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Asynchronous anti-noise hyper chaotic secure communication system based on dynamic delay and state variables switching
Physics Letters A 375 (2011) 2828–2835
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Asynchronous anti-noise hyper chaotic secure communication system based on dynamic delay and state variables switching Hongjun Liu a,b , Xingyuan Wang a,∗ , Quanlong Zhu a a b
Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, China Weifang Vocational College, Weifang 261041, China
a r t i c l e
i n f o
Article history: Received 22 January 2011 Received in revised form 15 May 2011 Accepted 13 June 2011 Available online 16 June 2011 Communicated by A.R. Bishop Keywords: Asynchronous Anti-noise Dynamic delay State variables switching Secure communication
a b s t r a c t This Letter designs an asynchronous hyper chaotic secure communication system, which possesses high stability against noise, using dynamic delay and state variables switching to ensure the high security. The relationship between the bit error ratio (BER) and the signal-to-noise ratio (SNR) is analyzed by simulation tests, the results show that the BER can be ensured to reach zero by proportionally adjusting the amplitudes of the state variables and the noise figure. The modules of the transmitter and receiver are implemented, and numerical simulations demonstrate the effectiveness of the system. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In recent years, the use of chaotic signals for information transmission attracts great attention of modern scientists, the methods of secure communication such as chaotic masking [1], chaotic modulation [2], chaos shift keying [3] and chaotic digital code-division multiple access (CDMA) [4] have been widely used. Based on these methods, scholars advance a lot of chaos-based secure communication schemes [5,6]. Further researches show that the security of chaotic masking is dissatisfactory [7], which requires the signal energy should be much lower than the chaotic signal energy, so it’s weak to resist noise jamming. Compares with the chaotic masking method, the chaos shift keying method performs better over noise jamming and parameter mismatch capacity, but poor in privacy and security [8], so it’s easily to be deciphered. Short [7,9] deciphered the chaotic communication systems and pointed out that the security of existing chaotic communication systems is almost as same as that of spread spectrum communication system. Recently, the scholars proposed many new chaos-based secure communication schemes. Chang designed a secure communication system to modulate each of delivered bit information to be a carrier signal in the continuous form, which is taken as a parameter of the unified chaotic system, and this guarantees the communication security more [10]. Li et al. designed a secure communication scheme based on chaotic maps and strong tracking filter (STF). The message is modulated by chaotic mapping and is output through a nonlinear function, and message is recovered dynamically by the STF with estimation of message. Simulation results demonstrate that STF can effectively recover the codes of the message from the noisy chaotic signals [11]. Wu et al. designed two different hyper chaotic secure communication schemes by using generalized function projective synchronization (GFPS), and the information signal can be recovered accurately by the receiver [12]. Channel noise is ubiquitous in the transmission of the masked signal, many experimental and numerical results show that noises play an important role in chaos synchronization in different ways [13], so the effect of noise should be taken into account when to evaluate the performance of a chaos communication scheme. Minai et al. proposed a method for the secure transmission of encrypted message using chaos and noises [14]. Wang et al. considered the robust demodulation problem when there are disturbances and noises in the channel [15]. Murali numerically investigated the secure communication based on the heterogeneous chaotic systems with channel noises and nonidentity of parameters [16]. Moskalenko et al. reported a secure communication based on the generalized synchronization, they
*
Corresponding author. E-mail address: [email protected] (X. Wang).
0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.06.029
H. Liu et al. / Physics Letters A 375 (2011) 2828–2835
(a)
2829
(b)
(c) Fig. 1. When r = 0.6, the projections of attractor of hyper chaotic Chen system.
use the subsidiary source of noise in the proposed scheme to provide the additional masking of the signal, and the results show high stability to noise [17]. For many communication systems based on chaotic maps, the time delay is usually fixed, and once one of the state variables is selected, it remains unchanged throughout the whole communication process [18,19], so it’s difficult to guarantee the high security. Coulon et al. designed multi-user receivers for a multiple-access system based on chaotic sequences on unknown asynchronous frequency-selective channels, they use a Differential Chaos Shift Keying (DCSK) modulation, the transmission channels are frequencyselective, and the channel characteristics (gains and delays) are unknown at the receiver side [20]. Kaddoum et al. proposed two systems for achieving synchronization in asynchronous multi-user chaos-based direct-sequence-CDMA (DS-CDMA), these synchronization processes are evaluated under the assumption of an additive white Gaussian noise channel together with multi-user interferences [21]. Kaddoum et al. further studied the BER performance of chaos-based DS-CDMA system over an m-distributed fading channel, the BER computation approach is generalized for asynchronous multi-user case [22]. Our Letter designs an asynchronous hyper chaotic secure communication system, which can possess high stability against noise, using dynamic delay and state variables switching to ensure the high security. We analyze the relationship between the BER and the SNR by many simulation tests, by proportionally adjusting the amplitudes of the state variables and the noise figure, the results show that the BER can be ensured to reach zero. The modules of the transmitter and receiver are implemented, and numerical simulations demonstrate the effectiveness of the system. 2. System description 2.1. The hyper chaotic Chen system The hyper chaotic Chen system can be described as follows [23].
