A new switching parameter varying optoelectronic delayed feedback model with computer simulation

In this paper, a new switching parameter varying optoelectronic delayed feedback model is proposed and analyzed by computer simulation. This model is switching between two parameter varying optoelectronic delayed feedback models based on chaotic pseudorandom sequences. Complexity performance results show that this model has a high complexity compared to the original model. Furthermore, this model can conceal the time delay effectively against the auto-correlation function, delayed mutual information and permutation information analysis methods, and can extent the key space, which greatly improve its security.

parameters can be used to extent the key space to resist brute-force attacks. This new model has varies applications in many fields such as computer secure communication, random sequence generator and cryptography.
The rest of this paper is organized as follows. In Section 2, the new switching model is introduced. The complexity performance of this model is introduced in Section 3. The time-delay concealment is analyzed in Section 4. Finally, Section 5 concludes the whole paper.
New switching parameter varying optoelectronic delayed feedback model. The original optoelectronic delayed feedback system in ref. 9 can be modeled by the following integro-differential delayed equation: where β = πgAGP/2V π , Φ = πV B /2V dc , θ = 1/2πf L , and τ = 1/2πf H . V π and V dc are radio-frequency and dc half-wave voltages of MZ modulator; V B is the biased voltage; P is the power of semiconductor laser source; T is the time-delay; g and G are the gain coefficients of photodiode and ratio-frequency amplifier, respectively. A is the overall attenuation of feedback loop; f L and f H are low and high cutoff frequencies, respectively. Model (1) has been proved to be unsecure 21,23 . Here, we propose a new switching model with varying parameters, which can be written as: where x 1 and x 2 are the output signals of sub-model 1 and 2. τ 1, 2 , θ 1, 2, , Φ 1, 2 , T, T 2 (t), β 1 (t) and β are the parameters of the two sub-models, respectively. r is the control coefficient of the logistic map with output y i . U is the switching threshold. u is the time interval, and operating floor (t/u) changes the completely discrete signal y i into signal y(t). y(t), a step signal with time interval u, is not continuous. z(t) is the final output signal of this model. As for sub-models 1 and 2 shown in Eq. (2), the parameters of sub-models 1 and 2 are both varying. For sub-model 1, the gain coefficient β is varying; and for sub-model 2, the time-delay T is varying. The varying strategy is as follows.
As explained ealier, the transfer frequency of output signal z(t) and the parameter varying signal β(t), T(t) are all determined by time interval u. The larger the time u is, the slower transfer frequency is take u = 5 ns as an example, set r = 3.9999, y 0 = 0.1, U = 0.5, β = 4.5, T = 30 ns, the switching signal z(t) and parameter varying signal β(t), T(t) are shown in Fig. 1.
Evidently, the tunable parameters of this new model include β, T, r, U and three initial values. Compare to the original model (1), the parameter r and initial value y 0 of digital chaotic map are newly introduced. Assume the precision is 10 −16 , the key space will increase more than 0.4 × 10 32 ≈ 2 105 approximately, which greatly extent the key space of original model. Furthermore, by analyzing the sensitivity on small key differences, we focus on the newly introduced parameters r and y 0 , and change these parameters by 10 −16 . With different parameters, we can get different trajectories. Then we get the differences between these two trajectories, the results are shown in Fig. 2. Figure 2 indicates that the trajectories are completely different with a small difference of parameter after about 100 ns, which means that the newly brought in secret keys are rather sensitive.
Chaotic trajectories. Figure 3 shows the trajectories of model (2) for β = 4.5, indicating that our model is chaotic under these parameters and has good pseudorandom property. Furthermore, based on Fig. 3, the trajectory has a drift during the first 50 μs before z(t) enters the chaotic attractor. As time extends, the region of z(t) gradually approaches the chaotic attractor. This drift is not relevant to the delay time but induced by the model (2). After this drift, the trajectory will distribute in the whole chaotic attractor, showing that the model has a good ergodicity.
Scientific RepoRts | 6:22295 | DOI: 10.1038/srep22295 Bifurcation and amplitude. Figure 4 shows the bifurcation behavior of model (2) with bifurcation parameter β. When β = 0, the trajectory is a fixed point. As β increases, the trajectory will soon become chaotic. The simulation result indicates that our model can enter chaos region with a lower β than model (1), and other recently proposed systems in refs 14,21,23. This feature will enlarge the chaotic key space of parameter β. Figure 5 shows the amplitude in the asymptotic regime at each value of the bifurcation parameter. As β increases, the amplitude will gradually increase, resulting in a larger range of signal z(t).
Approximate entropy. ApEn was first proposed by Pincus 25 to measure the complexity of time-series, which is closely related to the system distribution and correlation characteristics. The ApEn of the signals of this new model is shown in Fig. 6. As the gain coefficient β increases, the dynamical complexity will improve. As a result, ApEn will increase accordingly. From Fig. 6, we know that both ApEn of models (1) and (2)       entropy to measure the uncertainty of these ordering. PE is easily implemented and is computationally much faster than other comparable methods, such as Lyapunov exponents, while also being robust to noise 27 . As suggest in ref. 26, we select the ordinal pattern length m = 6, the embedding delay D = 2, and the length N of time series be 1.2 × 10 5 . Figure 7 compares the PE of model (1) and (2). From Fig. 7 we find out that the PE of our model (2) is larger than 0.99 when β > 4, indicating that the model (2) is relatively complex. By comparing the PE of model (2) and (1), the PE of model (2) is larger than the PE of model (1) when β ≦ 4. For β > 4, and the PE curves are very close with each other, because as β increases, the PE of original model (1) has already approached the ideal value 1. Although the gap is quite small, the PE has still been increased. Take β = 4.5 as an example, the PE of model (1) and (2)  Time-delay concealment. A secure optoelectronic delayed feedback system should conceal time-delay signature against other existing methods. Currently, ACF, DMI and PIA are the three most effective methods for identifying the time-delay. In this section, we will prove that our model (2) can effectively conceal the time-delay against these three methods.
ACF method. Figure 8 show the results of both models (1) and (2) by using ACF method for β = 4.5. For model (1), there is an obvious extrema at t = 30 ns, suggesting that the time-delay has been identified. While for model (2), there is no obvious extrema, showing that the time-delay has been concealed effectively.
The effect of parameter β on the time-delay concealment is further analyzed. First, we define the background Q ACF as  where SD is the standard deviation. For each β, z(β) is the output signal of model (2). The peak size at the relevant time-delay is defined as ACF Figure 9 shows the size of the peak at the relevant time-delay T = 30 ns from the background (red lines) Q ACF in ACF(x 1 ). Figure 9 shows that the peaks are not apparent from the background Q ACF once β increases to the approximate critical value 4.5. For β ≥ 4.5, the time-delay can be concealed using ACF method.
DMI method. Figure 10 shows the results of model (1) and (2) by using DMI method for β = 4.5. Figure 10 shows that there is an obvious peak at t = 30 ns, the exact time delay of model (1). While for model (2), there is no obvious extrema along the evolution, indicating that the time-delay cannot be identified.
We further analyze the influence of parameter β on the time-delay concealment. The background Q DMI is defined as

