Ubiquitous Computing and Communication Journal LORENZ-BASED CHAOTIC SECURE COMMUNICATION SCHEMES I.A. Kamil and O.A. Fakolujo Department of Electrical and Electronic Engineering University of Ibadan, Nigeria ismaila.kamil@ui.edu.ng ABSTRACT Secure communication systems employing chaos have recently attracted significant interest. This is partly due to their high unpredictability and simplicity of implementation over conventional secure communications systems. This study presents the implementation of four chaotic modulation techniques employing Lorenz system as chaos generator. The techniques are Chaotic Masking (CM), Chaos Shift Keying (CSK), Chaos On-Off Keying (COOK), and Differential Chaos Shift Keying (DCSK). Simulations were carried out using Simulink in Matlab environment to implement these techniques. A qualitative evaluation of the transmitted signal waveforms in all the cases considered showed that DCSK gives the highest level of security followed by CSK while COOK gives the least level of security. Keyword: Secure communication, Chaos, Lorenz system, Modulation 1. INTRODUCTION schemes are complex in hardware [3,4]. A secure communication is not only a system where privacy is Recent years have witnessed appreciable growth in ensured, it must also ensure the integrity of the personal communications most especially in the area transmitted message i.e. the exact information meant of mobile communication and the internet [1,2]. Data for the receiver is received. encryption and security are essential ingredients of personal communication that are recently receiving Chaos based secure communication has been of attention because of the need to ensure that the much interest in the recent time since it offers information being sent is not intercepted by an potential advantage over conventional methods due unwanted listener. Besides, these are very essential to its simplicity [3] and high unpredictability which for protecting the content integrity of a message as means higher security. Besides, analog well as its copyright [2]. implementation is possible [5]. A secure communication system as it is generally Many chaotic secure communication schemes called, transforms the information signal in such a have been reported in literature but only a few of way that only an authorized receiver who has a prior them have actually witnessed practical implementation. This paper attempts to model and knowledge of the transformation parameters can simulate four of these schemes using Simulink in receive the information. The security of this Matlab. The choice of Simulink was to bring the information is a measure of the difficulty schemes as close to practical implementation as encountered by an unauthorized interceptor who possible since each Simulink block can easily be attempts to decode it. There have been a good replaced by a practical unit. The four schemes number of approaches to secure communications considered were Chaotic Masking, Chaos On-Off reported in the literature, but most of the commonly Keying, Chaos Shift Keying and Differential Shift employed conventional encryption and security Keying. Volume 7 Number 2 Page 1248 www.ubicc.org UNIVERSITY OF IBADAN LIBRARY Ubiquitous Computing and Communication Journal 2. THEORY mask signal ̂() which is subtracted from the transmitted signal r(t) to obtain the recovered 2.1. Background message signal (). Chaos communication is rather a new field in the communication research. It evolved from the study Assuming a noise free channel and perfect of chaotic dynamical systems, not only in synchronization between the two chaotic systems, mathematics, but also in physics or electrical s(t)=r(t), c(t)= ̂() and m(t)= (). engineering somewhere at the beginning of 1990 [6]. For higher security of the message signal, Yang Prior to this period, the evolution of chaos has caused reported that the message signal is typically made much euphoria among the mathematicians and about 20dB to 30dB weaker than the chaotic signal physicists, while the engineering community has [13]. observed the development with skepticism. Chaotic signals are irregular, aperiodic, uncorrelated, broadband, and impossible to predict 2.3.2 Chaos Shift Keying over long times. These properties coincide with the In this modulation scheme, the message signal, requirements for signals applied in conventional which is a digital signal, is used to switch the communication systems, in particular spread- transmitted signal between two statistically similar spectrum communications, multi-user attractors () and () which are respectively communications, and secure communication. used to encode bit 0 and bit 1 of the message signal. The two attractors are generated by two chaotic 2.2. Chaotic System systems with the same structure but different parameters [13, 14]. The chaotic system employed in this work is the At the receiver end, the received signal is Lorenz system One of the earliest indications of correlated with a synchronized reproduction of any chaotic behaviour was developed by Edward N. of the two chaotic signals used in the transmitter. The Lorenz in the 60’s [7]. [8] stated that the Lorenz message signal is recovered by low-pass filtering and system was published as a model of two-dimensional