Journal of Circuits, Systems, and Computers, Vol. 10, Nos. 3 & 4 (2000) 147–158c©World Scientific Publishing Company

TDMA SECURE COMMUNICATION SCHEME BASED ON

SYNCHRONIZATION OF CHUA’S CIRCUITS∗

ZHENYA HE, KE LI and LUXI YANG

Dept. of Radio Engineering, Southeast University, Nanjing 210018, China

YUHUI SHI

Electronic Data Systems, MS 1206, Kokomo, IN 46902, USA

Received 24 April 2000Revised 1 June 2000

A novel Time Division Multiple Access (TDMA) secure communication scheme is pro-posed based on sporadic coupling chaos synchronization. Compared with conventionalchaos masking method, it has higher noise-resistibility, better security and frequencyefficiency, etc. Simulation results illustrate the effectiveness and efficiency of this method.Finally, we analyze factors which affect system ability and give some conclusions.

1. Introduction

Based on the drive-response synchronization method,1 chaos masking secure com-

munication scheme2,3 was proposed which can preclude the spectral analysis attack

by exploiting the broadband nature of chaotic carrier to mask information. However,

in order to keep the synchronization between transmitter and receiver, the chaotic

carrier to information signal ratio (CSR) should be greater than about 30 dB which

in turn enables the interceptor to disclose the information easily by dynamics-based

attacks.4,5 In addition, the receiver cannot recover information signal precisely be-

cause its approach in power to channel noise. Finally, as we know, the ideal chaotic

signal has infinite bandwidth, thus, cannot be transmitted in bandlimited channel.

All of these greatly restrain the application of chaos masking.

In Refs. 6–9, it was verified that if the driving period is less than a threshold, the

asymptotic stability of discrete time driven dynamical system is identical to that of

continuous time driven system. By analyzing the asymptotic stability of sporadic

coupling driven system, a novel chaos communication scheme is proposed which

can eliminate the drawbacks of conventional chaos masking, and a time division

multiple access (TDMA) secure communication system is constructed based on the

∗This work was partly supported by National Natural Science Foundation of China (Grant No.69735101) and the Grant for PhD Programme in Higher Education Institutions of MOE (No.98028630).

147

148 Z. He et al.

proposed scheme. It improves the security, frequency efficiency, etc. greatly as

compared with conventional chaos masking scheme. Computer simulations were

carried out to prove these statements.

2. Synchronization of Sporadic Coupling Chaotic Systems

Consider two identical N -dimensional chaotic systems as described below,

x = F (x) , (1)

x′ = F (x′) . (2)

Decompose the state vector x′ of Eq. (2) into two parts: u′ = [x′1, x′2, . . . , x

′m],

v′ = [x′m+1, x′m+2, . . . , x

′N ] and do the same decomposition to x and the vector field

F , that is, x = [u,v] and F = [Fu,Fv]. If we drive system of Eq. (2) using u

only at time-equidistant moments, then, synchronization of the systems described

by Eqs. (1) and (2) can be achieved provided that (i) subsystem v′ = Fv(u,v′) is

asymptotically stable when driven continuously by u(t); (ii) the time interval T is

less than a threshold TH which is determined by the drive signal.8 The corresponding

driven system can be described as follows,9

u′ = Fu(u′,v′) + δT (t) · (uT − u′T−) , (3a)

v′ = Fv(u′,v′) , (3b)

in which δT (t) =∑+∞

n=−∞ δ(t−nT ) is a pulse sampling sequence with period T ; uT is

a sampled version of driving signal u(t), uT = . . . ,u(−T ),u(0),u(T ),u(2T ), . . .;u′T− is the sampled sequence of u′(t) with sample times immediately prior to the

times nT . System of Eqs. (1) and (3) oscillate independently except at the equidis-

tant times t = nT when u′ is set to u(t). Obviously, Eq. (3) degrades to PC method

when T = 0.

