## Abstract

To solve the security problem of information transmission, we add a more complex key of variable RF amplifier gain to enhance the confidentiality of the chaotic optical communication system. In the system, the RF amplifier gain is variable. The numerical results indicate that the bit error rate of the eavesdropper is much higher than that of the authorized receiver. And the eavesdropper cannot decrease the BER by decreasing the mismatch of other parameters in the electro-optic oscillator gain. Such system can be used to realize communication with high level of privacy in the future.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

With the development of decryption technology, the security of optical fiber communication has been challenged. Because chaotic signals are pseudo-random and unpredictable, the application of chaos in optical communication has attracted the attention of scholars [1–4]. By using synchronous transceiver, hardware encryption is realized, which is of great significance in the field of secure communication. In 1990, chaos synchronization was realized in the circuit [5]. Since then, many researches have been conducted on chaotic communication system structure and chaotic synchronization [6–11]. The first field experiment of chaotic optical communication using commercial optical networks was carried out successfully in 2005 [12]. In recent years, chaotic communication systems to WDM (wavelength division multiplexing) optical networks has also been extensively studied for actual application [13–15], and chaotic communication in practice may be realized in the future.

At the same time, it has been proved that chaotic optical communication system is not absolutely safe [16]. Therefore, researchers began to search for a higher dimensional chaotic laser secure communication system, and several achievements have been obtained. The application of subcarrier modulation technology in chaotic optical communication system is proposed [17]. The feedback phase of short-cavity laser [18] and the feedback length of long-cavity laser [19] are used as keys to enhance the security of chaotic communication system. In addition, the chaotic communication scheme with time delay signature suppression is proposed recently [20,21]. However, there exists some defects in the above systems such as the complex system or lack of practicality. Therefore, it is of great significance to propose a new high-dimensional chaotic laser secure communication system.

In this paper, a more complex key of RF amplifier gain is added to the chaotic optical communication system to enhance the confidentiality based on the effect of the mismatch noise on chaotic communication system. In the system, a control circuit is introduced to adjust the gain of RF (radio frequency) amplifier. Based on the simulation results, we study the BER (bit error rate) of the authorized receiver and the eavesdropper, and then discuss the effect of other parameters on BER. The results show that the BER of Eve is about 6 orders of magnitude higher than that of Bob. And it is useless for Eve to decrease the BER by decreasing the mismatch of other parameters in the electro-optic oscillator gain. Such system is of great significance for improving the confidentiality in information transmission.

## 2. Principle

#### 2.1 System configuration

Fig. 1 shows the system configuration. Based on the optoelectronic feedback structure, the chaotic laser signal is generated by the nonlinear characteristic of Mach-Zehnder modulator. The message is loaded into the system through a 2 × 2 coupler at the emitter, and demodulated by the receiver which is synchronized with the emitter. At the emitter, the optoelectronic feedback loop consists of Mach-Zehnder modulator, photodetector and radio frequency amplifier. The receiver has similar structure except the open-loop. The RF amplifier gain controller controls the gain coefficients of RF amplifiers at the emitter and receiver synchronously. In this way, the RF amplifier gain changes linearly and periodically in a certain range.

When the system operates, the RF amplifier gain of the authorized receiver (Bob) changes synchronously with that of the emitter (Alice). Through the modulation of RF gain controller at transmitter and receiver, the RF amplifier gain of both is guaranteed to be the same. In this way, the introduction of RF amplifier gain variation will not affect the synchronization quality of Bob. For the eavesdropper (Eve), the RF amplifier gain of Alice is unknown, which will lead to the mismatch of the RF amplifier gain between the Eve and Alice. It makes the BER of Eve increase and thus affects the deciphering ability of Eve eventually. It should be note that although the key is more complex in this new chaotic communication system, in fact, the key data dose not increase much. In this way, there are many good mature transmission techniques to share key data between Alice and Bob securely. Therefore, the security of the chaotic optical communication system will be enhanced.

#### 2.2 BER model analysis

The confidentiality of communication systems is usually assessed by BER. Therefore, the BER model of this system will be analyzed in detail below. In this system, the electro-optic oscillator gain of the emitter *β* and the receiver *β*′ are related to the RF amplifier gain of the emitter *G* and the receiver *G*′ respectively, which can be expressed as [22]

*A*is the photoelectric conversion efficiency of PD in the optoelectronic feedback loop of the emitter,

*γ*is the attenuation coefficient in the optoelectronic feedback loop of the emitter,

*P*is the laser power at the emitter, and

*V*is the RF half-wave voltage at the emitter.

