

 dV (t ) ≤ |s(t )| γ − μγˆ (t ) + |s(t )|(γˆ (t ) − γ ) dt ≤ |s(t )|(1 − μ )γˆ (t ) ≤ −l (t ) ≤ 0.

(16)

The value of l (t ) = −|s(t )|(1 − μ )γˆ (t ) is positive with μ > 1. Thus, integrating above-mentioned equation from 0 to t yields:

V (0 ) ≥ V (t ) +



t 0

l (λ )dλ.

(17)

Since V (0 ) − V (t ) ≥ 0 is positive and limited and t limt→∞ 0 l (λ )dλ exists, according to the Barbalat’s lemma [41,42], we have:

lim l (t ) = lim |s(t )|(μ − 1 )γˆ (t ) = 0

t→∞

t→∞

(18)

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53

and since both (μ − 1 ) and γˆ (t ) are greater than zero, 2 it is concluded that s = 0. Moreover, replacing i=1 (· ) 2 with i=0 (· ) in the error calculation function, sat( s(t ) ) can be used in (12) instead of sgn(s(t)) to avoid the natural t  collision in which e0 (t ) = 0 x1 (tˆ) − y1 (tˆ) dtˆ. Finally, we have:



u(t ) =

2

i=0



hi ei+1 (t ) + | f (x, t ) − g(y, t )| + k + μγˆ (t )



× sat

s(t )





= ηsat



s(t )



 .

(19) 

Since u(t) is the control input and considering the fact that energy is calculated through integrating the signal power over a certain time period, the energy of the control input can be calculated using the following equation:

w(t ) =

 0

t

u2 ( ρ )d ρ .

(20)

2.3. Demodulation and data extraction Demodulation of the received signal and data extraction is discussed in this section. The equations of the slave system before synchronization via Genesio chaotic system was given in (5). Now, after synchronization, the output of the slave system is the signal sent by the transmitter, as discussed in Section 2.2:

y˙ 2 (t ) = xˆ3 (t )





y˙ 3 (t ) = I (t ) = −ca2 zˆ1 (t ) − zˆ2 (t ) + (ac−a2 )(c−a )zˆ1 (t ) +c (ac−a2 )zˆ2 (t ) − (ac − a2 )zˆ1 (t )zˆ3 (t )





g(y, t ) = −0.5y1 (t ) d (t ) = 0.2 cos(π t ).

ˆ (t )zˆ1 (t )zˆ3 (t ). +abm

(21)

Finally, using a reversible process for the extraction of data from the carrying chaos, we have:

 1 [I (t ) + ca2 zˆ1 (t ) − zˆ2 (t ) abˆz1 (t )zˆ3 (t ) −(ac−a2 )(c−a )zˆ1 (t ) −c (ac−a2 )zˆ2 (t ) +(ac − a2 )zˆ1 (t )zˆ3 (t )





−a2 zˆ3 (t ) zˆ1 (t ) − zˆ2 (t ) +aˆz12 (t )zˆ2 (t )].

(22)

ˆ (t ) is a very exact apSimulation results confirm that m proximation of m(t). 3. Simulation results Simulation results for the proposed system are given in this section. For the modulated system described in (4), the parameters a, b, and c are set to 8, 3 and 3, respectively. The initial condition vector is set to [1,2,1]. Moreover, the message signal m(t) is assumed to be as follows:

m(t ) = 5 sin(2π f1 t ) +6 sin(2π f2 t ) +12, f1 = 2 Hz, (23)

(24)

The sliding surface here can be described as: (25)

where h0 = 6, h1 = 6, and h2 = 11 are chosen. Finally the control signal in this simulation presented in (19) can be expressed in following form:

u(t ) =

+a2 zˆ3 (t ) zˆ1 (t ) − zˆ2 (t ) −aˆz12 (t )zˆ2 (t )

f2 = 1 Hz.