DMI
The peak size at the relevant time-delay T = 30 ns and the background (red lines) Q DMI are shown in Fig. 11. From Fig. 11 we know that, when β ≥ 4.5, all the peak sizes are located in the background Q DMI , indicating that the time-delay will be concealed for β ≥ 4.5.  PIA method. Figure 12 shows the PE(z) of the embedding delay T is shown for β = 4.5 with ordinal pattern lengths L = 3, 4, 5, 6 and 7, respectively (The selection of L is suggest in ref. 26). Figure 12 suggests that there is no local extrema in PE(z), and that the time-delay can be concealed.
In order to analyze the influence of parameter β on the time-delay concealment, we define the background Q PIA and the peak size at the relevant time-delay as   Scientific RepoRts | 6:22295 | DOI: 10.1038/srep22295 Figure 13 shows the peak size at relevant time-delay of 30 ns from the background (red lines) Q PIA in PE(x 1 ). As shown in Fig. 13, when β ≥ 4.5, all peak sizes are in the Q PIA interval. This result shows that the time-delay is not available by PIA method.
In summary, the time-delay signal of model (2) can be effectively concealed using the ACF, DMI and PIA methods for β ≥ 4.5.

Conclusion
In this paper, we propose a new switching parameter varying optoelectronic delayed feedback model with computer simulation. This model is switching between two parameter varying optoelectronic delayed feedback models based on chaotic pseudorandom number sequences. We use the trajectory diagram, bifurcation diagram, approximate entropy and permutation entropy to illustrate its complexity. The results show that this model is more complex than the original model (1). We prove that our new model can conceal the time delay effectively against the ACF, DMI and PIA methods. Furthermore, some tunable parameters are brought in, which can extent the key space to resist brute-force attack.
As we know, chaotic systems have a widely use in many different kinds of fields, especially in cryptography. This paper discusses and proposes a new chaotic model, which has a high dynamical complexity, large key space, and proper conceal of time-delay compared to other common methods. All these characteristics show that our chaotic model is relatively secure to be applied in chaotic cryptography, including chaotic secure communication and chaotic pseudorandom sequence generator. For future work, we will study the applications based on this new chaotic model.