- Email: [email protected]
A secure communication scheme using projective chaos synchronization
Chaos, Solitons and Fractals 22 (2004) 477–481 www.elsevier.com/locate/chaos
A secure communication scheme using projective chaos synchronization Zhigang Li b
a,*
, Daolin Xu
b
a Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA School of Mechanical and Production Engineering, Nanyang Technological University, Singapore 639798, Singapore
Accepted 24 February 2004
Abstract Most secure communication schemes using chaotic dynamics are based on identical synchronization. In this paper, we show the possibility of secure communication using projective synchronization (PS). The unpredictability of the scaling factor in projective synchronization can additionally enhance the security of communication. It is also showed that the scaling factor can be employed to improve the robustness against noise contamination. The feasibility of the communication scheme in high-dimensional chaotic systems, such as the hyperchaotic R€ ossler system, is demonstrated. Numerical results show the success in transmitting a sound signal through chaotic systems. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Since the seminal work by Pecora and Carrol [1], synchronization of chaotic systems has aroused much interest. In particular, chaos synchronization has been widely investigated for applications in secure communication [2–16]. The idea is that chaotic signal can be used as a carrier and transmitted together with an information signal to a receiver. At the receiver end chaos synchronization is employed to recover the information signal. Secure communication can be implemented through many ways. A simple method is to add the information signal to the chaotic carrier [2,4,6] and use the combined signal to drive the receiver. The combined signal cannot produce a complete synchronization in the receiver. In this case, error occurs in the recorded information signal. Either the information signal can only be recovered with limited accuracy or the ratio of information signal to chaotic signal is restricted. Another approach was discussed in [5,6,8], where the information signal is used to drive the transmitter and the scalar signal transmitted to the receiver is a function of the transmitter variables and information signal. If synchronization happens between the transmitter and receiver, information signal can be recovered exactly. The restriction of this method is that the signal that is input into the transmitter has to be chosen carefully to ensure that the transmitter and receiver remain chaotic. For digital signal, a parametric modulation method can be used and it has been examined by many researchers [2,3,7]. All these methods, however, either have inherent drawbacks or have been shown susceptible to attack by a determined intruder using predictive modeling and noise reduction methods from nonlinear dynamics [17–19]. In this paper, we introduce a new scheme that uses the idea of projective synchronization (PS) [20–25]. In this scheme, information signal can be transmitted in a form of any function pre-designed between a sender and a receiver, and recovered exactly. Since the function is arbitrary and the scaling factor in PS is unpredictable [22], the possibility for an interceptor to extract the information from the transmitted signal is therefore greatly reduced. The idea of the communication scheme will be addressed in the following part of the paper. In numerical applications, we shall provide
*
Corresponding author. E-mail address: [email protected] (Z. Li).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.02.019
478
Z. Li, D. Xu / Chaos, Solitons and Fractals 22 (2004) 477–481
an example to illustrate the implementation of the scheme for transmitting sound signals of a water flow. Furthermore, we shall discuss the robustness of the method against noise. As indicated in literature [15,18], one reason for decoding the transmitted signal is that the chaotic systems used in the communication system are low-dimensional ones, whose attractors have easily identifiable structure. This directs the attention of secure communication to higher-dimensional systems [26,27]. Therefore, we illustrate the application of our method to hyperchaotic R€ ossler system to show the feasibility of our method in high-dimensional systems. 2. Encryption scheme Our method is based on PS, which has been investigated in [20–25]. In PS, two identical chaotic systems are coupled through one of the state variables and the other variables synchronize up to a scaling factor. Generally, PS occurs in partially linear chaotic systems [20]. However, further studies [23,24] showed that it could be produced in any chaotic system if a control algorithm is applied to the response system. Consider a general form of chaotic system, for instance, of three dimensions X_ ¼ hðXÞ;
ð1Þ
where state vector X ¼ ðx; y; zÞ. Decompose X into a vector u ¼ ðx; yÞ and a scalar z, system (1) can be rewritten as u ¼ fðu; zÞ; z ¼ gðu; zÞ:
ð2Þ