Consider the well-known Chua’s circuit,10C1 · dv1/dτ = G(v2 − v1)− g(v1) ,

C2 · dv2/dτ = G(v1 − v2) + i3 ,

L · di3/dτ = −v2 ,

(4)

where v1, v2 are the voltages across capacitors C1, C2 respectively and i3 is the

current through inductor L; g(v1) = Gbv1 + 1/2 · (Ga−Gb)(|v1 +Bp|− |v1−Bp|) is

the v–i characteristic of the nonlinear resistor. By a simple transformation, Eq. (4)

can be rewritten in dimensionless form,

dx/dt = [α(x2 − x1 − g(x1)), x1 − x2 + x3, −βx2] ,

or the simplified form,

x = C(x) , (5)

TDMA Secure Communication Scheme 149

where x1 = v1/Bp, x2 = v2/Bp, x3 = i3/BpG, a = Ga/G, b = Gb/G, α = C2/C1,

β = C2/LG2, t = τG/C2. Using x1 in Eq. (5) as driving signal, we get the

corresponding driven system,

y = C(y) + [δT (t) · (x1T − y1T−) , 0 , 0]T . (6)

The conditional Lyapunov exponents (CLEs) of subsystem v′ = [y2, y3] are

(−0.5,−0.5), therefore, it is asymptotically stable. Numerically, it is found that

there exists a nonzero TH = 0.26 for Eq. (6). The synchronization error decreases

and increases exponentially for T < TH and T > TH respectively as time elapse.

3. TDMA Chaotic Secure Communication Scheme

From a sampling viewpoint, the driven system of Eq. (6) can be treated as a kind of

nonlinear interpolation on uT and produces the interpolated signal u′(t). The inter-

polation is successful provided that Eq. (6) is asymptotically stable for given T , i.e.,

u(t) = u′(t). Then, u(t) is determined uniquely by its samples uT . The spectrum of

δT (t) · u(t) is periodic with period fs = 1/T , so, the sampled chaotic signal can be

transmitted through an ideal low-pass channel. This enables the synchronization of

two chaotic system through bandlimited channel and thus improves the frequency

efficiency. The chaotic signal can be reproduced perfectly at the receiver despite its

infinite width of spectrum. Therefore, we can apply the chaos masking to digital

communications and construct a chaotic encoder-decoder pair as follows,x = C(x) + [δT (t) · (st − x1T−), 0, 0]T ,

st = Q(x1T− + si) ,

(7)y = C(y) + [δT (t) · (st − y1T−), 0, 0]T ,

si = st −Q(y1T−) ,

where Q(·) is quantize function, si = . . . , si(−T ), si(0), si(T ), si(2T ), . . . is the

information sequence and st the transmitted sequence. st is the received sequence

contaminated with noise. Due to the widely use of pulse-code modulation (PCM)

in digital communications, here, we adopt PCM in quantizing and coding. In con-

ventional chaos masking scheme, the information signal was seen as an additional

noise overlapped on the drive signal. Therefore, its power should be much lower

to ensure a preferable synchronization of the two ends. Nevertheless, the synchro-

nization error is still much larger which consequently leads to poorer signal-to-noise

ratio (SNR) of the recovered information.11 In the sporadic coupling system, we

drive the transmitter and receiver simultaneously with the overlapped and quan-

tized signal st at each driving time. The information signal was no longer seen as

noise but part of the drive signal. It guarantees the chaotic systems of the two ends

running under the same condition and keeps exact synchronization. So, the SNR

will be improved greatly.

150 Z. He et al.

Although the transmitted signal is no longer pure chaotic signal after being

overlapped with information and quantizing, it still remains some basic properties

of chaotic signal such as pseudo-randomness and noise-like power spectrum. For

secure communication system, we don’t care whether it is chaotic signal but whether

it can improve the security of the system, so the overlapping and quantizing are

reasonable and effective. The mixed signal will stay in the basin of attractor without

divergence as long as the power of information signal and quantizing error are less

than certain values, only in this case can the synchronization be achieved and held.

