# Secure Physical Layer Network Coding versus Secure Network Coding +
## 4 Network with Three Source Nodes
### Comparison  
All the proposed protocols are summarized in Table 5. Since the security of our interest is (S4), we compare the protocols except for the protocol given in Section 4.1.1. Only the protocol given in Section 4.2.1 satisfies Security (S4). To implement these protocols as wireless communication network, we compare the transmission rates of the protocols given in Sections 4.1 and 4.2 when each edge is given as the BPSK scheme of a single-input Gaussian channel (16), a two-input Gaussian channel (2), or a three-input Gaussian channel (2):  
\[Y=hX_{1}+hX_{2}+hX_{3}+N, \tag{21}\]  
where \(h\in\mathbb{C}\) are the channel fading coefficients, \(N\) is a complex Gaussian random variable with zero mean and a variance of one, and \(X_{i}\) is coded as \((-1)^{A_{i}}\) with \(A_{i}\in\mathbb{F}_{2}\). In this comparison, we make the same assumptions for \(h_{1}\), \(h_{2}\), and \(GT\) as the previous section. Additionally, we assume that ideal codes given in Section 3.4 are available, and that the mutual information rate triple  
\[\Big{(}\frac{I(Y;A_{1}A_{2}A_{3})_{\text{Equation(21)}}}{3},\frac{I(Y;A_{1}A_{ 2}A_{3})_{\text{Equation(21)}}}{3},\frac{I(Y;A_{1}A_{2}A_{3})_{\text{Equation(21) }}}{3}\Big{)}\]  
is available in the MAC channel (21) when three transmitters intend to send their own message to the receiver, where the random variables \(A_{1}\), \(A_{2}\), and \(A_{3}\) are independently subject to the uniform distribution [37]. Using this rate, we compare the secure NC protocol given in Section 4.1.2 and the secure PLNC protocol given in Section 4.2 because both protocols realize the secrecy for intermediate (untrusted) nodes. When any Gaussian MAC is not used, the secure NC protocol given in Section 4.1.2 requires five time slots at least as shown in Table 6. In particular, the edges \(e_{7}\), \(e_{8}\), and \(e_{9}\) need to send the information symbols twice as the remaining edges. Therefore, when the length of the transmitted message is \(G\), the first and second time slots need transfer time \(\frac{GT}{I(Y;A)_{\text{Equation(16)}}}\), and the remaining time slots need transfer time \(\frac{2 GT}{I(Y;A)_{\text{Equation(16)}}}\). Hence, the total transfer time is calculated to be \(\frac{8 GT}{I(Y;A)_{\text{Equation(16)}}}\). When we use the Gaussian MAC, the secure NC protocol given in Section 4.1.2 can be implemented with two time slots as in Table 7. The first time slot needs transfer time \(\frac{2 GT}{I(Y;A_{1},A_{2})_{\text{Equation(2)}}}\), and the second time slot needs transfer time \(\frac{6 GT}{I(Y;A_{1}A_{2}A_{3})_{\text{Equation(21)}}}\). Hence, the total transfer time is calculated to be \(\frac{6 GT}{I(Y;A_{1}A_{2}A_{3})_{\text{Equation(21)}}}+\frac{2 GT}{I(Y;A_{1},A_ {2})_{\text{Equation(2)}}}\). The secure PLNC protocol given in Section 4.2.1 can be implemented only with two time slots as in Table 8, where the pairs \((e_{1},e_{2})\), \((e_{4},e_{5})\), and \((e_{6},e_{7})\) are realized by the secure CAF based on the Gaussian MAC channel (2). The first time slot needs transfer \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Time Span** & **Time i** & **Time ii** & **Time iii** & **Time iv** & **Time v** \\ \hline Channel & \(e_{1},e_{2},e_{3}\) & \(e_{4},e_{5},e_{6}\) & \(e_{7}\) & \(e_{8}\) & \(e_{9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Secure NC with Gaussian MAC. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Protocol** & **Assumption** & **Security** & **Type** \\ \hline Section 4.1.1 & – & (S3) & Secure NC \\ Section 4.1.2 & – & (S3) (S4) & Secure NC \\ Section 4.2.1 & (A3) & (S4) & Secure PLNC with secure CAF \\ Section 4.2.2 & (A3) & (S3) (S4) & Secure PLNC with CAF and secure NC \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison for protocols in network given in Sections 4.1 and 4.2.  
time \(\frac{GT}{2I(Y;A_{1}\oplus A_{2})_{\text{Equation(2)}}-I(Y;A_{1},A_{2})_{\text{ Equation(2)}}}\), and the second time slot needs transfer time \(\frac{3 GT}{I(Y;A_{1}A_{2}A_{3})_{\text{Equation(21)}}}\).