## 4 Security Analysis and Numerical Simulation
### Finite-Size Analysis
We can find a specific covariance matrix \(\gamma_{AB_{1}}^{\epsilon_{PE}}\) that can successfully minimize the secret key rate by selecting appropriate parameters. The matrix is compatible with the exchanged data except with failure probability of parameter estimation \(\epsilon_{PE}\) and has the following form  
\[\gamma_{AB_{1}}^{\epsilon_{PE}}= \begin{bmatrix}(V_{A}+1)I&t_{min}Z_{N}\sigma _{z}\\ t_{min}Z_{N}\sigma_{z}&(t_{min}V_{A}+\sigma_{max}^{2})I\end{bmatrix} , \tag{18}\]  
where \(t_{min}\) is the minimum value of \(t\) and \(\sigma_{max}^{2}\) is the maximum value of \(\sigma^{2}\). Maximum-likelihood estimators \(\hat{t}\) and \(\hat{\sigma}^{2}\) are known for the normal linear mode [39; 41]:  
\[\hat{t}=\frac{\sum_{i=1}^{m}x_{i}y_{i}}{\sum_{i=1}^{m}x_{i}^{2}},\hat{\sigma}^ {2}=\frac{1}{m}\sum_{i=1}^{m}(y_{i}-\hat{t}x_{i})^{2}, \tag{19}\]  
and they obey the following distribution:  
\[\hat{t}\sim N(t,\frac{\sigma^{2}}{\sum_{i=1}^{m}x_{i}^{2}}),\frac{m\hat{\sigma }^{2}}{\sigma^{2}}\sim\chi^{2}(m-1), \tag{20}\]  
where \(t\) and \(\sigma^{2}\) are the authentic values of the parameters. Therefore, we can obtain the worst-case value of the parameters \(t\) and \(\sigma^{2}\) to maximize the value of the Holevo information between [PERSON] and [PERSON] with the statistics except with probability \(\epsilon_{PE}\), namely  
\[t_{min}\approx\sqrt{T_{sea}}-z_{\epsilon_{PE}/2}\sqrt{\frac{T_{sea}\epsilon+1 }{mV_{A}}}, \tag{21}\] \[\sigma_{max}^{2}\approx T_{sea}\epsilon+1+z_{\epsilon_{PE}/2} \frac{(T_{sea}\epsilon+1)\sqrt{2}}{\sqrt{m}},\]where \(z_{\epsilon_{PE}/2}\) is the probability density value corresponding to the upper fractile of the normal distribution and it satisfies \(1-erf(z_{\epsilon_{PE}/2}/\sqrt{2})/2=\epsilon_{PE}/2\). \(erf(x)\) is the error function which satisfies \(erf(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt\). Similar to the asymptotic case, the mutual information \(I(A:B)\) and the Holevo information \(S_{N}^{\epsilon_{PE}}(E:B)\) can be calculated by replacing the parameters \(T\) and \(\epsilon\) with \(t_{min}\) and \(\sigma_{max}^{2}\), respectively. Other basic parameter settings are shown in Table 2, and all the variances and noises are in shot noise unit. According to the above asymptotic safety analysis, there is little difference between homodyne detection and heterodyne detection in the discrete-modulated underwater CV-QKD system, so here we only consider homodyne detection in the finite-size scenario. We plot the functional relationship for the four-state and eight-state protocols between the secret key rate and the transmission distance under the four types seawater in finite-size scenario, and compare them with that in the asymptotic case, which are shown in Figures 11 and 12, respectively. The orange solid line, green solid line, purple solid line, blue solid line, and the red dash line in the subgraphs correspond to the block lengths of \(10^{8}\), \(10^{10}\), \(10^{12}\), and \(10^{14}\), and the asymptotic curves, respectively. We can know from the figures that the performance of the discrete-modulated CV-QKD protocols under four types of seawater in the finite-size scenario have the same trend, and their performance is not as good as those obtained in the asymptotic case. Although the transmission distance and secret key rate gradually increase with the increase of \(N\), and the transmission distance is very close to the maximum transmission distance in the asymptotic case when \(N=10^{14}\), the secret key rate is always lower than the value in the asymptotic case, because a part of the exchanged signals between Alice and Bob are used for parameter estimation. When comparing the sub-figures in Figures 11 and 12, it can be found that, with the decrease of complexity of the seawater, the performance of the protocol is  
Figure 11: (Color online.) The finite-size and the asymptotic secret key rate of the four-state underwater CV-QKD protocol.