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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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