Secure Communications Based on the Synchronization...

Advanced Studies in Theoretical Physics Vol. 9, 2015, no. 8, 379 - 394

HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2015.5228

Secure Communications Based on the

Synchronization of the New Lorenz-like

Attractor Circuit

Aceng Sambas1,3, Mustafa Mamat1,*, Mada Sanjaya WS2,3, Zabidin Salleh4 and Fatma Susilawati Mohamad5

1,5Department of Information Technology

Universiti Sultan Zainal Abidin, Kuala Terengganu, Malaysia *Corresponding author

2Department of Physics, Universitas Islam Negeri Sunan Gunung Djati

Bandung, Indonesia

3Bolabot Techno Robotic Institute Sanjaya Star Group Corp, Bandung, Indonesia

4Pusat Pengajian Informatik dan Matematik Gunaan

Universiti Malaysia Terengganu, Malaysia

Copyright © 2015 Aceng Sambas et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Synchronization of chaotic systems is important because it might be useful in some type of private communications. In this paper, the phenomenon of Chaos that produced in the case of autonomous New Lorenz-like attractor circuit, have been studied extensively. The initial study of this work also includes some of the most well-known tools of nonlinear dynamics, such as the Lyapunov exponents and the Poincaré map. Furthermore, the use of this type of chaotic circuit in the connection substitution synchronization method is presented in details. After conducting the analysis of the proposed synchronization scheme, the use of such an autonomous chaotic circuit as a signal modulation in secure communication systems, has been examined. Finally, numerical simulations by using MATLAB

380 Aceng Sambas et al. 2010, as well as implementation of circuit simulations by using MultiSIM 10.0, has been performed in this work. Keywords: New Lorenz-like attractors, bifurcation diagram, secure communication system

1. Introduction

Chaos theory has been established since the 1970s due to its applications in many different research areas, such as electronic circuits [1], ecology [2], biology [3], economy [4], bit generators [5], psychology [6], robotics [7], secure communication systems [8-11] etc. Generally, Chaos is an aperiodic, long-term behavior in a deterministic system that exhibits a sensitive dependence on initial conditions.

Since the Dutch scientist Christiaan Huijgens in 1665 noted the synchronizing behavior of pendulum clocks, many scientists have been investigating the synchronization of several dynamical systems. The landmark in the evolution of the chaotic synchronization’s theory is the study of the interaction between coupled chaotic systems by Pecora and Carroll [12], in which two identical chaotic systems with different initial conditions can be synchronized. Till now, various types of synchronization have been reported, namely complete synchronization [13], phase synchronization [14], generalized synchronization [15], anti-synchronization and anti-phase synchronization [16], lag synchronization [17], anticipating synchronization [18], projective synchronization [19] and inverse π-lag synchronization [20]. So, the phenomenon of chaotic synchronization has been intensively and extensively investigated due to its potential applications in a variety of areas, such as in secure communications [21, 22], encryption systems [23], chemical reactions [24], neuronal systems [25, 26] and economic models [27].

This paper focuses on the use of an autonomous New Lorenz-like chaotic attractor circuit in the case of a signal masking system. This new type of nonlinear system presents the same chaotic behavior as the well-known Lorenz system. Numerical simulations in MATLAB as well as analog circuit’s simulations in MultiSIM confirm the expected chaotic behavior of the system. Next, this system is used in the connection substitution synchronization scheme which is applicated to a secure communication system. The success of the proposed method has been confirmed by the extended simulation results.

The paper is organized as follows. In Section 2, numerical simulations in MATLAB 2010, Lyapunov exponent’s analysis, Poincaré map and MultiSIM 10.0 simulation results of the proposed nonlinear system, are obtained. The application of the connection substitution synchronization method in the case of coupled autonomous New Lorenz-like attractor circuits is presented in Section 3. In Section 4, the simulation in MATLAB 2010 as well as in MultiSIM 10.0 of a chaotic masking communication system, by using this new type of nonlinear circuit, is described in details. Finally, Section 5 contains the conclusion remarks.

