# Bias Current of Semiconductor Laser: An Unsafe Key for Secure Chaos Communication
## 3 Results  
### 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 \(I_{1}=1.6I_{\text{th}}\) and an amplitude feedback strength \(k_{1}=0.08\). 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 (\(\tau_{\text{RO}}\)) and the height indicated the visibility of the ROS. Figure 2: (**a**) Power spectrum and (**b**) the autocorrelation function (ACF) trace of the external-cavity semiconductor laser (ECL) output with a bias current \(I_{1}=1.6I_{\text{th}}\) and an amplitude feedback strength of 0.08. Arrows denote the \(f_{\text{RO}}\) and \(\tau_{\text{RO}}\), respectively. The inset plots the temporary waveform of the ECL output. Figure 3 plots the signature of relaxation oscillation, as a function of the bias current, which was separately obtained at different feedback strengths \(k_{1}=0.08\) (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 \(\tau_{\text{RO}}\). 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 \(\tau_{\text{RO}}\); (2) calculating the initial bias current \(I_{\text{E0}}\), and (3) decreasing \(I_{\text{E}}\) from \(I_{\text{E0}}\). First, \(\tau_{\text{RO}}\) was obtained from the power spectrum or the autocorrelation curve of the transmitter chaos carrier, by an eavesdropper; Figure 2. With this \(\tau_{\text{RO}}\), the initial bias current of eavesdropper (\(I_{\text{E0}}\)) 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 \(I_{\text{E0}}\), until the chaos was synchronized and then the hidden message was deciphered. The advantages of this method were as follows: \(I_{\text{E0}}\) 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.