A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS

Transcription

1 International Journal of Computational Cognition ( Volume 2, Number 2, Pages 81 13, June 24 Publisher Item Identifier S (4)125-4/$2. Article electronically published on April 23, 23 at Please cite this paper as: Tao Yang, A Survey of Chaotic Secure Communication Systems, International Journal of Computational Cognition ( Volume 2, Number 2, Pages 81 13, June 24. A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS TAO YANG This paper is dedicated to the beginning of a brand new discipline called Computational Cognition for its profound influences on future intelligent systems. Abstract. Secure communication using synchronization between chaotic systems(chaotic secure communication, for short) is a new concept of secure communication. The great potentials of this kind of hardware key secure communication systems had driven the progress of this field rapidly. Since 1992, chaotic secure communication had evolved four generations. In this paper, a detailed history of chaotic secure communication systems is given. The disadvantage of the first three generations of chaotic secure communication schemes is low efficiency of channel usage. To overcome this disadvantage, a chaotic communication scheme, which belongs to the fourth generation, using impulsive synchronization of chaotic systems is presented. In this paper, impulsive synchronization of two chaotic systems is reformulated as impulsive stabilization of a synchronization error system to the origin. Based on the theory of impulsive differential equations, we present theoretical results on the asymptotic synchronization of two chaotic systems by using synchronization impulses. An estimate of the upper bound of impulse interval is given for the purpose of asymptotic synchronization. The robustness of impulsive synchronization to additive channel noise and parameter mismatch is also studied. We conclude that impulsive synchronization is more robust than continuous synchronization. Combining both conventional cryptographic method and impulsive synchronization of chaotic systems, we propose a new chaotic secure communication scheme. We use this new chaotic secure communication scheme to transmit a speech signal. Computer simulation results based on Chua s oscillators are given. c 23 Yang s Scientific Research Institute, LLC. All rights reserved. Received by the editors April 2, 22 / final version received April 22, 23. Key words and phrases. Secure communication, chaos, impulsive system, spread spectrum communication. This work was completed with the support of Yang s Foundation of Initiative of Computational Cognition. c Yang s Scientific Research Institute, LLC. All rights reserved. 81

2 82 TAO YANG 1. Introduction For over one century, chaos is found to be harmful to most of engineering applications when only linear methods were used to guide engineering design. However, since the seminal paper of Ott, Grebogi and Yorke(OGY)[2], the scientific and engineering communities realized that chaos could be controlled. Recently, the engineering community began to seek the possible applications of chaos. The fact that discrete pseudo-chaotic systems had been used by the cryptographic community[19] for a long time to generate cipher keys leads to the initiation of applying chaos to secure communication. Since 1992, the chaotic secure communication systems have updated to its fourth generation. The first three generations share a same framework of chaotic synchronization, i.e., continuous chaotic synchronization. The main problem of this synchronizing framework is that the synchronization signal uses a bandwidth comparable to that of message signals, thus the efficiency of bandwidth usage is very low. To overcome this problem, the fourth generation, called impulsive methods, was presented. The fourth generation employed impulsive chaotic synchronization to synchronize chaotic transmitters and chaotic receivers. The synchronization signal for a third-order chaotic transmitter only used 94Hz bandwidth. This bandwidth is much smaller compared with 3kHz bandwidth needed for transmitting synchronization signals in the other three generations. To understand the theoretical backbone for the fourth generation, one should understand first impulsive effects and their mathematical models. We usually use state equations to describe the dynamics of continuous dynamical systems which evolve along time. And we also suppose that the state-variables are continuous for physical systems. But there do exist a class of dynamical systems whose state variables are subject to jumps (abrupt changes) at discrete time instants, which we call impulsive effects of dynamical systems. Processes of such characteristics can be found in many fields, such as: physics, mechanics, population dynamics, chemical technology, economics, biology and electrical engineering. To model this kind of system, we use the theory of impulsive differential equations[5, 36, 37]. Since the theory of impulsive differential equations is much difficult than its counterpart of ordinary differential equations it remained a pure mathematical theory for almost 3 years until recently its promising applications to impulsive control and synchronization of chaotic systems were found by the chaotic secure and spread spectrum communication group in University of California at Berkeley[1, 11, 12, 13, 14, 15]. Although the rigorous theory of impulsive synchronization only presented in the early of 1997[15], its

3 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS 83 applications had been developed in both chaotic secure communication and chaotic spread spectrum communication. More recently, its applications are led to a US patent by Tao Yang and Leon O. Chua at the University of California at Berkeley. 1 In this paper I show that impulsive method is much more robust to parameter mismatch than the other three generations. The degree of security of the fourth generation is higher than that of the third generation. I also investigate the stability of impulsive synchronization of two chaotic systems. First, this problem is reduced to the stability of the origin of a synchronization error system, which is described by an impulsive differential equation. Then the stability of the trivial solution of a kind of impulsive differential equation is studied. These results are used to study the conditions under which the impulsive synchronization between two Chua s oscillators[6], which are standard chaotic electronic circuits developed by the chaotic secure and spread spectrum communication group in University of California at Berkeley, is asymptotically stable. An estimate of the upper bound of the impulsive interval is also presented. Since only synchronization impulses are sent to the driven system in an impulsive synchronization scheme, the information redundancy in the transmitted signal is reduced. In this sense, even low-dimensional chaotic systems can provide high security. In this paper, we will use impulsive synchronization to develop a new framework for chaotic secure communication. The organization of this paper is as follows. In section 2, a detailed history of the first three generations of chaotic secure communication systems is presented. In section 3, a theory on the stability of impulsive differential equations is given. In section 4, a stability criterion for impulsive synchronization between two Chua s oscillators is presented. In section 5, simulation results of impulsive synchronization of Chua s oscillators are presented. In section 6, application of impulsive synchronization to secure communication is presented. In section 7, some concluding remarks are given. 2. History of chaotic secure communication Chaos is a very universal and robust phenomenon in many nonlinear systems. Although the great mathematician Pincaré had noted that some mechanical systems could behave chaotically[21], chaos did not attract wide attention until Lorenz published his paper in 1963[22]. In engineering community, chaos had been mixed with noise for a long time. In 198 s, the electrical engineers first time officially announced the existence of chaos 1 U.S.Patent 6,331,974 issued Dec. 18,21, Name of Technology: Chaotic digital codedivision multiplex access for wireless communication systems.