_{π}*A*′,

*γ*′,

*P*′ and

*V*′ of the receiver are corresponding to those of the emitter.

_{π}Based on the calculation of mismatch noise in [22], the synchronization error <*ε*^{2}(*G*, *G′*)>, which is related to the RF amplifier gain of the emitter and the receiver *G* and *G*′, can be expressed as

*β*(

*G*,

*G′*) =

*β′*(

*G′*) −

*β*(

*G*), Δ

*T*is the mismatch of the time delay, Δ

*τ*is the mismatch of the high cutoff response time, and

*τ*is the high cutoff response time.

Further, based on [22], the root-mean-square amplitude of mismatched noise affected by *G* and *G*′ can be expressed as

*ϕ*is the mismatch of the off-set phase, and Δ

*ϕ*=

*ϕ′*−

*ϕ*± π/2,

*ϕ′*and

*ϕ*are off-set phase of the modulator at the receiver and the emitter relatively. Then, the replicated chaos becomes the negative image of the emitter chaos. In this way, we can use combiner to cancel the chaotic component of the transmitted signal in receiver.

*K*and

*K'*are the current amplitudes of PD3 and PD4 respectively. Δ

*K*is their mismatch.

*K*and

*K'*can be expressed as

*P*is the laser power of the emitter,

*g*and

_{EDFA}*g*is the gain of EDFA and PD3,

_{APD}*α*is the loss in the transmission line,

_{T}*e*is the elementary charge,

*η*is the quantum efficiency,

*h*is the Planck constant,

*ν*is the laser frequency,

*P'*is the laser power of the receiver, and

*g'*is the gain of PD4. To realize the parameter match, the gain of EDFA g

_{APD}_{EDFA}is set as

*g*4

_{EDFA = }*/α*.

_{T}Under certain conditions, the probability density function (pdf) of the mismatch noise can be expressed as a function of Gaussian [22]. Therefore, the BER of the system, which is related to signal-to-noise ratio *u*(*G*, *G'*), can be expressed as

*α*represents the masking efficiency. When the gain of the RF amplifier varies continuously, the BER calculated from the Eqs. (1)-(5) also varies continuously. For a certain moment, the BER(

*G, G’*) is an instantaneous value, which can be expressed as BER

_{instantaneous}(

*G, G’, t*)

_{.}The actual BER of the system is defined as the average value of instantaneous BER over a period of time [23], which can be expressed as

From the above analysis, when the RF amplifier gain of Eve *G'* cannot keep the same value with the RF amplifier gain of the emitter *G*, Δ*β*(*G*, *G'*) will increase, which will increase the mismatch noise, therefore, the instantaneous BER and the BER will increase ultimately.

To analyze the instantaneous BER and the BER more clearly, simulation will be presented based on the model above in the next section. The instantaneous BER model with Eqs. (1)-(5) can be used to take numerical analysis on the instantaneous BER and the BER model with Eqs. (1)-(6) can be used to take numerical analysis on the BER. The analysis on the two types of BER can show the better confidentiality performance of such scheme we proposed more clearly and believably.

## 3. Simulation results and discussion

#### 3.1 Comparison of BER between Bob and Eve

Based on the BER model in the second chapter, we calculate the BER of Bob and Eve respectively. The simulation parameters of the system are set as: emitter power *P* = 5mW, mask efficiency *α* = 0.2, time delay mismatch Δ*T* = 1ps, APD current mismatch Δ*K* = 10^{−6}A, the quantum efficiency *η* = 0.75, the gain of APD *g _{APD}* = 100, receiver off-set phase

*ϕ′*=

*ϕ*+ Δ

*ϕ*± π/2 (

*ϕ*is emitter offset phase), offset phase mismatch Δ

*ϕ*= 0.02rad, high cutoff response time high

*τ*= 25ps, cutoff response time mismatch rate Δ

*τ*/

*τ*= 0.02.

In this section, we assume that the other parameters in the electro-optic oscillator gain match perfectly, thus the mismatch of the electro-optic oscillator gain Δ*β* is only determined by the mismatch of the RF amplifier gain. The RF amplifier gain of the emitter *G* is set in the range of 17dB to 19dB. The RF amplifier gain of Bob keeps the same with the emitter while the RF amplifier gain of Eve keeps unchanged at 18dB.