threshholding the synchronization error. The block convection in a horizontal layer of fluid heated from diagram representation of the scheme is shown in rd below. The original equations for this 3 –order non- Fig. 2. linear system are [9-12]: 2.3.3 Chaos On-Off Keying  = − +  Chaos On-Off Keying is similar to CSK in all respects except that only one chaotic signal is used in  =  −  −  (1) transmission of message signal. When the message  = − +  s i g n a l i s b i t 1 , t h e c h a o t i c s i g n a l is transmitted, but when the message signal is bit 0 no signal is where x, y and z are the variables and σ, r and b are transmitted. The same procedure is used in dimensionless parameters usually assumed positive. demodulating the received signal as in CSK as Varying the values of the parameters leads to series shown in Fig. 3. of bifurcation and eventually chaos. Typical parameter values are σ=10, b=8/3 and r=20 [9]. 2.3. Chaos Modulation Schemes Four modulation schemes considered in this paper are Chaotic Masking (CM), Chaos On-Off Keying c(t) s(t) r(t) (COOK), Chaos Shift Keying (CSK) and Differential chaotic channel chaotic Chaos Shift Keying (DCSK). m(t) messag 2.3.1 Chaotic Masking e ̂() In chaotic masking, two identical chaotic are used: () one at the transmitter end and the other at the recovered receiver. As shown in Fig. 1, the message signal m(t) message RECEIVER is added to the chaotic mask signal c(t) giving the TRANSMITTER transmitted signal s(t). The chaotic system at the Figure 1: Chaotic Masking receiver end produces another copy of the chaotic Volume 7 Number 2 Page 1249 www.ubicc.org UNIVERSITY OF IBADAN LIBRA chaotic synchronization RY Ubiquitous Computing and Communication Journal sample functions are correlated in the receiver and the decision is made by thresholding [15]. chaotic 3. SIMULATION system 1 channel chaotic 3.1. Lorenz system 0 system 0 Cuomo et al observed that a direct message implementation of Eq.(1) with an electronic circuit is signal difficult because the state variables in Eq.(1) occupy a wide dynamic range with values that exceed LPF and reasonable power supply limits [16]. However, this thresholding difficulty can be eliminated by a simple transformation of variables; specifically, for the TRANSMITTER recovered RECEIVER coefficients message σ, r, and b used, an appropriate transformation is u=x/10, v=y/10, and w=z/10. With this scaling, the Lorenz equations are transformed to: Figure 2: Chaos On-Off Keying  = ( − )  =  −  − 20 (2)  = 5 −  The above equation was implemented using Simulink with the parameter values taken as σ=16, chaotic 1() 1 () r=45.6, and b=4 .The time series for the three state system 1 channel chaotic 2() variables is shown in Fig. 5. chaotic system 2 () system 2 3.2. Self Synchronization of Lorenz system message signal The receiver is made up of two stable subsystems decomposed from the original system using Pecora LPF and & Carrol Scheme [16-19]. In the second approach thresholding using v as the drive signal, the first subsystem, (u′), is  () given by: TRANSMITTER recovered RECEIVER ′ = ( − ′) (3) message The second response subsystem, ( ′, ′), is given by: Figure 3: Chaos Shift Keying  = ′ − ′ − 20′′  ′ = 5′′ − ′ (4) 2.3.4 Differential Chaos Shift Keying message In Differential Chaos Shift Keying, no signal synchronization is required as in the other three schemes earlier described. The same chaotic signal -1 channel used at the transmitter (called reference signal) is ! transmitted and used to demodulate the message " chaotic !" # signal at the receiver end. This is illustrated in Fig. 4. # system Delay Delay In this scheme, every bit is transmitted two sample Block Block Clock, Tb functions. The first sample function serves as the Clock, Tb reference while the second one carries the information. Thus, bit 1 is sent by transmitting the TRANSMITTER RECEIVER reference signal twice in succession and bit 0 is sent by transmitting the reference signal followed by an inverted copy of the reference signal. The two Figure 4: Differential Chaos Shift Keying Volume 7 Number 2 Page 1250 www.ubicc.org UNIVERSIT chaotic synchronization Y chaotic synchronization OF IBADAN LIBRAR LPF an d thresho lding Y R ecovered m essage signa l Ubiquitous Computing and Communication Journal The complete response system is therefore given by: and for the receiver, the initial conditions were (0) = 250, (0) = 1 and (0) = 1. A ′ = ( − ′) parameter variation of 0.1 was also introduced ′ =  − ′ − 20′′ (5) b e t w een the transmitter and receiver systems. The time series and orbit difference for the two systems  ′ = 5′′ −  are as shown in Fig. 6. Since the two subsystems are stable,  3.3. Chaos Modulation Schemes ≈′ as t→∞. Thus synchronization is achieved. The four schemes earlier described were modeled The transmitter and the receiver systems were and simulated with Simulink using self-synchronized modeled with Simulink. For the transmitter, the Lorenz system. The simulation results are shown in initial conditions were u(0)=200, v(0)=1 and w(0)=1 Figs. 7 to 10. (ay) 5 0 -5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -4 x 10 (b) 5 0 -50 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -4 (c) x 1 0 4 2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -4 