⎧ x˙ 1 = a(x2 − x1 ) + x4 , ⎪ ⎪ ⎨ x˙ = dx − x x + cx , 2
1
1 3
⎪ x˙ = x1 x2 − bx3 , ⎪ ⎩ 3 x˙ 4 = x2 x3 + rx4 .
2
(1)
Here X = [ x1 x2 x3 x4 ]T are state variables, a, b, c , d and r are control parameters. When a = 35, b = 3, c = 12, d = 7 and 0.085 < r 0.798, the system evolves into hyper chaotic state [23]. According to the method presented by Ramasubramanian et al. [24], when r = 0.6 we obtain the Lyapunov exponents: λ1 = 0.567, λ2 = 0.126. It is obvious that the system exhibits a hyper chaotic behavior, the projections of the attractor are shown in Fig. 1. The state variables generated by the hyper chaotic Chen system will be used to mask the useful signal.
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Fig. 2. The distribution of x with different p in 5000 iterations.
2.2. The PWLCM system The PWLCM can be described in Eq. (2):
⎧ 0 xi < p , ⎨ xi / p , xi +1 = F p (xi ) = (xi − p )/(0.5 − p ), p xi < 0.5, ⎩ F p (1 − xi ), xi 0.5,
(2)
where xi ∈ [0, 1), when control parameter p ∈ (0, 0.5), Eq. (2) evolves into chaotic state [25], p can be served as the secret key. The PWLCM system has uniform invariant distribution and very good ergodicity, confusion and determinacy, so it can provide excellent random sequence, which is suitable for cryptosystem. The distribution of x with different p of the PWLCM system is shown in Fig. 2. We use the PWLCM system to generate two pseudo-random integer sequences V and D τ with different initial values and parameters. Suppose the size of original binary signal is n, n = 2 j, j = 1, 2, 3, . . . .
V = { v 1 , v 2 , . . . , v n/2 }, where
v i ∈ {1, 2, 3, 4},
(3)
⎧ 1, xi ∈ [0, 0.25), ⎪ ⎪ ⎨ 2, x ∈ [0.25, 0.5), i vi = i = 1 , 2 , . . . , n /2 ; ⎪ 3, xi ∈ [0.5, 0.75), ⎪ ⎩ 4, xi ∈ [0.75, 1), D τ = {d1 , d2 , . . . , dn/2 }, di ∈ {1, 2, . . . , 8},
(4)
⎧ 8, ⎪ ⎪ ⎪ 7, ⎪ ⎪ ⎪ ⎪ ⎪ 6, ⎪ ⎪ ⎪ ⎨ 5, di = ⎪ 4, ⎪ ⎪ ⎪ ⎪ ⎪ 3, ⎪ ⎪ ⎪ ⎪ 2, ⎪ ⎩ 1,
(6)
(5)
where
xi ∈ [0, 0.125), xi ∈ [0.125, 0.25), xi ∈ [0.25, 0.375), xi ∈ [0.375, 0.5), xi ∈ [0.5, 0.625),
i = 1 , 2 , . . . , n /2 .
xi ∈ [0.625, 0.75), xi ∈ [0.75, 0.875), xi ∈ [0.875, 1),
3. Design for secure communication scheme 3.1. Signal modulation Step 1. Suppose s(n) is the original binary signal, for each bit in s(n), we can get s (n) by Eq. (7).
s (n) = s(n) −
s(n) + 1 mod 2 .
After the conversion, each signal 0 in s(n) is converted to −1, so s (n) is composed of
(7)
−1 and 1. Step 2. After iterating the hyper chaotic Chen system for limited times, we begin to get X (t ), which is composed of x1 (t ), x2 (t ), x3 (t ) and x4 (t ), and they lie within different intervals. Step 3. Proportionally adjust the amplitudes of state variables X (t ) to the basically uniform interval of [min, max] to get Y (t ), i.e. convert x1 (t ), x2 (t ), x3 (t ) and x4 (t ) to y 1 (t ), y 2 (t ), y 3 (t ) and y 4 (t ) respectively by Eq. (8), where m is the minimum signal magnification factor.
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Fig. 3. The composition of y (t ).
Fig. 4. The relationship of y (t ), di and l(t ).
Fig. 5. The model of the transmitter for asynchronous communication system.
⎧ y 1 (t ) = x1 (t ) × 3m, ⎪ ⎪ ⎨ y (t ) = x (t ) × 3m, 2
2
(8)
⎪ y (t ) = (x3 (t ) − 20) × 4m, ⎪ ⎩ 3 y 4 (t ) = (x4 (t ) + 3) × m.
