A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS · secure communication. The great potentials of this kind of “hardware key” secure communication systems had driven the progress

International Journal of Computational Cognition (http://www.YangSky.com/yangijcc.htm)Volume 2, Number 2, Pages 81–130, June 2004Publisher Item Identifier S 1542-5908(04)10205-4/$20.00Article electronically published on April 23, 2003 at http://www.YangSky.com/ijcc22.htm. Please cite thispaper as: 〈Tao Yang, “A Survey of Chaotic Secure Communication Systems”, International Journalof Computational Cognition (http://www.YangSky.com/yangijcc.htm), Volume 2, Number 2, Pages 81–130,June 2004〉.

A SURVEY OF CHAOTIC SECURE COMMUNICATIONSYSTEMS

TAO YANG

This paper is dedicated to the beginning of a brand new discipline called ComputationalCognition for its profound influences on future intelligent systems.

Abstract. Secure communication using synchronization between chaoticsystems(chaotic secure communication, for short) is a new concept ofsecure communication. The great potentials of this kind of “hardwarekey” secure communication systems had driven the progress of thisfield rapidly. Since 1992, chaotic secure communication had evolvedfour generations. In this paper, a detailed history of chaotic securecommunication systems is given. The disadvantage of the first threegenerations of chaotic secure communication schemes is low efficiencyof channel usage. To overcome this disadvantage, a chaotic communi-cation scheme, which belongs to the fourth generation, using impul-sive synchronization of chaotic systems is presented. In this paper,impulsive synchronization of two chaotic systems is reformulated asimpulsive stabilization of a synchronization error system to the origin.Based on the theory of impulsive differential equations, we presenttheoretical results on the asymptotic synchronization of two chaoticsystems by using synchronization impulses. An estimate of the upperbound of impulse interval is given for the purpose of asymptotic syn-chronization. The robustness of impulsive synchronization to additivechannel noise and parameter mismatch is also studied. We concludethat impulsive synchronization is more robust than continuous syn-chronization. Combining both conventional cryptographic methodand impulsive synchronization of chaotic systems, we propose a newchaotic secure communication scheme. We use this new chaotic securecommunication scheme to transmit a speech signal. Computer sim-ulation results based on Chua’s oscillators are given. c©2003 Yang’sScientific Research Institute, LLC. All rights reserved.

Received by the editors April 20, 2002 / final version received April 22, 2003.Key words and phrases. Secure communication, chaos, impulsive system, spread spec-

trum communication.This work was completed with the support of Yang’s Foundation of Initiative of Com-

putational Cognition.

c©2003-2004 Yang’s Scientific Research Institute, LLC. All rights reserved.

81

82 TAO YANG

1. Introduction

For over one century, chaos is found to be harmful to most of engineeringapplications when only linear methods were used to guide engineering de-sign. However, since the seminal paper of Ott, Grebogi and Yorke(OGY)[2],the scientific and engineering communities realized that chaos could be con-trolled. Recently, the engineering community began to seek the possibleapplications of chaos. The fact that discrete pseudo-chaotic systems hadbeen used by the cryptographic community[19] for a long time to generatecipher keys leads to the initiation of applying chaos to secure communica-tion. Since 1992, the chaotic secure communication systems have updatedto its fourth generation.

The first three generations share a same framework of chaotic synchro-nization, i.e., continuous chaotic synchronization. The main problem of thissynchronizing framework is that the synchronization signal uses a bandwidthcomparable to that of message signals, thus the efficiency of bandwidth us-age is very low. To overcome this problem, the fourth generation, calledimpulsive methods, was presented. The fourth generation employed impul-sive chaotic synchronization to synchronize chaotic transmitters and chaoticreceivers. The synchronization signal for a third-order chaotic transmitteronly used ≤ 94Hz bandwidth. This bandwidth is much smaller comparedwith 30kHz bandwidth needed for transmitting synchronization signals inthe other three generations.

To understand the theoretical backbone for the fourth generation, oneshould understand first impulsive effects and their mathematical models.We usually use state equations to describe the dynamics of continuousdynamical systems which evolve along time. And we also suppose thatthe state-variables are continuous for physical systems. But there do ex-ist a class of dynamical systems whose state variables are subject to jumps(abrupt changes) at discrete time instants, which we call impulsive effects ofdynamical systems. Processes of such characteristics can be found in manyfields, such as: physics, mechanics, population dynamics, chemical technol-ogy, economics, biology and electrical engineering. To model this kind ofsystem, we use the theory of impulsive differential equations[5, 36, 37].

Since the theory of impulsive differential equations is much difficult thanits counterpart of ordinary differential equations it remained a pure mathe-matical theory for almost 30 years until recently its promising applicationsto impulsive control and synchronization of chaotic systems were found bythe chaotic secure and spread spectrum communication group in Universityof California at Berkeley[10, 11, 12, 13, 14, 15]. Although the rigorous the-ory of impulsive synchronization only presented in the early of 1997[15], its

A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS 83

applications had been developed in both chaotic secure communication andchaotic spread spectrum communication. More recently, its applicationsare led to a US patent by Tao Yang and Leon O. Chua at the University ofCalifornia at Berkeley.1

In this paper I show that impulsive method is much more robust to pa-rameter mismatch than the other three generations. The degree of securityof the fourth generation is higher than that of the third generation. I alsoinvestigate the stability of impulsive synchronization of two chaotic systems.First, this problem is reduced to the stability of the origin of a synchroniza-tion error system, which is described by an impulsive differential equation.Then the stability of the trivial solution of a kind of impulsive differentialequation is studied. These results are used to study the conditions un-der which the impulsive synchronization between two Chua’s oscillators[6],which are standard chaotic electronic circuits developed by the chaotic se-cure and spread spectrum communication group in University of Californiaat Berkeley, is asymptotically stable. An estimate of the upper bound ofthe impulsive interval is also presented.

Since only synchronization impulses are sent to the driven system in animpulsive synchronization scheme, the information redundancy in the trans-mitted signal is reduced. In this sense, even low-dimensional chaotic systemscan provide high security. In this paper, we will use impulsive synchroniza-tion to develop a new framework for chaotic secure communication.

The organization of this paper is as follows. In section 2, a detailed his-tory of the first three generations of chaotic secure communication systemsis presented. In section 3, a theory on the stability of impulsive differentialequations is given. In section 4, a stability criterion for impulsive synchro-nization between two Chua’s oscillators is presented. In section 5, simulationresults of impulsive synchronization of Chua’s oscillators are presented. Insection 6, application of impulsive synchronization to secure communicationis 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 nonlinearsystems. Although the great mathematician Pincare had noted that somemechanical systems could behave chaotically[21], chaos did not attract wideattention until Lorenz published his paper in 1963[22]. In engineering com-munity, chaos had been mixed with noise for a long time. In 1980’s, theelectrical engineers first time “officially” announced the existence of chaos

1U.S.Patent 6,331,974 issued Dec. 18,2001, Name of Technology: Chaotic digital code-division multiplex access for wireless communication systems.