4 84 TAO YANG in electrical systems. Since the noise-like behaviors of chaotic electronic circuits, electrical engineers felt uncomfortable to deal with them. It was physicists first showed in 199 that chaos could be controlled[2]. Then the synchronization between two identical chaotic systems was reported in 199[23]. In 1992, the electrical engineering community realized that chaos could used in secure communication systems[24, 25, 26] because chaos is extremely sensitive to initial conditions and parameters. The concept of chaotic hardware key for secure communication systems was then gradually realized by engineers and scientists. Since the great potential of applying chaos to secure communication systems, many groups over the world involved in the researches in this field. So far, chaotic communication systems have been updated to the fourth generation. In this paper, theory and structure of the fourth generation is presented. It is useful to provide the reader who is not involved in chaotic secure communication systems before a detailed history of these three generations The first generation. The first generation was developed in 1993 known as additive chaos masking[25] shown in Fig. 1(a) and chaotic shift keying[26] shown in Fig. 1(b). The additive chaos masking scheme shown in Fig. 1(a) consists of two identical chaotic systems in both the transmitter and the receiver. The chaotic mask denoted by c(t) is one of the state variables of the chaotic system 2 in the transmitter. The message signal m(t), which is typically 2 db to 3dB weaker than c(t) is added into the chaotic mask signal and gives the transmitted signal s(t). Since the chaotic signal c(t) is very complex and m(t) is much smaller than c(t), one may hope that the message signal m(t) can not be separated from s(t) without knowing the exact c(t). To give the reader a hands-on experience on chaotic secure communication systems, an example of additive chaotic masking scheme is given as follow. From Fig. 1(a) we can see that a chaotic synchronization block is needed in the receiver. Chaotic synchronization is a generalization of carrier synchronization in the normal communication systems but it is very different from the latter. We use Chua s oscillators to demonstrate the chaotic synchronization. A Chua s Oscillator is shown in Fig. 2(a). 2 For an autonomous system, at least three state variables are involved to generate chaos. For a non-autonomous system, at least two state variables and an independent input are involved to generate chaos. In some cases, c(t) may be a function of more than one state variable.

5 chaotic synchronization chaotic synchronization A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS 85 chaotic system c(t) s(t) channel r(t) chaotic system c(t) ~ m(t) transmitter message signal m(t) ~ recovered message signal receiver (a) chaotic system chaotic system 1 c (t) c (t) 1 s(t) r(t) chaotic c(t) ~ channel system m(t) transmitter message signal e(t) LPF and thresholding m(t) ~ receiver recovered message signal (b) Figure 1. The block diagrams of the first generation of chaotic secure communication systems. (a) The additive chaos masking scheme. (b) The chaotic shift keying scheme, also known as chaotic switching scheme. The state equations for the Chua s oscillator are given by dv 1 dt = 1 [ G(v2 v 1 ) f(v 1 ) ] C 1 dv 2 dt = 1 [ ] (1) G(v1 v 2 ) + i 3 C 2 di 3 dt = 1 [ ] v2 R i 3 L

6 86 TAO YANG i R R R G b i R voltage vs. current characteristic of Chua s diode v R C 1 v 1 v 2 C2 L i3 G a 1 1 v R Chua s Diode (a) (b) G b Figure 2. (a) The Chua s oscillator. (b) The characteristics of the Chua s diode. where v 1, v 2 and i 3 are the voltage across C 1, the voltage across C 2 and the current through L, respectively. We set G = 1 R. The term R i 3 is added to account for the small resistance of the inductor in the physical circuit. The piece-wise linear v-i characteristic f(v 1 ) of the Chua s diode, is given by (2) f(v 1 ) = G b v (G a G b ) ( v 1 + E v 1 E ) where E is the breakpoint voltage of the Chua s diode as shown in Fig. 2(b). If we consider the Chua s oscillator in Eq.(1) as the transmitter, then the following Chua s oscillator is receiver dṽ 1 dt = 1 [ G(ṽ2 ṽ 1 ) f(ṽ 1 ) ] C 1 dṽ 2 dt = 1 [ ] (3) G(v1 ṽ 2 ) + ĩ 3 C 2 dĩ 3 dt = 1 [ ] ṽ2 R ĩ 3 L where ṽ 1, ṽ 2 and ĩ 3 are state variables of the receiver. Observe that in the second equation of Eq.(3), we substitute ṽ 1 with v 1. In this sense, v 1 is the transmitted signal. We can prove that under some conditions, the Chua s oscillators in Eqs.(1) and (3) can be synchronized. The simulation results are shown in Fig. 3. The solid lines show the state variables of the transmitter and the dotted lines show those of the receiver. We can see that the state variables of the receiver approach those of the transmitter within.5ms though the initial states of these two circuits are much different. After we understand what is chaotic synchronization, we can then hide a small message signal m(t) into the transmitted signal s(t) such that the

7 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS 87 4 Generated by /taoyang/c/chuasyn.c 3 2 v1 and ^v time x 1 3 (a) Generated by /taoyang/c/chuasyn.c v2 and ^v time x 1 3 (b) Figure 3. Synchronization of two Chua s oscillators. (a) The synchronizing process of v 1 and ṽ 1. (b) The synchronizing process of v 2 and ṽ 2. (c) The synchronizing process of i 3 and ĩ 3. receiver in Eq.(3) can be rewritten as 3 dṽ 1 dt = 1 [ G(ṽ2 ṽ 1 ) f(ṽ 1 ) ] C 1 dṽ 2 dt = 1 [ ] (4) G((v1 + m(t)) ṽ 2 ) + ĩ 3 C 2 dĩ 3 dt = 1 [ ] ṽ2 R ĩ 3 L 3 In this case, we assume a noiseless channel is used such that r(t) = s(t) = v1 (t)+m(t) is satisfied.