Step 4. Set the detection threshold by deleting those values within the interval of [−k, k] from Y (t ), where k (k 6) is the noise figure, then pack Y (t ) to get Y p (t ). Step 5. Set A = min − max, E = max − min, the subtraction scope of Y p (t ) at any two times is [ A , E ], set B , C , D ∈ ( A , E ), then A , B , C , D , E separate the interval [ A , E ] into four intervals of [ A , B ), [ B , C ), [C , D ) and [ D , E ]. Step 6. For each pair of s (2i − 1) and s (2i ) in s (n), randomly select one of the state variables y v i (t ) ∈ Y p (t ) by v i ∈ V and get two continuous values to generate y (t ). The composition of y (t ) is shown in Fig. 3, for each y v i (i ), the symbol “–” with finite size denotes the value from one of the state variables that remain unchanged. The two continuous symbols “o” denote the values of state variable that will be used to mask two continuous signals from s (n). Step 7. Set t i = (2(i − 1) + 1 + sum( D [1 : i ])) × T s , i = 1, 2, . . . , n/2, τ = T s . For any y (t i ), y (t i + τ ) ∈ y (t ), if | y (t i )| − | y (t i + τ )| ∈ [−k, k], we need to adjust their values to make | y (t i )| − | y (t i + τ )| ∈ / [−k, k], otherwise the signal would not be correctly recovered by the receiver. Step 8. Get y (t ) by Eq. (9).
⎧ y (t ) = −m| y (t )|, ⎪ ⎪ ⎨ y (t ) = −m| y (t )|, ⎪ y (t ) = m| y (t )|, ⎪ ⎩ y (t ) = m| y (t )|,
y (t + τ ) = −m| y (t + τ )|, y (t + τ ) = m| y (t + τ )|,
y (t + τ ) = −m| y (t + τ )|, y (t + τ ) = m| y (t + τ )|,
if | y (t )| − | y (t + τ )| ∈ [ A , B ), if | y (t )| − | y (t + τ )| ∈ [ B , C ), if | y (t )| − | y (t + τ )| ∈ [C , D ),
(9)
if | y (t )| − | y (t + τ )| ∈ [ D , E ].
Step 9. From t = t 0 , we mask the signal s (n) into y (t ) to get l(t ) by Eq. (10). The relationship of y (t ), di and l(t ) are shown in Fig. 4.
l(t ) = s (n) y (t ).
(10)
After the process, l(t ) becomes the chaotic signal with the original signal, which can be only transmitted in the physical channel without noise. Step 10. For the noise occurs in physical channel inevitably, here we add white Gaussian noise w (t ) to l(t ) to imitate the channel noises, the transmitted signal can be generated by Eq. (11).
l (t ) = l(t ) + w (t ). The model of the transmitter for asynchronous communication system is shown in Fig. 5.
(11)
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Fig. 6. The model of the receiver for asynchronous communication system.
3.2. Signal demodulation The model of the receiver for the asynchronous communication system is shown in Fig. 6. When the receiver receives the signal l (t ), delays it to get l (t + τ ), t ∈ (2 jT s , (2 j + 1) T s ), j = 0, 1, 2, 3, . . . , τ = T s . If |l (t )| − |l (t + τ )| ∈ [ A , B ), and l (t ) < 0, l (t + τ ) < 0, we can decode the signal sˆ (n) = 11. The algorithm of the comparison model is as follows. After the demodulation process, the receiver can recover the signal sˆ (n) = s(n). 4. Numerical simulations 4.1. Initial values and parameters For the hyper chaotic Chen system, we set the parameters a = 35, b = 3, c = 12, d = 7 and 0.085 < r 0.798 [23], the initial values x01 , x02 , x03 and x04 are served as the keys. To ensure a large divergence of a chaotic trajectory from the initial condition, the iteration times should be relatively large but not too much, so we iterate the system 200 times firstly. For the PWLCM system, we set parameters p ∈ (0, 0.5) to generate the sequences of V and D τ , the initial values of z V and z D τ are served as the keys. 4.2. The relationship between BER and SNR We analyze the relationship between the BER and the SNR by many simulation tests, the results show that the BER can be ensured to reach zero by proportionally adjusting the amplitudes of the state variables and eliminating the noise jamming. Here the SNR specifies the signal-to-noise ratio per sample, in dB. We set the state variables magnification factor m = 2, and noise figure k = 6, and use three groups of initial values of the hyper chaotic Chen system to test the system. The relationship between BER and SNR are shown in Fig. 7, we can find that the BER decrease with the increase of SNR, when SNR −5, the BER can be ensured to reach zero, so the information signal can be completely recovered from the transmitted signal. 4.3. The magnified state variables For the original state variables X (t ), we firstly adjust and enlarge their amplitudes to a basically uniform interval, then delete those values within the interval of [−k, k], or they may be covered by the white Gaussian noise, for example, at one time, if the modulated chaotic signal is k/2, the noise is −k/2, then the result of addition is zero, which would lead to the bit error. The results of state variables of y 1 (t ), y 2 (t ), y 3 (t ) and y 4 (t ) in Y p (t ) are shown in Fig. 8.
H. Liu et al. / Physics Letters A 375 (2011) 2828–2835
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Fig. 7. The relationship between SNR and BER.
(a)
(b)
(c)
(d)
Fig. 8. The state variables of y 1 (t ), y 2 (t ), y 3 (t ) and y 4 (t ) in Y p (t ).
4.4. The signals of y (t ), l(t ) and l (t ) Suppose the size of original signal s(n) is 280. The modulated chaotic signal y (t ) is shown in Fig. 9(a), which is a combination of four amplified state variables generated by Eq. (1) according to the dynamical time delay, from the figure we can find that the result is still chaotic. The signal l(t ) without noise and the transmitted signal l (t ) with white Gaussian noise are shown in Fig. 9(b). 4.5. The original signal, converted signal and demodulated signal Suppose the original signal s(n) = 101110111010010101001011010101101010, here we set the state variables magnification factor m = 2, the noise figure k = 6, and the signal-to-noise ratio snr ∈ [−5, 10], the original signal can be recovered successfully. The original signal s(n), the converted signal s (n) and the demodulated signal sˆ (n) by receiver are shown in Fig. 10, from the results we can find that sˆ (n) = s(n).
