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A novel electro-optic chaos source is proposed on the basis of the reverse-time chaos theory and an analog-digital hybrid feedback loop. The analog output of the system can be determined by the numeric states of shift registers, which makes the system robust and easy to control. The dynamical properties as well as the complexity dependence on the feedback parameters are investigated in detail. The correlation characteristics of the system are also studied. Two improving strategies which were established in digital field and analog field are proposed to conceal the time-delay signature. The proposed scheme has the potential to be used in radar and optical secure communication systems.
© 2016 Optical Society of America
1. Introduction
Chaos system has attracted considerable attention in the past twenty years due to its widespread applications, such as secure communications [1,2], physical random number generation [3,4], and chaotic radar [5]. The design of optical chaos source is of high importance, since it is the fundamental basis of these applications. There are mainly two types of chaos systems, namely the analog chaos and digital chaos. Analog chaos systems, especially the optical ones, can generate broadband noise like signals with high dynamical complexity. Such characteristics are well aligned with security requirement in physical encryption systems. However, the synchronization robustness of analog chaos systems is still practically a big challenge in the real world applications [6–8]. Moreover, the sensitivity of the system parameters and the synchronization robustness are contradictory, which means that the safety and the reliability requirements are hard to meet simultaneously. Digital chaos system, which is realized in computing hardware, can overcome the problem suffered by analog system. However, the calculation speed is limited by the computational complexity of the algorithm and the performance of the digital signal process (DSP), thus restricts the digital chaos from being used in broadband communication systems. In view of the complementary properties of the analog and digital systems, some hybrid schemes have been proposed and demonstrated. As discussed in [9–11], pseudo-random binary sequences (PRBS) or digital chaos is combined with analog electro-optical chaos systems. The time delay signatures (TDS) of the analog systems are concealed by injecting digital signal, meanwhile the periodicity of digital signals can be masked simultaneously. Recently, a secure communication scheme based on a chaotic laser has also been proposed and demonstrated by constructing an independent dual-loop feedback which is phase modulated by PRBS [12]. The TDS is suppressed and the key space is enhanced due to the digital modulation. In [13], a hybrid model is established to solve the problem of robust synchronization of uncertain continuous chaos and dynamical degradation of digital chaos. Hybrid chaotic systems have also been applied in solving the nondeterministic polynominal problems [14] and the neuromorphic spike processing technique [15].
The conventional electro-optical oscillator described by the Ikeda equation has been studied for many years, which is a closed-loop system containing a nonlinear media and a linear filter [16]. In last decade, various physical structures have been proposed to implement the electro-optical oscillator [16–20]. In this paper, we present an alternative way to design a hybrid chaos source by utilizing the concept of reverse-time chaos [21], which could also be considered as an electro-optical oscillatory system. The theory of reverse-time chaos is a creative outlet that may bridge the gap between analog chaos and digital random sequence. We have realized the reverse-time chaos in optical domain by designing an optical matched filter [22]. However, this scheme still needs to use an independent random or pseudo-random binary source as the input, and the dynamical complexity is still relatively low compared with the conventional optical chaos systems. Recently, a novel hybrid chaos circuit in which a binary feedback is integrated has been proposed and investigated [23–25]. Here, we will report an electro-optical chaos source that combines the reverse-time chaos and a hybrid feedback loop with which high complexity binary sequence and optical chaos signal can be obtained simultaneously under an overall frame.
2. Scheme setup and principle
The principle of our scheme is shown in Fig. 1(a). It consists of four parts: shift registers (SR), the reverse-time chaos generator (RTCG), the nonlinear transformation module (NLTM) and the sampling-quantification module (SQM). The sequence generated from the SR is used as the input of the RTCG. The obtained reverse-time chaos signal is then post-processed by the NLTM and SQM, before fed back into the SR.