84 TAO YANG

in electrical systems. Since the noise-like behaviors of chaotic electroniccircuits, electrical engineers felt uncomfortable to deal with them. It wasphysicists first showed in 1990 that chaos could be controlled[2]. Thenthe synchronization between two identical chaotic systems was reported in1990[23]. In 1992, the electrical engineering community realized that chaoscould used in secure communication systems[24, 25, 26] because chaos isextremely sensitive to initial conditions and parameters. The concept ofchaotic hardware key for secure communication systems was then graduallyrealized by engineers and scientists.

Since the great potential of applying chaos to secure communication sys-tems, many groups over the world involved in the researches in this field.So far, chaotic communication systems have been updated to the fourthgeneration. In this paper, theory and structure of the fourth generation ispresented. It is useful to provide the reader who is not involved in chaoticsecure communication systems before a detailed history of these three gen-erations.

2.1. The first generation. The first generation was developed in 1993known as additive chaos masking[25] shown in Fig. 1(a) and chaotic shiftkeying[26] shown in Fig. 1(b). The additive chaos masking scheme shown inFig. 1(a) consists of two identical chaotic systems in both the transmitterand the receiver. The chaotic mask denoted by c(t) is one of the statevariables of the chaotic system2 in the transmitter. The message signalm(t), which is typically 20 dB to 30dB weaker than c(t) is added into thechaotic mask signal and gives the transmitted signal s(t). Since the chaoticsignal c(t) is very complex and m(t) is much smaller than c(t), one mayhope that the message signal m(t) can not be separated from s(t) withoutknowing the exact c(t).

To give the reader a hands-on experience on chaotic secure communi-cation systems, an example of additive chaotic masking scheme is givenas follow. From Fig. 1(a) we can see that a chaotic synchronization blockis needed in the receiver. Chaotic synchronization is a generalization of“carrier synchronization” in the normal communication systems but it isvery different from the latter. We use Chua’s oscillators to demonstrate thechaotic synchronization. A Chua’s Oscillator is shown in Fig. 2(a).

2For an autonomous system, at least three state variables are involved to generatechaos. For a non-autonomous system, at least two state variables and an independentinput are involved to generate chaos. In some cases, c(t) may be a function of more thanone state variable.


chaotic system

message signal

c(t)

m(t)

channels(t) chaotic

systemr(t)

chao

tic s

ynch

roni

zatio

n

c(t)~

m(t)

recoveredmessagesignal

~

receiver

transmitter

(a)

message signal

m(t)

chaotic system

chao

tic s

ynch

roni

zatio

nchannel

s(t) r(t) c(t)~

m(t)


~receiver

transmitter

chaotic system 0

c (t)0

c (t)chaotic system 1

1

e(t)

LPF andthresholding

(b)

Figure 1. The block diagrams of the first generation ofchaotic secure communication systems. (a) The additivechaos 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

dv1

dt=

1C1

[G(v2 − v1)− f(v1)

]

dv2

dt=

1C2

[G(v1 − v2) + i3

]

di3dt

=1L

[−v2 −R0i3]

(1)

86 TAO YANG

L

iCC

2

R

1vR

Chua’sDiode

iR

v v

voltage vs. currentcharacteristic ofChua’s diode

(a) (b)

−1 1

iR

vR0

3

1 2

G b

Ga

G b

R0

Figure 2. (a) The Chua’s oscillator. (b) The characteris-tics of the Chua’s diode.

where v1, v2 and i3 are the voltage across C1, the voltage across C2 and thecurrent through L, respectively. We set G = 1

R . The term R0i3 is added toaccount for the small resistance of the inductor in the physical circuit. Thepiece-wise linear v-i characteristic f(v1) of the Chua’s diode, is given by

f(v1) = Gbv1 +12(Ga −Gb)

(|v1 + E| − |v1 − E|)(2)

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

dv1

dt=

1C1

[G(v2 − v1)− f(v1)

]

dv2

dt=

1C2

[G(v1 − v2) + i3

]

di3dt

=1L

[−v2 −R0i3]

(3)

where v1, v2 and i3 are state variables of the receiver.Observe that in the second equation of Eq.(3), we substitute v1 with v1.

In this sense, v1 is the transmitted signal. We can prove that under someconditions, 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 statevariables 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 thetransmitter within 0.5ms though the initial states of these two circuits aremuch different.

After we understand what is chaotic synchronization, we can then hidea small message signal m(t) into the transmitted signal s(t) such that the


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10−3

−3

−2

−1

0

1

2

3

4

time

v1 a

nd ^

v1

Generated by /taoyang/c/chuasyn.c

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10−3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time

v2 a

nd ^

v2


(b)

Figure 3. Synchronization of two Chua’s oscillators. (a)The synchronizing process of v1 and v1. (b) The synchro-nizing process of v2 and v2. (c) The synchronizing processof i3 and i3.

receiver in Eq.(3) can be rewritten as3

dv1

dt=

1C1

[G(v2 − v1)− f(v1)

]

dv2

dt=

1C2

[G((v1 + m(t))− v2) + i3

]

di3dt

=1L

[−v2 −R0i3]

(4)

3In this case, we assume a noiseless channel is used such that r(t) = s(t) = v1(t)+m(t)is satisfied.

88 TAO YANG

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10−3

−4

−3

−2

−1

0

1

2

3

4x 10

−3

time

i3 a

nd ^

i3


(c)Figure 3 (Continued).

Given m(t) is weak enough, the synchronization between the transmitterand the receiver can be maintained because synchronization is a kind ofrobust phenomenon. In Fig. 4 we present the simulation results for additivechaotic mask method. Figure 4(a) shows the weak message signal. Figure4(b) shows the transmitted signal. Since the message signal is very weakwe can not see anything related to it in the transmitted signal. In thissense, 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 transientprocess, a (non-perfect) synchronization is achieved and the message signalis recovered with many noises.

This scheme was proved that it could not be used under practical condi-tions because of the following drawbacks. Since the message signal is typ-ically 20dB to 30dB weaker than the “chaotic mask”, this method is verysensitive to channel noise and parameter mismatch between the chaotic sys-tems in the transmitter and the receiver. Furthermore, this scheme has avery low degree of security[8].

Chaotic shift keying shown in Fig. 1(b) also known as chaotic switchingwas designed to transmit digital message signal. In this scheme, the messagesignal, which is a digital signal, is used to switch the transmitted signalbetween two statistically similar chaotic attractors, which are respectivelyused to encode bit 0 and bit 1 of the message signal. These two attractorsare generated by two chaotic systems with the same structure and differentparameters. At the receiver end, the received signal is used to drive a


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−3

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−3

time

mes

sage

sig

nal


(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−3

−3

−2

−1

0

1

2

3

time

tran

smitt

ed s

igna

l


(b)

Figure 4. Simulation results of additive mask scheme us-ing two Chua’s oscillators. (a) The weak message signalm(t). (b) The transmitted signal s(t). (c) The recoveredsignal m(t).

chaotic system, which is identical to any of the two chaotic systems inthe transmitter. The message signal is recovered by low-pass filtering andthen thresholding the synchronization error signal e(t), which is depicted inFig. 1(b).