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(a)
(b) Fig. 9. The signals of y (t ), l(t ) and l (t ).
(a)
(b)
(c) Fig. 10. The signals of s(n), s (n) and sˆ (n).
5. Security analysis In the modulation process, the switching process from the chaotic signal X (t ) to l (t ) is as follows.
X (t ) → Y (t ) → Y p (t ) → y (t ) → y (t ) → l(t ) → l (t ). After the processes we get the transmitted signal l (t ), which is the combination of the chaotic signal, the modulated signal s (n) and the white Gaussian noise. Even though the attackers can intercept the signal l (t ), it is impossible to demodulate the chaotic signal if the initial values, the values of B , C , D and Eq. (9) are unknown. Here we find that four intervals of [ A , B ), [ B , C ), [C , D ) and [ D , E ] are suitable by trial and error, or the hardware complexity of the communication system will be greatly enhanced. For the dynamic behavior of hyper chaotic system is more difficult to predict than the low dimension chaotic system, it can greatly improve the security of the secure communication system, and the higher the dimension of chaotic system, the higher the confidentiality of secure communication. In the physical channel with noise, as long as the noise amplitude interval can be detected, the noise figure and the amplification factor of the chaotic signal can be adjusted to ensure the BER to reach zero.
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6. Conclusion We design an asynchronous anti-noise hyper chaotic secure communication system, using dynamic delay and state variables switching to ensure the high security. The relationship between the BER and the SNR is analyzed by many simulation tests, the results show that the BER can be ensured to reach zero by proportionally adjusting the amplitudes of the state variables and the noise figure. The modules of the transmitter and receiver are implemented, and numerical simulations demonstrate the effectiveness of the system. Furthermore, we will study how to mix the useful signal in the unwanted signal to further enhance the security. Acknowledgements The research is supported by the National Natural Science Foundation of China (Nos. 60973152 and 60573172), the Superior University Doctor Subject Special Scientific Research Foundation of China (No. 20070141014), the National Natural Science Foundation of Liaoning province (No. 20082165). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
L. Kocarev, K.S. Halle, K. Eckert, L.O. Chua, International Journal of Bifurcation and Chaos 2 (5) (1992) 709. T. Yang, L.O. Chua, IEEE Transactions on Circuits and Systems I 43 (9) (1996) 817. H. Dedieu, M.P. Kennedy, IEEE Transactions on Circuits and Systems II 40 (10) (1993) 634. T. Yang, L.O. Chua, International Journal of Bifurcation and Chaos 7 (12) (1997) 2789. X.Y. Wang, C.F. Duan, Journal on Communications 26 (6) (2005) 106. A.I. Panas, T. Yanag, International Journal of Bifurcation and Chaos 8 (3) (1998) 639. K.M. Short, International Journal of Bifurcation and Chaos 4 (4) (1994) 959. X.M. Wang, J.S. Zhang, Physics Letters A 357 (4–5) (2006) 323. K.M. Short, International Journal of Bifurcation and Chaos 6 (2) (1996) 367. W.D. Chang, Digital Signal Processing 19 (4) (2009) 693. X.J. Li, Z.G. Xu, D.H. Zhou, Physics Letters A 372 (44) (2008) 6627. X.J. Wu, H. Wang, H.T. Lu, Nonlinear Analysis: Real World Applications 12 (2) (2011) 1288. Y.H. Sun, J.D. Cao, G. Feng, Physics Letters A 372 (33) (2008) 5442. A.A. Minai, T.D. Pandian, Chaos 8 (3) (1998) 621. X.F. Wang, Z.Q. Wang, International Journal of Bifurcation and Chaos 13 (1) (2003) 227. K. Murali, Physics Letters A 272 (3) (2000) 184. O.I. Moskalenko, A.A. Koronovskii, A.E. Hramov, Physics Letters A 374 (29) (2010) 2925. S.K. Changchien, C.K. Huang, H.H. Nien, H.W. Shieh, Chaos, Solitons & Fractals 39 (4) (2009) 1578. D.M. Li, Z.D. Wang, J. Zhou, J.A. Fang, J.J. Ni, Chaos, Solitons & Fractals 38 (4) (2008) 1217. M. Coulon, D. Roviras, Signal Processing 90 (2) (2010) 587. G. Kaddoum, D. Roviras, P. Chargé, Danièle Fournier-Prunaret, Signal Processing 89 (5) (2009) 807. G. Kaddoum, M. Coulon, D. Roviras, P. Chargé, Signal Processing 90 (11) (2010) 2923. T.G. Gao, Z.Q. Chen, G.R. Chen, International Journal of Modern Physics C 17 (4) (2006) 471. K. Ramasubramanian, M.S. Sriram, Physica D: Nonlinear Phenomena 139 (1–2) (2000) 72. B. Alatas, E. Akin, A.B. Ozer, Chaos, Solitons & Fractals 40 (4) (2009) 1715.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Asynchronous anti-noise hyper chaotic secure communication system based on dynamic delay and state variables switching Hongjun Liu a,b , Xingyuan Wang a,∗ , Quanlong Zhu a a b
Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, China Weifang Vocational College, Weifang 261041, China
a r t i c l e
i n f o
Article history: Received 22 January 2011 Received in revised form 15 May 2011 Accepted 13 June 2011 Available online 16 June 2011 Communicated by A.R. Bishop Keywords: Asynchronous Anti-noise Dynamic delay State variables switching Secure communication