Based on the principle, a typical implementation in the form of the electro-optical circuit is shown in Fig. 1(b). The output x1 from SR is a binary sequence. After amplified, x1 is used to modulate a directly modulated laser (DML) which is biased in the linear region. The optical output of DML E(t) could be simply expressed by the equation
where ω denotes the frequency of DML and ϕ0 is the constant phase shift. S(t) is the modulation signal which is determined by the sequence x1(n). For simplicity, the relationship between the S(t) and x1 could be approximately given as a linear equation:where A is the amplitude. T is the duration of the unit pulse. n denotes the serial number of binary sequence. Thus, the electric field of E(t) can be simply expressed byThe obtained optical pulses are then delivered to a microwave photonic filter (MPF). The shape of the MPF is specially designed to obtain a shift dynamic. The MPF together with the RF1 and DML forms the RTCG. As a result, the optical reverse-time chaos signal can be produced. According to [22], the process could be written as
Here, w and β are constant parameters of the MPF. x2 represents the electric field of optical reverse-time chaos signal.Then the chaotic signal is post-processed by the NLTM which is composed of a photodetector (PD1), a RF driver (RF2), a laser diode (LD), a polarization controller (PC), and a Mach-Zehnder modulator (MZM). The electric input of MZM is amplified by the RF2 which means that the amplitude of the input can span several Vpi (half-wave voltage) of the modulator’s transfer function. Therefore, MZM is working in its nonlinear region and can be seen as a nonlinear noninvertible transformation [26]. After captured by PD2, high complexity chaos signal x3 can be obtained
P2 is the average power of the input light which is emitted from LD. γ1 and γ2 are the responsivity of PD1 and PD2, respectively. a2 is the gain induced by the RF2 in the NLTM. Vpi is the half-wave voltage of the MZM and φ is the bias phase.To characterize the strength of the nonlinear effect, a normalized gain parameter G is introduced and
where (.)pk-pk means the peak to peak value. The gain G represents the ratio of the peak-peak value of electric input of MZM to Vpi.Finally, an analog-digital converter (ADC) with M bits precision is used as the SQM. One bit from each sample is selected and fed back into the SR to close the feedback circle. For simplicity, here we define this bit as feedback-bit (FB). The extraction process and the SR can be implemented by a DSP.
where δ(t) is the sampling function with sample-rate f = 1/T. x is an independent variable and g(x) is the quantification function, round(.) denoting the nearest integer of the variable. f(x) represents the bit-extraction process, and the mod operation is used to calculate the remainder. M is the number of bits of ADC, B representing the position of extracted bit. τ denotes the time-delay which is determined by the device latency and the length of the SR.The mathematical model of the system can be seen as a modified Ikeda equation after some simplifications. From the analog system point of view, the circuit is an electro-optic delayed system with an analog-digital hybrid feedback loop. The time-delay and initial state of system can be influenced by the SR. From the digital viewpoint, the system can also be considered as a nonlinear feedback shift register of which the nonlinear part is implemented by an analog transformation. The digital part of system is composed of SR and ADC, so the digital computation cost could be as low as regular shift registers. Therefore, high complexity analog signal and digital sequences can be produced simultaneously with low computational cost. It is worth noting that the optical domain filtering and the nonlinear transformation are both deterministic processes. This fact means that the analog signal can be reproduced with the knowledge of the digital sequence. Digital signals are much easier to store, transmit and process than analog signals. Therefore, the proposed scheme could be a more robust way to be used in chaos based applications.
3. The fundamental characteristic of the hybrid system
In order to investigate the dynamic property of this hybrid system, some numerical simulations have been conducted by using MATLAB. The Eqs. (1) - (9) are solved by 4th-order Runge-Kutta method and the time step is set as dt = 1/160 ns. The device latency is ignored, so the time-delay of the system is entirely determined by the SR. The length of the SR is denoted by N. The lag of one-bit shift register in SR is T = 0.1ns, so the time-delay of system is τ = N∙T. The MPF parameters w and β are set as 2kπ/T and ln2/T, respectively. The bias phase of MZM is φ = 0. Finally, an 8-bit ADC is used to quantize the analog signal x3.
Figure 2 shows the dynamic behavior of the system via the temporal waveforms of x3 in a large time scale and a short one. Here, time delay of the SR is set as τ = N∙T = 3ns where N = 30 T = 0.1ns, and the least significant bit is selected as the FB. The gain G being small, the system could be considered as an open loop. After the transient oscillation, the system will enter into a steady state. When G increases to 0.28, the periodic state can be observed. As shown in Fig. 2(a), the waveform is synthesized by the periodic bipolar pulse sequences and a local intensity oscillation. The waveform in each period is shown in Fig. 2(e) and the period length is about 5.9 ns. When G = 0.4, the waveform still exhibits a periodic state in the large time scale, as depicted in Fig. 2(b), and the period length is 5.5ns, decreasing slightly. However, the local oscillation in each period becomes complicated with the increasing of G, as shown in Fig. 2(f). Then continuing to increase G, the amplitude of pulse will get increased and the period length decreases. When G = 2, the waveform becomes non-periodic. Due to the boundedness of the transfer function of the MZM, the peak to peak value of x3 will be restricted in the interval of [0, γ2P2] where the value of γ2P2 is set as 1. As G is further increased, the nonlinear effect of the MZM could be apparent and the waveform becomes chaotic, as shown in Figs. 2(d) and 2(h). The bifurcation is obtained through a sweep to the normalized gain G, which is displayed in Fig. 3.