Time x 10 Figure 5: Lorenz system time series (a) u (b) v (c) w (a ) y 5 v' v 0 -5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ((ab)) -4x 10 2 0 -2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time -4 (b) x 10 Figure 6: Self synchronization of two Lorenz systems using v as drive signal with different initial conditions and parameter values, (a) Time series of v and v’ (b) Synchronization Error. Volume 7 Number 2 Page 1251 www.ubicc.org U (vN'-v) v/v ' w v uIVERSITY OF IBADAN LIBRARY Ubiquitous Computing and Communication Journal 5 (a) 0 -5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5 -3x 10 ( b ) 0 -5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5 -3 (c) x 100 -5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -3 0.5 x 10 (d) 0 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1.5 -3x 10 (e) 10.5 0-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time -3x 10 Figure 7: Chaotic Masking using Lorenz systems (a) chaotic signal (b) transmitted signal (c) recovered message signal with synchronization error (d) transmitted message signal (e) recovered message signal 5 0 (a) -50 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -3 x 10 10 (b) 0 -100 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -3x 10 (c) 1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -3 1.5 x 10 1 (d) 0.5 0 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -3 x 10 1.5 (e) 1 0.5 0 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time -3 x 10 Figure 8: Chaos Shift Keying using Lorenz systems (a) transmitted signal (b) correlated signal (c) thresholded and filtered signal (d) transmitted message signal (e) recovered message signal Volume 7 Number 2 Page 1252 www.ubicc.org UNIVERSITY OF IBADAN LIBRARY Ubiquitous Computing and Communication Journal 5 (a) 0 -5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -3 x 10 5 0 (b) -5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -3 x 10 2 (c) 1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -3 x 10 1.5 (d) 1 0.5 0 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -3 x 101.5 (e) 10.5 0 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -3 x 10 Tim e Figure 9: Chaos On-Off Keying using Lorenz systems (a) transmitted signal (b) correlated signal (c) thresholded and filtered signal (d) transmitted mes sage signal (e) recovered message signal 5 (a) 0 -50 0.001 0.002 0.003 0.004 0.00 5 0.006 0.007 0.008 0.009 0.01 (b) 5 0 -5 0 0.001 0.002 0.003 0.004 0.00 5 0.006 0.007 0.008 0.009 0.01 (c) 10 0 -10 0 0.001 0.002 0.003 0.004 0.00 5 0.006 0.007 0.008 0.009 0.01 (d) 100 -10 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 2 (e) 0 -20 0.001 0.002 0.003 0.004 0.00 5 0.006 0.007 0.008 0.009 0.01 (f) 1.5 0. 1 05 -0.5 0 0.001 0.002 0.003 0.004 0.00 5 0.006 0.007 0.008 0.009 0.01 (g) 1.5 0.105 -0.5 0 0.001 0.002 0.003 0.004 0.00 5 0.006 0.007 0.008 0.009 0.01 Time Figure 10: Differential Chaos Shift Keying using Lorenz systems (a) chaotic signal (b) transmitted signal (c) correlated signal (d) thresholded signal (e) filtered signal (f) trans mitted message signal (g) recovered message signal Volume 7 Number 2 Page 1253 www.ubicc.org UNIVERSITY OF IBADAN LIBRARY Ubiquitous Computing and Communication Journal 4. DISCUSSIONS [7] E. N. Lorenz, "Deterministic non-periodic flow," Journal of Atmo. Sci., vol. Vol. 20, pp. The results obtained in Fig 6 showed that a pp. 130-141, 1963. difference in initial conditions and slight parameter [8] P. G. Drazin, Nonlinear Systems Cambridge variation that would otherwise cause the two chaotic University Press., 1992. systems to produce divergent time series, had no [9] Y. Gauthier, "Application of the Lorenz effect when the two were synchronized using self Chaotic System to Secure Communication synchronization approach. and Encryption," Carleton University., 1998. Figs. 7, 8, 9 and 10 confirmed the effectiveness of [10] http://en.wikipedia.org/wiki/Lorenz_attractor, the four modulation schemes as the message signals "Lorenz Attractor," Wikipedia, the free were recovered at the receiver. The transmitted signal encyclopedia, 2008. waveforms confirmed the security of the chaos [11] C. Sparrow, "The Lorenz Equations," Chaos, modulation schemes. It could be observed that DCSK ed. A.V. Holden, Princeton Univ. Press, provided the highest security followed by chaotic 1986. masking. COOK provided the lowest level of security. [12] E. Sánchez and M. A. Maltiás, "Transition to The data transmission rate of DCSK was however Chaotic Rotating Waves in Arrays of twice those of others. Coupled Lorenz Oscillators," Int. Jour. of Bifurc. & Chaos, vol. Vol. 9, pp. pp. 2335- 5. CONCLUSSION 2343, 1999. [13] T. Yang, "A Survey of Chaotic Secure We have discussed in this paper the use of Simulink Communication Systems," Int. Jour. of to demonstrate various chaotic secure communication Comp. Cognition, vol. Vol. 2, pp. pp 81-130, schemes. We have assumed an ideal noiseless 2004. communication channel in this study. Further work is [14] H. Yu and H. Leung, "A Comparative Study on going to demonstrate same for a practical noisy of Different Chaos Based Spread Spectrum channel. Communication Systems," Proc. IEEE Int. Symp. Cct. & Syst. 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