90 TAO YANG

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−3

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−3

time

reco

vere

d re

sult



−0.5 0 0.5 1 1.5 2 2.5−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

V1(V)

V2(

V)

−0.5 0 0.5 1 1.5 2 2.5−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

v1(V)

v2(V

)

(a) (b)

Figure 5. Chaotic attractors of Chua’s oscillators used inchaotic shift keying. Both of them are shown in the v1-v2

plane. (a) The chaotic attractor for encoding bit “1”. (b)The chaotic attractor for encoding bit “0”.

To demonstrate this scheme, we also use Chua’s oscillators as the trans-mitter and the receiver. In the transmitter, two Rossler-like chaotic at-tractors are used to encode bit “1” and bit “0”, respectively. The param-eters for encoding bit “1” are: R = 1000Ω, R0 = 20Ω, Ga = −1.139mS,


Gb = −0.711mS, E = 1V , L = 12mH, C1 = 17nF , and C2 = 178nF .The corresponding chaotic attractor is shown in Fig. 5(a). The parame-ters for encoding bit “0” are: R = 1000Ω, R0 = 20Ω, Ga = −1.139mS,Gb = −0.711mS, E = 1V , L = 12.5mH, C1 = 17.5nF , and C2 = 197nF .The corresponding chaotic attractor is shown in Fig. 5(b). These two sets ofparameters generate two different but statistically similar chaotic attractors.The voltage v1 is transmitted to the receiver. The chaotic synchronizationblock is something similar though not to be necessary the same to that usedin Fig. 1(a)4.

The simulation results are shown in Fig. 6. Figure 6(a) shows the binarymessage signal with a 10ms cycle length. Figure 6(b) shows the synchroniza-tion error of the authorized receiver corresponding to bit “0”. Figure 6(c)shows the synchronization error of the authorized receiver corresponding tobit “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, ithas a low degree of security[7] if the chaotic attractors are too far away in thebifurcation space. However, since this is the first scheme of chaotic digitalcommunication systems, there still exist many possibilities of improving it.

2.2. The second generation. The second generation was proposed during1993 to 1995 known as chaotic modulation. This generation used two differ-ent ways to modulate message signals into chaotic carriers. The first methodcalled chaotic parameter modulation[27] shown in Fig. 7(a) used message sig-nals to change parameters of the chaotic transmitter. The second methodcalled chaotic non-autonomous modulation[28] shown in Fig. 7(b) used themessage signal to change the phase space of the chaotic transmitter.

In Fig. 7(a) the message signal m(t) is used to modulate some parame-ters of the chaotic system in the transmitter such that its trajectories keepchanging in different chaotic attractors. Since the bifurcation space of achaotic system is very complex, it is very difficult to figure out the way ofthe changes of the parameters even through the intruder knows some partialknowledge of the structure of the chaotic system in the transmitter. At thereceiver end an adaptive controller is used to adaptively tune the param-eters of the chaotic system such that the synchronization error approach

4Since the chaotic synchronization blocks in different chaotic secure communicationsystems are usually not the same because there are so many choices of chaotic systems.Different chaotic systems may used different synchronization schemes. The synchro-nization of chaotic system is now a very active research area in both the scientific andengineering communities. In the rest of this section, I will not emphasize the differ-ence between the synchronization block used in different chaotic communication systems,instead I provide many references for those who want to know the technical details.

92 TAO YANG

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0

0.2

0.4

0.6

0.8

1

Time(s)

(a)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

Time(s)

e1(V

)

(b)

Figure 6. The simulation results of chaotic shift keying.(a) The binary message signal m(t). (b) Synchronizationerror of Chua’s oscillator with the parameters correspond-ing to bit “0”. (c) Synchronization error of Chua’s oscillatorwith the parameters corresponding to bit “1”.

zero. By doing this, the output of the adaptive controller can recover themessage signal. The simulation results are shown in Fig. 8. In this sim-ulation, three message signals are used to tune three different parameters


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Time(s)

e1(V

)


of the chaotic system in the transmitter. Since the chaotic system keepschanging its attractors, the waveform of the transmitted signal as shown inFig. 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 recoveredmessage 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 chaoticnon-autonomous modulation shown in Fig. 7(b) used the message signal toperturb chaotic attractor directly in the phase space. Unlike in chaoticparameter modulation where the transmitter is switched among differenttrajectories in different chaotic attractors, the transmitter in chaotic non-autonomous modulation is switched among different trajectories of the samechaotic attractor. Theoretically, chaotic non-autonomous modulation is anerror free scheme. The second generation improved the degree of securityto some degree but was still found unsatisfactory[18, 29, 33, 34, 35].

2.3. The third generation. The third generation shown in Fig. 9 wasproposed in 1997[9] for the purpose of improving the degree of security toa much higher level than the first two generations. We call this generationas chaotic cryptosystem. In this generation, the combination of the classicalcryptographic technique and chaotic synchronization is used to enhance thedegree of security. So far, this generation has the highest security in allthe chaotic secure communication systems had been proposed and has notyet been broken. In the chaotic cryptosystem the plain text signal p(t) is

94 TAO YANG

chaotic system

message signal

m(t)

channels(t) chaotic

systemr(t)

chao

tic s

ynch

roni

zatio

n

~


receiver

transmitter

parameters

parameters

s(t)

e(t)

adaptivecontroller

~m(t)

(a)

chaotic system

message signal

m(t)

channels(t) chaotic

systemr(t)

chao

tic s

ynch

roni

zatio

n

~


receiver

transmitter

s(t)

~m(t)

(b)

Figure 7. The block diagrams for the second generationof chaotic secure communication systems. (a) The chaoticparameter modulation. (b) The chaotic non-autonomousmodulation.

encrypted by a encryption rule with a key signal, k(t), which is generatedby the chaotic system in the transmitter. The scrambled signal is usedfurther to drive the chaotic system such that the chaotic dynamics is changed


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2−4

−3

−2

−1

0

1

2

3

4

time(s)

x

(a)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2−9.5

−9.4

−9.3

−9.2

−9.1

−9

−8.9

−8.8

−8.7

−8.6

x 10−4

time(s)

a an

d a1

(b)

Figure 8. Using the chaotic parameter modulation totransmit three signals simultaneously. (a) The transmittedsignal s(t). (b) The first message signal and its recoveredversion. (c) The second message signal and its recoveredversion. (d) The third message signal and its recoveredversion.

continuously in a very complex way5. Then another state variable of the

5Observe that the third generation had some “modulation” properties provided bythe second generation.

96 TAO YANG

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2−5.2

−5.15

−5.1

−5.05

−5

−4.95

−4.9

−4.85

x 10−4

time(s)

b an

d b1

(c)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1x 10

−5

time(s)

c an

d c1

(d)Figure 8 (Continued).

chaotic system in the transmitter is transmitted to through public channelwhich can be accessed by the intruder. Since the intruder can not get accessto the chaotic hardware key, it is very difficult to find p(t) out from s(t). Atthe receiver, the received signal r(t) = s(t)+n(t), where n(t) is the channelnoise, is used to synchronize both of the chaotic systems in transmitterand the receiver. After the chaotic synchronization had been achieved, thesignal k(t) and y(t) can be recovered at the receiver with some noises asdenoted by k(t) and y(t). By feeding k(t) and y(t) into the decryption


Plain Text

Public Channel

IntruderPlain Text

p(t) Encryption

Chaotic System Chaotic System

k(t)k(t) y(t)

p(t)y(t)

~ ~

~

s(t) s(t)+n(t)

Encrypter Decrypter

DecryptionRule e(.) Rule d(.)