Let si be the sampled speech signal with 4 kHz bandwidth and 8 kHz sampling

ratio, then, it can be transmitted securely after being coded with chaotic signal

which is also set to the same sampling ratio by adjusting the circuit’s parameters.

It is found that for a given range of driving period T , the power of si should be much

lower than that of carrier or the trajectory of chaotic system will diverge and lead to

the out of control of the whole system. This is mainly due to the narrow attraction

range of Chua’s circuit and the small driving period. Consider the addition of sion chaotic signal as a kind of disturbance to chaotic trajectory, if the period of the

disturbance is much smaller so as to leave little time for the system to modify its

orbit, the trajectory will walk away from the attractor eventually. However, too low

a power of si will diminish the relative quantize precision of si and result in serious

distortion of recovered speech signal at the receiving end.

In Ref. 7, it is observed that despite the presence of a positive CLE of Eq. (2)

with time continuously driving, it is still possible to achieve asymptotic stability of

Eq. (3) for certain T > 0 values. By calculating the CLEs of the driven system in

different time period, it is found that being driven by x2, the subsystem v′ = [y1, y3]

of Chua’s circuit is asymptotically stable for certain range of T , i.e., T ∈ (0, 0.83) ∪(1.23, 1.48). Therefore, system of Eq. (7) can be modified as follows,

x = C(x) + [0, δT (t) · (st − x2T−), 0]T ,

st = Q(x2T− + si) ,

(8)y = C(y) + [0, δT (t) · (st − y2T−), 0]T ,

si = st −Q(y2T−) .

Due to the much higher value of TH , the power of si can also be increased greatly

without loss of chaotic state. And the higher the power of si, the better the SNR

performance is. However, too high a power of si will result in the leakage of its spec-

tral information, therefore, to reach a tradeoff between the system performance and

security, we introduce into si a scale factor r which is multiplied with si in advance

to produce a moderate CSR. On the basis of Eq. (8), we propose a novel secure

communication scheme shown in Fig. 1. Furthermore, a TDMA chaotic secure

communication system can be constructed through multiplexing. It improves the

frequency efficiency greatly as compared with conventional analog chaos masking.

TDMA Secure Communication Scheme 151

)(ˆ Tsi

Σ

Σ

x (t)2

s (t)i

s (T)t

s (T )t

~

+

++

y (t)2y (T-)2

Chaoticsignal source

Samplingand

quantizing

Speechsignal source

PCMencoder

Digitalchannel

Noisesource

PCMdecoder

Drivenchaotic system

Samplingand

quantizing

Controllogic

Fig. 1. Secure communication scheme of single speech channel.

In the follows, we will show the efficiency of the proposed scheme (Scheme II)

compared with conventional chaos masking scheme2 (Scheme I) by computer simula-

tion. Suppose that the circuit’s parametersR = 1721 Ω, C1 = 12.1 pF, C2 = 121 pF,

L = 22.7 mH, Ga = −0.76 mS, Gb = −0.41 mS, Bp = 1 V, we get the correspond-

ing dimensionless parameters α = 10.0, β = 15.788, a = −1.31, b = −0.71. By

exploiting x1 as the drive signal for Scheme I, the corresponding CSR is 30.2 dB;

For Scheme II, we use x2 as drive signal with CSR = 11.1 dB for r = 0.7. Given

drive period T = 0.6, the sampling ratio of chaotic signal fs = G/TC2 = 8000 Hz.

The quantizer is logistic PCM in A law (quantized level L = 256). Figure 2 is the

simulation results of a segment of speech signal, the phrase “good afternoon” (see

Fig. 2(a)). Figures 2(b) and 2(c) are the transmitted signals for Scheme I and II

respectively. Figure 2(d) is the power spectra of transmitted signals and speech

signal, from which we can see that speech signal can be embedded appropriately

in chaotic carriers of both schemes and thus preclude the spectral analysis attack.