Fig. 2 The dynamic behavior of waveforms. (a), (b), (c) and (d) correspond to G = 0.28, 0.4, 2 and 10 respectively using the ideal filter. (e), (f), (g), (h) show their details. (i), (j), (k) and (l) correspond to G = 0.25, 0.4, 1.8 and 10 using the experimnetal response.
When G is larger than 2, it shows a big difference between the waveforms of x2 and x3, due to the strong nonlinear effect. The projections in phase space of x2 and x3 are shown in Figs. 4(a) and 4(b) when G = 10. In order to display the detail of attractor clearly, different number of points are used to plot the figures. In Fig. 4(a), the number of plotting points is 20000. In Fig. 4(b), 600 points are used and the linear interpolation algorithm is adopted to construct a smooth attractor due to the finite sample rate in our simulation. The attractor of x3 could be much more dense than x2 under the same condition.
Fig. 4 (a) corresponds to the attractor of x2 using the ideal filter; (b) and (c) correspond to the attractors of Fig. 2(h) and Fig. 2(l).
As discussed in [22], in real world applications, the shape of the optical filter can be slightly differently from the ideal one as described in Eq. (4). Here we replace the model of ideal filter with a non-ideal one, which is performed by importing the experimental data of the observed realistic filter into the VPItransmissionMaker 9.0 software. The shape of such filter was measured and stored by an optical spectrum analyzer, as demonstrated in our previous work [22]. Figures 2(i - l) show the waveforms of x3 under different status. The generated waveforms are similar to the ideal ones, despite a certain degree of distortion. As a result, the attractor of Fig. 2(l) is also slightly different from the ideal one, as shown in Figs. 4(b) and 4(c). The spectrums corresponding to Fig. 2(h) and Fig. 2(l) are depicted in Fig. 5 and their effective bandwidths [27] are 7.3 and 6.9 GHz respectively.
4. Complexity analysis
Since the complexity is a fundamental characteristic that significantly influences the security performance of an encryption strategy, the discussion about the dynamical complexity of the proposed system is necessary. The complexity of the analog signal and the binary sequences are evaluated by permutation entropy (PE) and Lempel-Ziv complexity (LZC), respectively. In our system, the complexity of signal is mainly related to three parameters: the normalized gain G, the position of the FB and the length of SR N. Here we focus on the theoretical performance of the scheme, and the following simulations are conducted by using the ideal model, which is described by Eqs. (1) - (9).
As shown in Fig. 6, we can witness the influence of G and N to the complexity of both the digital sequence and the analog chaotic waveform. Here T is set as 0.1ns and the least significant bit is used to complete the feedback. When G is small, e.g. G≤1, the LZC and the PE stay at low values even for large SR length N, because in this parameter regime the system is in the periodic state, which corresponds to low dynamical complexity. When G is increased to 2, the simulation results show that the complexity (including the LZC and PE) is raising with the increase of N and has a tendency to saturate at large SR length. As G is further increased, the LZC and PE are enlarged simultaneously when the other parameters are the same. However, the tendency of the complexity versus N remains unchanged. Meanwhile the saturation threshold of N is getting smaller. These phenomena could be attributed to the fact that the MZM is working in a strong nonlinear region when the feedback gain G is large enough.
Then we consider the influence of the position of the FB on the condition of G = 10, T = 0.1ns. Figure 7 shows the signal complexity for 6≤ N ≤ 35, and B is defined as the position of the FB. B = 1 corresponds to the least significant bit and the most significant bit is represented by B = 8. The tendencies of the LZC and PE versus N and B are coincident approximately. The complexities of both the binary sequence and the analog signal are raising with the increase of N, and approach saturation at large N. The saturation thresholds for B = 1~5 can be observed in Fig. 7. If we further increase N, the thresholds for B = 6~8 can also be obtained, as shown in Table 1. With the increase of B, the threshold for N is increasing and the complexity is decreasing. This could be attributed to the fact that the entropy rates for sequences of least significant bits more closely approach the metric entropy of a chaotic system [28].
Table 1. The measured complexity for B = 6~8
As discussed above, the chaos signal x3 and the binary sequence x1 both can maintain a high complexity. When the system parameters are set appropriately, the PE of the analog chaos could reach 0.98 which is close to the theoretical maximum value. Besides, the LZC of binary sequence x1 could be a little larger than 1. When N = 26, B = 1 and G = 10, the value of LZC equals to 1.01, where the LZC of high complexity binary sequence, such as the key stream of the “Rabbit” algorithm [29], is 1.024. When being applied in secure systems, the safety could be ensured in some degrees.
