# Wireless-Channel Key Distribution Based on Laser Synchronization
## 3 Experimental Results
### Secure Key Generation Rate  
In this section, we analyze the key generation rate of the scheme based on the previous assumptions. The secret bit rate, which is defined as the ratio of the number of secret bits to the raw samples, can be estimated using the following formula [20]:  
\[R_{b}=\frac{1}{M}\left[\left(1-\frac{M_{E}}{M}\right)(1-I_{E})-h\left(R_{fail} \right)\right], \tag{2}\]  
where \(M_{E}\) is the number of parameter-matched lasers that [PERSON] can manipulate, \(I_{E}\) is the information per bit that [PERSON] knows about [PERSON] or [PERSON]'s retained bits when [PERSON]'s random keying parameter \(V_{E}\) does not match that of [PERSON] and [PERSON], \(R_{fail}\) is the BER caused by imperfect synchronization between [PERSON] and [PERSON], and \(h(\cdot)\) is the binary entropy function defined by \(h(x)=-x\mathrm{log}(x)-(1-x)\mathrm{log}(1-x)\). \(1/M\) represents the probability of [PERSON] and [PERSON] matching the parameter \(V_{A}=V_{B}\), while \(1-M_{E}/M\) represents the probability for [PERSON] to match a different parameter from [PERSON] and [PERSON] when \(V_{A}=V_{B}\). \(h(R_{fail})\) estimates the rate loss caused by the error bits. When the conditions  
\[M_{E}<M,\ I_{E}<1,\ h\left(R_{fail}\right)<(1-M_{E}/M)(1-I_{E}),\]  
are satisfied, the value of \(R_{b}\) will be positive, indicating that secure key distribution is possible. Next, we will calculate \(R_{b}\) and the key rate based on experimental results and some reasonable assumptions. If [PERSON] and [PERSON] introduce a binary random keying parameter, the number of possible values of the parameter set is \(M=2\). Considering the sensitivity of the laser's inner parameters to synchronization, we assume that [PERSON] is unable to obtain a third laser with matching parameters [26], resulting in \(M_{E}=0\). Therefore, [PERSON] can only try to extract information from the drive signal. We further assume [PERSON] can obtain bits \(X_{E,i}\) by sampling and quantizing the drive signal using a single threshold [34], and retained those for \(i\) such that \(V_{A,i}=V_{B,i}\). In this case, \(I_{E}\) can be estimated by  
\[I_{E}=I(Y_{A,B};Y_{E}),\]  
Figure 4: Effects of external parameters mismatch on laser synchronization: (**a**) center wavelength mismatch, (**b**) bias current mismatch, and (**c**) injection power mismatch. Original settings: center wavelength \(\lambda_{A}=\lambda_{B}=1549.559\) nm, bias current \(I_{A}=24.20\) mA, \(I_{B}=23.50\) mA, injection power \(P_{|A}=P_{|B}=300\) μW. where \(Y_{E}\) is [PERSON]'s retained bits, \(I(X;Y)\) is the mutual information between random bits \(X\) and \(Y\), defined as  
\[I(X;Y)=\sum_{x=0}^{1}\sum_{y=0}^{1}P(x,y)\text{log}_{2}\,\frac{P(x,y)}{P(x)P(y)}, \tag{4}\]  
where \(P(x)\) is the probability of \(X=x\), \(P(y)\) is the probability of \(Y=y\), and \(P(x,y)\) is the joint probability of \(X=x\) and \(Y=y\). In practical operation, due to the limited number of available antennas, the drive signal used for estimating \(I_{E}\) is obtained by partitioning a portion of [PERSON]'s received signal. A robust dual-threshold quantization method [34] is used to reduce the BER of between [PERSON] and [PERSON]. We set the upper threshold \(V_{th,u}\) and the lower threshold \(V_{th,l}\) as  
\[ \begin{array}{l}V_{th,u}=m+\sigma\text{C}_{+},\\ V_{th,l}=m-\sigma\text{C}_{-},\end{array}  \tag{5}\]  
where \(m\) and \(\sigma\) are the mean and standard deviation of the temporal waveform, and \(\text{C}_{+}\) and \(\text{C}_{-}\) are the threshold coefficients. A sampling point will be quantized to bit \"1\" if its voltage is larger than \(V_{th,u}\), and quantized to bit \"0\" if its voltage is smaller than \(V_{th,l}\); otherwise, it will be discarded. To guarantee a nearly uniform distribution of the quantized bit stream, \(\text{C}_{+}\) and \(\text{C}_{-}\) are tuned to satisfy  
\[\int_{-\infty}^{V_{th,l}}p[x(t)]dx(t)=\int_{V_{th,u}}^{+\infty}p[x(t)]dx(t), \tag{6}\]  
where \(p[x(t)]\) is the probability density function of the temporal waveform \(x(t)\).