8 88 TAO YANG 4 x 1 3 Generated by /taoyang/c/chuasyn.c i3 and ^i time x 1 3 (c) Figure 3 (Continued). Given m(t) is weak enough, the synchronization between the transmitter and the receiver can be maintained because synchronization is a kind of robust phenomenon. In Fig. 4 we present the simulation results for additive chaotic mask method. Figure 4(a) shows the weak message signal. Figure 4(b) shows the transmitted signal. Since the message signal is very weak we can not see anything related to it in the transmitted signal. In this sense, we have hided the message signal using the noise-like chaotic signal. The recovered results are shown in Fig 4(c). Observe that after a transient process, a (non-perfect) synchronization is achieved and the message signal is recovered with many noises. This scheme was proved that it could not be used under practical conditions because of the following drawbacks. Since the message signal is typically 2dB to 3dB weaker than the chaotic mask, this method is very sensitive to channel noise and parameter mismatch between the chaotic systems in the transmitter and the receiver. Furthermore, this scheme has a very low degree of security[8]. Chaotic shift keying shown in Fig. 1(b) also known as chaotic switching was designed to transmit digital message signal. In this scheme, the message signal, which is a digital signal, is used to switch the transmitted signal between two statistically similar chaotic attractors, which are respectively used to encode bit and bit 1 of the message signal. These two attractors are generated by two chaotic systems with the same structure and different parameters. At the receiver end, the received signal is used to drive a

9 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS 89 2 x 1 3 Generated by /taoyang/c/chuasyn.c message signal time x 1 3 (a) Generated by /taoyang/c/chuasyn.c transmitted signal time x 1 3 (b) Figure 4. Simulation results of additive mask scheme using two Chua s oscillators. (a) The weak message signal m(t). (b) The transmitted signal s(t). (c) The recovered signal m(t). chaotic system, which is identical to any of the two chaotic systems in the transmitter. The message signal is recovered by low-pass filtering and then thresholding the synchronization error signal e(t), which is depicted in Fig. 1(b).

10 9 TAO YANG 2 x 1 3 Generated by /taoyang/c/chuasyn.c recovered result time x 1 3 (c) Figure 4 (Continued) V2(V) v2(v) V1(V) (a) v1(v) (b) Figure 5. Chaotic attractors of Chua s oscillators used in chaotic shift keying. Both of them are shown in the v 1 -v 2 plane. (a) The chaotic attractor for encoding bit 1. (b) The chaotic attractor for encoding bit. To demonstrate this scheme, we also use Chua s oscillators as the transmitter and the receiver. In the transmitter, two Rössler-like chaotic attractors are used to encode bit 1 and bit, respectively. The parameters for encoding bit 1 are: R = 1Ω, R = 2Ω, G a = 1.139mS,

11 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS 91 G b =.711mS, E = 1V, L = 12mH, C 1 = 17nF, and C 2 = 178nF. The corresponding chaotic attractor is shown in Fig. 5(a). The parameters for encoding bit are: R = 1Ω, R = 2Ω, G a = 1.139mS, G b =.711mS, E = 1V, L = 12.5mH, C 1 = 17.5nF, and C 2 = 197nF. The corresponding chaotic attractor is shown in Fig. 5(b). These two sets of parameters generate two different but statistically similar chaotic attractors. The voltage v 1 is transmitted to the receiver. The chaotic synchronization block is something similar though not to be necessary the same to that used in Fig. 1(a) 4. The simulation results are shown in Fig. 6. Figure 6(a) shows the binary message signal with a 1ms cycle length. Figure 6(b) shows the synchronization error of the authorized receiver corresponding to bit. Figure 6(c) shows the synchronization error of the authorized receiver corresponding to bit 1. We can easily recover the binary signal (a) by moving averaging, thresholding (b) or (c). This scheme is very robust to noise and parameter mismatch. However, it has a low degree of security[7] if the chaotic attractors are too far away in the bifurcation space. However, since this is the first scheme of chaotic digital communication systems, there still exist many possibilities of improving it The second generation. The second generation was proposed during 1993 to 1995 known as chaotic modulation. This generation used two different ways to modulate message signals into chaotic carriers. The first method called chaotic parameter modulation[27] shown in Fig. 7(a) used message signals to change parameters of the chaotic transmitter. The second method called chaotic non-autonomous modulation[28] shown in Fig. 7(b) used the message signal to change the phase space of the chaotic transmitter. In Fig. 7(a) the message signal m(t) is used to modulate some parameters of the chaotic system in the transmitter such that its trajectories keep changing in different chaotic attractors. Since the bifurcation space of a chaotic system is very complex, it is very difficult to figure out the way of the changes of the parameters even through the intruder knows some partial knowledge of the structure of the chaotic system in the transmitter. At the receiver end an adaptive controller is used to adaptively tune the parameters of the chaotic system such that the synchronization error approach 4 Since the chaotic synchronization blocks in different chaotic secure communication systems are usually not the same because there are so many choices of chaotic systems. Different chaotic systems may used different synchronization schemes. The synchronization of chaotic system is now a very active research area in both the scientific and engineering communities. In the rest of this section, I will not emphasize the difference between the synchronization block used in different chaotic communication systems, instead I provide many references for those who want to know the technical details.

12 92 TAO YANG Time(s) (a) e1(v) Time(s) (b) Figure 6. The simulation results of chaotic shift keying. (a) The binary message signal m(t). (b) Synchronization error of Chua s oscillator with the parameters corresponding to bit. (c) Synchronization error of Chua s oscillator with the parameters corresponding to bit 1. zero. By doing this, the output of the adaptive controller can recover the message signal. The simulation results are shown in Fig. 8. In this simulation, three message signals are used to tune three different parameters

13 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS e1(v) Time(s) (c) Figure 6 (Continued). of the chaotic system in the transmitter. Since the chaotic system keeps changing its attractors, the waveform of the transmitted signal as shown in Fig. 8(a) is much more complex than a normal chaotic signal. In Figs.8(b) to (d) we show the three original message signals and the three recovered message signals. Observe that after a transient process of synchronization, the message signals are recovered with some cross talks and small delays. Instead of changing the parameters of the chaotic transmitter, the chaotic non-autonomous modulation shown in Fig. 7(b) used the message signal to perturb chaotic attractor directly in the phase space. Unlike in chaotic parameter modulation where the transmitter is switched among different trajectories in different chaotic attractors, the transmitter in chaotic nonautonomous modulation is switched among different trajectories of the same chaotic attractor. Theoretically, chaotic non-autonomous modulation is an error free scheme. The second generation improved the degree of security to some degree but was still found unsatisfactory[18, 29, 33, 34, 35] The third generation. The third generation shown in Fig. 9 was proposed in 1997[9] for the purpose of improving the degree of security to a much higher level than the first two generations. We call this generation as chaotic cryptosystem. In this generation, the combination of the classical cryptographic technique and chaotic synchronization is used to enhance the degree of security. So far, this generation has the highest security in all the chaotic secure communication systems had been proposed and has not yet been broken. In the chaotic cryptosystem the plain text signal p(t) is

