Secure Computer Communication Based on Chaotic Rössler Oscillators

We describe a communication scheme based on chaotic Rössler oscillators for transmission of secure messages via computers. The computers are synchronized through one of the channels via one of the variables of the Rössler system, while an information signal is transmitted through another channel by adding the message to another system variable. This scheme provides more stable communication because the information signal does not enter to the receiver and hence does not cause an error in synchronization. The method is tested with different types of information signals: audio, text, and image. INTRODUCTION One of the great achievements of the chaos theory is its application in secure communications. The chaos communication is based on synchronization of chaotic systems under suitable conditions if any one drives the other of the systems. Since Pecora and Carroll [1-3] have demonstrated that chaotic systems can be synchronized, the research in synchronization of couple chaotic circuits is carried out intensively and some interesting applications such as communication with chaos have come out of that research. Nowadays there exists a great interest in the use of nonlinear differentials equations, such as the Rössler and other oscillators, for application to secure communication [1-15]. The implementation of coupled chaotic systems to secure communication is based on synchronization of the chaotic signals. Chaos is understood as an irregular motion with no defined frequencies and a broaden Fourier transform spectrum. Deterministic chaotic systems have the property of being sensible to initial conditions. Trajectories of a chaotic system starting from very near initial conditions in phase space tend to diverge exponentially. Nevertheless it was demonstrated that certain chaotic systems can be connected such that their chaotic movements are synchronized [11-13]. The phenomenon of synchronization has been discovered in 1665 by Christian Huygens. Now synchronization is understood as an adjustment of the rhythm of oscillators [11]. Synchronization of coupled Rössler oscillators has been studied by Pecora and Carroll who have demonstrated the possibility for synchronous motion in this system [12]. Recently, synchronization of bistable chaotic Rössler oscillators has been demonstrated [16-18]. In this Letter we describe a scheme for secure communication based on two coupled chaotic Rössler oscillators. First, we analyze separately the dynamics of a single oscilla*Address correspondence to this author at the Universidad de Guadalajara, Centro Universitario de los Lagos, Lagos de Moreno, Jalisco, Mexico; E-mail: [email protected] tor when a control parameter is varied and then we investigate synchronization of the coupled system. Bifurcation diagrams of the output signal with respect to the control parameter are constructed. We demonstrate how secure communication can be realized by using two channels; the system is synchronized via one of the channel and an information signal is sent and recovered via another channel. We show that such a scheme improves synchronization of the coupled system. COMPUTATIONAL MODEL The computational model used is based on a chaotic Rössler oscillator. The master and slave oscillators can be described by the following model equations dx1 dt = x2 x3, (1) dx2 dt = x1 + A1x2 , (2) dx3 dt = A3 + x3 x1 A2 ( ), (3) dx1 ́ dt = x2 x3 ́, (4) dx2 ́ dt = x1 ́+A1x2 , (5) dx3 ́ dt = A3 + x3 ́ x1 ́ A2 ( ), (6) where x1, x2, x3 and x1 ́, x2 ́, x3 ́ are the state variables of the master and slave oscillator, respectively, A1 = = A3 = 0.2 are fixed parameters and A2 is a control parameter varied from 0 to 10. The system (1)-(6) is solved using a 6 th order RungeKutta algorithm for different values of A2. DYNAMICS BEHAVIOR OF SINGLE OSCILLATOR The dynamic behavior of the master system exhibits complex behavior defined by the control parameter A2. Fig. (1) shows the bifurcation diagram of the peak values of x1 of 42 The Open Electrical and Electronic Engineering Journal, 2008, Volume 2 García-López et al. the master oscillator. The diagram represents the well-know cascade of period-doubling bifurcations leading to chaos at A2 = 4.2. Fig. (1). Bifurcation diagram of master oscillator. TWO ELECTRONIC CIRCUITS In this work we apply a novel communication system proposed in [15] for secure computer communications. Master and slave systems are given by the differential equations (1) – (6). The scheme uses two variables x1 and x2. x1 of the master oscillator is used for message transmission and x1 ́ of the slave oscillator for message recuperation, i.e. x1sent = x1master + SInformation , (7) slave sent n Informatio x x S 1 1 = . (8) TWO COUPLED CHAOTIC RÖSSLER OSCILLATORS Fig. (2) shows the time series of the variables x2 and x2 ́ of the master and slave oscillators and the phase space plot of x2 and x2 ́ where one can see that the chaotic oscillators are completely synchronized. Fig. (2). Chaotic time series of master and slave variables and x2 versus x2 ́ demonstrated complete synchronization. The master and slave oscillators are identical. A small information signal is added to the chaotic signal of the master oscillator. If the oscillators are completely synchronized it becomes possible to recover the message. When the master and slave systems are completely synchronized, every variable of the slave system follows completely the same trajectory as the corresponding variable of the master system. SECURE COMMUNICATION WITH A CHAOTIC SYSTEM In Fig. (3) we show a schematic diagram of the message encoding system. Fig. (3). Diagram of the message encoding process. The codification process begins from transformation of a message to decimal numbers and then to binary numbers. After this conversion the message is a set of zeros and ones (Fig. 4). Then this signal of information is added to the chaotic signal of one of the variables of the master system. Fig. (4). The message first is encoded to the decimal numbers and then to the binary numbers. Fig. (5) illustrates the information encoding process. MESSAGE RECOVERING Fig. (6) shows the decoding process. The information signal is the rest of the transmitted signal over variable x1 ́ of the slave system (8). The procedure shown in Table 1 allows us to obtain the message in the form of binary numbers. These numbers are reordered to the rows of 8 bits, that are equivalent to ASCII characters, and thus the computer interprets each row and sends it to a screen in the form of characSecure Computer Communication Based The Open Electrical and Electronic Engineering Journal, 2008, Volume 2 43 ters which can form an electronic mail and can be sent to other computers. Fig. (5). Chaotic signal of the master oscillator, signal to transmit, and the transmitted signal with information. Fig. (6). Diagram of decoding process. Table 1. Recovered and Reordered Data to form ASCII Code RECOVERED SIGNAL INFORMATION VECTOR WHOLE PART MATRIX ASCII CODE