Fig. 2 shows the BER of both Bob and Eve. The BER of Bob is always below 10^{−7} and remains unchanged, because the RF amplifier gain of Bob is the same with Alice, which makes Δ*β* = 0 in different RF amplifier gain of the emitter. For Eve, the BER decreases first and then increases with the increase of RF amplifier gain of the emitter. That is due to the change of Δ*β* between Eve and the emitter with the increase of *G*. When G = 17dB or G = 19dB, the BER of Eve is higher than 10^{−1}. When G < 17.75dB or G > 18.23dB, the BER of Eve is higher than 10^{−2}. The results indicate that the BER of Eve is sensitive enough to the mismatch of RF gain, and that also means it is much more difficult for Eve to intercept the message. When *G* = 18dB, the mismatch of the electro-optic oscillator gain Δ*β* between Alice and Eve is zero, because the RF amplifier gain of Eve matches with that of the emitter perfectly in this case. However, as shown in Fig. 2(b), it is worth noting that the BER of Eve does not reach the valley value when *G* = 18dB. That is because that the minimum synchronization error is affected by Δ*β*/*β* and Δ*τ*/*τ* commonly [24]. Since Δ*τ*/*τ* in the simulation is not zero, an amount in good concordance of Δ*β* can induce a minimum BER of Eve. From Fig. 2(a), by comparing the two curves, the BER of Eve is much higher than that of Bob over most of the RF amplifier gain, and the maximum difference is about 7 orders of magnitude. It shows that the change of the RF amplifier gain indeed improves the security of the system significantly.

When the system operates, we set the RF amplifier gain of the emitter changing linearly from 17dB to 19dB, and then back to 17dB, whose cycle is 40s as shown in Fig. 3. Fig. 3 shows the instantaneous BER of Bob and Eve and the BER of Eve in two cycles. According to the figure, with the increase of time, the instantaneous BER of Bob remains unchanged, while the instantaneous BER of Eve changes periodically with the periodic change of RF amplifier gain of the emitter. In each cycle, firstly, with *G* increasing from 17dB to 19dB, the instantaneous BER of Eve decreases first and then increases. Then, the instantaneous BER of Eve decreases again and then increases with G decreasing from 19dB to 17dB. The above analysis is based on the instantaneous BER at each moment. To better understand the deciphering ability of Eve in continues information transmission, we further give the variation of BER of Eve, as shown in Fig. 3. It is obvious that the BER of Eve remains above 10^{−2} and fluctuates slightly.

To further study the BER of Bob and Eve, we compare their BER in 120s.And the results are showed in Fig. 4. Fig. 4(a) shows that the BER of Bob is below 10^{−7}, while the BER of Eve fluctuates slightly at first, then stabilizes gradually after 40s and maintains about 10^{−1}, which is about 6 orders of magnitude higher than the BER of Bob.

Fig. 4(b) shows the BER and the instantaneous BER of Eve in 40s. From the figure, the BER of Eve decreases first, then increases, and then decreases and increases finally in the first 40 seconds. That is affected by the variation trend of the instantaneous BER of Eve. As shown in Eq. (6), the BER is the integration of the instantaneous BER in *T* divided by *T*. The instantaneous BER of Eve reaches the valley value in the first period when the time is about 10s. Before that, BER of Eve decreases with the instantaneous BER of Eve. After that, the instantaneous BER of Eve increases, but it is still below BER of Eve. Thus, BER of Eve still decreases. Until 14.55s, the instantaneous BER of Eve and BER of Eve become the same. And after 14.55s, BER of Eve begins to increase with the instantaneous BER of Eve. So, due to the accumulated effect of instantaneous BER in the previous time, the extreme point of BER of Eve shifts to the right compared to the instantaneous BER of Eve in a period, as shown in Fig. 4(b). With the increase of time, the influence of the instantaneous BER on BER is weakening. Thus, the jitter of BER of Eve becomes smaller and the curve finally tends to be stable.