14 chaotic synchronization chaotic synchronization 94 TAO YANG chaotic system s(t) channel r(t) chaotic system ~ s(t) parameters m(t) message signal transmitter (a) e(t) adaptive controller ~m(t) recovered message signal parameters receiver chaotic system s(t) channel r(t) chaotic system ~ s(t) m(t) message signal transmitter (b) ~m(t) recovered message signal receiver Figure 7. The block diagrams for the second generation of chaotic secure communication systems. (a) The chaotic parameter modulation. (b) The chaotic non-autonomous modulation. encrypted by a encryption rule with a key signal, k(t), which is generated by the chaotic system in the transmitter. The scrambled signal is used further to drive the chaotic system such that the chaotic dynamics is changed

15 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS x time(s) x 1 4 (a) a and a time(s) (b) Figure 8. Using the chaotic parameter modulation to transmit three signals simultaneously. (a) The transmitted signal s(t). (b) The first message signal and its recovered version. (c) The second message signal and its recovered version. (d) The third message signal and its recovered version. continuously in a very complex way 5. Then another state variable of the 5 Observe that the third generation had some modulation properties provided by the second generation.

16 96 TAO YANG x b and b c and c time(s) x 1 5 (c) time(s) (d) Figure 8 (Continued). chaotic system in the transmitter is transmitted to through public channel which can be accessed by the intruder. Since the intruder can not get access to the chaotic hardware key, it is very difficult to find p(t) out from s(t). At the receiver, the received signal r(t) = s(t) + n(t), where n(t) is the channel noise, is used to synchronize both of the chaotic systems in transmitter and the receiver. After the chaotic synchronization had been achieved, the signal k(t) and y(t) can be recovered at the receiver with some noises as denoted by k(t) and ỹ(t). By feeding k(t) and ỹ(t) into the decryption

17 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS 97 Plain Text p(t) Encryption Rule e(.) k(t) y(t) Intruder p(t) ~ Decryption Rule d(.) ~ ~ k(t) y(t) Plain Text Chaotic System Public Channel Chaotic System Encrypter s(t) s(t)+n(t) Decrypter Figure 9. The block diagram for the third generation of chaotic secure communication systems. rule at the receiver, the plain text signal can be recovered with some noises as p(t). The simulation result is shown in Fig. 1. Figure 1(a) shows the transmitted signal s(t), from which one can not observe the embedded plain signal. Figure 1(b) shows the recovered and decrypted result at the receiver. Observe that after the transient process of synchronization, the plain text signal is recovered. To show the high security of this scheme, the unmasking method provided in [18] is used to decode the plain text signal. Figure 1(c) shows the unmasked signal ỹ(t) by the intruder, from which it is impossible to retrieve the plain text signal as shown in Fig. 1(d) Remarks. To implement practical chaotic secure communication systems, two critical technical problems should be first solved for archiving chaotic synchronization. The first one is the parameter mismatch between the chaotic transmitter and receiver. This problem was solved by using adaptive synchronization[4, 3]. The other problem was the nonlinearity of the channel. This problem was first solved in [31, 3]. However, the channel model used in [31, 3] was far from a real channel. Another possible method was presented in [32] to provide a channel-independent scheme. Although all the examples provided in this paper work in base band, the corresponding RF system can be built based on the same principles. Also, for simplicity all the examples are demonstrated under the analogue communication framework. However, in the practical implementations, the chaotic secure communication systems are built as digital secure communication systems using chaotic carriers. So far, there existed at least two different ways to build chaotic digital secure communication systems. The first one was to digitize the transmitted signal s(t) and then send it via classic digital communication techniques. Since the robustness of the adaptive chaotic synchronization used in the second and third generations, the quantizing error can only introduce a small portion of total noises in the recovered signal. The second one was dedicated to transmit digital information signals as in

18 98 TAO YANG 3 2 Amplitude(v) Time(s) (a) Amplitude(v) Time(s) (b) Figure 1. Simulation results of chaotic cryptosystem. (a) The transmitted signal s(t). (b) The recovered and then decrypted signal p(t). (c) The recovered encrypted signal using the method in [18]. (d) The decrypted result of that shown in (c). chaotic shift keying. Although so far only a few prototypes of chaotic secure communication systems have been built as RF digital communication systems, the tendency in almost all groups in the world is focused on wireless chaotic digital secure communication systems. However, the main principles presented in this section are still the backbones of all these systems. One can see that chaotic synchronization is the critical technology for building a chaotic secure communication scheme. In fact, it is the evolution of chaotic synchronization technology to trigger that of the chaotic secure communication systems. The first generation based on the simplest chaotic synchronization, which was from feedback control theory. The second and

19 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS Amplitude(v) Time(s) (c) Amplitude(v) Time(s) (d) Figure 1 (Continued). the third generations based on the adaptive chaotic synchronization, which was from adaptive control theory. In principle, the chaotic synchronization used in the first three generations are continuous synchronization. In 1997 a brand new chaotic synchronization technology called impulsive synchronization was invented based on impulsive control theory[15, 14, 13, 11, 12]. By applying impulsive synchronization, the fourth generation of chaotic secure communication system was presented. In the rest parts of this paper, the theory and structure of the fourth generation of chaotic secure communication systems are presented. 3. Basic Theory of Impulsive Differential Equations To make this paper self-contained and since the theory of impulsive differential equation is not well-known to the scientific community, we address some basic results in this section which is useful in this paper.