In PS, two identical system X1 (drive) and X2 (response) are coupled through the scalar variable z (i.e., the two systems share the same variable z). For partially linear systems, PS may automatically happen provided that the systems satisfy the stability conditions [21,25]. For any nonpartially linear system, PS could be generated if a control is applied to the response system [24]. In PS, the state vectors u1 and u2 synchronize up to a constant ratio, i.e., limt!1 jau1 u2 j ¼ 0, where a is called scaling factor. A sketch designed for our communication scheme using PS is shown in Fig. 1. System X1 is used as the transmitter and X2 as the receiver. The signal m0 ðtÞ, which is formulated as a function of the information signal mðtÞ as well as the state variables of the transmitter, defined by m0 ðtÞ ¼ F ½x1 ; y1 ; z; mðtÞ, is added to the variable x1 (or y1 ). Both the combined signal U ðtÞ ¼ x1 þ m0 ðtÞ and the scalar variable z are transmitted to the receiver. At the receiver end, variable z serves as a driving signal that enables the systems X1 and X2 to synchronize together. With the occurrence of PS, u2 will approach to au1 . The scaling factor a could be a predefined value or any desired value to be directed by a feedback control [22–24], such that m0 ðtÞ can be extracted through a simple transformation m0 ðtÞ ¼ U ðtÞ x2 =a. If the function F is invertible, the information signal mðtÞ can be recovered by mðtÞ ¼ F 1 ½x2 ; y2 ; m0 ðtÞ.
3. Numerical experiments and discussion In the following, we will illustrate the communication scheme discussed above using the Lorenz system [28], which is given by x_ ¼ rðy xÞ; y_ ¼ ðl zÞx y;
ð3Þ
z_ ¼ xy qz;
x1 + X1
y1 z
Transmitter (Drive system)
m′(t)= F [x1 , y1 , z , m(t)]
x2
U (t)= x1 + m ′(t) X2
1α
−
+ m ′(t)
F −1
m(t)
y2 z
Receiver (Response system)
Fig. 1. Communication scheme based on projective synchronization of chaotic systems.
Z. Li, D. Xu / Chaos, Solitons and Fractals 22 (2004) 477–481
479
where the parameters are set to r ¼ 10, l ¼ 60, and q ¼ 8=3 so that the system behaves chaotically. The information signal mðtÞ is the sound signal of a water flow and the function F is given by m0 ðtÞ ¼ y1 þ mðtÞ. The scaling factor a is set to be 5. Fig. 2 depicts the numerical results for the Lorenz system. The information signal mðtÞ and the transmitted signal m0 ðtÞ are shown in (a) and (b) respectively. The recovered information signal, which is denoted by m ðtÞ, is shown in (c). Fig. 2(d) displays the error between the original information signal and the recovered one. In (d), it is easy to find that the information signal mðtÞ is recovered exactly after a short transient. In the method presented in [2,4], information signal is added to chaotic carrier and the combined signal is used to drive the receiver. Since the driving signal is not purely chaotic the receiver cannot exactly synchronize with the transmitter. The dynamic error eðtÞ ¼ x2 x1 will persist as t ! 1 and the information signal cannot be recovered accurately. To reduce the error eðtÞ, the ratio between the information signal jmðtÞj and chaotic carrier is required to be very small. This limitation makes their communication method vulnerable to the presence of the noise inherent in the communication system. In contrast, our scheme avoids such problem because the driving signal is z that is purely chaotic. In this case, synchronization can be completely realized and the information signal can be recovered accurately. Furthermore, this scheme offers much more flexibility in signal transmission and makes a third party difficult to break the information signal not only because the function F is arbitrary and the variation of both U ðtÞ and z is erratic but also the scaling factor is unknown to the interceptor. The proposed scheme has sound capability against noise contamination in information channel. As discussed before, the communication scheme does not use a combined signal to drive the receiver so that the selection of the magnitude of the information signal does not affect the recovery process (synchronization). Thus we can easily increase the ratio of information signal to noise by amplifying the information signal mðtÞ with a multiple k. In this case, transmitted signal is m0 ðtÞ ¼ F ½x1 ; y1 ; z; k; mðtÞ. The effect of noise n in the information channel can be greatly reduced because mðtÞ will be recovered by m ðtÞ ¼ ½kmðtÞ þ n=k. If k ! 1, m ðtÞ ! mðtÞ. Nevertheless, the noise effect in the z channel that drives the receiver is unavoidable, which may decline the quality of synchronization. An example of the noise effect in the transmission of the sound signal (see Fig. 2a) of a water flow is depicted in Fig. 3. The multiple is set at k ¼ 1 in (a) and k ¼ 100 in (b). The variance of noise is 0.001 and added to both the U ðtÞ channel and the z channel. Comparing with the results in Fig. 3a and b, it is seen that the effect of noise is greatly depressed with a large multiple k ¼ 100. It is worthwhile mentioning that the multiple k cannot be set too large for the sake of security in practice. Otherwise, the chaotic carrier cannot completely mask the information signal. It has been suggested that secure communication can be improved by using hyperchaotic systems due to the increased unpredictability and the much more complicated structure of the attractors. To explore the feasibility of the
Fig. 2. Transmission of a sound signal through the chaotic carrier produced by the Lorenz system: (a) sound signal mðtÞ of a water flow; (b) transmitted signal m0 ðtÞ ¼ y1 þ mðtÞ; (c) recovered signal m ðtÞ and (d) error between the recovered and the original information signal.