a b s t r a c t This Letter designs an asynchronous hyper chaotic secure communication system, which possesses high stability against noise, using dynamic delay and state variables switching to ensure the high security. The relationship between the bit error ratio (BER) and the signal-to-noise ratio (SNR) is analyzed by simulation tests, the results show that the BER can be ensured to reach zero by proportionally adjusting the amplitudes of the state variables and the noise figure. The modules of the transmitter and receiver are implemented, and numerical simulations demonstrate the effectiveness of the system. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In recent years, the use of chaotic signals for information transmission attracts great attention of modern scientists, the methods of secure communication such as chaotic masking [1], chaotic modulation [2], chaos shift keying [3] and chaotic digital code-division multiple access (CDMA) [4] have been widely used. Based on these methods, scholars advance a lot of chaos-based secure communication schemes [5,6]. Further researches show that the security of chaotic masking is dissatisfactory [7], which requires the signal energy should be much lower than the chaotic signal energy, so it’s weak to resist noise jamming. Compares with the chaotic masking method, the chaos shift keying method performs better over noise jamming and parameter mismatch capacity, but poor in privacy and security [8], so it’s easily to be deciphered. Short [7,9] deciphered the chaotic communication systems and pointed out that the security of existing chaotic communication systems is almost as same as that of spread spectrum communication system. Recently, the scholars proposed many new chaos-based secure communication schemes. Chang designed a secure communication system to modulate each of delivered bit information to be a carrier signal in the continuous form, which is taken as a parameter of the unified chaotic system, and this guarantees the communication security more [10]. Li et al. designed a secure communication scheme based on chaotic maps and strong tracking filter (STF). The message is modulated by chaotic mapping and is output through a nonlinear function, and message is recovered dynamically by the STF with estimation of message. Simulation results demonstrate that STF can effectively recover the codes of the message from the noisy chaotic signals [11]. Wu et al. designed two different hyper chaotic secure communication schemes by using generalized function projective synchronization (GFPS), and the information signal can be recovered accurately by the receiver [12]. Channel noise is ubiquitous in the transmission of the masked signal, many experimental and numerical results show that noises play an important role in chaos synchronization in different ways [13], so the effect of noise should be taken into account when to evaluate the performance of a chaos communication scheme. Minai et al. proposed a method for the secure transmission of encrypted message using chaos and noises [14]. Wang et al. considered the robust demodulation problem when there are disturbances and noises in the channel [15]. Murali numerically investigated the secure communication based on the heterogeneous chaotic systems with channel noises and nonidentity of parameters [16]. Moskalenko et al. reported a secure communication based on the generalized synchronization, they
*
Corresponding author. E-mail address: [email protected] (X. Wang).
0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.06.029
H. Liu et al. / Physics Letters A 375 (2011) 2828–2835
(a)
2829
(b)
(c) Fig. 1. When r = 0.6, the projections of attractor of hyper chaotic Chen system.
use the subsidiary source of noise in the proposed scheme to provide the additional masking of the signal, and the results show high stability to noise [17]. For many communication systems based on chaotic maps, the time delay is usually fixed, and once one of the state variables is selected, it remains unchanged throughout the whole communication process [18,19], so it’s difficult to guarantee the high security. Coulon et al. designed multi-user receivers for a multiple-access system based on chaotic sequences on unknown asynchronous frequency-selective channels, they use a Differential Chaos Shift Keying (DCSK) modulation, the transmission channels are frequencyselective, and the channel characteristics (gains and delays) are unknown at the receiver side [20]. Kaddoum et al. proposed two systems for achieving synchronization in asynchronous multi-user chaos-based direct-sequence-CDMA (DS-CDMA), these synchronization processes are evaluated under the assumption of an additive white Gaussian noise channel together with multi-user interferences [21]. Kaddoum et al. further studied the BER performance of chaos-based DS-CDMA system over an m-distributed fading channel, the BER computation approach is generalized for asynchronous multi-user case [22]. Our Letter designs an asynchronous hyper chaotic secure communication system, which can possess high stability against noise, using dynamic delay and state variables switching to ensure the high security. We analyze the relationship between the BER and the SNR by many simulation tests, by proportionally adjusting the amplitudes of the state variables and the noise figure, the results show that the BER can be ensured to reach zero. The modules of the transmitter and receiver are implemented, and numerical simulations demonstrate the effectiveness of the system. 2. System description 2.1. The hyper chaotic Chen system The hyper chaotic Chen system can be described as follows [23].