#### 3.2 The Effect of other parameters on BER

In practical chaotic communication systems, as shown in Eq. (2) and (3), chaotic encryption system is chiefly effect by the mismatch of electro-optic oscillator gain Δ*β*, the mismatch of the time delay Δ*T*, the mismatch of the high cutoff response time Δ*τ*, and the mismatch of the off-set phase Δ*β*. However, RF amplifier gain *G* will only affect the mismatch of electro-optic oscillator gain Δ*β*. So, the mismatch of electro-optic oscillator gain Δ*β* will be analyzed further. As shown in Eq. (1), besides the RF amplifier gain, the electro-optic oscillator gain is also determined by several parameters: the photoelectric conversion efficiency of PD in the optoelectronic feedback loop, the laser power and the RF half-wave voltage. In many cases, these parameters of emitter and receiver may not match perfectly. The combined effect of such mismatch and the variation of the RF amplifier gain will affect the BER of the system. Therefore, we will analyze the three parameters respectively.

Firstly, we study the effect of photoelectric conversion efficiency of PD in the optoelectronic feedback loop. Fig. 5 shows BER of Bob and Eve in 120s under different photoelectric conversion efficiency of the receiver *A′*. Assuming that the photoelectric conversion efficiency of the emitter *A* is 0.65 and change tendency of the RF amplifier gain is the same as that in section3.1, which changes periodically between 17dB and 19dB. As shown in Fig. 5(a), BER of Bob increases with the mismatch between *A* and *A′*. That is because the mismatch of photoelectric conversion efficiency will increase the mismatch of electro-optic oscillator gain. For each photoelectric conversion efficiency of the receiver *A′*, BER of Bob does not change with the time. It indicates that BER of Bob is not affected by the change of the RF amplifier gain of the emitter. From Fig. 5(b), for different photoelectric conversion efficiency of Eve, BERs have intensive oscillation in the first, and then becomes stable gradually. It can be noted that with the increase of *A′,* the valley value of BER of Eve will shift to the right, and the oscillation extent of BER of Eve will also decrease.

Fig. 5(c) shows the BER of Eve versus the RF amplifier gain of the emitter *G* under different *A′*. With the increase of *A′*, the curve of the BER of Eve shifts to the right. When *A′* are 0.62,0.64,0.66 and 0.68, the minimum BERs of Eve appear at 17.78dB, 17.91dB, 18.05dB and 18.18dB respectively. And the valley BERs of Eve have close values under different *A′*. That is because the RF amplifier gain and the photoelectric conversion efficiency in the feedback loop are two factors of the electro-optic oscillator gain, as shown in Eq. (1). In terms of the mismatch of the electro-optic oscillator gain Δ*β*, the mismatch of the RF amplifier gain Δ*G* and the mismatch of the photoelectric conversion efficiency in the feedback loop Δ*A* can compensate each other under certain conditions. According to Eq. (1), Δ*β* can be expressed as

When the other parameters match perfectly, which means *γ* = *γ*′, *P = P*′ and *V _{π}* =

*V*′. Then Δ

_{π}*β*can be expressed as

In the simulation, RF amplifier gain of Eve *G′* is 18dB and the photoelectric conversion efficiency in the feedback loop of the emitter *A* is 0.65. *G′* and *A* are constant, therefore, according to Eq. (8), when the photoelectric conversion efficiency in the feedback loop of Eve *A′* increases, RF amplifier gain of the emitter *G* should increase to make Δ*β* close to zero. That explains why the curve of the BER of Eve shifts to the right with the increase of *A′* in Fig. 5(c).

In terms of BER of Eve, it is affected by the BER at each moment in the previous time. As shown in Fig. 5(b), with the increase of *A′*, due to the right shift of the BER of Eve at each moment, the valley value of BER shifts to the right. And since it takes longer time to reach the valley BER, the oscillation amplitude of BER of Eve decreases with the increase of *A′*. It can explain the change of BER of Eve in the first 20s under different *A′* as shown in Fig. 5(b).

In addition, Fig. 5(b) shows that BERs of Eve have close steady values under different *A′* over time, about 10^{−1}. When the time is long enough, different BER at each moment has little effect on BER so that BER of Eve tends to be stable gradually. On the other hand, the curves of BERs of Eve have close valley values and half-widths as shown in Fig. 5(c), therefore, the steady values of BERs for each *A′* are close.

In summary, the mismatch of the photoelectric conversion efficiency in the optoelectronic feedback loop will increase BER of Bob, while it has little effect on BER of Eve when the time is long enough. Thus, in practical application, the mismatch of photoelectric conversion efficiency in the optoelectronic feedback loop should be controlled small enough to obtain better communication efficiency. For the eavesdropper, the ability of eavesdropping cannot be improved through reducing the mismatch of the photoelectric conversion efficiency in the optoelectronic feedback loop.