At the receiver, the Genesio system described by the differential equations in (5) with the parameters a = 6, b = 2.92, and c = 1.2 and the initial condition vector of [−1,1,−1] are applied. Also in the Genesio system, we have:

s(t ) = e3 (t ) + h2 e2 (t ) + h1 e1 (t ) + h0 e0 (t )

y˙ 1 (t ) = xˆ2 (t )

ˆ (t ) = m

Fig. 2. Information signal.



|11e3 (t ) + 6e2 (t ) + 6e1 (t ) + |a e1 (t )

+b e2 (t ) + c e3 (t )



+ F(x(t ))

−y21

  s(t ) (t ) | +0.2+15γˆ (t ) sat 

(26)

with k = 0.2, μ = 15, and  = 0.3. Fig. 2 shows the information signal m(t). The modulation procedure of the information signal in the master chaotic system and the slave system states before the application of the controller and synchronization are illustrated in Fig. 3. The differences between the states of the transmitted signal and those of the slave system are clearly presented in this figure. Figs. 4–6 demonstrate the synchronization between the modulated transmitted signal and the received signal with the help of the controller. According to Figs. 4–6 it is clear that the time required for the synchronization of the transmitter and receiver is very short. The required time for the synchronization in each state and the minimum, maximum, and average synchronization time of all three states are given in Table 1. It can be inferred from this table that the synchronization time is very short even at its maximum value, i.e. t = 1.127. Fig. 7 shows the control input. The comparison of these figures confirms that the synchronization is successfully performed and the controller is capable of providing such

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A. Kajbaf et al. / Chaos, Solitons and Fractals 84 (2016) 49–57

Fig. 3. States of the chaotic modulation and the slave system before synchronization.

Fig. 4. First states of the chaotic modulation and the slave system after synchronization.

Fig. 6. Third states of the chaotic modulation and the slave system after synchronization.

Fig. 7. Control input for synchronization after using the modulation of the information. Table 1 Synchronization times and their minimum, maximum and average values.

Fig. 5. Second states of the chaotic modulation and the slave system after synchronization.

States of transmitter and receiver

1

2

3

Synchronization time (s) Minimum time (s) Maximum time (s) Average time (s)

0.750

1.127 0.571 1.127 0.816

0.571

synchronization. Adaptive parameter γˆ (t ) is also shown in Fig. 8. Finally, in Fig. 9, the original message signal, the recovered message signal at the receiver output and the error of the message extraction are shown together. Fig. 10 presents both the original and recovered signals. It can be deduced from these figures that the estimation of the information signal is performed with low error level and high accuracy, and the largest possible scope of

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55

information is extracted considering the fast synchronization of the transmitter and receiver. 4. Security analysis and discussion

Fig. 8. Adaptive parameter γˆ (t ).

Chaotic signals cannot be easily predicted due to their pseudo-noise and spread-spectrum nature and the issues discussed in the introduction. In the proposed method, the message signal is injected to the chaotic signal based on the already explained method. Since the chaotic nature is preserved after modulation, it is challenging to extract the message. Furthermore, since the signal modulated using the Chen system is not directly transmitted through the channel and is somehow encrypted via the explained mapping, malicious message extraction using some of the detection techniques such as identification-based breaking is impossible. On the other hand, since chaotic systems are very sensitive to the initial conditions, system parameters and the mapping changes. From the security viewpoint, initial conditions, system parameters and all of the mapping equations that might be considered as parameters, can be used as a secret key for the telecommunication system. The key length is indeed very important to the achieved security level. Assuming that the malicious attacker is aware of the initial conditions, the length of key P = {P1 , . . . , P6 } is defined as below:

P1 = a, P2 = b, P3 = c, P4 = z1 , P5 = z2 , P6 = z3

Fig. 9. Original and recovered message signals and the extraction error.

(27)

System sensitivities to each of these parameters are shown in Table 2. According to this table, the key length equals 103 × 106×3 ×107×2 = 1035 . Since the current exhaustive search machines are from maximum length of 2100 and 2100 < 1035 , the security of key used for the message extraction is ensured. Due to the application of a control signal as the control input and the direct extraction without the need for solving dynamic equations to find the output, the system design enjoys low complexity and computational cost. This feature is very helpful for the practical implementations and design of the commercial electronic systems. In our proposed model, common information between the transmitter and receiver such as chaotic system parameters and initial conditions is not required for the purpose of synchronization. This fact helps to remove the requirement of a key for synchronization. Although this need for key is not considered a limitation and in some cases is a contributing factor to the system security, it is not required in the synchronization stage of our proposed method due to its high security level and high capability of the controller in the synchronization of the transmitter and receiver without the need for the chaotic system parameters of the transmitter.