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Z. Li, D. Xu / Chaos, Solitons and Fractals 22 (2004) 477–481
0
-2
(a)
log | m*(t) - m (t)|
log|m*(t) - m (t)|
k=1 -2 -4 -6
0
1
2
-4 -6 -8
3
(b)
k = 100
t
0
1
2
t
3
Fig. 3. The effect of noise on the recovery of information signal. Noise variance is 0.001 and contaminates both the information channel and z channel. Errors of recovered signal in the cases: (a) k ¼ 1, and (b) k ¼ 100, where k is a multiple assigned to information signal.
communication scheme in high-dimensional cases, we use hyperchaotic R€ ossler system as the transmitter and receiver to transmit the sound signal of a water flow as done in Fig. 2. The hyperchaotic R€ ossler system [29] is given by x_ ¼ y z; y_ ¼ x þ ay þ w;
ð4Þ
z_ ¼ b þ xz; w_ ¼ cz þ dw;
where the parameters are set as a ¼ 0:25, b ¼ 3, c ¼ 0:5, d ¼ 0:05. Fig. 4 shows the information signal in (a) and the transmitted signal in (b). The recovered information signal and the error between the initial information signal and the recovered one are depicted in (c) and (d). It is easy to find that the information signal can still be recovered accurately.
0.4 0.4
(c)
(a)
0.2 m (t)
m(t)
0.2 0.0
-0.2
-0.2 -0.4
0.0
-0.4 0
2
4
6
8
0
10
2
4
30
log |m*(t) - m (t)|
*
m (t)
-30 -60
10
20
30 t
8
10
40
50
(d)
0
(b)
0
-90 0
6 t
t
-6 -12 -18 0
1
2
3
t
Fig. 4. Transmission of a sound signal through the chaotic carrier produced by the hyperchaotic R€ ossler system: (a) sound signal mðtÞ. of a water flow, which is the same as that in Fig. 2; (b) transmitted signal m0 ðtÞ ¼ y1 þ mðtÞ; (c) recovered signal m ðtÞ and (d) error between the recovered and the original information signal. For the sake of clarity, (a) and (c) are depicted only up to t ¼ 10.
Z. Li, D. Xu / Chaos, Solitons and Fractals 22 (2004) 477–481
481
4. Conclusion In conclusion, we have presented a scheme based on the idea of projective synchronization for secure communication. The information signal transmitted between a sender and a receiver can be recovered accurately through simple operation and the security of information can be guaranteed because the function F could be arbitrary and the scaling factor a in projective synchronization is hardly predictable to the interceptor. In addition, we discussed the robustness of the communication scheme against noise. The effect of noise in information recovery process can be greatly reduced by amplifying the magnitude of the information signal. Finally, we showed that the scheme could be applied in highdimensional chaotic systems, such as the hyperchaotic R€ ossler system. Numerical results showed the successfulness in transmitting a sound signal through chaotic carriers.