Figure 9. The block diagram for the third generation ofchaotic secure communication systems.

rule at the receiver, the plain text signal can be recovered with some noisesas p(t). The simulation result is shown in Fig. 10. Figure 10(a) showsthe transmitted signal s(t), from which one can not observe the embeddedplain signal. Figure 10(b) shows the recovered and decrypted result at thereceiver. Observe that after the transient process of synchronization, theplain text signal is recovered. To show the high security of this scheme, theunmasking method provided in [18] is used to decode the plain text signal.Figure 10(c) shows the unmasked signal y(t) by the intruder, from which itis impossible to retrieve the plain text signal as shown in Fig. 10(d).

2.4. Remarks. To implement practical chaotic secure communication sys-tems, two critical technical problems should be first solved for archivingchaotic synchronization. The first one is the parameter mismatch betweenthe chaotic transmitter and receiver. This problem was solved by usingadaptive synchronization[4, 30]. The other problem was the nonlinearity ofthe channel. This problem was first solved in [31, 3]. However, the channelmodel used in [31, 3] was far from a real channel. Another possible methodwas presented in [32] to provide a channel-independent scheme.

Although all the examples provided in this paper work in base band, thecorresponding RF system can be built based on the same principles. Also,for simplicity all the examples are demonstrated under the analogue commu-nication framework. However, in the practical implementations, the chaoticsecure communication systems are built as digital secure communicationsystems using chaotic carriers. So far, there existed at least two differentways to build chaotic digital secure communication systems. The first onewas to digitize the transmitted signal s(t) and then send it via classic digi-tal communication techniques. Since the robustness of the adaptive chaoticsynchronization used in the second and third generations, the quantizing er-ror 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

98 TAO YANG

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01−3

−2

−1

0

1

2

3

Time(s)

Am

plitu

de(v

)

(a)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time(s)

Am

plitu

de(v

)

(b)

Figure 10. Simulation results of chaotic cryptosystem.(a) The transmitted signal s(t). (b) The recovered andthen decrypted signal p(t). (c) The recovered encryptedsignal using the method in [18]. (d) The decrypted resultof that shown in (c).

chaotic shift keying. Although so far only a few prototypes of chaotic securecommunication systems have been built as RF digital communication sys-tems, the tendency in almost all groups in the world is focused on wirelesschaotic digital secure communication systems. However, the main principlespresented in this section are still the backbones of all these systems.

One can see that chaotic synchronization is the critical technology forbuilding a chaotic secure communication scheme. In fact, it is the evolutionof chaotic synchronization technology to trigger that of the chaotic securecommunication systems. The first generation based on the simplest chaoticsynchronization, which was from feedback control theory. The second and


0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

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Time(s)

Am

plitu

de(v

)

(c)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02−0.4

−0.3

−0.2

−0.1

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0.2

0.3

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Time(s)

Am

plitu

de(v

)


the third generations based on the adaptive chaotic synchronization, whichwas from adaptive control theory. In principle, the chaotic synchronizationused in the first three generations are “continuous” synchronization. In 1997a brand new chaotic synchronization technology called impulsive synchro-nization was invented based on impulsive control theory[15, 14, 13, 11, 12].By applying impulsive synchronization, the fourth generation of chaotic se-cure communication system was presented. In the rest parts of this paper,the theory and structure of the fourth generation of chaotic secure commu-nication systems are presented.

3. Basic Theory of Impulsive Differential Equations

To make this paper self-contained and since the theory of impulsive dif-ferential equation is not well-known to the scientific community, we addresssome basic results in this section which is useful in this paper.

100 TAO YANG

Consider the general nonlinear system

x = f(t,x)(5)

where f : R+ × Rn 7→ Rn is continuous, x ∈ Rn is the state variable, and

x ∆=dxdt

.

Consider a discrete set τi of time instants, where

0 < τ1 < τ2 < ... < τi < τi+1 < ...,

τi →∞ as i →∞Let

U(i,x) = ∆x|t=τi

∆= x(τ+i )− x(τ−i )(6)

be the “jump” in the state variable at the time instant τi. Then this impul-sive system is described by

x = f(t,x), t 6= τi

∆x = U(i,x), t = τi

x(t+0 ) = x0, t0 ≥ 0, i = 1, 2, ...(7)

This is called an impulsive differential equation[1]. To study the stability ofthe impulsive differential equation (7) we use the following definitions andtheorems[1].

Definition 1: Let V : R+ ×Rn 7→ R+, then V is said to belong to classV0 if

1. V is continuous in (τi−1, τi]× Rn and for each x ∈ Rn, i = 1, 2, ...,

lim(t,y)→(τ+

i ,x)V (t,y) = V (τ+

i ,x)(8)

exists;2. V is locally Lipschitzian in xDefinition 2: For (t,x) ∈ (τi−1, τi]× Rn, we define

D+V (t,x) ∆= limh→0

sup1h

[V (t + h,x + hf(t,x))− V (t,x)](9)

Since the system in Eq.(7) is an nth-order impulsive differential equation,instead of studying the stability of Eq.(7), it is convenient to study that ofa first-oder impulsive differential equation which is given by the followingdefinition.

Definition 3: Comparison systemLet V ∈ V0 and assume that

D+V (t,x) ≤ g(t, V (t,x)), t 6= τi

V (t,x + U(i,x)) ≤ ψi(V (t,x)), t = τi(10)


where g : R+ × R+ 7→ R is continuous and ψi : R+ 7→ R+ is nondecreasing.Then the system

w = g(t, w), t 6= τi

w(τ+i ) = ψi(w(τi))

w(t+0 ) = w0 ≥ 0(11)

is called the comparison system of Eq.(7).Definition 4:

Sρ = x ∈ Rn| ||x|| < ρ(12)

where ‖ · ‖ denotes the Euclidean norm on Rn.Definition 5: A function α is said to belong to class K if α ∈ C[R+,R+],

α(0) = 0 and α(x) is strictly increasing in x.Assumptions: f(t, 0) = 0, U(i, 0) = 0 and g(t, 0) = 0 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 setτi.

Theorem 1 shows how can the stability of an nth-order impulsive differ-ential equation be equivalent to that of a first-order impulsive differentialequation, i.e., the comparison system.

Theorem 1(Theorem 3.2.1, page 139, [1]): Assume that the followingthree conditions are satisfied:

1. V : R+ × Sρ 7→ R+, ρ > 0, V ∈ V0, D+V (t,x) ≤ g(t, V (t,x)), t 6= τi.2. there exists a ρ0 > 0 such that x ∈ Sρ0 implies that x + U(i,x) ∈ Sρ0

for all i and V (t,x + U(i,x)) ≤ ψi(V (t,x)), t = τi, x ∈ Sρ0 .3. β(||x||) ≤ V (t,x) ≤ α(||x||) on R+ × Sρ,

where α(·), β(·) ∈ K.Then the stability properties of the trivial solution of the comparison sys-

tem (11) imply the corresponding stability properties of the trivial solutionof (7).

