TDMA SECURE COMMUNICATION SCHEME BASED ON ... - CiteSeerX
Journal of Circuits, Systems, and Computers, Vol. 10, Nos. 3 & 4 (2000) 147–158 c 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 2000 Revised 1 June 2000 A novel Time Division Multiple Access (TDMA) secure communication scheme is proposed based on sporadic coupling chaos synchronization. Compared with conventional chaos masking method, it has higher noise-resistibility, better security and frequency efficiency, 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 communication 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 because 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).
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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) ,
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Decompose the state vector x0 of Eq. (2) into two parts: u0 = [x01 , x02 , . . . , x0m ], v 0 = [x0m+1 , x0m+2 , . . . , x0N ] 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˙ 0 = Fv (u, v 0 ) 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˙ 0 = Fu (u0 , v 0 ) + δT (t) · (uT − u0T − ) ,
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P+∞ 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 ), . . .}; u0T − is the sampled sequence of u0 (t) with sample times immediately prior to the times nT . System of Eqs. (1) and (3) oscillate independently except at the equidistant times t = nT when u0 is set to u(t). Obviously, Eq. (3) degrades to PC method when T = 0. Consider the well-known Chua’s circuit,10 C1 · dv1 /dτ = G(v2 − v1 ) − g(v1 ) , (4) C2 · dv2 /dτ = G(v1 − v2 ) + i3 , L · di3 /dτ = −v2 , where v1 , v2 are the voltages across capacitors C1 , C2 respectively and i3 is the current through inductor L; g(v1 ) = Gb v1 + 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)
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where x1 = v1 /Bp , x2 = v2 /Bp , x3 = i3 /Bp G, a = Ga /G, b = Gb /G, α = C2 /C1 , β = C2 /LG2 , t = τ G/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 0 = [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 u0 (t). The interpolation is successful provided that Eq. (6) is asymptotically stable for given T , i.e., u(t) = u0 (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 ) , st − y1T − ), 0, 0]T , y˙ = C(y) + [δT (t) · (˜
(7)
sˆi = s˜t − 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. s˜t 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 conventional 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 synchronization 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 quantized 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.
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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 si on 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 0 = [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 ) , st − y2T − ), 0]T , y˙ = C(y) + [0, δT (t) · (˜
(8)
sˆi = s˜t − 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 spectral 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.
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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 simulation. Suppose that the circuit’s parameters R = 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 corresponding 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/T C2 = 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 forward to attack on it in which the dynamics-based methods in Refs. 4 and 5 are much
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effective. Hopefully the using of sporadic driving instead of time-continuously driving 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
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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 information 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
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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 correlation 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 addition, 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.
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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 probability 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 σt2 = (L2 − 1)/3 · ∆2 Pe in which ∆ is quantized interval. Due
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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/σt2 = 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 significant bit (MSB) of the code. The distortion will desynchronize the chaotic systems temporarily and thus decrease the SNR. Fortunately, the probability of large amplitude 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 communications, 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
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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 transmission 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 inherent noise resistibility than analog ones. References 1. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems”, Phys. Rev. Lett. 64 (1990) 821–823. 2. L. Kocarev, K. S. Halle, K. Eckert, and L. O. Chua, “Experimental demonstration of secure communication via chaotic synchronization”, Int. J. Bifur. Chaos 2 (1992) 1011–1020. 3. K. M. Cuomo, A. V. Oppenheim, and S. H. Strogatz, “Synchronization of Lorenzbased chaotic circuits with applications to communications”, IEEE Trans. CAS(II) 40 (1993) 626–632. 4. G. Perez and H. A. Cerdeira, “Extracting messages masked by chaos”, Phys. Rev. Lett. 74 (1995) 1970–1973. 5. K. M. Short, “Steps toward unmasking secure communications”, Int. J. Bifur. Chaos 4 (1994) 959–977. 6. S. Fahy and D. R. Hanann, “Transition from chaotic to nonchaotic behavior in randomly driven systems”, Phys. Rev. Lett. 69 (1992) 761–764. 7. R. E. Amritkar and N. Gupte, “Synchrony of chaotic orbits: Effect of finite time step”, Phys. Rev. E 47 (1993) 3889–3895. 8. T. Stojakovski, L. Kocarev, U. Parlitz, and R. Harris, “Sporadic driving of dynamical systems”, Phys. Rev. E 55 (1997) 4035–4048.
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9. T. Stojakovski, L. Kocarev, and U. Parlitz, “Digital coding via chaotic systems”, IEEE Trans. CAS(I) 44 (1997) 562–565. 10. R. N. Madan (ed.), Chua’s Circuit: A Paradigm for Chaos, World Scientific, Singapore, 1993. 11. K. M. Cuomo and A. V. Oppenheim, “Robustness and signal recovery in a synchronized chaotic system”, Int. J. Bifur. Chaos 3 (1993) 1629–1638. 12. Z. He, K. Li, L. Yang, and Y. Shi, “A robust digital secure communication scheme based on sporadic coupling chaos synchronization”, IEEE Trans. CAS(I) 47 (2000) 397–402.