Figure 2(e) is the power spectra of recovered signal. Notice that the recovered signal

in Scheme I includes considerable energy at high frequencies, the speech recovery

can be improved by lowpass filtering. The final speech to reconstructing noise ratio

SNR = 13.6 dB which is rather poor in practice. The SNR for Scheme II is 26.5 dB.

Figure 2(f) gives the SNR of both schemes under different CSR. We note that the

SNR of Scheme II decreases as CSR increases due to quantization, but, in general

it is much higher than that of Scheme I especially in case of low CSR.

4. Security Analysis

For conventional chaos masking scheme, many successful approaches were put for-

ward to attack on it in which the dynamics-based methods in Refs. 4 and 5 are much

152 Z. He et al.

0 2 4 6Freq. (kHz)

-70

-50

-30

-10

10

Pow

er s

pect

rum

(dB

)

Transmitted signalof Scheme I

Transmitted signalof Scheme II

Original speech signal

.

0 2 4 6Freq. (kHz)

-70

50

30

10

10

Pow

er s

pect

rum

(dB

)

Recovered speech of Scheme I

Recovered speech of Scheme II

Original speech signal

0 10 20 30 40CSR (dB)

0

10

20

30

SN

R (

dB)

Scheme I

Scheme II

.

0 0.25 0.5 0.75 1t

(s ec )

-1

-0.5

0

0.5

1

0 10 20 30t

&(ms' )

-4

-2

0

2

4

( ) *

.

0 2000 4000 6000 8000-1

-0.5

0

0.5

1

.

(a) (b)

(c) (d)

(e) (f )

Fig. 2. Simulation results of speech signal.

effective. Hopefully the using of sporadic driving instead of time-continuously driv-

ing will give the system the inherent robustness to such attacks.12 Here, we compare

and analyze the ability of Scheme I and II under the attack of these two methods

respectively.

4.1. Return map method

According to Ref. 4, given just one of the variables in the chaotic system, say x(t),

one can properly construct a return map where the dynamics is attracted to an

TDMA Secure Communication Scheme 153

almost 1D set. Starting from some arbitrary point in time, define mi as the time

when x(t) reaches its ith local maximum, and Xi as the value of x at that moment.

Similarly, define nj as the time when x reach its jth local minimum, and Yj the

value of x at that moment. We can construct the return map Xi versus Yj which

has almost 1D attractor, i.e., smooth lines as in Fig. 3(a).

As Scheme I requires time continuously driving and that the power of informa-

tion should be much lower than that of chaotic carrier, the information signal can

be seen as a kind of distortion to the chaotic trajectory which will result in the

deviation of map points from the original “pure” attractor constructed from pure

chaotic signal, i.e., the lines are broadened into diffuse stripes. By measuring the

distance between the present position of the points in the attractor and the place

they should have appeared in the absence of a message, and taking into account

to which side of it the point has moved, the information signal can be successfully

extracted.

Due to the sporadic driving employed in Scheme II and a much high power of

information signal, the dynamical feature of the chaotic system embraced in carrier

was reduced greatly, i.e., the maxima and minima of the transmitted signal no

-2 0 2 4-4

-2

0

2

.

-0.2 0.2 0.6 1-1

-0.6

-0.2

0.2

.

0 0.2 0.4 0.6 0.8 1-0.5

-0.3

-0.1

0.1

0.3

0.5

.

0 0.2 0.4 0.6 0.8 1-0.5

-0.3

-0.1

0.1

0.3

0.5

.

(a)

Xn

Yn

(c)

Xn

Yn

t (s)

t(s)

(b)

(d)

Fig. 3. Experimental result of the attach of return map method on Scheme I and II.