As mentioned above, the electro-optic oscillator gain *β* has three influence factors: the photoelectric conversion efficiency of PD in the optoelectronic feedback loop, the laser power and the RF half-wave voltage. The effect of the photoelectric conversion efficiency of PD in the optoelectronic feedback loop has already been analyzed, and next we investigate the effect of two other parameters of *β* on BER.

Referring to the effect of photoelectric conversion efficiency mismatch on Δ*β,* which has been analyzed above, the effect of laser power mismatch on Δ*β* can be analyzed in a similar way. For Bob, the mismatch of the laser power will lead to the increase of Δ*β* and BER eventually. For Eve, Δ*β* is affected by the laser power mismatch and the RF amplifier gain mismatch comprehensively. In addition, the laser power will affect not only the electro-optic oscillator gain (Δ*β*), but also the current amplitude of PD3 and PD4, as showed in Eq. (4). The mismatch of the laser power will increase the mismatch between current amplitude of PD3 and PD4 (Δ*K*), and then increase the BER of the system. That effect will both happen in Bob and Eve.

Fig. 6 shows BER of Bob and Eve at different laser power of the receiver *P'*, while the laser power of the emitter *P* = 5 mw. As can be seen from Fig. 6(a), BER of Bob increases with the mismatch of the laser power. Fig. 6(b) shows that when the laser power of Eve increases, the oscillation amplitude of BER decreases obviously at first. For a certain *P'*, the oscillation extent of BER is large at first 20s, and then stabilizes gradually. And finally, BERs for different *P'* tend to be close.

Fig. 7 shows BERs of Bob and Eve under different RF half-wave voltage of the receiver. The RF half-wave voltage of the emitter is set as *V _{π}* = 4.2v. From Fig. 7(a), BER of Bob increases with the increase of the mismatch of RF half-wave voltage. And from Fig. 7(b), when

*V*increases, the oscillation extent of BER of Eve increases, and the valley value of BER shifts to the left. Furthermore, for a certain

_{π}'*V*, the oscillation extent of BER is large at first 20s, and then stabilizes gradually. And finally, BERs for different

_{π}'*V*tend to be the close, it shows that the eavesdropper also cannot decrease BER through decreasing the mismatch of the RF half-wave voltage.

_{π}'## 4. Conclusion

In summary, a high-dimensional chaotic optical communication system is designed to improve the confidentiality by using the change of the RF amplifier gain as a key. And the BER model considering the effect of the key is established. Based on the BER model, we simulate and compare the instantaneous BER and BER of Bob and Eve with the change of RF amplifier gain of the emitter. When the RF amplifier gain of the emitter changes from 17dB to 19dB, the BER of Bob is always below 10^{−7} while the BER of Eve can reach 10^{−1}. The results of BER show that when the oscillation is stabilized, BER of Eve is about 10^{−1}, which is 6 orders of magnitude higher than that of Bob. These results indicate that such system has a significant effect on increasing the BER of eavesdroppers. It should also be note that since the key of variable RF amplifier gain is dynamic, the changing speed of RF amplifier gain and the changing law of RF amplifier gain will also increase the dimension of the key further, making the key more complex and the key space larger. That means it has become more difficult for the eavesdropper Eve to extract message from the received data.

Furthermore, we also study other factors that may affect the confidentiality when the RF amplifier gain changes, including the photoelectric conversion efficiency of PD in the optoelectronic feedback loop, the laser power and RF half-wave voltage. The results show that when the mismatch of the three parameters increases, BER of Bob increases significantly, while BER of Eve has different oscillation extents and extreme points in the early few periods, and then stabilizes gradually at about 10^{−1}. It indicates that the mismatch of the three parameters should be controlled little enough to make the system operates in a good state. Moreover, the eavesdropper cannot decrease the BER by decreasing the mismatch of the three parameters. Thus, the system can enhance the confidentiality of chaotic communication effectively.

## Funding

Joint Funds of Space Science and Technology (6141B060307); Suzhou Technology Innovative for Key Industries Program of China (SYG201729); National Natural Science Foundation of China (61205045); Fundamental Research Funds for the Central Universities (021314380152).

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