References [1] Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–4. [2] Kocarev L, Halle KS, Eckert K, Chua LO, Parlitz U. Experimental demonstration of secure communications via chaotic synchronization. Int J Bifurcat Chaos 1992;2:709–13. [3] Parlitz U, Chua L, Kocarev LJ, Halle KS, Shang A. Transmission of digital signals by chaotic synchronization. Int J Bifurcat Chaos 1992;2:973–7. [4] Cuomo KM, Oppenheim AV. Circuit implementation of synchronized chaos with applications to communications. Phys Rev Lett 1993;71:65–8. [5] Lu J, Wu XQ, Lu JH. Synchronization of a unified chaotic system and the application in secure communication. Phys Lett A 2002;305:365–70. [6] Cuomo KM, Oppenheim AV, Strogatz SH. Robustness and signal recovery in a synchronized chaotic system. Int J Bifurcat Chaos 1993;3:1629. [7] Feki M, Robert B, Gelle G, Colas M. Secure digital communication using discrete-time chaos synchronization. Chaos, Solitons & Fractals 2003;18:881–90. [8] Kocarev L, Parlitz U. General approach for chaotic synchronization with applications to communication. Phys Rev Lett 1995;74:5028–31. [9] Parlitz U, Kocarev L, Stojanovski T, Preckel H. Encoding messages using chaotic synchronization. Phys Rev E 1996;53:4351–61. [10] Yang T, Chua LO. Channel-independent chaotic secure communication. Int J Bifurcat Chaos 1996;6:2653–60. [11] Xiao JH, Hu G, Qu ZL. Synchronization of spatiotemporal chaos and its application to multichannel pread-spectrum communication. Phys Rev Lett 1996;77:4162–5. [12] Boccaletti S, Farini A, Arecchi FT. Adaptive synchronization of chaos for secure communication. Phys Rev E 1997;55:4979–81. [13] Carroll TL, Pecora LM. Synchronizing hyperchaotic volume-preserving maps and circuits. IEEE Trans Circuits Systems I–– Fundament Theory Appl 1998;45:656–9. [14] Minai AA, Anand T. Synchronization of chaotic maps through a noisy coupling channel with application to digital communication. Phys Rev E 1999;59:312–20. [15] Sundar S, Minai AA. Synchronization of randomly multiplexed chaotic systems with application to communication. Phys Rev Lett 2000;85:5456–9. [16] Garcia-Ojalvo J, Roy R. Spatiotemporal communication with synchronized optical chaos. Phys Rev Lett 2001;86:5204–7. [17] Short KM. Steps toward unmasking secure communications. Int J Bifurcat Chaos 1994;4:959–77. [18] Perez G, Cerdeira HA. Extracting messages masked by chaos. Phys Rev Lett 1995;74:1970–3. [19] Short KM. Unmasking a modulated chaotic communications scheme. Int J Bifurcat Chaos 1996;6:367–75. [20] Mainieri R, Rehacek J. Projective synchronization in three-dimensional chaotic systems. Phys Rev Lett 1999;82:3042–5. [21] Li ZG, Xu DL. Stability criterion for projective synchronization in three-dimensional chaotic systems. Phys Lett A 2001;282:175– 9. [22] Xu DL. Control of projective synchronization in chaotic systems. Phys Rev E 2001;63:027201. [23] Xu DL, Li ZG, Bishop SR. Manipulating the scaling factor of projective synchronization in three-dimensional chaotic systems. Chaos 2001;11:439–42. [24] Xu DL, Li ZG. Controlled projective synchronization in nonpartially-linear chaotic systems. Int J Bifurcat Chaos 2002;12:1395– 402. [25] Xu DL, Ong WL, Li ZG. Criteria for the occurrence of projective synchronization in chaotic systems of arbitrary dimension. Phys Lett A 2002;305:167–72. [26] Kocarev L, Parlitz U, Stojanovski T. An application of synchronized chaotic dynamic arrays. Phys Lett A 1996;217:280–4. [27] Carroll TL, Heagy JF, Pecora LM. Transforming signals with chaotic synchronization. Phys Rev E 1996;54:4676–80. [28] Lorenz EN. Deterministic non-periodic flow. J Atmos Sci 1963;20:130–41. [29] R€ ossler OE. An equation for hyperchaos. Phys Lett A 1976;71:155–67.