20 1 TAO YANG (5) Consider the general nonlinear system ẋ = f(t, x) where f : R + R n R n is continuous, x R n is the state variable, and ẋ = dx dt. Consider a discrete set {τ i } of time instants, where (6) Let < τ 1 < τ 2 <... < τ i < τ i+1 <..., τ i as i U(i, x) = x t=τi = x(τ + i ) x(τ i ) be the jump in the state variable at the time instant τ i. Then this impulsive system is described by ẋ = f(t, x), t τ i (7) x = U(i, x), t = τ i x(t + ) = x, t, i = 1, 2,... This is called an impulsive differential equation[1]. To study the stability of the impulsive differential equation (7) we use the following definitions and theorems[1]. Definition 1: Let V : R + R n R +, then V is said to belong to class V if 1. V is continuous in (τ i 1, τ i ] R n and for each x R n, i = 1, 2,..., (8) lim V (t, y) = V (τ + (t,y) (τ + i,x) i, x) exists; 2. V is locally Lipschitzian in x Definition 2: For (t, x) (τ i 1, τ i ] R n, we define (9) D + V (t, x) = lim sup 1 [V (t + h, x + hf(t, x)) V (t, x)] h h Since the system in Eq.(7) is an n th -order impulsive differential equation, instead of studying the stability of Eq.(7), it is convenient to study that of a first-oder impulsive differential equation which is given by the following definition. Definition 3: Comparison system Let V V and assume that { D + V (t, x) g(t, V (t, x)), t τ (1) i V (t, x + U(i, x)) ψ i (V (t, x)), t = τ i

21 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS 11 where g : R + R + R is continuous and ψ i : R + R + is nondecreasing. Then the system ẇ = g(t, w), t τ i (11) w(τ + i ) = ψ i(w(τ i )) w(t + ) = w is called the comparison system of Eq.(7). Definition 4: (12) S ρ = {x R n x < ρ} where denotes the Euclidean norm on R n. Definition 5: A function α is said to belong to class K if α C[R +, R + ], α() = and α(x) is strictly increasing in x. Assumptions: f(t, ) =, U(i, ) = and g(t, ) = for all i. Remark: With the above assumptions we find that the trivial solutions of Eqs. (7) and (11) are identical for all times except at the discrete set {τ i }. Theorem 1 shows how can the stability of an n th -order impulsive differential equation be equivalent to that of a first-order impulsive differential equation, i.e., the comparison system. Theorem 1(Theorem 3.2.1, page 139, [1]): Assume that the following three conditions are satisfied: 1. V : R + S ρ R +, ρ >, V V, D + V (t, x) g(t, V (t, x)), t τ i. 2. there exists a ρ > such that x S ρ implies that x + U(i, x) S ρ for all i and V (t, x + U(i, x)) ψ i (V (t, x)), t = τ i, x S ρ. 3. β( x ) V (t, x) α( x ) on R + S ρ, where α( ), β( ) K. Then the stability properties of the trivial solution of the comparison system (11) imply the corresponding stability properties of the trivial solution of (7). In next theorem, we present the stability criterion for a first-order impulsive differential equation, which is the general form of the comparison system of a synchronization error system we will study in this paper. Theorem 2(Corollary , page 142, [1]): Let g(t, w) = λ(t)w, λ C 1 [R +, R + ], ψ i (w) = d i w, d i for all i. Then the origin of system (7) is asymptotically stable if the conditions (13) and (14) are satisfied. λ(τ i+1 ) + ln(γd i ) λ(τ i ), for all i, where γ > 1 λ(t)

22 12 TAO YANG 4. Impulsive synchronization of Chua s oscillators In this section, we study the impulsive synchronization of two Chua s oscillators. The dimensionless form of a Chua s oscillator is given by[6] (15) ẋ = α(y x f(x)) ẏ = x y + z ż = βy γz where α, β and γ are three constants and f(x) is the piecewise-linear characteristic of the Chua s diode, which is given by (16) f(x) = bx + 1 (a b)( x + 1 x 1 ) 2 where a < b < are two constants. Let x = (x, y, z), then we can rewrite the Chua s oscillator equation into the form (17) where (18) A = α α β γ ẋ = Ax + Φ(x), Φ(x) = αf(x) In the impulsive synchronization schemes, there are two Chua s oscillators. One of them is called the driving system and the other is called the driven system. In an impulsive synchronization configuration, the driving system is given by Eq.(15). The driven system is given by (19) x = A x + Φ( x) where x = ( x, ỹ, z) is the state variables of the driven system. At discrete instants, τ i, i = 1, 2,..., the state variables of the driving system are transmitted to the driven system and then the state variables of the driven system are subject to jumps at these instants. The driven system is described by the impulsive differential equation { x = A x + Ψ( x), t τ (2) i x t=τi = Be, i = 1, 2,... where B is a 3 3 matrix, and e = (e x, e y, e z ) = (x x, y ỹ, z z) is the synchronization error. If we define αf(x) + αf( x) (21) Ψ(x, x) = Φ(x) Φ( x) =

23 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS 13 then the error system of the impulsive synchronization is given by { ė = Ae + Ψ(x, x), t τi (22) e t=τi = Be, i = 1, 2,... We use the following theorem to guarantee that our impulsive synchronization is asymptotically stable. Theorem 3: Let d 1 be the largest eigenvalue of (I + B )(I + B), where B is a symmetric matrix. Assume the spectral radius ρ of I + B satisfies ρ(i + B) 1. Let q be the largest eigenvalue of (A + A ) and assume the impulses are equidistant from each other and separated by an interval. If (23) q + 2 αa 1 ln(ξd 1), ξ > 1 then the impulsive synchronization of two Chua s oscillators is asymptotically stable. Proof: Let us construct the Lyapunov function V (t, e) = e e. For t τ i, we have (24) D + V (t, e) = e Ae + e A e + e Ψ(e) + Ψ (e)e qe e + 2 α f(x) f( x) e x qe e + 2 αa e 2 x (q + 2 αa )e e = (q + 2 αa )V (t, e) Condition 1 of Theorem 1 is satisfied with g(t, w) = (q + 2 αa )w. Since B is symmetric we know (I + B) is also symmetric. By using Euclidean norm we have (25) ρ(i + B) = I + B Given any ρ > and e S ρ, we have (26) e + Be I + B e = ρ(i + B) e e The last inequality follows from ρ(i + B) 1. Consequently, e + Be S ρ. For t = τ i, we have (27) V (τ i, e + Be) = (e + Be) (e + Be) = e (I + B )(I + B)e d 1 V (τ i, e) Condition 2 of Theorem 1 is satisfied with ψ i (w) = d 1 w. We can see that condition 3 of Theorem 1 is also satisfied. It follows from Theorem 1 that