154 Z. He et al.

longer represent the real character of chaotic trajectory. Therefore, return map of

the transmitted signal will not be attracted to the neighbor of “pure” attractor

but distribute almost uniformly in the whole space. Obviously, the original signal

cannot be extracted from this map. Figure 3 is the experimental result. Figure 3(a)

is the “pure” attractor derived from x1 of Chua’s circuit. The recovered speech

by constructing return map from transmitted signal in Scheme I (see Fig. 2(b))

was shown in Fig. 3(b). Although there is some background noise when playing,

the meaning of the signal can be recognized clearly and the result can be further

improved by selecting a more suitable map. As shown in Fig. 3(c), the map points

of return map derived from the transmitted signal in Scheme II (see Fig. 2(c))

do not gather beside the pure attractor but distribute uniformly. Therefore, the

reconstructed signal includes no information of original speech (see Fig. 3(d)).

4.2. Nonlinear forecasting method

We know that the chaotic dynamical systems generally exhibit regular geometric

structures if viewed in suitable phase space. These structures can be used to predict

the behavior of the chaotic carrier so that the hidden information can be extracted.

According to Ref. 5, the phase space of the chaotic system is reconstructed firstly

from a piece of intercepted signal. By dividing the phase space into several local

regions, each region will have short pieces of trajectory which can be studied as a

group to determine the local behavior in just that patch of the phase space. After

studying all the patches separately, it is possible to build up the global behavior of

the dynamical system.

Similar to that in the return map method, it is required that the synchronized

system should be driven continuously to ensure a much strong short-time correla-

tion of the transmitted signal and that CSR should be greater than about 20–30 dB.

Only in this case can the phase space be reconstructed much precisely. The sporadic

driving employed in Scheme II leads to poor correlation of consecutive points in time

series and it contains little dynamical characteristics of the chaotic system. In addi-

tion, the much higher power of information signal results in serious self-intersections

in reconstructed attractor. All of these preclude a proper reconstruction of phase

space thus a correct recovery of the hidden information. Figure 4(a) is reconstructed

attractor from the transmitted signal in Scheme I. The reconstructed space is of 3D.

By calculating the mutual information of the signal, the first minimum turned out

to be six time steps and then time-delay in construction is 6 as well. Figure 4(b) is

the prediction result of speech. Although not perfect, most of the key features are

still presented. Figure 4(c) is the reconstructed phase space from the transmitted

signal in Scheme II. It contains little dynamical property of the chaotic system and

of cause the information cannot be extracted correctly.

TDMA Secure Communication Scheme 155

-4 -2 0 2 4-4

-2

0

2

4

. 0 0.2 0.4 0.6 0.8 1t (s)

-1

-0.5

0

0.5

1

.

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

.

(a) (b)

(c)

Fig. 4. Experimental result of the attack of nonlinear forecasting on Scheme I and II.

5. Discussions and Conclusions

In PCM communication system, the error of the reconstructed speech signal is

mainly induced by quantizing error and error code in channel in which the later will

affect the synchronization of chaotic systems in the transmitter and receiver.

The quantization SNR of logistic PCM is much higher than that of linear PCM as

a result of the more frequently occurring of small signal amplitudes than large ones

in speech signal. Because of the uniform distribution of chaotic signal amplitude,

the mixed signal distributes uniformly as well in case of large CSR. So, the linear

PCM for mixed signal performances better than logistic PCM. But, the difference

is small in case of small CSR, so, one can choose flexibly in practical system design

(see Fig. 5). In addition, although the bigger the r is, the smaller the quantization

error is, too high a power of speech will result in the divergence of chaotic trajectory

and quickly drop of SNR. According to the request of high quality (long-distance)

telephone with SNR > 25 dB, the scale factor r = 0.6 ∼ 1.0 is appropriate in this

case. And for r > 1.0, the system will diverge.