A secure communication scheme using projective chaos synchronization Zhigang Li b
a,*
, Daolin Xu
b
a Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA School of Mechanical and Production Engineering, Nanyang Technological University, Singapore 639798, Singapore
Accepted 24 February 2004
Abstract Most secure communication schemes using chaotic dynamics are based on identical synchronization. In this paper, we show the possibility of secure communication using projective synchronization (PS). The unpredictability of the scaling factor in projective synchronization can additionally enhance the security of communication. It is also showed that the scaling factor can be employed to improve the robustness against noise contamination. The feasibility of the communication scheme in high-dimensional chaotic systems, such as the hyperchaotic R€ ossler system, is demonstrated. Numerical results show the success in transmitting a sound signal through chaotic systems. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Since the seminal work by Pecora and Carrol [1], synchronization of chaotic systems has aroused much interest. In particular, chaos synchronization has been widely investigated for applications in secure communication [2–16]. The idea is that chaotic signal can be used as a carrier and transmitted together with an information signal to a receiver. At the receiver end chaos synchronization is employed to recover the information signal. Secure communication can be implemented through many ways. A simple method is to add the information signal to the chaotic carrier [2,4,6] and use the combined signal to drive the receiver. The combined signal cannot produce a complete synchronization in the receiver. In this case, error occurs in the recorded information signal. Either the information signal can only be recovered with limited accuracy or the ratio of information signal to chaotic signal is restricted. Another approach was discussed in [5,6,8], where the information signal is used to drive the transmitter and the scalar signal transmitted to the receiver is a function of the transmitter variables and information signal. If synchronization happens between the transmitter and receiver, information signal can be recovered exactly. The restriction of this method is that the signal that is input into the transmitter has to be chosen carefully to ensure that the transmitter and receiver remain chaotic. For digital signal, a parametric modulation method can be used and it has been examined by many researchers [2,3,7]. All these methods, however, either have inherent drawbacks or have been shown susceptible to attack by a determined intruder using predictive modeling and noise reduction methods from nonlinear dynamics [17–19]. In this paper, we introduce a new scheme that uses the idea of projective synchronization (PS) [20–25]. In this scheme, information signal can be transmitted in a form of any function pre-designed between a sender and a receiver, and recovered exactly. Since the function is arbitrary and the scaling factor in PS is unpredictable [22], the possibility for an interceptor to extract the information from the transmitted signal is therefore greatly reduced. The idea of the communication scheme will be addressed in the following part of the paper. In numerical applications, we shall provide
*
Corresponding author. E-mail address: [email protected] (Z. Li).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.02.019
478
Z. Li, D. Xu / Chaos, Solitons and Fractals 22 (2004) 477–481
an example to illustrate the implementation of the scheme for transmitting sound signals of a water flow. Furthermore, we shall discuss the robustness of the method against noise. As indicated in literature [15,18], one reason for decoding the transmitted signal is that the chaotic systems used in the communication system are low-dimensional ones, whose attractors have easily identifiable structure. This directs the attention of secure communication to higher-dimensional systems [26,27]. Therefore, we illustrate the application of our method to hyperchaotic R€ ossler system to show the feasibility of our method in high-dimensional systems. 2. Encryption scheme Our method is based on PS, which has been investigated in [20–25]. In PS, two identical chaotic systems are coupled through one of the state variables and the other variables synchronize up to a scaling factor. Generally, PS occurs in partially linear chaotic systems [20]. However, further studies [23,24] showed that it could be produced in any chaotic system if a control algorithm is applied to the response system. Consider a general form of chaotic system, for instance, of three dimensions X_ ¼ hðXÞ;
ð1Þ
where state vector X ¼ ðx; y; zÞ. Decompose X into a vector u ¼ ðx; yÞ and a scalar z, system (1) can be rewritten as u ¼ fðu; zÞ; z ¼ gðu; zÞ:
ð2Þ