24 14 TAO YANG the asymptotic stability of the origin of the synchronization error system in Eq.(22) is implied by that of the following comparison system (28) (29) From Eq.(23), we have τi+1 ω = (q + 2 αa )ω, t τ i ω(τ i ) = d 1 ω(τ i ) ω(t ) = ω τ i (q + 2 αa )dt + ln(ξd 1 ), ξ > 1 and λ(t) = q +2 αa. It follows from Theorem 2 that the trivial solution of Eq.(22) is asymptotically stable. Theorem 3 also gives an estimate for the upper bound max of ; namely, max = ln(ξd 1) (3), ξ 1 + q + 2 αa Observe that the upper bound given by Eq.(3) is sufficient but not necessary. Consequently, we can only say that we have a predicted stable region, which is usually smaller than the actual stable region because we can not assert that all other regions are unstable. 5. Simulation results of impulsive synchronization In the following simulations, we choose the parameters of Chua s oscillator as α = 15, β = 2, γ =.5, a = 12 7, b = A fourth-order Runge-Kutta with step size 1 5 is used. The initial conditions are given by (x(), y(), z()) = ( ,.6617, ) and ( x(), ỹ(), z()) = (,, ). The trajectories of the driving system are shown in Fig. 11, which is the Chua s double scroll attractor Simulation 1: strong coupling. In this simulation, we choose the matrix B as B = k (31) 1 1 where the impulsive coupling is strong. It follows from Theorem 3 that ρ(i + B) 1 should be satisfied, which implies that 2 k. By using this B matrix, it is easy to see that (32) d 1 = (k + 1) 2

25 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS y x 2 2 z 2 Figure 11. The double scroll attractor. We have (33) A = , A + A = from which we find q = Then an estimate of the boundaries of the stable region is given by (34) (lnξ + ln(k + 1)2 ), 2 k q + 2 αa Figure 12 shows the stable region for different ξ s. The entire region below the curve corresponding to ξ = 1 is the predicted stable region. When ξ, the stable region shrinks to a line k = 1. The simulation results are shown in Fig. 13. Figure 13(a) shows instability for k = 1.5 and = 1. The solid waveform, the dash-dotted waveform and the dotted waveform correspond to e x (t), e y (t) and e z (t), respectively. Figure 13(b) shows stable results within the stable region for k = 1.5 and

26 16 TAO YANG Delta xi= stable region k Figure 12. Estimate of the boundaries of stable regions with different ξ s used in simulation 1. =.2. One can see that the system asymptotically approaches the origin with a settling time 6 of about.5. However, the true stable region is larger than that predicted in Fig. 12. In order to demonstrate this fact, we show in Fig. 13(c) the stable results for k = 1.5 and =.5. We can also see that the system asymptotically approaches the origin with a settling time of about 1.4 which is much larger than that shown in Fig. 13(b) Simulation 2: weak coupling. In this simulation, we choose the matrix B as B = k (35) Here the settling time is defined as Ts when the synchronization error e(t) < 1 3 for any time t T s. Since the low resolution of the printer, we can not determine T s from the figures. We can find T s from the data.

27 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS time (a) time (b) Figure 13. Simulation results. (a) Unstable results outside the stable region. (b) Stable results inside the predicted stable region. (c) Stable results outside the predicted stable region. where the impulsive coupling is much weaker than that chosen in simulation 1.

28 18 TAO YANG time (c) Figure 13 (Continued). It is easy to see that { (k + 1) d 1 = 2, (k + 1) 2.81 (36).81, elsewhere (37) An estimate of the boundaries of the stable region is given by ln ξ + ln(k + 1)2, (k + 1) 2.81 q + 2 αa ln ξ + ln(.81), elsewhere q + 2 αa, 2 k Figure 14 shows the stable region. The entire region below the curve corresponding to ξ = 1 is the predicted stable region. In this case, is always bounded. It seems that we can not control the system to the origin with an arbitrarily prescribed speed because ξ has to satisfy 1 < ξ < This is different from the case shown in Fig. 12, where any value of ξ > 1 is possible. The simulation results are shown in Fig. 15. Again, the solid waveform, the dash-dotted waveform and the dotted waveform correspond to e x (t), e y (t) and e z (t), respectively. Figure 15(a) shows the instability results for k = 1 and =.4. Figure 15(b) shows the stable results in the stable region for k = 1 and = The control system asymptotically approaches the origin with a settling time of about.5. Also, the true stable region is larger than that predicted in Fig. 14. To demonstrate this

29 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS 19 4 x 1 4 xi= Delta stable region k Figure 14. Estimate of the boundaries of stable region used in simulation 2. fact, we show in Fig. 15(c) the stable results for k = 1 and =.1. We can also see that the system asymptotically approaches the origin with a settling time equal approximately to 1, which is much larger than that shown in Fig. 15(b) Effects of channel noise. Let us study next the robustness of the impulsive synchronization to additive channel noise. In the following simulations, we choose the matrix B as (38) B = Figure 16 shows the simulation results when the signal-to-noise ratio(snr) and the time interval of the impulses are given respectively by SNR=2dB and =.1. Figure 16(a) shows the noise (the red waveform) in impulses x(τ i ) and the synchronization error(the blue waveform) x x. Figure 16(b) shows the noise (the red waveform) in impulses y(τ i ) and the synchronization

30 11 TAO YANG time (a) time (b) Figure 15. Simulation results. (a) Unstable results outside the stable region. (b) Stable results in the stable region. (c) Stable results outside the stable region. error(the blue waveform) y ỹ. Figure 16(c) shows the noise (the red waveform) in impulses z(τ i ) and the synchronization error(the blue waveform) z z. From the above simulation results we find that the synchronization errors are comparable to the noise in the synchronization impulses.