In next theorem, we present the stability criterion for a first-order im-pulsive differential equation, which is the general form of the comparisonsystem of a synchronization error system we will study in this paper.

Theorem 2(Corollary 3.2.1., page 142, [1]): Let g(t, w) = λ(t)w, λ ∈C1[R+,R+], ψi(w) = diw, di ≥ 0 for all i. Then the origin of system (7) isasymptotically stable if the conditions

λ(τi+1) + ln(γdi) ≤ λ(τi), for all i, where γ > 1(13)

and

λ(t) ≥ 0(14)

are satisfied.

102 TAO YANG

4. Impulsive synchronization of Chua’s oscillators

In this section, we study the impulsive synchronization of two Chua’soscillators. The dimensionless form of a Chua’s oscillator is given by[6]

x = α(y − x− f(x))y = x− y + zz = −βy − γz

(15)

where α, β and γ are three constants and f(x) is the piecewise-linear char-acteristic of the Chua’s diode, which is given by

f(x) = bx +12(a− b)(|x + 1| − |x− 1|)(16)

where a < b < 0 are two constants.Let x> = (x, y, z), then we can rewrite the Chua’s oscillator equation

into the form

x = Ax + Φ(x)(17)

where

A =

−α α 01 −1 10 −β −γ

, Φ(x) =

−αf(x)

00

(18)

In the impulsive synchronization schemes, there are two Chua’s oscilla-tors. One of them is called the driving system and the other is called thedriven system. In an impulsive synchronization configuration, the drivingsystem is given by Eq.(15). The driven system is given by

˙x = Ax + Φ(x)(19)

where x = (x, y, z) is the state variables of the driven system.At discrete instants, τi, i = 1, 2, ..., the state variables of the driving

system are transmitted to the driven system and then the state variables ofthe driven system are subject to jumps at these instants. The driven systemis described by the impulsive differential equation

˙x = Ax + Ψ(x), t 6= τi

∆x|t=τi = −Be, i = 1, 2, ...(20)

where B is a 3× 3 matrix, and e> = (ex, ey, ez) = (x− x, y− y, z− z) is thesynchronization error. If we define

Ψ(x, x) = Φ(x)− Φ(x) =

−αf(x) + αf(x)

00

(21)


then the error system of the impulsive synchronization is given by

e = Ae + Ψ(x, x), t 6= τi

∆e|t=τi= Be, i = 1, 2, ...

(22)

We use the following theorem to guarantee that our impulsive synchro-nization is asymptotically stable.

Theorem 3: Let d1 be the largest eigenvalue of (I + B>)(I + B), whereB is a symmetric matrix. Assume the spectral radius ρ of I + B satisfiesρ(I + B) ≤ 1. Let q be the largest eigenvalue of (A + A>) and assume theimpulses are equidistant from each other and separated by an interval ∆. If

0 ≤ q + 2|αa| ≤ − 1∆

ln(ξd1), ξ > 1(23)

then the impulsive synchronization of two Chua’s oscillators is asymptoti-cally stable.

Proof: Let us construct the Lyapunov function V (t, e) = e>e. For t 6= τi,we have

D+V (t, e) = e>Ae + e>A>e + e>Ψ(e) + Ψ>(e)e

≤ qe>e + 2|α||f(x)− f(x)|ex

≤ qe>e + 2|αa|e2x

≤ (q + 2|αa|)e>e

= (q + 2|αa|)V (t, e)(24)

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

ρ(I + B) = ||I + B||(25)

Given any ρ0 > 0 and e ∈ Sρ0 , we have

||e + Be|| ≤ ||I + B||||e|| = ρ(I + B)||e|| ≤ ||e||(26)

The last inequality follows from ρ(I + B) ≤ 1. Consequently, e+ Be ∈ Sρ0 .For t = τi, we have

V (τi, e + Be) = (e + Be)>(e + Be)(27)

= e>(I + B>)(I + B)e≤ d1V (τi, e)

Condition 2 of Theorem 1 is satisfied with ψi(w) = d1w. We can see thatcondition 3 of Theorem 1 is also satisfied. It follows from Theorem 1 that

104 TAO YANG

the asymptotic stability of the origin of the synchronization error system inEq.(22) is implied by that of the following comparison system

ω = (q + 2|αa|)ω, t 6= τi

ω(τi) = d1ω(τi)ω(t0) = ω0 ≥ 0

(28)

From Eq.(23), we have∫ τi+1

τi

(q + 2|αa|)dt + ln(ξd1) ≤ 0, ξ > 1(29)

and λ(t) = q+2|αa| ≥ 0. It follows from Theorem 2 that the trivial solutionof Eq.(22) is asymptotically stable. ¤

Theorem 3 also gives an estimate for the upper bound ∆max of ∆; namely,

∆max =∣∣∣ ln(ξd1)q + 2|αa|

∣∣∣, ξ −→ 1+(30)

Observe that the upper bound given by Eq.(30) is sufficient but not neces-sary. Consequently, we can only say that we have a predicted stable region,which is usually smaller than the actual stable region because we can notassert that all other regions are unstable.

5. Simulation results of impulsive synchronization

In the following simulations, we choose the parameters of Chua’s os-cillator as α = 15, β = 20, γ = 0.5, a = − 120

7 , b = − 757 . A fourth-order

Runge-Kutta with step size 10−5 is used. The initial conditions are given by(x(0), y(0), z(0)) = (−2.121304,−0.066170, 2.881090) and (x(0), y(0), z(0)) =(0, 0, 0). The trajectories of the driving system are shown in Fig. 11, whichis the Chua’s double scroll attractor.

5.1. Simulation 1: strong coupling. In this simulation, we choose thematrix B as

B =

k 0 00 −1 00 0 −1

(31)

where the impulsive coupling is “strong”. It follows from Theorem 3 thatρ(I + B) ≤ 1 should be satisfied, which implies that −2 ≤ k ≤ 0. By usingthis B matrix, it is easy to see that

d1 = (k + 1)2(32)


−2

0

2

−2

0

2

−0.4

−0.2

0

0.2

0.4

zx

y

Figure 11. The double scroll attractor.

We have

A =

−15 15 01 −1 10 −20 −0.5

, A + A> =

−30 16 016 −2 −190 −19 −1

(33)

from which we find q = 20.162180. Then an estimate of the boundaries ofthe stable region is given by

0 ≤ ∆ ≤ − (lnξ + ln(k + 1)2)q + 2|αa| ,−2 ≤ k ≤ 0(34)

Figure 12 shows the stable region for different ξ’s. The entire regionbelow the curve corresponding to ξ = 1 is the predicted stable region. Whenξ −→∞, the stable region shrinks to a line k = −1.

The simulation results are shown in Fig. 13. Figure 13(a) shows instabil-ity for k = −1.5 and ∆ = 1. The solid waveform, the dash-dotted waveformand the dotted waveform correspond to ex(t), ey(t) and ez(t), respectively.Figure 13(b) shows stable results within the stable region for k = −1.5 and

106 TAO YANG

−2 −1.5 −1 −0.5 00

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

k

Del

ta

xi= 1 2 10 10 2 1

stable region

Figure 12. Estimate of the boundaries of stable regionswith different ξ’s used in simulation 1.

