# Secure Physical Layer Network Coding versus Secure Network Coding +
## 3 Butterfly Network
### Comparison  
To implement the above discussed protocols as wireless communication networks, we compare the transmission rates of the protocols given in Sections 3.2 and 3.3 when each edge is given as the BPSK scheme of a two-input Gaussian channel as (2) or a single-input Gaussian channel:  
\[Y=hX+N, \tag{15}\]  
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 \(X\) is coded as \((-1)^{A}\) with \(A\in\mathbb{F}_{2}\). Hence, (16) is rewritten as:  
\[Y=h(-1)^{A}+N, \tag{16}\]  
In this comparison, for simplicity, we assume that \(h_{1}=h_{2}=h\). We assume that \(T\) is the time period of transmitting one Gaussian signal on each edge. Additionally, we assume that ideal codes are available as follows. The mutual information rate \(I(Y;A)_{\text{Equation}(16)}\) is achievable over the channel (16), the rate \(I(Y;A_{1}\oplus A_{2})_{\text{Equation}(2)}\) is achievable for CAF in the channel (2), and the rate \(2I(Y;A_{1}\oplus A_{2})_{\text{Equation}(2)}-I(Y;A_{1},A_{2})_{\text{Equation}(2)}\) is available for secure CAF in the channel (2). Notice that the relation \(I(Y;A_{2}|A_{1})_{\text{Equation}(2)}=I(Y;A_{1}|A_{2})_{\text{Equation}(2)}\) holds in this case. In addition, the mutual information rate pair \((I(Y;A_{1}A_{2})_{\text{Equation}(2)}/2,I(Y;A_{1}A_{2})_{\text{Equation}(2)}/2)\) is available in the MAC channel (2) when both transmitters intend to send their own message to the receiver. (Generally, the symmetric rate \((I(Y;A_{1},A_{2})/2,I(Y;A_{1},A_{2})/2)\) is achievable when the symmetric rate \((I(Y;A_{1},A_{2})/2,I(Y;A_{1},A_{2})/2)\) belongs to the interval between \((I(Y;A_{1}),I(Y;A_{2}|A_{2}))\) and \((I(Y;A_{1}|A_{2}),I(Y;A_{2}))\). Our case with \(h_{1}=h_{2}=h\) satisfies this condition.) In the above discussion, the random variables \(A_{1}\), \(A_{2}\), and \(A\) are subject to the uniform distribution independently. The secure NC protocol given in Section 3.2 needs to avoid a crossed line when Gaussian MAC is not used. Now, we consider how much time is needed for this protocol. In this protocol, we need to repeat several processes, each of which is composed of the encoding, wireless communication, and decoding. In the protocol given in Section 3.2, the first step can make the simultaneous transmissions on \(e_{1}\) and \(e_{4}\). However, the simultaneous transmission on \(e_{1}\) cannot be performed simultaneously in order to avoid the cross line on the receiving on \(V_{3}\). Hence, the second step makes the simultaneous transmissions on \(e_{2}\) and \(e_{7}\). We say that the time period for the first step is the time slot of Time i, and the time period for the second step is the time slot of Time ii. That is, each time span for the process composed of the encoding, wireless communication, and decoding is called a time slot. Now, to evaluate the required number of time slots, we assume that all players have only one transmitting antenna, which can broadcast the transmitting signal. Then, we find that the whole network has five time slots at least as presented in Table 2. When the length of the transmitted message is \(G\), the transfer time for each time slot is \(\frac{GT}{I(Y;A)_{\text{Equation}(16)}}\). Therefore, the total transfer time in this case is calculated to be \(\frac{5 GT}{I(Y;A)_{\text{Equation}(16)}}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Protocol** & **Assumption** & **Security** & **Type** \\ \hline Section 3.2 & (A1) & (S1) (S2) & Secure NC \\ Section 3.3.1 & (A2) & (S2) & Secure PLNC with secure CAF \\ Section 3.3.2 & (A1) (A2) & (S1) (S2) & Secure PLNC with CAF and secure NC \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison for protocols in butterfly network. When we use the Gaussian MAC, the secure NC protocol given in Section 3.2 can be implemented with three time slots as Table 3 because \(V_{4}\) broadcasts the information to \(e_{5}\) and \(e_{6}\).