⎧ x˙ 1 = a(x2 − x1 ) + x4 , ⎪ ⎪ ⎨ x˙ = dx − x x + cx , 2
1
1 3
⎪ x˙ = x1 x2 − bx3 , ⎪ ⎩ 3 x˙ 4 = x2 x3 + rx4 .
2
(1)
Here X = [ x1 x2 x3 x4 ]T are state variables, a, b, c , d and r are control parameters. When a = 35, b = 3, c = 12, d = 7 and 0.085 < r 0.798, the system evolves into hyper chaotic state [23]. According to the method presented by Ramasubramanian et al. [24], when r = 0.6 we obtain the Lyapunov exponents: λ1 = 0.567, λ2 = 0.126. It is obvious that the system exhibits a hyper chaotic behavior, the projections of the attractor are shown in Fig. 1. The state variables generated by the hyper chaotic Chen system will be used to mask the useful signal.
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Fig. 2. The distribution of x with different p in 5000 iterations.
2.2. The PWLCM system The PWLCM can be described in Eq. (2):
⎧ 0 xi < p , ⎨ xi / p , xi +1 = F p (xi ) = (xi − p )/(0.5 − p ), p xi < 0.5, ⎩ F p (1 − xi ), xi 0.5,
(2)
where xi ∈ [0, 1), when control parameter p ∈ (0, 0.5), Eq. (2) evolves into chaotic state [25], p can be served as the secret key. The PWLCM system has uniform invariant distribution and very good ergodicity, confusion and determinacy, so it can provide excellent random sequence, which is suitable for cryptosystem. The distribution of x with different p of the PWLCM system is shown in Fig. 2. We use the PWLCM system to generate two pseudo-random integer sequences V and D τ with different initial values and parameters. Suppose the size of original binary signal is n, n = 2 j, j = 1, 2, 3, . . . .
V = { v 1 , v 2 , . . . , v n/2 }, where
v i ∈ {1, 2, 3, 4},
(3)
⎧ 1, xi ∈ [0, 0.25), ⎪ ⎪ ⎨ 2, x ∈ [0.25, 0.5), i vi = i = 1 , 2 , . . . , n /2 ; ⎪ 3, xi ∈ [0.5, 0.75), ⎪ ⎩ 4, xi ∈ [0.75, 1), D τ = {d1 , d2 , . . . , dn/2 }, di ∈ {1, 2, . . . , 8},
(4)
⎧ 8, ⎪ ⎪ ⎪ 7, ⎪ ⎪ ⎪ ⎪ ⎪ 6, ⎪ ⎪ ⎪ ⎨ 5, di = ⎪ 4, ⎪ ⎪ ⎪ ⎪ ⎪ 3, ⎪ ⎪ ⎪ ⎪ 2, ⎪ ⎩ 1,
(6)
(5)
where
xi ∈ [0, 0.125), xi ∈ [0.125, 0.25), xi ∈ [0.25, 0.375), xi ∈ [0.375, 0.5), xi ∈ [0.5, 0.625),
i = 1 , 2 , . . . , n /2 .
xi ∈ [0.625, 0.75), xi ∈ [0.75, 0.875), xi ∈ [0.875, 1),
3. Design for secure communication scheme 3.1. Signal modulation Step 1. Suppose s(n) is the original binary signal, for each bit in s(n), we can get s (n) by Eq. (7).
s (n) = s(n) −
s(n) + 1 mod 2 .
After the conversion, each signal 0 in s(n) is converted to −1, so s (n) is composed of
(7)
−1 and 1. Step 2. After iterating the hyper chaotic Chen system for limited times, we begin to get X (t ), which is composed of x1 (t ), x2 (t ), x3 (t ) and x4 (t ), and they lie within different intervals. Step 3. Proportionally adjust the amplitudes of state variables X (t ) to the basically uniform interval of [min, max] to get Y (t ), i.e. convert x1 (t ), x2 (t ), x3 (t ) and x4 (t ) to y 1 (t ), y 2 (t ), y 3 (t ) and y 4 (t ) respectively by Eq. (8), where m is the minimum signal magnification factor.
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Fig. 3. The composition of y (t ).
Fig. 4. The relationship of y (t ), di and l(t ).
Fig. 5. The model of the transmitter for asynchronous communication system.
⎧ y 1 (t ) = x1 (t ) × 3m, ⎪ ⎪ ⎨ y (t ) = x (t ) × 3m, 2
2
(8)
⎪ y (t ) = (x3 (t ) − 20) × 4m, ⎪ ⎩ 3 y 4 (t ) = (x4 (t ) + 3) × m.
Step 4. Set the detection threshold by deleting those values within the interval of [−k, k] from Y (t ), where k (k 6) is the noise figure, then pack Y (t ) to get Y p (t ). Step 5. Set A = min − max, E = max − min, the subtraction scope of Y p (t ) at any two times is [ A , E ], set B , C , D ∈ ( A , E ), then A , B , C , D , E separate the interval [ A , E ] into four intervals of [ A , B ), [ B , C ), [C , D ) and [ D , E ]. Step 6. For each pair of s (2i − 1) and s (2i ) in s (n), randomly select one of the state variables y v i (t ) ∈ Y p (t ) by v i ∈ V and get two continuous values to generate y (t ). The composition of y (t ) is shown in Fig. 3, for each y v i (i ), the symbol “–” with finite size denotes the value from one of the state variables that remain unchanged. The two continuous symbols “o” denote the values of state variable that will be used to mask two continuous signals from s (n). Step 7. Set t i = (2(i − 1) + 1 + sum( D [1 : i ])) × T s , i = 1, 2, . . . , n/2, τ = T s . For any y (t i ), y (t i + τ ) ∈ y (t ), if | y (t i )| − | y (t i + τ )| ∈ [−k, k], we need to adjust their values to make | y (t i )| − | y (t i + τ )| ∈ / [−k, k], otherwise the signal would not be correctly recovered by the receiver. Step 8. Get y (t ) by Eq. (9).