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 2000 Revised 1 June 2000 A novel Time Division Multiple Access (TDMA) secure communication scheme is proposed based on sporadic coupling chaos synchronization. Compared with conventional chaos masking method, it has higher noise-resistibility, better security and frequency efficiency, 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 communication 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 because 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).
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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) ,
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Decompose the state vector x0 of Eq. (2) into two parts: u0 = [x01 , x02 , . . . , x0m ], v 0 = [x0m+1 , x0m+2 , . . . , x0N ] 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˙ 0 = Fv (u, v 0 ) 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˙ 0 = Fu (u0 , v 0 ) + δT (t) · (uT − u0T − ) ,
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v˙ 0 = Fv (u0 , v 0 ) ,
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P+∞ 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 ), . . .}; u0T − is the sampled sequence of u0 (t) with sample times immediately prior to the times nT . System of Eqs. (1) and (3) oscillate independently except at the equidistant times t = nT when u0 is set to u(t). Obviously, Eq. (3) degrades to PC method when T = 0. Consider the well-known Chua’s circuit,10 C1 · dv1 /dτ = G(v2 − v1 ) − g(v1 ) , (4) C2 · dv2 /dτ = G(v1 − v2 ) + i3 , L · di3 /dτ = −v2 , where v1 , v2 are the voltages across capacitors C1 , C2 respectively and i3 is the current through inductor L; g(v1 ) = Gb v1 + 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) ,
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where x1 = v1 /Bp , x2 = v2 /Bp , x3 = i3 /Bp G, a = Ga /G, b = Gb /G, α = C2 /C1 , β = C2 /LG2 , t = τ G/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 .
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x˙ = C(x) + [δT (t) · (st − x1T − ), 0, 0]T , st = Q(x1T − + si ) , st − y1T − ), 0, 0]T , y˙ = C(y) + [δT (t) · (˜
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sˆi = s˜t − 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. s˜t 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 conventional 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 synchronization 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 quantized 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.
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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 si on 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 0 = [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, (
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x˙ = C(x) + [0, δT (t) · (st − x2T − ), 0]T , st = Q(x2T − + si ) , st − y2T − ), 0]T , y˙ = C(y) + [0, δT (t) · (˜
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sˆi = s˜t − 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 spectral 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.
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In the follows, we will show the efficiency of the proposed scheme (Scheme II) compared with conventional chaos masking scheme2 (Scheme I) by computer simulation. Suppose that the circuit’s parameters R = 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 corresponding 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/T C2 = 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 forward to attack on it in which the dynamics-based methods in Refs. 4 and 5 are much
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effective. Hopefully the using of sporadic driving instead of time-continuously driving 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
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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 information 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
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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 correlation 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 addition, 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.
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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 probability 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 σt2 = (L2 − 1)/3 · ∆2 Pe in which ∆ is quantized interval. Due
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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/σt2 = 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 significant bit (MSB) of the code. The distortion will desynchronize the chaotic systems temporarily and thus decrease the SNR. Fortunately, the probability of large amplitude 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 communications, 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
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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 transmission 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 inherent noise resistibility than analog ones. References 1. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems”, Phys. Rev. Lett. 64 (1990) 821–823. 2. L. Kocarev, K. S. Halle, K. Eckert, and L. O. Chua, “Experimental demonstration of secure communication via chaotic synchronization”, Int. J. Bifur. Chaos 2 (1992) 1011–1020. 3. K. M. Cuomo, A. V. Oppenheim, and S. H. Strogatz, “Synchronization of Lorenzbased chaotic circuits with applications to communications”, IEEE Trans. CAS(II) 40 (1993) 626–632. 4. G. Perez and H. A. Cerdeira, “Extracting messages masked by chaos”, Phys. Rev. Lett. 74 (1995) 1970–1973. 5. K. M. Short, “Steps toward unmasking secure communications”, Int. J. Bifur. Chaos 4 (1994) 959–977. 6. S. Fahy and D. R. Hanann, “Transition from chaotic to nonchaotic behavior in randomly driven systems”, Phys. Rev. Lett. 69 (1992) 761–764. 7. R. E. Amritkar and N. Gupte, “Synchrony of chaotic orbits: Effect of finite time step”, Phys. Rev. E 47 (1993) 3889–3895. 8. T. Stojakovski, L. Kocarev, U. Parlitz, and R. Harris, “Sporadic driving of dynamical systems”, Phys. Rev. E 55 (1997) 4035–4048.
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9. T. Stojakovski, L. Kocarev, and U. Parlitz, “Digital coding via chaotic systems”, IEEE Trans. CAS(I) 44 (1997) 562–565. 10. R. N. Madan (ed.), Chua’s Circuit: A Paradigm for Chaos, World Scientific, Singapore, 1993. 11. K. M. Cuomo and A. V. Oppenheim, “Robustness and signal recovery in a synchronized chaotic system”, Int. J. Bifur. Chaos 3 (1993) 1629–1638. 12. Z. He, K. Li, L. Yang, and Y. Shi, “A robust digital secure communication scheme based on sporadic coupling chaos synchronization”, IEEE Trans. CAS(I) 47 (2000) 397–402.