∆ = 0.002. One can see that the system asymptotically approaches theorigin with a settling time6 of about 0.05. However, the true stable regionis 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 ∆ = 0.05. We canalso see that the system asymptotically approaches the origin with a settlingtime of about 1.4 which is much larger than that shown in Fig. 13(b).

5.2. Simulation 2: weak coupling. In this simulation, we choose thematrix B as

B =

k 0 00 −0.1 00 0 −0.1

(35)

6Here the settling time is defined as Ts when the synchronization error |e(t)| < 10−3

for any time t ≥ Ts. Since the low resolution of the printer, we can not determine Ts

from the figures. We can find Ts from the data.


0 1 2 3 4 5 6 7 8−4

−3

−2

−1

0

1

2

3

4

time

(a)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−3

−2

−1

0

1

2

3

time

(b)

Figure 13. Simulation results. (a) Unstable results out-side the stable region. (b) Stable results inside the pre-dicted stable region. (c) Stable results outside the predictedstable region.

where the impulsive coupling is much weaker than that chosen in simulation1.

108 TAO YANG

0 0.5 1 1.5−3

−2

−1

0

1

2

3

time


It is easy to see that

d1 =

(k + 1)2, (k + 1)2 ≥ 0.810.81, elsewhere

(36)

An estimate of the boundaries of the stable region is given by

0 ≤ ∆

≤

− ln ξ + ln(k + 1)2

q + 2|αa| , (k + 1)2 ≥ 0.81

− ln ξ + ln(0.81)q + 2|αa| , elsewhere

, −2 ≤ k ≤ 0(37)

Figure 14 shows the stable region. The entire region below the curvecorresponding to ξ = 1 is the predicted stable region. In this case, ∆ isalways bounded. It seems that we can not control the system to the originwith an arbitrarily prescribed speed because ξ has to satisfy 1 < ξ < 100

81 .This is different from the case shown in Fig. 12, where any value of ξ > 1 ispossible.

The simulation results are shown in Fig. 15. Again, the solid waveform,the dash-dotted waveform and the dotted waveform correspond to ex(t),ey(t) and ez(t), respectively. Figure 15(a) shows the instability results fork = −1 and ∆ = 0.4. Figure 15(b) shows the stable results in the stableregion for k = −1 and ∆ = 3 × 10−4. The control system asymptoticallyapproaches the origin with a settling time of about 0.05. Also, the truestable region is larger than that predicted in Fig. 14. To demonstrate this


−2 −1.5 −1 −0.5 00

0.5

1

1.5

2

2.5

3

3.5

4x 10

−4

k

Del

ta

stable region

xi=1.0

Figure 14. Estimate of the boundaries of stable regionused in simulation 2.

fact, we show in Fig. 15(c) the stable results for k = −1 and ∆ = 0.01.We can also see that the system asymptotically approaches the origin witha settling time equal approximately to 1, which is much larger than thatshown in Fig. 15(b).

5.3. Effects of channel noise. Let us study next the robustness of theimpulsive synchronization to additive channel noise. In the following simu-lations, we choose the matrix B as

B =

−1 0 00 −1 00 0 −1

(38)

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=20dBand ∆ = 0.1. Figure 16(a) shows the noise (the red waveform) in impulsesx(τi) and the synchronization error(the blue waveform) x− x. Figure 16(b)shows the noise (the red waveform) in impulses y(τi) and the synchronization

110 TAO YANG

0 0.5 1 1.5 2 2.5 3 3.5 4−3

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0

1

2

3

4

time

(a)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

time

(b)

Figure 15. Simulation results. (a) Unstable results out-side the stable region. (b) Stable results in the stable re-gion. (c) Stable results outside the stable region.

error(the blue waveform) y− y. Figure 16(c) shows the noise (the red wave-form) in impulses z(τi) and the synchronization error(the blue waveform)z − z. From the above simulation results we find that the synchronizationerrors are comparable to the noise in the synchronization impulses.


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−3

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−1

0

1

2

3

time


For comparison, we also presented the corresponding results when a con-tinuous synchronization is used. The driven system of the continuous syn-chronization is given by

˙x = α(y − x− f(x))˙y = (x + n(t))− y + z˙z = −βy − γz

(39)

where x is the first state variable of the driving system and n(t) is the ad-ditive channel noise. When n(t) = 0, the driving system and the drivensystem can be synchronized. When the additive noise is added in the trans-mitted signal with SNR = 20dB, the synchronization errors are shown inFig. 16(d). Observe that the synchronization error x − x of continuousscheme is bigger than that of the impulsive scheme.

Given a SNR level, the synchronization errors of the impulsive schemebecome bigger if ∆ increases. In our simulations we find if ∆ = 1 thenthere exist some big synchronization error peaks. The simulation resultsare shown in Fig. 17 with SNR=20dB. Figure 17(a) shows the noise (thered waveform) in the impulses x(τi) and the synchronization error(the bluewaveform) x − x. Figure 17(b) shows the noise (the red waveform) in theimpulses y(τi) and the synchronization error(the blue waveform) y− y. Fig-ure 17(c) shows the noise (the red waveform) in the impulses z(τi) and thesynchronization error(the blue waveform) z− z. From the simulation resultswe find that the synchronization errors are comparable to the noise most of

112 TAO YANG

360 365 370 375 380 385 390 395 400−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

time

(a)

Figure 16. Simulation results of the impulsive synchro-nization and the continuous synchronization when channelnoise is added. (a) Noise in impulses x(τi)(red) and thesynchronization error x− x(blue) of the impulsive synchro-nization. (b) Noise in impulses y(τi)(red) and the synchro-nization error y− y(blue) of the impulsive synchronization.(c) Noise in impulses z(τi)(red) and the synchronization er-ror z − z(blue) of the impulsive synchronization. (d) Syn-chronization errors x− x(blue), y− y(red) and z− z(green)of the continuous synchronization.

the time. However, when the ∆ becomes too big, e.g., ∆ = 6, then most ofthe time we observe that the synchronization errors are much bigger thanthe noise.

From the above simulation we can conclude that if ∆ is small enough,e.g, ∆ = 0.1, then the impulsive synchronization is more robust than thecontinuous synchronization. This is because in continuous synchronizationthe synchronization error system is continuously disturbed by the continuouschannel noise and more often pushed to some local instabilities than inimpulsive synchronization. This in turn results in more frequent large peaksof synchronization errors in continuous synchronization.