⎧ y (t ) = −m| y (t )|, ⎪ ⎪ ⎨ y (t ) = −m| y (t )|, ⎪ y (t ) = m| y (t )|, ⎪ ⎩ y (t ) = m| y (t )|,
y (t + τ ) = −m| y (t + τ )|, y (t + τ ) = m| y (t + τ )|,
y (t + τ ) = −m| y (t + τ )|, y (t + τ ) = m| y (t + τ )|,
if | y (t )| − | y (t + τ )| ∈ [ A , B ), if | y (t )| − | y (t + τ )| ∈ [ B , C ), if | y (t )| − | y (t + τ )| ∈ [C , D ),
(9)
if | y (t )| − | y (t + τ )| ∈ [ D , E ].
Step 9. From t = t 0 , we mask the signal s (n) into y (t ) to get l(t ) by Eq. (10). The relationship of y (t ), di and l(t ) are shown in Fig. 4.
l(t ) = s (n) y (t ).
(10)
After the process, l(t ) becomes the chaotic signal with the original signal, which can be only transmitted in the physical channel without noise. Step 10. For the noise occurs in physical channel inevitably, here we add white Gaussian noise w (t ) to l(t ) to imitate the channel noises, the transmitted signal can be generated by Eq. (11).
l (t ) = l(t ) + w (t ). The model of the transmitter for asynchronous communication system is shown in Fig. 5.
(11)
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Fig. 6. The model of the receiver for asynchronous communication system.
3.2. Signal demodulation The model of the receiver for the asynchronous communication system is shown in Fig. 6. When the receiver receives the signal l (t ), delays it to get l (t + τ ), t ∈ (2 jT s , (2 j + 1) T s ), j = 0, 1, 2, 3, . . . , τ = T s . If |l (t )| − |l (t + τ )| ∈ [ A , B ), and l (t ) < 0, l (t + τ ) < 0, we can decode the signal sˆ (n) = 11. The algorithm of the comparison model is as follows. After the demodulation process, the receiver can recover the signal sˆ (n) = s(n). 4. Numerical simulations 4.1. Initial values and parameters For the hyper chaotic Chen system, we set the parameters a = 35, b = 3, c = 12, d = 7 and 0.085 < r 0.798 [23], the initial values x01 , x02 , x03 and x04 are served as the keys. To ensure a large divergence of a chaotic trajectory from the initial condition, the iteration times should be relatively large but not too much, so we iterate the system 200 times firstly. For the PWLCM system, we set parameters p ∈ (0, 0.5) to generate the sequences of V and D τ , the initial values of z V and z D τ are served as the keys. 4.2. The relationship between BER and SNR We analyze the relationship between the BER and the SNR by many simulation tests, the results show that the BER can be ensured to reach zero by proportionally adjusting the amplitudes of the state variables and eliminating the noise jamming. Here the SNR specifies the signal-to-noise ratio per sample, in dB. We set the state variables magnification factor m = 2, and noise figure k = 6, and use three groups of initial values of the hyper chaotic Chen system to test the system. The relationship between BER and SNR are shown in Fig. 7, we can find that the BER decrease with the increase of SNR, when SNR −5, the BER can be ensured to reach zero, so the information signal can be completely recovered from the transmitted signal. 4.3. The magnified state variables For the original state variables X (t ), we firstly adjust and enlarge their amplitudes to a basically uniform interval, then delete those values within the interval of [−k, k], or they may be covered by the white Gaussian noise, for example, at one time, if the modulated chaotic signal is k/2, the noise is −k/2, then the result of addition is zero, which would lead to the bit error. The results of state variables of y 1 (t ), y 2 (t ), y 3 (t ) and y 4 (t ) in Y p (t ) are shown in Fig. 8.
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Fig. 7. The relationship between SNR and BER.
(a)
(b)
(c)
(d)
Fig. 8. The state variables of y 1 (t ), y 2 (t ), y 3 (t ) and y 4 (t ) in Y p (t ).
4.4. The signals of y (t ), l(t ) and l (t ) Suppose the size of original signal s(n) is 280. The modulated chaotic signal y (t ) is shown in Fig. 9(a), which is a combination of four amplified state variables generated by Eq. (1) according to the dynamical time delay, from the figure we can find that the result is still chaotic. The signal l(t ) without noise and the transmitted signal l (t ) with white Gaussian noise are shown in Fig. 9(b). 4.5. The original signal, converted signal and demodulated signal Suppose the original signal s(n) = 101110111010010101001011010101101010, here we set the state variables magnification factor m = 2, the noise figure k = 6, and the signal-to-noise ratio snr ∈ [−5, 10], the original signal can be recovered successfully. The original signal s(n), the converted signal s (n) and the demodulated signal sˆ (n) by receiver are shown in Fig. 10, from the results we can find that sˆ (n) = s(n).