For uniform quantizer with natural binary coding (NBC), suppose that the prob-

ability of each quantized level is the same, the bit-error-rate (BER) Pe is much small

and there is only one error code in a n-bit codeblock, the power of noise generated

from error code is σ2t = (L2 − 1)/3 · ∆2Pe in which ∆ is quantized interval. Due

156 Z. He et al.

0.0 0.4 0.8 1.2r0

10

20

30

40

(dB

)

Logistic PCMLinear PCM

CSRSNR

Fig. 5. SNR and CSR versus r for Logarithmic PCM and Linear PCM.

to the nearly uniform distribution of the transmitted signal st, its full load power

S = L2∆2/12, so, the signal-to-noise ratio of st is SNRt = S/σ2t = L2/4(L2− 1)Pe.

In baseband PCM relay system, it is easy to limit the BER within 10−6, and the

corresponding SNRt is 54 dB. In this case, the channel noise impacts little influence

on transmitted signal and then on reconstructed speech signal.

For transmitted signal with large amplitude, the error is half of the signal’s

amplitude at most after NBC decoding if error code occurs for the most signifi-

cant bit (MSB) of the code. The distortion will desynchronize the chaotic systems

temporarily and thus decrease the SNR. Fortunately, the probability of large am-

plitude signal is not high because of the uniform distribution of st. Moreover, the

time needed to re-synchronizing is very short (for T = 0.6, about 1.3 ms at most).

Therefore, the influence of noise on transmitted signal and thereafter on the speech

is fairly small. The adoption of Folded Binary Coding (FBC) instead of NBC and

the use of channel coding will further decrease the error power. In wireless commu-

nications, however, due to the much worse transmission condition of radio channels

as compared with wired ones, the error code will greatly desynchronize the chaotic

systems and the desynchronization error will be the dominant factor of the error in

the received information. The recovered information will be unacceptable in this

case. So, another kind of coding and masking scheme should be adopted accordingly

to against such case, it is not the focus of this letter. And this will be considered

in our future work.

From the abovementioned analysis and simulations, we can see that Scheme II

has the following advantages compared with conventional chaos masking,

(1) Improves the security of the system and precludes the prediction-based attacks

effectively;

(2) Enhances the SNR at the receiving end greatly;

(3) Improves the frequency efficiency by exploiting time division multiplexing.

Scheme I gains security at the cost of bandwidth, that is, at least 24 kHz

bandwidth is needed for one channel signal, of which only 4 kHz was occupied

by speech. This is a large waste of frequency resource and the system is still

TDMA Secure Communication Scheme 157

rather frail under prediction-based attacks. In Scheme II, the overlapping of

chaotic drive signal on speech adds no additional bandwidth. So, the system

has the same frequency efficiency as general TDMA system.

As we have known, the only purpose of introducing chaotic signal into trans-

mission in chaos masking schemes is to improve the transmission security by

masking the spectrum of information signal, therefore, the power of chaotic

signal should be much greater than that of information signal, i.e., CSR will

always be greater than zero. And inevitably this will lead to redundancy in

transmission. For conventional chaos masking scheme, CSR should be in the

magnitude of 30 dB to ensure the security of transmission which means lots of

redundancy exists. In the proposed scheme, however, CSR is decreased greatly

to around 10 dB without loss of security, thus, the redundancy is meanwhile

greatly eliminated.

(4) Easy of implementation. Practically, instead of constructing a new system,

this scheme can be realized by appending chaotic signal generators at the both

ends of common PCM system and making simple adjustment. Therefore, it is

quite suitable for civil communication systems with low production cost and

increasing security requirement. From Fig. 1, we can see that the differences of

this system to common PCM system only are that before PCM coding and after

PCM decoding, so, it is identical in selecting the frame structure and multiplex

structure, frame acquiring and frame synchronization, etc. which simplifies the

system design greatly.

(5) Robustness to channel noise. The digital communication systems provide in-

herent noise resistibility than analog ones.

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