In PS, two identical system X1 (drive) and X2 (response) are coupled through the scalar variable z (i.e., the two systems share the same variable z). For partially linear systems, PS may automatically happen provided that the systems satisfy the stability conditions [21,25]. For any nonpartially linear system, PS could be generated if a control is applied to the response system [24]. In PS, the state vectors u1 and u2 synchronize up to a constant ratio, i.e., limt!1 jau1 u2 j ¼ 0, where a is called scaling factor. A sketch designed for our communication scheme using PS is shown in Fig. 1. System X1 is used as the transmitter and X2 as the receiver. The signal m0 ðtÞ, which is formulated as a function of the information signal mðtÞ as well as the state variables of the transmitter, defined by m0 ðtÞ ¼ F ½x1 ; y1 ; z; mðtÞ, is added to the variable x1 (or y1 ). Both the combined signal U ðtÞ ¼ x1 þ m0 ðtÞ and the scalar variable z are transmitted to the receiver. At the receiver end, variable z serves as a driving signal that enables the systems X1 and X2 to synchronize together. With the occurrence of PS, u2 will approach to au1 . The scaling factor a could be a predefined value or any desired value to be directed by a feedback control [22–24], such that m0 ðtÞ can be extracted through a simple transformation m0 ðtÞ ¼ U ðtÞ x2 =a. If the function F is invertible, the information signal mðtÞ can be recovered by mðtÞ ¼ F 1 ½x2 ; y2 ; m0 ðtÞ.
3. Numerical experiments and discussion In the following, we will illustrate the communication scheme discussed above using the Lorenz system [28], which is given by x_ ¼ rðy xÞ; y_ ¼ ðl zÞx y;
ð3Þ
z_ ¼ xy qz;
x1 + X1
y1 z
Transmitter (Drive system)
m′(t)= F [x1 , y1 , z , m(t)]
x2
U (t)= x1 + m ′(t) X2
1α
−
+ m ′(t)
F −1
m(t)
y2 z
Receiver (Response system)
Fig. 1. Communication scheme based on projective synchronization of chaotic systems.
Z. Li, D. Xu / Chaos, Solitons and Fractals 22 (2004) 477–481
479
where the parameters are set to r ¼ 10, l ¼ 60, and q ¼ 8=3 so that the system behaves chaotically. The information signal mðtÞ is the sound signal of a water flow and the function F is given by m0 ðtÞ ¼ y1 þ mðtÞ. The scaling factor a is set to be 5. Fig. 2 depicts the numerical results for the Lorenz system. The information signal mðtÞ and the transmitted signal m0 ðtÞ are shown in (a) and (b) respectively. The recovered information signal, which is denoted by m ðtÞ, is shown in (c). Fig. 2(d) displays the error between the original information signal and the recovered one. In (d), it is easy to find that the information signal mðtÞ is recovered exactly after a short transient. In the method presented in [2,4], information signal is added to chaotic carrier and the combined signal is used to drive the receiver. Since the driving signal is not purely chaotic the receiver cannot exactly synchronize with the transmitter. The dynamic error eðtÞ ¼ x2 x1 will persist as t ! 1 and the information signal cannot be recovered accurately. To reduce the error eðtÞ, the ratio between the information signal jmðtÞj and chaotic carrier is required to be very small. This limitation makes their communication method vulnerable to the presence of the noise inherent in the communication system. In contrast, our scheme avoids such problem because the driving signal is z that is purely chaotic. In this case, synchronization can be completely realized and the information signal can be recovered accurately. Furthermore, this scheme offers much more flexibility in signal transmission and makes a third party difficult to break the information signal not only because the function F is arbitrary and the variation of both U ðtÞ and z is erratic but also the scaling factor is unknown to the interceptor. The proposed scheme has sound capability against noise contamination in information channel. As discussed before, the communication scheme does not use a combined signal to drive the receiver so that the selection of the magnitude of the information signal does not affect the recovery process (synchronization). Thus we can easily increase the ratio of information signal to noise by amplifying the information signal mðtÞ with a multiple k. In this case, transmitted signal is m0 ðtÞ ¼ F ½x1 ; y1 ; z; k; mðtÞ. The effect of noise n in the information channel can be greatly reduced because mðtÞ will be recovered by m ðtÞ ¼ ½kmðtÞ þ n=k. If k ! 1, m ðtÞ ! mðtÞ. Nevertheless, the noise effect in the z channel that drives the receiver is unavoidable, which may decline the quality of synchronization. An example of the noise effect in the transmission of the sound signal (see Fig. 2a) of a water flow is depicted in Fig. 3. The multiple is set at k ¼ 1 in (a) and k ¼ 100 in (b). The variance of noise is 0.001 and added to both the U ðtÞ channel and the z channel. Comparing with the results in Fig. 3a and b, it is seen that the effect of noise is greatly depressed with a large multiple k ¼ 100. It is worthwhile mentioning that the multiple k cannot be set too large for the sake of security in practice. Otherwise, the chaotic carrier cannot completely mask the information signal. It has been suggested that secure communication can be improved by using hyperchaotic systems due to the increased unpredictability and the much more complicated structure of the attractors. To explore the feasibility of the
Fig. 2. Transmission of a sound signal through the chaotic carrier produced by the Lorenz system: (a) sound signal mðtÞ of a water flow; (b) transmitted signal m0 ðtÞ ¼ y1 þ mðtÞ; (c) recovered signal m ðtÞ and (d) error between the recovered and the original information signal.
