# Secure Physical Layer Network Coding versus Secure Network Coding +
## 3 Butterfly Network
### Comparison
When the length of the transmitted message is \(G\), the first time slot requires transfer time \(\frac{2 GT}{I(Y;A_{1},A_{2})_{\text{Equation}(2)}}\), and the second and third time slots require transfer time \(\frac{GT}{I(Y;A_{1})_{\text{Equation}(16)}}\). Hence, the total transfer time is calculated to be \(\frac{2 GT}{I(Y;A)_{\text{Equation}(16)}}+\frac{2 GT}{I(Y;A_{1},A_{2})_{\text{ Equation}(2)}}\). When we design the whole process as in Table 4, the first and third time slots require transfer time \(\frac{2 GT}{I(Y;A_{1},A_{2})_{\text{Equation}(2)}}\), and the second time slot requires transfer time \(\frac{GT}{I(Y;A)_{\text{Equation}(16)}}\). Hence, the total transfer time is \(\frac{GT}{I(Y;A)_{\text{Equation}(16)}}+\frac{4 GT}{I(Y;A_{1},A_{2})_{\text{ Equation}(2)}}\), which is larger than \(\frac{2 GT}{I(Y;A)_{\text{Equation}(16)}}+\frac{2 GT}{I(Y;A_{1},A_{2})_{\text{ Equation}(2)}}\) because \(\frac{I(Y;A_{1},A_{2})_{\text{Equation}(2)}}{2}\leq I(Y;A)_{\text{Equation}(16)}\). The secure PLNC protocol given in Section 3.3.1 can be performed only with three time slots as in Table 4, where the pairs \((e_{1},e_{2})\), \((e_{4},e_{5})\), and \((e_{6},e_{7})\) are realized by secure CAF based on the Gaussian MAC channel (2). The first and third time slots require transfer 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 requires transfer time \(\frac{GT}{I(Y;A)_{\text{Equation}(16)}}\). The total transfer time is calculated to be \(\frac{2 GT}{2I(Y;A_{1}\oplus A_{2})_{\text{Equation}(2)}-I(Y;A_{1},A_{2})_{ \text{Equation}(2)}}+\frac{GT}{I(Y;A)_{\text{Equation}(16)}}\). Secure PLNC protocol given in Section 3.3.2 can also be implemented only with three time slots as in Table 4. The first and third time slots require transfer time \(\frac{GT}{I(Y;A_{1}\oplus A_{2})_{\text{Equation}(2)}}\), and the second time slot requires transfer time \(\frac{GT}{I(Y;A)_{\text{Equation}(16)}}\). The total transfer time is calculated to be \(\frac{2 GT}{I(Y;A_{1}\oplus A_{2})_{\text{Equation}(2)}}+\frac{GT}{I(Y;A)_{ \text{Equation}(16)}}\). Figure 3 gives the numerical comparison among the following time periods:  
\[\frac{2 GT}{2I(Y;A_{1}\oplus A_{2})_{\text{Equation}(2)}-I(Y;A_{1},A_{2})_{\text{Equation}(2)}}+\frac{GT}{I(Y;A)_{\text{Equation}(16)}}, \tag{17}\] \[\frac{4 GT}{I(Y;A)_{\text{Equation}(16)}},\quad\frac{2 GT}{I(Y;A)_{ \text{Equation}(16)}}+\frac{2 GT}{I(Y;A_{1},A_{2})_{\text{Equation}(2)}},\] \[\frac{2 GT}{I(Y;A_{1}\oplus A_{2})_{\text{Equation}(2)}}+\frac{GT }{I(Y;A)_{\text{Equation}(16)}}.\]  
When \(h\rightarrow\infty\), these values converge to:  
\[\frac{5 GT}{\log 2^{*}}\frac{4 GT}{\log 2^{*}}\frac{10 GT}{\log 2^{*}}\frac{3 GT}{ \log 2^{*}}, \tag{18}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Time Slot** & **Time i** & **Time ii** & **Time iii** \\ \hline Channel & \((e_{1},e_{2})\) & \(e_{3}\) & \((e_{4},e_{5})\),\((e_{6},e_{7})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Secure PLNC with Gaussian MAC. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Time Slot** & **Time i** & **Time ii** & **Time iii** & **Time iv** & **Time v** \\ \hline Channel & \(e_{1},e_{4}\) & \(e_{2},e_{7}\) & \(e_{3}\) & \(e_{5}\) & \(e_{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Secure NC without Gaussian MAC. respectively. The secure NC protocol given in Section 3.2 requires a shorter transfer time for the transmission than the secure PLNC protocol given in Section 3.3 in this comparison. Since the difference is not so extensive, the secure PLNC protocol given in Section 3.3.1 is useful when it is not easy to prepare secure shared randomness between two source nodes. In fact, when the direct communication between two distinct source nodes is not available, we often use the butterfly network. In this case, such a secure shared randomness requires an additional cost. However, the secure PLNC protocol given in Section 3.3.2 has no advantage over the secure NC protocol with the MAC channel.