Bias Current of Semiconductor Laser: An Unsafe Key for Secure Chaos Communication

Abstract

In this study, we have proposed and numerically demonstrated that the bias current of a semiconductor laser cannot be used as a key for optical chaos communication, using external-cavity lasers. This is because the chaotic carrier has a signature of relaxation oscillation, whose period can be extracted by the first side peak of the carrier’s autocorrelation function. Then, the bias current can be approximately cracked, according to the well-known relationship between the bias current and relaxation period of a solitary laser. Our simulated results have shown that the cracked current eavesdropper could successfully crack an encrypted message, by means of a unidirectional locking injection or a bidirectional coupling. In addition, the cracked bias current was closer to the real value as the bias current increased, meaning that a large bias current brought a big risk to the security.

1. Introduction

The secure optical chaos communication process has received considerable attention due to its excellent features, such as hardware encryption, high transmission rate, long transmission distance, and compatibility with the existing fiber networks. The first field experiment of optical chaos communication was demonstrated in the commercial optical networks of Athens, which achieved a rate of 1 Gb/s with a transmission distance over 120 km [1]. Considering the robustness and cost, external-cavity semiconductor laser (ECL) is a promising chaotic transceiver, due to its simple structure, which is capable of integration. Photonics integration of chaotic ECL has become a research focus and some integrated chaotic semiconductor lasers have recently been reported [2,3,4,5].

Chaos-based communication can be realized only when the parameters of chaotic transceivers are matched. A parameter match means that the parameter values of a chaotic transmitter and receiver can ensure synchronization, and realize message encoding and decoding [6,7,8]. Thus, the parameters of chaotic lasers are generally considered to be key in optical chaos communication [9]. Multi-user communication is the trend of secure chaos communication. Current semiconductor integration technology can manufacture massive lasers with matched internal parameters, which means that the laser internal parameters are public. Therefore, for ECLs like chaotic transceivers, the controllable external parameters, including bias current, external-cavity length, and feedback strength should be selected as the keys, to ensure security. For example, Paul et al. proposed the external-cavity length as a key [10]. However, this is unsafe because the laser chaotic oscillation contains external-cavity resonances, leading to signature of feedback time delay, which exposes the external cavity length [11,12]. Many efforts have been made to suppress or eliminate the time delay signature, to enhance security by increasing the complexity of feedback cavity, such as double-mirror feedback [13], polarization-rotated feedback [14], fiber Bragg grating (FBG) feedback [15], chirped fiber Bragg grating (CFBG) feedback [16], random grating feedback [17], and feedback phase modulation [18]. Nevertheless, from the viewpoint of integration, the external cavity length is fixed, which is then also unsuitable for acting as a key, once the ECL is integrated. By comparison, the laser bias current is easy to adjust. However, for a solitary laser, the bias current is related to the relaxation oscillation frequency (f_RO). Therefore, the safety of using a bias current as a key, is worthy of a detailed investigations.

In this study, we numerically analyzed the relaxation oscillation signature (ROS) in a chaotic laser, as a function of the bias current, and then used it to crack the optical chaos communication, based on external-cavity lasers. The risk of a bias current in chaos communication was also analyzed.

2. Theoretical Model

Figure 1 shows the schematic diagram of the optical chaos communication system, with a pair of mutually-coupled, authorized external-cavity lasers (SL₁ and SL₂). Two kinds of eavesdroppers were considered. Eavesdropper Eve_A was disguised as an authorized transceiver which was bidirectionally coupled with the transmitter SL₁ (in this way, Eve_A could not only eavesdrop the message but could also send false information to SL₁). Eavesdropper Eve_B simply tapped the transmitted signal from SL₁ and unidirectionally injected into the eavesdropping laser SL_EB. Note that the ECLs of the communication users and eavesdroppers had the same structure and the same semiconductor lasers. In addition, we simulated the spectra of SL₁ with and without considering SL₂. It was found that the relaxation oscillation frequency did not show any obvious change. For brevity, we omitted the equations of SL₂ in this manuscript.