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(a)
(b) Fig. 9. The signals of y (t ), l(t ) and l (t ).
(a)
(b)
(c) Fig. 10. The signals of s(n), s (n) and sˆ (n).
5. Security analysis In the modulation process, the switching process from the chaotic signal X (t ) to l (t ) is as follows.
X (t ) → Y (t ) → Y p (t ) → y (t ) → y (t ) → l(t ) → l (t ). After the processes we get the transmitted signal l (t ), which is the combination of the chaotic signal, the modulated signal s (n) and the white Gaussian noise. Even though the attackers can intercept the signal l (t ), it is impossible to demodulate the chaotic signal if the initial values, the values of B , C , D and Eq. (9) are unknown. Here we find that four intervals of [ A , B ), [ B , C ), [C , D ) and [ D , E ] are suitable by trial and error, or the hardware complexity of the communication system will be greatly enhanced. For the dynamic behavior of hyper chaotic system is more difficult to predict than the low dimension chaotic system, it can greatly improve the security of the secure communication system, and the higher the dimension of chaotic system, the higher the confidentiality of secure communication. In the physical channel with noise, as long as the noise amplitude interval can be detected, the noise figure and the amplification factor of the chaotic signal can be adjusted to ensure the BER to reach zero.
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6. Conclusion We design an asynchronous anti-noise hyper chaotic secure communication system, using dynamic delay and state variables switching to ensure the high security. The relationship between the BER and the SNR is analyzed by many simulation tests, the results show that the BER can be ensured to reach zero by proportionally adjusting the amplitudes of the state variables and the noise figure. The modules of the transmitter and receiver are implemented, and numerical simulations demonstrate the effectiveness of the system. Furthermore, we will study how to mix the useful signal in the unwanted signal to further enhance the security. Acknowledgements The research is supported by the National Natural Science Foundation of China (Nos. 60973152 and 60573172), the Superior University Doctor Subject Special Scientific Research Foundation of China (No. 20070141014), the National Natural Science Foundation of Liaoning province (No. 20082165). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
L. Kocarev, K.S. Halle, K. Eckert, L.O. Chua, International Journal of Bifurcation and Chaos 2 (5) (1992) 709. T. Yang, L.O. Chua, IEEE Transactions on Circuits and Systems I 43 (9) (1996) 817. H. Dedieu, M.P. Kennedy, IEEE Transactions on Circuits and Systems II 40 (10) (1993) 634. T. Yang, L.O. Chua, International Journal of Bifurcation and Chaos 7 (12) (1997) 2789. X.Y. Wang, C.F. Duan, Journal on Communications 26 (6) (2005) 106. A.I. Panas, T. Yanag, International Journal of Bifurcation and Chaos 8 (3) (1998) 639. K.M. Short, International Journal of Bifurcation and Chaos 4 (4) (1994) 959. X.M. Wang, J.S. Zhang, Physics Letters A 357 (4–5) (2006) 323. K.M. Short, International Journal of Bifurcation and Chaos 6 (2) (1996) 367. W.D. Chang, Digital Signal Processing 19 (4) (2009) 693. X.J. Li, Z.G. Xu, D.H. Zhou, Physics Letters A 372 (44) (2008) 6627. X.J. Wu, H. Wang, H.T. Lu, Nonlinear Analysis: Real World Applications 12 (2) (2011) 1288. Y.H. Sun, J.D. Cao, G. Feng, Physics Letters A 372 (33) (2008) 5442. A.A. Minai, T.D. Pandian, Chaos 8 (3) (1998) 621. X.F. Wang, Z.Q. Wang, International Journal of Bifurcation and Chaos 13 (1) (2003) 227. K. Murali, Physics Letters A 272 (3) (2000) 184. O.I. Moskalenko, A.A. Koronovskii, A.E. Hramov, Physics Letters A 374 (29) (2010) 2925. S.K. Changchien, C.K. Huang, H.H. Nien, H.W. Shieh, Chaos, Solitons & Fractals 39 (4) (2009) 1578. D.M. Li, Z.D. Wang, J. Zhou, J.A. Fang, J.J. Ni, Chaos, Solitons & Fractals 38 (4) (2008) 1217. M. Coulon, D. Roviras, Signal Processing 90 (2) (2010) 587. G. Kaddoum, D. Roviras, P. Chargé, Danièle Fournier-Prunaret, Signal Processing 89 (5) (2009) 807. G. Kaddoum, M. Coulon, D. Roviras, P. Chargé, Signal Processing 90 (11) (2010) 2923. T.G. Gao, Z.Q. Chen, G.R. Chen, International Journal of Modern Physics C 17 (4) (2006) 471. K. Ramasubramanian, M.S. Sriram, Physica D: Nonlinear Phenomena 139 (1–2) (2000) 72. B. Alatas, E. Akin, A.B. Ozer, Chaos, Solitons & Fractals 40 (4) (2009) 1715.