480
Z. Li, D. Xu / Chaos, Solitons and Fractals 22 (2004) 477–481
0
-2
(a)
log | m*(t) - m (t)|
log|m*(t) - m (t)|
k=1 -2 -4 -6
0
1
2
-4 -6 -8
3
(b)
k = 100
t
0
1
2
t
3
Fig. 3. The effect of noise on the recovery of information signal. Noise variance is 0.001 and contaminates both the information channel and z channel. Errors of recovered signal in the cases: (a) k ¼ 1, and (b) k ¼ 100, where k is a multiple assigned to information signal.
communication scheme in high-dimensional cases, we use hyperchaotic R€ ossler system as the transmitter and receiver to transmit the sound signal of a water flow as done in Fig. 2. The hyperchaotic R€ ossler system [29] is given by x_ ¼ y z; y_ ¼ x þ ay þ w;
ð4Þ
z_ ¼ b þ xz; w_ ¼ cz þ dw;
where the parameters are set as a ¼ 0:25, b ¼ 3, c ¼ 0:5, d ¼ 0:05. Fig. 4 shows the information signal in (a) and the transmitted signal in (b). The recovered information signal and the error between the initial information signal and the recovered one are depicted in (c) and (d). It is easy to find that the information signal can still be recovered accurately.
0.4 0.4
(c)
(a)
0.2 m (t)
m(t)
0.2 0.0
-0.2
-0.2 -0.4
0.0
-0.4 0
2
4
6
8
0
10
2
4
30
log |m*(t) - m (t)|
*
m (t)
-30 -60
10
20
30 t
8
10
40
50
(d)
0
(b)
0
-90 0
6 t
t
-6 -12 -18 0
1
2
3
t
Fig. 4. Transmission of a sound signal through the chaotic carrier produced by the hyperchaotic R€ ossler system: (a) sound signal mðtÞ. of a water flow, which is the same as that in Fig. 2; (b) transmitted signal m0 ðtÞ ¼ y1 þ mðtÞ; (c) recovered signal m ðtÞ and (d) error between the recovered and the original information signal. For the sake of clarity, (a) and (c) are depicted only up to t ¼ 10.
Z. Li, D. Xu / Chaos, Solitons and Fractals 22 (2004) 477–481
481
4. Conclusion In conclusion, we have presented a scheme based on the idea of projective synchronization for secure communication. The information signal transmitted between a sender and a receiver can be recovered accurately through simple operation and the security of information can be guaranteed because the function F could be arbitrary and the scaling factor a in projective synchronization is hardly predictable to the interceptor. In addition, we discussed the robustness of the communication scheme against noise. The effect of noise in information recovery process can be greatly reduced by amplifying the magnitude of the information signal. Finally, we showed that the scheme could be applied in highdimensional chaotic systems, such as the hyperchaotic R€ ossler system. Numerical results showed the successfulness in transmitting a sound signal through chaotic carriers.