5.4. Effects of parameter mismatch. The parameter mismatch is anyparameter difference between the driving system and the driven system. Therobustness of impulsive synchronization to parameter mismatch is studied


360 365 370 375 380 385 390 395 400−8

−6

−4

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0

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4

6

8x 10

−3

time

(b)

360 365 370 375 380 385 390 395 400−0.06

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sync

hron

izat

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in the following simulations. Figure 18 shows the synchronization errorswhen different parameter mismatches are used. Figure 18(a) shows theresults when only 1% mismatch exists in parameter α with ∆ = 0.1. Figure18(b) shows the results when 10% mismatch exists in the parameter α with∆ = 0.1. Observe that the impulsive synchronization is robust enoughto parameter mismatch. We also show the results when the continuoussynchronization scheme is used. Figure 18(c) shows the results when a1% mismatch exists in the parameter α and continuous synchronization

114 TAO YANG

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sync

hron

izat

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erro

rs


is used. Observe that there exist some very big synchronization errors inx − x. Figure 18(d) shows the results when a 10% mismatch exists in theparameter α and the continuous synchronization is used. Observe that thereexist some very big synchronization errors in x− x. From above we can seethat impulsive synchronization is less sensitive to parameter mismatchesthan continuous synchronization.

6. The fourth generation of chaotic secure communicationsystem

Since the publication of several chaotic cryptanalysis results in low-dimensionalchaos-based secure communication systems[7, 8, 16, 18], there were concernsthat such communication schemes may not be secure enough. To overcomethis objection, one approach is to exploit hyperchaos-based7 secure com-munication systems, but such systems may introduce more difficulties tosynchronization.

On the other hand, we can enhance the security of low-dimensional chaos-based secure communication schemes by combining conventional crypto-graphic schemes with a chaotic system[9]. To overcome the low security ob-jections against low-dimensional continuous chaos-based schemes, we mayuse the following two methods. The first method is to make the transmittedsignal more complex. The second method is to reduce the redundancy inthe transmitted signal. In [9] we have presented a method to combine a

7A hyperchaotic system has at least two positive Lyapunov exponents.


360 365 370 375 380 385 390 395 400−0.2

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

time

(a)

360 365 370 375 380 385 390 395 400

−0.02

0

0.02

0.04

0.06

0.08

0.1

time

(b)

Figure 17. Simulation results of the impulsive synchro-nization when channel noise is added and ∆ is big. (a)Noise in impulses x(τi)(red) and the synchronization er-ror x − x(blue). (b) Noise in impulses y(τi)(red) and thesynchronization error y − y(blue). (c) Noise in impulsesz(τi)(red) and the synchronization error z − z(blue).

conventional cryptographic scheme with low-dimensional chaos to obtain avery complex transmitted signal. The impulsive synchronization presented

116 TAO YANG

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0

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time


in this paper offers a very promising approach of reducing the redundancyin transmitted signals.

6.1. A simple system in baseband. In this section, we combine the re-sults in [9] and impulsive synchronization to give a new chaotic secure com-munication scheme. The block diagram of this scheme is shown in Fig. 19.From Fig. 19 we can see that this chaotic secure communication systemconsists of a transmitter and a receiver. In both the transmitter and thereceiver, there exist two identical chaotic systems. Also, two identical con-ventional cryptographic schemes are embedded in both the transmitter andthe receiver. Let us now consider the details of each block in Fig. 19. Thetransmitted signal consists of a sequence of time frames. Every frame hasa length of T seconds and consists of two regions. In Fig. 20 we showthe concept of a time frame and its components. The first region of thetime frame is a synchronization region consisting of synchronization im-pulses. The synchronization impulses are used to impulsively synchronizethe chaotic systems in both transmitter and receiver. The second regionis the scrambled signal region where the scrambled signal is contained. Toensure synchronization, we have T < ∆max. Within every time frame, thesynchronization region has a length of Q and the remaining time intervalT −Q is the scrambled signal region.

The composition block in Fig. 19 is used to combine the synchronizationimpulses and the scrambled signal into the time frame structure shown inFig. 20. The simplest combination method is to substitute the beginning


380 381 382 383 384 385 386 387 388 389 390−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

time

sync

hron

izat

ion

erro

rs

(a)

Figure 18. Simulation results of the impulsive synchro-nization and the continuous synchronization when param-eter mismatches exist. (a) A 1% mismatch exists in α.The synchronization errors of the impulsive synchroniza-tion: x − x(blue), y − y(red) and z − z(green). (b) A10% mismatch exists in α. The synchronization errors ofthe impulsive synchronization: x− x(blue), y − y(red) andz − z(green). (c) A 1% mismatch exists in α when thecontinuous synchronization is used. The synchronizationerrors x− x(blue), y− y(red) and z − z(green). (d) A 10%mismatch exists in α when the continuous synchronizationis used. The synchronization errors x− x(blue), y− y(red)and z − z(green).

Q seconds of every time frame with synchronization impulses. Since Qis usually very small compared with T , the processing time for packing amessage signal is negligible. The decomposition block is used to separate thesynchronization region and the scrambled signal region within each frameat the receiver end. Then the separated synchronization impulses are usedto make the chaotic system in the receiver to synchronize with that in thetransmitter. The stability of this impulsive synchronization is guaranteedby our results in Section 4.

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sync

hron

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(b)

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−1

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1

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rs


In the transmitter and the receiver, we use the same cryptographic schemeblock for purposes of bi-directional communication. In a bi-directional com-munication scheme, every cellular phone should function both as a receiverand a transmitter. Here, the key signal is generated by the chaotic system.The cryptographic scheme is as follows[9]:


360 365 370 375 380 385 390 395 400−2.5

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We use a continuous n-shift cipher to encrypt the plain signal(messagesignal). The n-shift cipher is given by

e(p(t)) = f1

(...f1

(f1

︸︷︷︸n

(p(t), k(t)

), k(t)

), ..., k(t)

)

︸︷︷︸n

= y(t)(40)

where h is chosen such that p(t) and k(t) lie within (−h, h). Here, p(t) andk(t) denote the plain signal and the key signal, respectively, and y(t) denotesthe encrypted signal. The key signal k(t) is chosen as a state variable of thechaotic system. The notation f1(·, ·) denotes a scalar nonlinear function oftwo variables defined as follow:

f1(x, k) =

(x + k) + 2h, −2h ≤ (x + k) ≤ −h(x + k), −h < (x + k) < h

(x + k)− 2h, h ≤ (x + k) ≤ 2h(41)

This function is shown in Fig. 21.The corresponding decryption rule is the same as the encryption rule

p(t) = d(y(t)) = e(y(t))

= f1

(...f1

(f1

︸︷︷︸n

(y(t),−k(t)

),−k(t)

), ...,−k(t)

)

︸︷︷︸n

.(42)

To decode the encrypted signal, the same key signal should be used.

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messagesignal

cryptographicscheme

chaoticsystem

scrambledsignal

synchronizationimpulses

transmittedsignal

messagesignal

cryptographicscheme

chaoticsystem

scrambledsignal


decomposition

composition

Transmitter

Receiver

Figure 19. Block diagram of the impulsive-synchronization based chaotic secure communicationsystem.

The simulation results are as follows. The synchronization region is lo-cated in the initial 1% of every time frame. We choose the frame lengthas T = 1s. In the synchronization region of every time frame, we transmitthe impulses of the three state-variables of the Chua’s oscillators. The pa-rameter of the encrypted signal is chosen as h = 0.4. A continuous 10-shiftcipher was used. We choose x and x as the key signals and normalized themto fall within the amplitude range [−0.4, 0.4].