Table 2 Sensitivity to parameters. Parameter

Fig. 10. Original and recovered message signals.

P1 = a P2 = b P3 = c P4 = z1 P5 = z2 P6 = z3

Sensitivity 10−6 No. of possibilities 106

10−3 103

10−7 107

10−6 106

10−6 106

10−7 107

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5. Conclusion In this paper, a secure chaotic communication scheme based on the modulation of the information signal and the efficient synchronization of two non-identical systems using an adaptive sliding mode controller with one integral error term was proposed. This proposed scheme is very robust and efficient in the face of uncertainties, external distortions, and non-linear input. Moreover, the controller system applied at the receiver is very simple and low-cost despite the complexity of the signal sent by transmitter. Also, due to the type of chaotic modulation as well as the simplicity of the employed mapping scheme, the proposed method can be used for any other system in the transmitter providing that the desired mapping is extracted proportional to the type of chaotic system. However, due to the suggested scheme, it is inevitable to use the chaotic systems with a structure of Eq. (5) in the receiver, such as the Genesio system, and Duffing–Holmes damped spring system. Simulation results confirmed that high security level and fast synchronization of transmitter and receiver systems are achievable in the proposed scheme. As a consequence, the largest possible scope of information can be extracted at the information recovery process. Moreover, the information signal is recovered at the receiver side with proper accuracy and low error level. As a suggestion for future works, chaotic systems from higher dimensions can be used, and the controller capability at synchronization of such systems can be further studied. Furthermore, the system robustness against channel noise can also be examined using proper processing blocks to achieve a system robust to noise in addition to low complexity and high security. References [1] Yamada T, Fujisaka H. Stability theory of synchronized motion in coupled-oscillator systems. II: The mapping approach. Prog Theor Phys 1983;70(5):1240–8. doi:10.1143/PTP.70.1240. [2] Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–4. doi:10.1103/PhysRevLett.64.821. [3] Halle KS, Wu CW, Itoh M, Chua LO. Spread spectrum communication through modulation of chaos. Int J Bifurcation Chaos 1993;03(02):469–77. doi:10.1142/S0218127493000374. [4] Lin S-L, Tung P-C. A new method for chaos control in communication systems. Chaos Solitons Fractals 2009;42(5):3234–41. http://dx.doi. org/10.1016/j.chaos.2009.04.054. [5] Lin J-S, Huang C-F, Liao T-L, Yan J-J. Design and implementation of digital secure communication based on synchronized chaotic systems. Digital Signal Process 2010;20(1):229–37. http://dx.doi.org/10. 1016/j.dsp.2009.04.006. [6] Wang H, Zhu X-J, Gao S-W, Chen Z-Y. Singular observer approach for chaotic synchronization and private communication. Commun Nonlin Sci Numer Simul 2011;16(3):1517–23. http://dx.doi.org/10.1016/j. cnsns.2010.06.021. [7] Fallahi K, Leung H. A chaos secure communication scheme based on multiplication modulation. Commun Nonlin Sci Numer Simul 2010;15(2):368–83. http://dx.doi.org/10.1016/j.cnsns.2009.03.022. [8] Dachselt F, Schwarz W. Chaos and cryptography. IEEE Trans Circuits Syst I: Fundam Theory Appl 2001;48(12):1498–509. doi:10.1109/TCSI. 2001.972857. [9] Sheikhan M, Shahnazi R, Garoucy S. Hyperchaos synchronization using PSO-optimized RBF-based controllers to improve security of communication systems. Neural Comput Appl 2013;22(5):835–46. doi:10.1007/s00521-011-0774-4. [10] Zhu F. Observer-based synchronization of uncertain chaotic system and its application to secure communications. Chaos Solitons Fractals 2009;40(5):2384–91. http://dx.doi.org/10.1016/j.chaos.2007.10.052.

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