The ECLs were modeled by the following Lang–Kobayashi equations [19].

$$\frac{d{E}_1}{dt}=\frac{1+i\alpha }{2}\left[\frac{g(N_1-N_0)}{1+\epsilon {\left|E_1\right|}^2}-\frac{1}{{\tau }_p}\right]E_1+\frac{k_1}{{\tau }_{in}}{E}_1\left(t-\tau \right)e^{-i{\omega }_1\tau }+\frac{k_{\mathrm{A}}}{{\tau }_{in}}{E}_{\mathrm{A}}\left(t-{\tau }_c\right)e^{-i\left({\omega }_{\mathrm{A}}{\tau }_{\mathrm{c}}-\Delta \omega t\right)},$$

(1)

$$\frac{d{E}_{\mathrm{A},\mathrm{B}}}{dt}=\frac{1+i\alpha }{2}\left[\frac{g(N_{\mathrm{A},\mathrm{B}}-N_0)}{1+\epsilon {\left|E_{\mathrm{A},\mathrm{B}}\right|}^2}-\frac{1}{{\tau }_p}\right]E_{\mathrm{A},\mathrm{B}}+\frac{k_1}{{\tau }_{in}}{E}_{\mathrm{A},\mathrm{B}}\left(t-\tau \right)e^{-i{\omega }_{\mathrm{A},\mathrm{B}}\tau }+\frac{k_{\mathrm{A},\mathrm{B}}}{{\tau }_{in}}{E}_1\left(t-{\tau }_c\right)e^{-i\left({\omega }_{\mathrm{A},\mathrm{B}}\tau -\Delta \omega t\right)},$$

(2)

$$\frac{d{N}_{1,\mathrm{A},\mathrm{B}}}{dt}=\frac{I_{1,\mathrm{A},\mathrm{B}}}{qV}-\frac{N_{1,\mathrm{A},\mathrm{B}}}{{\tau }_N}-\frac{g(N_{1,\mathrm{A},\mathrm{B}}-N_0)}{1+\epsilon {\left|E_{1,\mathrm{A},\mathrm{B}}\right|}^2}{\left|E_{1,\mathrm{A},\mathrm{B}}\right|}^2,$$

(3)

where E(t) is the complex amplitude of optical field and N(t) represents the corresponding carrier density. The subscripts ‘1′, ‘A’, and ‘B’, represent the legal user, Eve_A, and Eve_B, respectively. I is the bias current. k is the amplitude feedback strength. τ = 5 ns is the feedback delay time. I_th = 12 mA is the laser threshold current. k_A,B = 0.447 is the amplitude coupling strength. Note that, we set k_A = 0 in the Eve_B simulation. τ_c = 19 ns represents the coupling delay time. Δω = ω₁ − ω_A,B = 0 denote the detuning angular frequency of the legal user’s laser and the eavesdropper’s laser. The other intrinsic parameters are listed as follows—transparency carrier density N₀ = 0.5 × 10⁵ μm⁻³, differential gain g = 2.125 × 10⁻³ μm³ns⁻¹, gain saturation parameter ε = 1 × 10⁻⁵ μm³, carrier lifetime τ_N = 2.2 ns, photon lifetime τ_p = 1.6 ps, linewidth enhancement factor α = 6.0, round-trip time in laser cavity τ_in = 7.3 ps, active layer volume V = 100 μm³, and the elementary charge q = 1.602 × 10⁻¹⁹ C. The fourth-order Runge–Kutta method, with a step of 2.5 ps was used to solve these equations in the simulation.

The relaxation frequency of the solitary laser without external feedback could be calculated according to the following formula [20]

$$f_{\mathrm{RO}}=\frac{1}{2\mathsf{\pi }}{\left(\frac{\left(I/I_{\mathrm{th}}-1\right)}{{\tau }_{\mathrm{N}}{\tau }_{\mathrm{p}}}(1+g{N}_0{\tau }_{\mathrm{p}})\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}.$$

(4)

For a bias current I = 1.6I_th, the used laser had a relaxation frequency of 2.35 GHz.