Tim

e

Fra

me

Num

ber

12

34

5

0T

2T3T

4T5T

(n−

1)T

nT

QT

−Q

Syn

chro

niza

tion

impu

lses

(syc

hron

izat

ion

regi

on)

Scr

ambl

ed

sign

alre

gion

Figure 20. Illustration of the concept of a time-frame andits components.

Figure 22 shows the simulation results of the above proposed secure com-munication system for transmitting a speech signal. Figure 22(a) shows the

122 TAO YANG

−2h −h

−h

f1

2hh0

(x,k)

(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. Figure22(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 addi-tive channel noise with SNR = 16dB. Figure 22(d) shows the spectrogramsfor the scrambled speech signal and the additive channel noise. We can seethat the structure of the signal in Fig. 22(b) was totally covered by analmost uniformly distributed noise-like spectrum. Figure 22(e) shows thewaveforms of the descrambled speech signal. Figure 22(f) shows the spec-trograms of the descrambled speech signal. We can see that some noiseswere introduced into the recovered results due to the channel noise, andthat the spectrograms became a little blur. But the structure of the speechsignal was perfectly recovered.

6.2. An illustrative example: secure digital cellular phone. In thisdigital communication era, the implementation of the whole system pre-sented in this section should be digital instead of analog. Let us revisit thebaseband block diagram in Fig. 19. Since the parameter mismatch of theChua’s oscillators(the chaotic systems) can be well controlled during the


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time (S)

NING YI ER SANG

(a)

Figure 22. The simulation results. (a) The time-domainwaveform of the speech signal. (b) The spectrogram forthe original speech signal. (c) The time-domain waveformof the scrambled speech signal. (d)The spectrogram of thescrambled speech signal. (e)The time-domain waveform ofthe descrambled speech signal. (f) The spectrogram of thedescrambled speech signal.

manufacturing process, each mobile station will have a copy of Chua’s oscil-lator as the hardware key8. The detailed block diagram of a secure digitalcellular phone system is shown in Fig. 23.

In the system shown in Fig. 23, there are two kinds of analogue signalsneeded to be transmitted to the receiver. The first one is the speech signals(t) and the second one is the synchronization impulse x(t). To ensure thehigh degree of security, the digitized speech signal, s(i), is scrambled by adigital “stream encipher”. The key stream for the “stream encipher” is adigitized output, k(i), of the chaotic system at the transmitter end. Notethat the output k(t) of the chaotic system is usually a nonlinear combinationof all state variables of the chaotic system. Thus k(t) is more complex thanany individual state variable. The output of the “stream encipher” block isa cipher text stream c(i). Since the key stream k(i) can not be sent througha public channel, in the system shown in Fig. 23, k(i) must be regenerated

8In fact, the chaotic system itself can also be implemented digitally by using a DSPchip.

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Time (x0.01s)

Fre

quen

cy

50 100 150 200 250 300 350 400

20

40

60

80

100

120

(b)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time (S)


at the receiver end. To do this, the two chaotic systems in the transmitterand the receiver should be identical and synchronized. Impulsive synchro-nization is used in this system. To transmit the synchronization impulsesfrom 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 “compositionand 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.


Time (x0.01s)

Fre

quen

cy

50 100 150 200 250 300 350 400

20

40

60

80

100

120

(d)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time (S)

NING YI ER SANG

(e)Figure 22 (Continued).

At the receiver end, the output of the “demodulation” block, d′(t), isusually different from d(t) at the transmitter end due to the noise anddistortions in the channel. d′(t) is then fed into the “decoding and decom-position” block whose outputs are the digitized synchronization impulsesx(i) and the recovered cipher text stream c(i). We can get an BER from10−3 to 10−6 for both x(i) and c(i). Since x(i) is usually transmitted at abit-rate less than 100bps, we can offer more resources to make the BER forx(i) under 10−6. By doing this, we can get a good synchronization between

126 TAO YANG

Time (x0.01s)

Fre

quen

cy

50 100 150 200 250 300 350 400

20

40

60

80

100

120

(f)Figure 22 (Continued).

the two chaotic systems. If 12-bit A/D and D/A converters are used, thequantization error will be much less than the robust boundary of impulsivesynchronization. x(i) is then fed into a D/A converter followed by an “im-pulsive synchronization controller” to synchronize the chaotic circuit in thereceiver with that in the transmitter. After the impulsive synchronizationhas been achieved, the key stream k(i) can be regenerated at the receiverend and finally the cipher text stream can be unmasked.

The security of the whole system depends on the parameters and struc-tures of the chaotic systems used in the receiver and the transmitter. Formilitary applications, the structures of chaotic systems can be changed byplugging in different “hardware keys”(circuits) to achieve extremely highsecurity. For commercial applications, the structures of chaotic systems willkeep the same for each mobile station. However, different parameters ofchaotic systems can be assigned to different mobile stations to achieve a highdegree of security. Since the digitized synchronization impulses are sampledat around 1Hz, even if an intruder can recover all digitized synchronizationimpulses, the dynamics of the chaotic system can not be reconstructed byusing synchronization impulses without knowing the parameters. Thus thekey stream x(i) can not be reconstructed from synchronization impulses.

7. Concluding Remarks

Since the critical technology for the fourth generation of chaotic securecommunication system is impulsive synchronization, in this paper we have


speechsignal A/D

s(t) streamencipher

s(i) compositionand coding

c(i)modulation

d(i)

RF carrier

chan

nel

demodulation

A/Dk(t)

k(i)


x(t)

A/D

digitizedsynchroni−zationimpulses

x(i)

x(i)

decoding anddecomposition

d’(t)streamdecipher

c(i)

digitizedsynchroni−zationimpulses

impulsivesynchroni−zationcontrol

chaoticsystem

chaoticsystem D/A

speechsignal D/A

A/Dk(t) k(i) x(

i)

x(i)x(t)

s(i)s(t)

receiver

transmitter

Figure 23. Block diagram of secure cellular phone systembased on the fourth generation of chaotic secure communi-cation scheme.

presented a theory of impulsive synchronization of chaotic systems. An es-timate of the upper bound of the impulse interval ∆ is also presented. Sinceall of our results are based on solid theoretical analysis and proofs, the re-sults in this paper provide a framework and foundation for future works.We then use this theory to impulsively synchronize Chua’s oscillators. Anapplication of impulsive chaotic synchronization to secure communicationis presented. The chaotic secure communication scheme presented here isa combination of a conventional cryptographic method and impulsive syn-chronization.

128 TAO YANG

We present some simulation results to show the robustness of the impul-sive synchronization to additive channel noise and parameter mismatches.We find that the impulsive synchronization has the same degree of robust-ness to additive channel noise as that of the continuous synchronization.Furthermore, we find that impulsive synchronization is much more robustto parameter mismatches than continuous synchronization. This observa-tion is particularly significant when we consider the 1% to 10% fabricatingerrors in VLSI chips.

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Department of Electrical Engineering and Computer Sciences, Yang’s Sci-entific Research Institute, Tucson, Arizona, USA. (http://www.YangSky.Com)

E-mail address: [email protected]

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A SURVEY OF CHAOTIC SECURE COMMUNICATION SYSTEMS · secure communication. The great potentials of this kind of “hardware key” secure communication systems had driven the progress