5. Correlation
The cross correlation function (CCF) of the generated chaos signal with different initial values has been investigated to verify the random characteristic of the system. The CCF is defined by
where stands for the time average, y1(t) and y2(t) represent two generated chaos time series x3 with different initial values of SR, and ∆t denotes the time lag. The result is shown in Fig. 8(a). The two different initial values of the SR are set as 101010101010101010101010101010 and 111010101010101010101010101010 respectively. The values of CCFs are around zero for all the time-shift length Δt. It means the generated chaos signal is independent when the initial value of SR is slightly changed, which indicates a good random feature.Another problem that we focus on here is the self-correlation properties of the output signal. For conventional delayed feedback optical chaos systems, the TDS is a crucial issue that significantly influences the performance of chaos based applications. From the viewpoint of the secure communication, with the information of TDS, the chaotic carrier could be reconstructed by the attacker, and the dimension of the key space could be reduced [30]. These facts will cause a significant degradation of the security level. For random bit generation, the existence of the TDS will also limit the choices of sampling periods and affect the statistical performance [31]. Unfortunately, this problem is universal for most of the optical chaos systems due to its intrinsic self correlation. Such correlation can be easily identified by computing the auto-correlation function (ACF) and delayed mutual information (DMI) of the chaos time series. In the proposed system, the time delay τ is induced by the SR and the latency of the analog devices. For simplicity, the device latency is ignored in our simulation. Here we set N = 30, T = 0.1ns and G = 10. The ACF and the DMI of the time series x3 are calculated and depicted in Fig. 8. As shown in Fig. 8(b), there is a sunken area around the time-delay point in the ACF curve, which can be seen as the TDS of the system. Although such TDS is quite weak, it does exist. The TDS is more obvious when B is large, as displayed in Fig. 9. We can also observe periodic peaks in the ACF, as can be seen in the inset of Fig. 8(b). The time interval between these peaks corresponds to the time shift of a single SR. As for the DMI, no obvious TDS can be observed, as shown in Fig. 8(c).
As discussed above, the TDS can be witnessed in Fig. 8(b). Some feasible improving schemes are proposed here to conceal the TDS. The first enhancing strategy is established in digital field, as shown in Fig. 10(a). The output sequence of a linear feedback shift register (LFSR) is used to perform the XOR operation with x1. The XOR operation is a nonlinear process which has also been widely adopted in random bits generation to improve the statistical characteristic of the output sequence [32]. The ACF is shown in Fig. 11(a) and the length of LFSR is set as 3 here. Although the lag of one-bit shift register still could be obtained, the overall TDS vanishes into the background. Note that the introduction of the LFSR not only could conceal the TDS of the system, but also can improve the complexity of chaos signal. On the condition of B = 1, G = 4 and N = 30, the LZC of the x2 equals to 1.1 and the PE value of x3 could reach 0.96 with the XOR operation, where the corresponding results are 1.0 and 0.94 in the original scheme. Moreover, the digital field operation will not introduce additional analog noise. Therefore, the robustness of the system will not be affected.
The second strategy is performed in the analog way that two MZMs are used in parallel as the NLTM, as shown in Fig. 10(b). It means that the nonlinear transformation induced by the MZM is applied to the signal x2 twice. Here, the G of two MZMs are set as 10. As shown in Fig. 11(b), the TDS can be concealed entirely. Compared to the digital field strategy, the analog method makes a better effect in concealing the TDS. However, the system structure could be more complicated and additional analog noise could be introduced due to the extra analog processing devices.
6. Conclusion
In this paper, we have proposed a novel chaos system whereby an optical reverse-time chaos source and a hybrid feedback loop are integrated. The dynamic behavior of the hybrid system and the complexity dependence on system parameters have been studied in detail. The correlation characteristic and the TDS of the generated chaotic signal have also been investigated. Two improving schemes are proposed to conceal the intrinsic TDS of the system. In the proposed scheme, high complexity binary sequence and optical chaos signal can be obtained simultaneously under an overall frame. It could be a novel structure to generate broadband and robust chaos signals with high dynamical complexity. Due to the development of the ultrafast photonic analog-to-digital converters [33,34] and optical shift register technology [35–37], the hybrid system structure could be established in an all-optical way, and has the potential to be used in radar and optical secure communication systems.
Funding
National 863 Program of China (2015AA016904); National Natural Science Foundation of China (NSFC) (61505061 and 61675083).
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