3. Results

3.1. Principles of the Cracking Process

When moderate optical feedback was applied, the laser generated chaotic oscillation. Figure 2a plots the RF spectrum of laser intensity chaos, which was obtained with a fixed bias current I₁ = 1.6I_th and an amplitude feedback strength k₁ = 0.08. The spectrum obviously had a dominant peak around the relaxation frequency. This meant that the chaotic carrier had a signature of laser relaxation oscillation. More interestingly, the relaxation oscillation frequency or period could be clearly extracted from the autocorrelation function (ACF) of the temporal waveform which was the inverse Fourier transform of the power spectrum. As shown in Figure 2b, the ACF trace had a side peak closest to the main peak. The location of this side peak was 0.367 ns, corresponding to a frequency of 2.72 GHz, which was the relaxation frequency of the laser with feedback. Note that the slight increase of the relaxation frequency was caused by the optical feedback [20]. Therefore, the signature of relaxation oscillation was quantitatively characterized by the side peak of ACF—the location read the relaxation oscillation period (τ_RO) and the height indicated the visibility of the ROS.

Figure 3 plots the signature of relaxation oscillation, as a function of the bias current, which was separately obtained at different feedback strengths k₁ = 0.08 (circles), 0.1 (triangles), and 0.12 (squares). Figure 3a plots the location of the ACF side peak and also plots the solitary laser’s relaxation period, in black line, which was calculated from Equation (4). Compared to the solitary laser, the external feedback light reduced the relaxation period. The stronger was the amplitude feedback strength, the greater was the decrease of τ_RO. However, the reduction was quite small. Figure 3b depicts the height of relaxation oscillation as a function of the bias current. The greater the bias current or lower the amplitude feedback strength, the more pronounced were the observed relaxation oscillation characteristics. This indicated that one could easily identify the ROS from the ACF of laser intensity chaos.

According to the rule of the aforementioned relaxation oscillation characteristics, the cracking process was implemented with the following formulas (Equations (4)–(6)). It consisted of three main stages: (1) extracting the relaxation oscillation period τ_RO; (2) calculating the initial bias current I_E0, and (3) decreasing I_E from I_E0. First, τ_RO was obtained from the power spectrum or the autocorrelation curve of the transmitter chaos carrier, by an eavesdropper; Figure 2. With this τ_RO, the initial bias current of eavesdropper (I_E0) could be calculated using Equation (4)—the formula for relaxation oscillation in the solitary laser without external feedback. Based on the principle of relaxation oscillation in Figure 3a, the bias current of the eavesdroppers was gradually reduced from I_E0, until the chaos was synchronized and then the hidden message was deciphered. The advantages of this method were as follows: I_E0 could be obtained immediately from the relaxation oscillation period, which narrowed the range of the crack space. On the other hand, the optical feedback light reduced the relaxation oscillation period in the chaotic laser, which indicated the crack direction. As a result, the eavesdropper could crack the secret keys faster than the brute-force attack, using our proposed method.

$$s(f)={\left|\mathrm{FT}\left\{P(t)\right\}\right|}^2,$$

(5)

$$f_{\mathrm{RO}}=\mathrm{find}(s(f)=\mathrm{maximum}),$$

(6)

where P(t) is the intensity time-series of chaotic laser and FT{} denotes Fourier transform.

3.2. Cracking Results

In the simulation, chaos masking was adopted to encrypt the message (binary pseudorandom sequences), for its simple structure. The electrical message was applied on an electro-optical modulator, to modulate a continuous-wave semiconductor laser, with a data rate of 2.5 Gb/s, of which the wavelength and polarization was identical to the transmitter laser, and then the generated optical message was masked into the optical chaos carrier, through an optical coupler. The external modulation index was 0.05. Furthermore, the decoded messages were obtained by a fourth-order low-pass Butterworth filter. We estimated the bit error rate (BER) of the deciphered data, by calculating the Q factor of the eye diagram. The BER threshold of 1.8 × 10⁻³ was used to evaluate the quality of the chaotic communication [21]. That is, the message could be decoded when BER was lower than the BER threshold. Here, we set the bias current of the transmitter as I₁ = 1.6I_th. The eavesdropper extracted the τ_RO of 0.367 ns from the chaos carrier, and the I_E0 was considered to be 1.8I_th, according to Equation (4).