References [1] Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–4. [2] Kocarev L, Halle KS, Eckert K, Chua LO, Parlitz U. Experimental demonstration of secure communications via chaotic synchronization. Int J Bifurcat Chaos 1992;2:709–13. [3] Parlitz U, Chua L, Kocarev LJ, Halle KS, Shang A. Transmission of digital signals by chaotic synchronization. Int J Bifurcat Chaos 1992;2:973–7. [4] Cuomo KM, Oppenheim AV. Circuit implementation of synchronized chaos with applications to communications. Phys Rev Lett 1993;71:65–8. [5] Lu J, Wu XQ, Lu JH. Synchronization of a unified chaotic system and the application in secure communication. Phys Lett A 2002;305:365–70. [6] Cuomo KM, Oppenheim AV, Strogatz SH. Robustness and signal recovery in a synchronized chaotic system. Int J Bifurcat Chaos 1993;3:1629. [7] Feki M, Robert B, Gelle G, Colas M. Secure digital communication using discrete-time chaos synchronization. Chaos, Solitons & Fractals 2003;18:881–90. [8] Kocarev L, Parlitz U. General approach for chaotic synchronization with applications to communication. Phys Rev Lett 1995;74:5028–31. [9] Parlitz U, Kocarev L, Stojanovski T, Preckel H. Encoding messages using chaotic synchronization. Phys Rev E 1996;53:4351–61. [10] Yang T, Chua LO. Channel-independent chaotic secure communication. Int J Bifurcat Chaos 1996;6:2653–60. [11] Xiao JH, Hu G, Qu ZL. Synchronization of spatiotemporal chaos and its application to multichannel pread-spectrum communication. Phys Rev Lett 1996;77:4162–5. [12] Boccaletti S, Farini A, Arecchi FT. Adaptive synchronization of chaos for secure communication. Phys Rev E 1997;55:4979–81. [13] Carroll TL, Pecora LM. Synchronizing hyperchaotic volume-preserving maps and circuits. IEEE Trans Circuits Systems I–– Fundament Theory Appl 1998;45:656–9. [14] Minai AA, Anand T. Synchronization of chaotic maps through a noisy coupling channel with application to digital communication. Phys Rev E 1999;59:312–20. [15] Sundar S, Minai AA. Synchronization of randomly multiplexed chaotic systems with application to communication. Phys Rev Lett 2000;85:5456–9. [16] Garcia-Ojalvo J, Roy R. Spatiotemporal communication with synchronized optical chaos. Phys Rev Lett 2001;86:5204–7. [17] Short KM. Steps toward unmasking secure communications. Int J Bifurcat Chaos 1994;4:959–77. [18] Perez G, Cerdeira HA. Extracting messages masked by chaos. Phys Rev Lett 1995;74:1970–3. [19] Short KM. Unmasking a modulated chaotic communications scheme. Int J Bifurcat Chaos 1996;6:367–75. [20] Mainieri R, Rehacek J. Projective synchronization in three-dimensional chaotic systems. Phys Rev Lett 1999;82:3042–5. [21] Li ZG, Xu DL. Stability criterion for projective synchronization in three-dimensional chaotic systems. Phys Lett A 2001;282:175– 9. [22] Xu DL. Control of projective synchronization in chaotic systems. Phys Rev E 2001;63:027201. [23] Xu DL, Li ZG, Bishop SR. Manipulating the scaling factor of projective synchronization in three-dimensional chaotic systems. Chaos 2001;11:439–42. [24] Xu DL, Li ZG. Controlled projective synchronization in nonpartially-linear chaotic systems. Int J Bifurcat Chaos 2002;12:1395– 402. [25] Xu DL, Ong WL, Li ZG. Criteria for the occurrence of projective synchronization in chaotic systems of arbitrary dimension. Phys Lett A 2002;305:167–72. [26] Kocarev L, Parlitz U, Stojanovski T. An application of synchronized chaotic dynamic arrays. Phys Lett A 1996;217:280–4. [27] Carroll TL, Heagy JF, Pecora LM. Transforming signals with chaotic synchronization. Phys Rev E 1996;54:4676–80. [28] Lorenz EN. Deterministic non-periodic flow. J Atmos Sci 1963;20:130–41. [29] R€ ossler OE. An equation for hyperchaos. Phys Lett A 1976;71:155–67.