31 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS time (c) Figure 15 (Continued). For comparison, we also presented the corresponding results when a continuous synchronization is used. The driven system of the continuous synchronization is given by (39) x = α(ỹ x f( x)) ỹ = (x + n(t)) ỹ + z z = βỹ γ z where x is the first state variable of the driving system and n(t) is the additive channel noise. When n(t) =, the driving system and the driven system can be synchronized. When the additive noise is added in the transmitted signal with SNR = 2dB, the synchronization errors are shown in Fig. 16(d). Observe that the synchronization error x x of continuous scheme is bigger than that of the impulsive scheme. Given a SNR level, the synchronization errors of the impulsive scheme become bigger if increases. In our simulations we find if = 1 then there exist some big synchronization error peaks. The simulation results are shown in Fig. 17 with SNR=2dB. Figure 17(a) shows the noise (the red waveform) in the impulses x(τ i ) and the synchronization error(the blue waveform) x x. Figure 17(b) shows the noise (the red waveform) in the impulses y(τ i ) and the synchronization error(the blue waveform) y ỹ. Figure 17(c) shows the noise (the red waveform) in the impulses z(τ i ) and the synchronization error(the blue waveform) z z. From the simulation results we find that the synchronization errors are comparable to the noise most of

32 112 TAO YANG time (a) Figure 16. Simulation results of the impulsive synchronization and the continuous synchronization when channel noise is added. (a) Noise in impulses x(τ i )(red) and the synchronization error x x(blue) of the impulsive synchronization. (b) Noise in impulses y(τ i )(red) and the synchronization error y ỹ(blue) of the impulsive synchronization. (c) Noise in impulses z(τ i )(red) and the synchronization error z z(blue) of the impulsive synchronization. (d) Synchronization errors x x(blue), y ỹ(red) and z z(green) of the continuous synchronization. the time. However, when the becomes too big, e.g., = 6, then most of the time we observe that the synchronization errors are much bigger than the noise. From the above simulation we can conclude that if is small enough, e.g, =.1, then the impulsive synchronization is more robust than the continuous synchronization. This is because in continuous synchronization the synchronization error system is continuously disturbed by the continuous channel noise and more often pushed to some local instabilities than in impulsive synchronization. This in turn results in more frequent large peaks of synchronization errors in continuous synchronization Effects of parameter mismatch. The parameter mismatch is any parameter difference between the driving system and the driven system. The robustness of impulsive synchronization to parameter mismatch is studied

33 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS x time (b).6.4 synchronization errors time (c) Figure 16 (Continued). in the following simulations. Figure 18 shows the synchronization errors when different parameter mismatches are used. Figure 18(a) shows the results when only 1% mismatch exists in parameter α with =.1. Figure 18(b) shows the results when 1% mismatch exists in the parameter α with =.1. Observe that the impulsive synchronization is robust enough to parameter mismatch. We also show the results when the continuous synchronization scheme is used. Figure 18(c) shows the results when a 1% mismatch exists in the parameter α and continuous synchronization

34 114 TAO YANG.6.4 synchronization errors time (d) Figure 16 (Continued). is used. Observe that there exist some very big synchronization errors in x x. Figure 18(d) shows the results when a 1% mismatch exists in the parameter α and the continuous synchronization is used. Observe that there exist some very big synchronization errors in x x. From above we can see that impulsive synchronization is less sensitive to parameter mismatches than continuous synchronization. 6. The fourth generation of chaotic secure communication system Since the publication of several chaotic cryptanalysis results in low-dimensional chaos-based secure communication systems[7, 8, 16, 18], there were concerns that such communication schemes may not be secure enough. To overcome this objection, one approach is to exploit hyperchaos-based 7 secure communication systems, but such systems may introduce more difficulties to synchronization. On the other hand, we can enhance the security of low-dimensional chaosbased secure communication schemes by combining conventional cryptographic schemes with a chaotic system[9]. To overcome the low security objections against low-dimensional continuous chaos-based schemes, we may use the following two methods. The first method is to make the transmitted signal more complex. The second method is to reduce the redundancy in the transmitted signal. In [9] we have presented a method to combine a 7 A hyperchaotic system has at least two positive Lyapunov exponents.

35 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS time (a) time (b) Figure 17. Simulation results of the impulsive synchronization when channel noise is added and is big. (a) Noise in impulses x(τ i )(red) and the synchronization error x x(blue). (b) Noise in impulses y(τ i )(red) and the synchronization error y ỹ(blue). (c) Noise in impulses z(τ i )(red) and the synchronization error z z(blue). conventional cryptographic scheme with low-dimensional chaos to obtain a very complex transmitted signal. The impulsive synchronization presented

36 116 TAO YANG time (c) Figure 17 (Continued). in this paper offers a very promising approach of reducing the redundancy in transmitted signals A simple system in baseband. In this section, we combine the results in [9] and impulsive synchronization to give a new chaotic secure communication scheme. The block diagram of this scheme is shown in Fig. 19. From Fig. 19 we can see that this chaotic secure communication system consists of a transmitter and a receiver. In both the transmitter and the receiver, there exist two identical chaotic systems. Also, two identical conventional cryptographic schemes are embedded in both the transmitter and the receiver. Let us now consider the details of each block in Fig. 19. The transmitted signal consists of a sequence of time frames. Every frame has a length of T seconds and consists of two regions. In Fig. 2 we show the concept of a time frame and its components. The first region of the time frame is a synchronization region consisting of synchronization impulses. The synchronization impulses are used to impulsively synchronize the chaotic systems in both transmitter and receiver. The second region is the scrambled signal region where the scrambled signal is contained. To ensure synchronization, we have T < max. Within every time frame, the synchronization region has a length of Q and the remaining time interval T Q is the scrambled signal region. The composition block in Fig. 19 is used to combine the synchronization impulses and the scrambled signal into the time frame structure shown in Fig. 2. The simplest combination method is to substitute the beginning

37 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS 117 synchronization errors time (a) Figure 18. Simulation results of the impulsive synchronization and the continuous synchronization when parameter mismatches exist. (a) A 1% mismatch exists in α. The synchronization errors of the impulsive synchronization: x x(blue), y ỹ(red) and z z(green). (b) A 1% mismatch exists in α. The synchronization errors of the impulsive synchronization: x x(blue), y ỹ(red) and z z(green). (c) A 1% mismatch exists in α when the continuous synchronization is used. The synchronization errors x x(blue), y ỹ(red) and z z(green). (d) A 1% mismatch exists in α when the continuous synchronization is used. The synchronization errors x x(blue), y ỹ(red) and z z(green). Q seconds of every time frame with synchronization impulses. Since Q is usually very small compared with T, the processing time for packing a message signal is negligible. The decomposition block is used to separate the synchronization region and the scrambled signal region within each frame at the receiver end. Then the separated synchronization impulses are used to make the chaotic system in the receiver to synchronize with that in the transmitter. The stability of this impulsive synchronization is guaranteed by our results in Section 4.