Secure communications based on the synchronization … 381

2. The New Lorenz-like System

In this work the new Lorenz-like system, which is presented by Li et al in 2008 [28], is used. This is a three-dimensional autonomous nonlinear system that is described by the following system of ordinary differential equations:

x = a(y - x)

y = abx - axz

z = xy - cz

(1)

Where T 3( , , ) x y z R is the state variables of the system, while a, b and c are

the system’s parameters. As it is shown in equation (1) there are six terms on the right-hand side of differential equations, but the system’s nonlinearity only relies on the two quadratic terms xy and xz. 2.1 Numerical Simulations

In this section, all the numerical simulations are carried out using the MATLAB 2010. The fourth-order Runge-Kutta method is used to solve numerically the system of differential equations (1).

2 2.5 3 3.5 4 4.5 5 5.5 6-4

-2

0

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4

6

8

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c

max

(X)

Figure 1: Bifurcation diagram of x vs. the control parameter c, for the specific values (a = 5, b = 4), with MATLAB 2010.

A bifurcation occurs when a small change made in the parameter value (the

bifurcation parameter) of a system causes a sudden qualitative or topological change in its dynamical behavior. In dynamical systems, a bifurcation diagram shows the possible long term values (equilibrium points, periodic orbits or chaotic behavior) of a system as a function of a bifurcation parameter. Figure 1 shows a possible bifurcation diagram for system (1) in the range of 2 ≤ c ≤ 6. Specifically, for 2 c 3.3, Figure 2 displays a chaotic region, whereas for 3.3 c 6 the system displays a periodic behavior.

382 Aceng Sambas et al.

In Figures 2 (a)-(c), the projections of the phase space orbit onto the x-y plane, the x-z plane and the y-z plane, are shown respectively. The parameters and initial conditions of the New Lorenz-like system (1) are chosen as (a, b, c) = (5, 4, 2.5) and (x0, y0, z0) = (0.1, 0.1, 0.1), so that the system shows the expected chaotic behavior. So, it can be clearly observed that the phase portraits, especially onto the x-z plane are similar to that of the family of Lorenz systems. This is happened because the proposed system has all the characteristics of the above mentioned system’s family, such as the symmetry and invariance under the transformation S: (x, y, z) → (–x, –y, z), the dissipativity, the diffeomorphism and the topological equivalence.

Furthermore, the time-series of signals x, y and z, for the same set of parameters and initial conditions, are shown in Figures 2(d)-(f). Especially, the signal x has the well-known pattern of all the signals produced by the systems of Lorenz systems. In details, the signal x spirals outward from one of the symmetric

equilibria +

P = ( bc , bc ,b) and

P = (- bc ,- bc ,b) for some times and then

switches to spiraling outward from the other equilibrium point. It is also known from the theory of nonlinear dynamics that for a three

dimensional system, like this, there has been three Lyapunov exponents (λ1, λ2, λ3). In more details, for a 3D continuous dissipative system the values of the Lyapunov exponents are useful for distinguishing among the various types of orbits. So, the possible spectra of attractors, of this class of dynamical systems, can be classified in four groups, based on Lyapunov exponents [29]. (λ1, λ2, λ3) → (–, –, –): a fixed point (λ1, λ2, λ3) → (0, –, –): a limit point (λ1, λ2, λ3) → (0, 0, –): a 2-torus (λ1, λ2, λ3) → (+, 0, –): a strange attractor

Therefore, the last configuration is the only possible third-order chaotic system. In this case, a positive Lyapunov exponent reflects a “direction” of stretching and folding and therefore determines chaos in the system.

So, in Figure 3 the dynamics of the proposed system’s Lyapunov exponents for the variation of the parameter [2,6]c , is shown. For 2 c 3.3 a strange

attractor is displayed as the system has one positive Lyapunov exponent, while for values of 3.3 c 6 is a transition to a periodic behavior as the system has three negative Lyapunov exponents.