Figure 4 gives the eavesdropping results, including the chaotic temporal waveforms and the corresponding eye diagrams of the outputs of SL₁ and Eve, with different bias current I_E = 1.8I_th and 1.616I_th. For the eavesdropper Eve_A, when the I_EA declined to 1.616I_th, the chaos synchronization was established because of the matched bias current between the SL₁ and SL_EA. As a result, the opened eye diagram and the BER of 3.12 × 10⁻⁵ meant that Eve_A had already decoded the message under this scenario, shown in Figure 4(a1,a2). It is worth nothing that the cracking could be achieved by only reducing the bias current of 2.2 mA, with several attempts by Eve_A.

For the eavesdropper Eve_B, as shown in Figure 4b, the message was decoded with a BER of 1.875 × 10⁻⁵ and the system was cracked by utilizing the bias current of I_EB = 1.8I_th. The reason was that Eve_B achieved a high-quality chaos synchronization with SL₁, through a unidirectional injection. Compared with Eve_A, Eve_B directly cracked the system, with a bias current of 1.8I_th. The results also proved that the security of bidirectionally-coupled synchronization was higher than the unidirectionally-coupled synchronization, in the optical chaos communication [22].

To better qualify the bias current crack range of this communication system, a more careful analysis has been carried out in Figure 5. Figure 5a shows the BER as a function of the bias current mismatches (ΔI = I_E − I₁). BER threshold is marked with red dash line. It is obvious that cracked ΔI values ranged from −0.25 to 0.25. Additionally, as the I_E decreased, the BER gradually decreased to a minimum, and then rose to an unchanged value. The point where BER reached a minimum meant that the I_EA was the closest value to I₁. Thus, Eve_A broke this communication system without knowing the bias current and the cracked area resembled the shape of the letter ‘V’, with the bias current mismatches.

As shown in Figure 5b, the BER of SL₁ and Eve_B was always over the BER threshold, which indicated a better eavesdropping, compared with Eve_A. Unlike the bidirectionally-coupled synchronization in the Eve_A, a small mismatch induced a dramatic loss, and the unidirectional injection synchronization in Eve_B scheme showed a better robustness. Unfortunately, this robustness increased the possibility of the optical chaos communication system being cracked [22].

4. Discussion

In our system, the bias current of eavesdropper was determined by the τ_RO of temporal waveform. However, as can be seen from Figure 3a, with an increasing bias current, the cracked bias current was closer to the real value, which meant that the larger the bias current, the more dangerous the secure optical chaos communication becomes. In addition, a transmitter using the bias current of the ECL as a key, was proposed in the mutually-coupled laser system in our scheme, and the τ_RO was extracted from the chaos carrier. Thereafter, the bias current was an unsafe key in the optical chaos communication. However, τ_RO could be eliminated in some chaos generation methods, such as delayed self-interference [23], optical heterodyning of two ECLs [24], and short-cavity VCSEL [25]. In these methods, it was suitable to use the bias current as a key, because an eavesdropper could not achieve the τ_RO from the chaotic waveform. Hence, eliminating the ROS of ECL could be the direction for future development.

5. Conclusions

In summary, we have analyzed the security of the bias current used as a key in secure optical chaos communication. The τ_RO and bias current of ECL have been studied in detail. With an increase in bias current, the τ_RO of ECL always approaches that of a solitary laser. Due to this relationship, two eavesdropping scenarios have been proposed and the results have demonstrated that the bias current used as a key was unsafe in the chaos secure communication, based on the synchronization with the mutually coupled chaotic laser. Results showed that, without the knowledge of the bias current, the eavesdropper could intercept the data from the legal user.