38 118 TAO YANG synchronization errors time 1.5 (b) synchronization errors time (c) Figure 18 (Continued). In the transmitter and the receiver, we use the same cryptographic scheme block for purposes of bi-directional communication. In a bi-directional communication scheme, every cellular phone should function both as a receiver and a transmitter. Here, the key signal is generated by the chaotic system. The cryptographic scheme is as follows[9]:

39 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS synchronization errors time Figure 18 (Continued). (d) We use a continuous n-shift cipher to encrypt the plain signal(message signal). The n-shift cipher is given by ( ( ) ) ) (4) e(p(t)) = f 1...f 1 (f 1 p(t), k(t), k(t),..., k(t) = y(t) } {{ } } {{ } n n where h is chosen such that p(t) and k(t) lie within ( h, h). Here, p(t) and k(t) denote the plain signal and the key signal, respectively, and y(t) denotes the encrypted signal. The key signal k(t) is chosen as a state variable of the chaotic system. The notation f 1 (, ) denotes a scalar nonlinear function of two variables defined as follow: (41) f 1 (x, k) = (x + k) + 2h, 2h (x + k) h (x + k), h < (x + k) < h (x + k) 2h, h (x + k) 2h This function is shown in Fig. 21. The corresponding decryption rule is the same as the encryption rule (42) p(t) = d(y(t)) = e(y(t)) ( ( ) ) ) = f 1...f 1 (f 1 y(t), k(t), k(t),..., k(t). } {{ } } {{ } n n To decode the encrypted signal, the same key signal should be used.

40 12 TAO YANG message signal Transmitter cryptographic scheme chaotic system scrambled signal synchronization impulses composition transmitted signal message signal Receiver cryptographic scheme scrambled signal decomposition chaotic system synchronization impulses Figure 19. Block diagram of the impulsivesynchronization based chaotic secure communication system. The simulation results are as follows. The synchronization region is located in the initial 1% of every time frame. We choose the frame length as T = 1s. In the synchronization region of every time frame, we transmit the impulses of the three state-variables of the Chua s oscillators. The parameter of the encrypted signal is chosen as h =.4. A continuous 1-shift cipher was used. We choose x and x as the key signals and normalized them to fall within the amplitude range [.4,.4].

41 Frame Number Time T 2T 3T 4T 5T A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS 121 (n 1)T nt Synchronization impulses (sychronization region) Q T Q Scrambled signal region Figure 2. Illustration of the concept of a time-frame and its components. Figure 22 shows the simulation results of the above proposed secure communication system for transmitting a speech signal. Figure 22(a) shows the

42 122 TAO YANG f 1 (x,k) h 2h h h 2h (x+k) h Figure 21. Nonlinear function used in the continuous shift cipher. waveforms of the sampled speech of four Chinese digits NING (zero) YI (one) ER (two) SANG (three). The sampling rate is 8K. Figure 22(b) shows the spectrograms of the original speech signal in Fig. 22(a), from which we can see the structure of the speech signal. Figure 22(c) shows the waveforms of the received scrambled speech signal and the additive channel noise with SNR = 16dB. Figure 22(d) shows the spectrograms for the scrambled speech signal and the additive channel noise. We can see that the structure of the signal in Fig. 22(b) was totally covered by an almost uniformly distributed noise-like spectrum. Figure 22(e) shows the waveforms of the descrambled speech signal. Figure 22(f) shows the spectrograms of the descrambled speech signal. We can see that some noises were introduced into the recovered results due to the channel noise, and that the spectrograms became a little blur. But the structure of the speech signal was perfectly recovered An illustrative example: secure digital cellular phone. In this digital communication era, the implementation of the whole system presented in this section should be digital instead of analog. Let us revisit the baseband block diagram in Fig. 19. Since the parameter mismatch of the Chua s oscillators(the chaotic systems) can be well controlled during the

43 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS NING YI ER SANG Time (S) (a) Figure 22. The simulation results. (a) The time-domain waveform of the speech signal. (b) The spectrogram for the original speech signal. (c) The time-domain waveform of the scrambled speech signal. (d)the spectrogram of the scrambled speech signal. (e)the time-domain waveform of the descrambled speech signal. (f) The spectrogram of the descrambled speech signal. manufacturing process, each mobile station will have a copy of Chua s oscillator as the hardware key 8. The detailed block diagram of a secure digital cellular phone system is shown in Fig. 23. In the system shown in Fig. 23, there are two kinds of analogue signals needed to be transmitted to the receiver. The first one is the speech signal s(t) and the second one is the synchronization impulse x(t). To ensure the high degree of security, the digitized speech signal, s(i), is scrambled by a digital stream encipher. The key stream for the stream encipher is a digitized output, k(i), of the chaotic system at the transmitter end. Note that the output k(t) of the chaotic system is usually a nonlinear combination of all state variables of the chaotic system. Thus k(t) is more complex than any individual state variable. The output of the stream encipher block is a cipher text stream c(i). Since the key stream k(i) can not be sent through a public channel, in the system shown in Fig. 23, k(i) must be regenerated chip. 8 In fact, the chaotic system itself can also be implemented digitally by using a DSP

44 124 TAO YANG 2 4 Frequency Time (x.1s) (b) Time (S) (c) Figure 22 (Continued). at the receiver end. To do this, the two chaotic systems in the transmitter and the receiver should be identical and synchronized. Impulsive synchronization is used in this system. To transmit the synchronization impulses from the transmitter to the receiver, the analogue impulse samples, x(t), are digitized as x(i). Then x(i) is combined with c(i) at the composition and coding block. The output of the composition and coding block, d(i), is then sent to the modulation block. Based on different system settings, the modulation block may be that used in TDMA or CDMA.

45 A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS Frequency Time (x.1s) (d) NING YI ER SANG Time (S) (e) Figure 22 (Continued). At the receiver end, the output of the demodulation block, d (t), is usually different from d(t) at the transmitter end due to the noise and distortions in the channel. d (t) is then fed into the decoding and decomposition block whose outputs are the digitized synchronization impulses x(i) and the recovered cipher text stream c(i). We can get an BER from 1 3 to 1 6 for both x(i) and c(i). Since x(i) is usually transmitted at a bit-rate less than 1bps, we can offer more resources to make the BER for x(i) under 1 6. By doing this, we can get a good synchronization between