digital speech communication by truncated chaotic ...

International Journal of Bifurcation and Chaos, Vol. 13, No. 3 (2003) 691–701 c World Scientific Publishing Company

Int. J. Bifurcation Chaos 2003.13:691-701. Downloaded from www.worldscientific.com by UNIVERSITY OF WISCONSIN-MADISON on 10/23/17. For personal use only.

DIGITAL SPEECH COMMUNICATION BY TRUNCATED CHAOTIC SYNCHRONIZATION Y. ZHANG∗ , G. H. DU, Y. M. HUA and J. J. JIANG∗ Institute of Acoustics, State Key Laboratory of Modern Acoustics, Nanjing University, Nanjing 210093, P.R. China ∗ Department of Surgery, Division of Otolaryngology Head and Neck Surgery, University of Wisconsin Medical School, Madison, WI 53792-7375, USA Received October 19, 2001; Revised January 8, 2002 The method of truncated chaotic synchronization is suggested for digital speech communication. The sequence, composed by truncating some byte segments from the low-dimensional chaotic attractor, demonstrates complicated dynamics and high dimension. Predicting the dynamics of the truncated map through the local approximation method becomes very difficult. However, synchronization lets the message be exactly recovered. The synchronization condition is deduced to synchronize two systems of the truncated one-way coupled ring maps. The sensitivity of synchronization to parameter mismatch is discussed for the security in communication. For the mismatch of parameters or keys, the probability of decoding the message through guessing the keys is below 2−200 . The scheme of truncated chaotic synchronization has strong security in encoding message. Keywords: Truncated chaotic synchronization; speech communication.

1. Introduction

digital speech communication have been investigated in our recent research. However, some researchers have found that the masked message could be approximately extracted by employing nonlinear dynamic forecasting or other techniques [Short & Parker, 1998; Yang et al., 1998; Zhou & Lai, 1999]. The primary reason was that the chaotic masking signal could be used to reconstruct the phase space of the chaotic transmitter. The message was decoded if the perturbation of the message to the dynamics of the transmitter was detectable. Therefore, for the secure encoding in speech digital communication, how to avoid the dynamics of the transmitter being detected is very important. Ni et al. [1996] and Deng et al. [1999] applied the method of truncation to encode speech signal. The truncated chaotic sequence, composed by truncating some byte segments from

Chaos theory presents scientists with new insights into speech science. The unpredictable and noiselike characteristic of chaos has promised its application in modeling vocal fold [Titze, 1993; Herzel, 1995a, 1995b; Jiang et al., 2001], speech signal processing [Narayanan & Alwen, 1995; Kumar & Mullick, 1996; Hu et al., 2000], and speech communication [Cuomo & Oppenheim, 1993; Dai et al., 1998, 1999; Zhang et al., 1998, 1999]. In recent years, chaotic communication has been an active subject. Many approaches of chaotic synchronization have been proposed [Pecora & Carroll, 1990; Pyragas, 1993; Kocarev & Parlitz, 1995; Yang et al., 1997; Zhang et al., 2001]. The message encoded by a broadband chaotic masking signal can be recovered well by using the synchronization technique. The applications of chaotic synchronization in 691


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the output of the low-dimensional logistic map, showed complex topological property and pseudorandom characteristics. However, at that stage, the dynamic characteristic of the truncated solution had not been quantitatively determined. It was unknown why the dynamics of the truncated map could avoid being detected by nonlinear forecasting methods. Particularly, they had not presented an effective synchronization method to recover the message. The initial conditions of the transmitter and receiver should be exactly identical. Otherwise, for the extreme sensitivity of chaotic systems, a slight error would be amplified into the remarkably distorted result so that performing communication becomes impossible. This greatly limited the application of their scheme in digital secure communication. In this paper, a scheme based on chaotic synchronization is applied into speech communication. The method of truncated chaotic synchronization is suggested to synchronize two truncated systems. XOR (exclusive-or) operation between the speech signal and the binary truncated signal is performed as the encryption algorithm. Because the truncated map has very complex dynamics and high dimension, the local approximation forecasting method cannot predict its dynamics. Furthermore, this synchronization scheme of m-TOCRML (truncated one-way coupled ring map lattice with length m) systems is used to perform digital speech communication in the local area network of Nanjing University. The synchronization condition is given to exactly recover the speech message, despite the different initial conditions of two TOCRML systems. The sensitivity of chaotic synchronization to parameter mismatch is investigated. The probability of decoding the message through guessing the keys is demonstrated to be much smaller than 2 −200 .