Another useful tool in the study of nonlinear systems is the Poincaré map, which is simply a map that showing a pattern from its time-series data. It is not a time-series map, yet it allows transversal changes of time-series data in each time of iteration. In this way each element of displayed data can no longer be viewed differently from time-series that represent them. In fact, every time-series data has been inherent with the data that graphically represented [30]. Figures 4(a)-(c) shows the Poincaré section map using MATLAB 2010, for a = 5, b = 4, c = 2.5.

So, for the chosen value of c (c = 2.5), the system has a chaotic behavior. A single trajectory plotted in the phase plane intersects itself many times, and the portrait soon becomes very messy. However, if one plots the first returns on the

Secure communications based on the synchronization … 383 Poincaré section, then a strange attractor is formed that demonstrates some underlying structure as shown in Figures 4(a)-(c). It must be noted that the chaotic attractor will have different forms on different Poincaré sections. This strange attractor has a fractal structure.

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Figure 2: Numerical simulation results using MATLAB 2010, for a = 5, b = 4, c = 2.5:

(a) x-y plane, (b) x-z plane, (c) y-z plane, (d) time series of signal x, (e) time series of signal y and (f) time series of signal z

2 2.5 3 3.5 4 4.5 5 5.5 6-1.5

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c

Lya

pu

no

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xpo

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x

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Figure 3: Lyapunov exponents versus the parameter [2,6]c , with MATLAB 2010


5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.85

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Poincare Map Analysis New lorenz-Like Circuit

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(a) (b) (c)

Figure 4: A gallery of Poincaré maps for system (1), when a = 5, b = 4, c = 2.5: (a) show the plot given the maxima of x(n + 1) against those of x(n); (b) show the plot given the

maxima of y(n + 1) against those of y(n); (c) show the plot given the maxima of z(n + 1) against those of z(n), with MATLAB 2010.

2.2 Analog Circuit Simulations

A circuit schematic that closely realizes the scaled Lorenz system is shown in Figure 5. The voltages at the nodes labeled x, y, and z correspond to the states of the equation system (1). The operational amplifiers and associated circuitry perform the basic operations of addition, subtraction, and integration. The nonlinear terms in the equation are implemented with the analog multipliers AD633 [26]. The occurrence of the chaotic attractor can be clearly seen in Figures 5(a)–(c), for the same values of system’s parameters. By comparing Figures 2(a)-(c) and Figures 6(a)-(c) a good qualitative agreement between the numerical integration of system (1) by using MATLAB 2010, and the circuit’s simulation by using MultiSIM 10.0, can be concluded.

3. Connection Substitution Chaotic Synchronization Scheme

Based on Master-Slave approach, a New Lorenz-like equation before connection substitution synchronized method is:

),( 1,111 zyxfx

),( 2,222 zyxfx

(2)

New Lorenz-like circuit equations in a synchronization scheme are as follows:

),( 1,111 zyxfx

),( 2,212 zyxfx

(3)


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Figure 5: Schematic of the proposed New Lorenz-Like circuit

(a) (b) (c)

Figure 6: Various projections of the chaotic attractor

using MultiSIM in (a) x-y plane, (b) x-z plane and (c) y-z plane


The following master-slave (connection substitution method) configuration, as described below:

2212

2112

122

1111

1111

111

)(

)(

czyxz

zaxabxy

xyax

Slave

czyxz

zaxabxy

xyax

Master

(4)

The asymptotic synchronized situation is defined as:

0)()(lim 21

tytyt (5)

3.1 Numerical Simulations

Numerical simulations in MATLAB 2010, by solving the coupling system (4) with the fourth-order Runge-Kutta method, are used to describe the dynamics of the connection substitution chaotic synchronization scheme. Figure 7 shows the chaotic synchronization phase portrait and the error numerical results.

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Error Chaotic Synchronization

(a) (b)

Figure 7: (a) Synchronization phase portrait of y2 versus y1 and (b) synchronization error (y2 – y1) versus time t, by using the

connection substitution technique, in MATLAB 2010

3.2 Analog Circuit Simulation in MultiSIM 10.0

Synchronization of chaotic systems is the key issue in symmetric chaos-based secure communication schemes. It is a phenomenon that may occur when two, or

Secure communications based on the synchronization … 387 more, chaotic oscillators are coupled. This paper presents in this Section the study of circuit’s simulation by using MultiSIM 10.0. For this reason, the drive and response circuits were constructed. So, in Figure 8 the implementation of the connection substitution synchronization scheme of coupled New Lorenz-Like circuits’, with MultiSIM 10.0, is displayed. Finally, in Figure 9 the simulation results of this scheme which confirms the case of chaotic synchronization are shown.