2. The Truncated Chaotic Synchronization Method The autonomous discrete dynamics of the transmitter Zn+1 = F(Zn ) can always be rewritten as a nonautonomous system: Xn+1 = F(Xn , sn )

(1)

where X ∈ Rm determines the m-dimensional state of the transmitter, and sn is the transmitted signal with the driving function: sn = h[j][k](Xn ) ⊕ in or sn+1 = h[j][k](Xn , sn ) ⊕ in . The message in is

the speech signal with 16 bit resolution. It is incorporated into the transmitted signal s n through ⊕ operation. ⊕ denotes the XOR (exclusive-or) operator. For the synchronization technique of active– passive decomposition [Kocarev & Parlitz, 1995], the discrete dynamics of the driving signal s n can be defined as the differentiable function satisfying sn = h(Xn ) or sn+1 = h(Xn , sn ). The security of the encoder defined by this method is questionable. The delay coordinates of sn can be used to reconstruct the phase space [Short, 1998]. The dynamics of the transmitter can thus be approximately predicted by nonlinear dynamic forecasting. Therefore, to avoid the dynamics of the system being detected in the reconstructed phase space, we introduce the method of truncation [Ni et al., 1996; Deng et al., 1999] to design the driving function h [j][k](Xn ). In the digital system, a decimal float point data h(•) can be represented as the binary number with multiple bytes. Then, the binary number h [j][k](Xn ) can be produced by the following two steps: (i) truncating the jth and kth byte segments from the binary solution of h(•); (ii) combining these two bytes into a new binary number h[j][k](Xn ). Obviously, the truncated signal sn contains very little information about the dynamics of the transmitter. Consequently, the complete dynamics of the transmitter cannot be predicted from sn . The receiver is an identical system Yn+1 = F(Yn , sn )

(2)

driven by the transmitted signal sn . In the receiver, the message can be recovered as i 0n = h[j][k](Yn ) ⊕ (h[j][k](Xn ) ⊕ in ). For the difference en = Yn − Xn , if the difference equation en+1 = F(Yn , sn ) − F(Xn , sn ) has a global asymptotically stable zero solution, then we can synchronize systems (1), (2), i.e. en = 0 when n is significantly large [He & Vaidya, 1992]. It can be checked numerically by calculating the conditional Lyapunov exponents from the difference dynamics, en+1 = DX F(Xn )en

(3)

where DX F(Xn ) is the Jacobian matrix at Xn . Synchronization occurs only if all conditional Lyapunov exponents are negative. In this scheme, truncation and XOR operator are used as the encryption algorithm. The strategy of chaotic synchronization, as the decryption algorithm, is employed to synchronize two truncated systems

Digital Speech Communication by Truncated Chaotic Synchronization 693

and recover the message. We therefore call it Truncated Chaotic Synchronization (TCS). For two synchronous systems, limn→∞ ken+1 k = 0 will be yielded so that the message can be recovered as: i0n = h[j][k](Yn ) ⊕ (h[j][k](Xn ) ⊕ in ) = (h[j][k](Yn ) ⊕ h[j][k](Xn )) ⊕ in = 0 ⊕ in = in . A specific case of synchronizing the truncated systems has been reported by Zhang et al. [1999]. However, in this paper truncated chaotic synchronization, proposed as a universal method, gives a more general investigation and have more comprehensive applications in chaotic communication.


3. Dynamic Characteristics of the Truncated Time Series To illustrate the security of the truncated solution in communication, we investigate the reconstructed phase space, correlation dimension and local approximation of Hénon map: xn+1 = 1 − 1.4x2n + yn , yn+1 = 0.3xn and its truncated map xn+1,[3][2] . For a time series {xn , n = 1, 2, . . . , N ; xn ∈ R}, a time-delay vector Xn {xn , xn+τ , . . . , xn+(d−1)τ } can be used to reconstruct the phase space. According to Takens embedding theorem [Packard et al., 1980; Takens, 1981], the reconstructed attractor will be diffeomorphically equivalent to the original attractor if the embedding dimension d satisfies d > 2D + 1 (where D is the dimension of system). We can then investigate the original dynamics in the reconstructed phase space. The simple structure of the reconstructed phase space of Hénon map is displayed in Fig. 1(a). However, if we truncate the second, third byte segments from x n , the reconstructed phase space of the truncated map shows quite a different topological property. The truncated map illustrates the surprisingly complicated dynamics. In Fig. 1(b), a cloud of points in the reconstructed phase space is characterized by the absence of any structure. There is no information about the encoder, i.e. Hénon map, presented in the plot. To describe the dimension of Hénon map and the truncated map, the correlation dimension D2 proposed by Grassberger and Procaccia [1983] is used: log 2 C(N, r) r→0 N →∞ log2 r

D2 = lim lim

(4)

where r is the size of cube. C(N, r) is the correlation integral, which can be calculated from a time

series Xn with length N : N N X X 1 θ(r − kXi − Xj k) C(N, r) = N (N − 1) i=1 j=1,j6=i

(5) and θ(r) is the Heaviside function satisfying θ(r) = n 1 r≥0 . Figure 2(a) illustrates the dependence of 0 r