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MasterSlave

Figure 8: Schematic of the connection substitution chaotic synchronization

of coupled New Lorenz-Like circuits

4. Application to Secure Communication Systems

A true square wave is a signal of periodic recurrence made up of an infinite number of odd harmonics of the fundamental frequency. The general equation of the square wave can be written [31]:

1

)12(sin12

14)(

n

s xnn

tm (6)


(a) (b)

Figure 9: (a) Synchronization phase portrait y2 versus y1 and (b) Time series of signal y, with MultiSIM 10.0

The message signal ms (t) is a square wave and tx . The sum of the signal ms(t) and the chaotic signal mNewLorenz–Like(t), produced by the New Lorenz-Like circuit, is a new encryption signal mencryption, which is given by Eq.(7).

Encryption S NewLorenz Like( ) (t) (t)

m t m m (7)

The signal mNewLorenz–Like(t) is one of the parameters of equation (1). After finishing the encryption process the original signal can be recovered with the following procedure.

New_Signal Encryption NewLorenz Like( ) (t) (t)

m t m m (8)

So, mNew_Signal(t) is the original signal and must be the same with ms(t). Due to

the fact that the input signal can be recovered from the output signal, it turns out that it is possible to implement a secure communication system using the proposed chaotic system.

4.1 Numerical Simulations Figures 10 (a)-(c) show the MATLAB 2010 numerical simulation results for

the proposed chaotic masking communication scheme.


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Figure 10: MATLAB 2010 simulation of New Lorenz-Like circuit masking communication system when amplitude 0.5 V and frequency 2KHz:

(a) Information signal, (b) Chaotic masking transmitted signal, (c) Retrieved signal 4.2 Analog Circuit Simulation in MultiSIM 10.0

The principle of a chaos-based secure communication scheme is the information signal, which is masked by a chaotic signal at the transmitter, and then sent it to the receiver by the public channel. Finally, the encrypted signal is decrypted at the receiver. In this scheme, the key issue is that the two identical chaos generators in the transmitter and the receiver end need to be synchronized [32]. Figure 11 shows the MultiSIM 10.0 simulation results for the masking signal communication system.

(a) (b) (c)

Figure 11: MultiSIM 10.0 outputs of New Lorenz-Like circuit masking communication

systems when amplitude 0.5 V and frequency 2KHz: (a) Information signal, (b) Chaotic masking transmitted signal, (c) Retrieved signal


Also, in the proposed masking scheme, the square wave signal of amplitude 0.5 V and frequency 2 KHz is added to the synchronizing driving chaotic signal in order to regenerate the original driving signal at the receiver. Thus, as it can be shown from Fig.11(c), the message signal has been perfectly recovered by using the signal masking approach through the synchronization of chaotic New Lorenz-Like circuits. Furthermore, simulation results with Multisim 10.0 have shown that the performance of chaotic New Lorenz-Like circuits in chaotic masking and message recovery is very satisfactory. Finally, Figure 12 shows the circuit schematic of implementing the New Lorenz-Like circuit chaotic masking communication scheme.

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Figure 12: New Lorenz-Like circuit masking communication circuit.

5. Conclusion In this paper the chaotic synchronization of two identical New Lorenz-Like circuits system has been investigated by implementing connection substitution technique. The proposed method of synchronization between chaotic circuits can be applied successfully to a secure communication scheme. Chaos synchronization and chaos masking were realized by using MATLAB 2010 and

Secure communications based on the synchronization … 391 MultiSIM 10.0 programs. The comparison between MATLAB 2010 and MultiSIM 10.0 simulation results demonstrate the effectiveness of the proposed secure communication scheme.

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