## 4 Performance Evaluation and Discussion  
### Secrecy Capacity  
The definitions of terms and symbols used in Section 4.1 are listed in Table 2.  
The sender and the receiver are communicating, and their communications are tapped by a cyberterrorist. The maximum transmission rate between the sender and the receiver when the cyberterrorist cannot receive any information is called the secrecy capacity. [PERSON] showed that for discrete memoryless channels, the secrecy capacity is the difference between the channel capacity of the receiver and the cyberterrorist [10,31]. This result has been generalized to Gaussian channels by [PERSON] [32]. We use the secrecy capacity as an evaluation index of the wiretapping resistance. If \(C_{y}\) is the channel capacity of the receiver under attack and \(C_{z}\) is the channel capacity due to wiretapping by the cyberterrorist, the secrecy capacity, \(C\) can be calculated by the following Equation [31]. \[C=C_{y}-C_{z} \tag{8}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Term/Symbol** & **Definition** \\ \hline Sender & Terminal sending information to the receiver \\ Receiver & Terminal receiving information from the sender \\ Cyberterrorist & Terminal that can be wiretapped and attacked simultaneously \\ \(C_{y}\) & Channel capacity of the receiver under attack \\ \(C_{z}\) & Channel capacity due to wiretapping by the cyberterrorist \\ \(C\) & Secrecy capacity \\ \(X\) & Random sequence sent by the sender \\  & \(X=\{x_{1}=0,\ x_{2}=1\}\) \\ \(Y\) & Received sequence by the receiver under attack \\  & \(Y=\{y_{1}=0,\ y_{2}=1\}\) \\ \(Z\) & Wiretapped sequence by the cyberterrorist \\  & \(Z=\{z_{1}=0,\ z_{2}=1\}\) \\ \(D\) & Errors not corrected by decoder \\ \(H(X)\) & Entropy of \(X\) \\ \(H(Y)\) & Entropy of \(Y\) \\ \(H(Z)\) & Entropy of \(Z\) \\ \(P(x_{i})\) & Probability of \(x_{i}\) \\ \(P(y_{i})\) & Probability of \(y_{i}\) \\ \(P(z_{i})\) & Probability of \(z_{i}\) \\ \(I(X;Y)\) & Mutual information between \(X\) and \(Y\) \\ \(I(X;Z)\) & Mutual information between \(X\) and \(Z\) \\ \(P_{eq}\) & Probability of different bits of \(X\) and \(Y\) \\ \(P_{ez}\) & Probability of different bits of \(X\) and \(Z\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Definitions of terms and symbols. Information transmitted by the sender is represented by \(X=\{x_{1}=0,\ x_{2}=1\}\), information received by the receiver under attack is represented by \(Y=\{y_{1}=0,\ y_{2}=1\}\), and information obtained by the cybererrorist through wiretapping is represented by \(Z=\{z_{1}=0,\ z_{2}=1\}\). The entropies of \(X,\ Y\) and \(Z\) are represented by \(H(X)\), \(H(Y)\), and \(H(Z)\), respectively. The entropy \(H(X)\) is given by the probability \(P(x_{i})\) of \(x_{i}\) as Equation (9). \[H(X)=-\sum_{i=1}^{2}P(x_{i})\log_{2}(P(x_{i})) \tag{9}\]  
\(H(Y)\) and \(H(Z)\) can be calculated in the same way as Equation (9). The amount of information shared by the receiver with the sender can be calculated by the mutual information \(I(X;Y)\) as follows. \[I(X;Y)=H(Y)-H(Y|X) \tag{10}\]  
In this paper, \(Y\) corresponds to the received information corrected by the shortened RA code. If \(D\) is the error that could not be corrected, \(Y\) can be expressed as Equation (11). \[Y=X+D \tag{11}\]  
If \(X\) and \(D\) are independent, Equation (10) can be transformed into Equation (12). \[I(X;Y)=H(Y)-H(X+D|X)=H(Y)-H(D) \tag{12}\]  
If \(P(x_{1})=P(x_{2})=1/2\), \(P(y_{1})=P(y_{2})=1/2\), and \(D\) is a random error with an error rate \(P_{ey}\), the mutual information \(I(X;Y)\) is given by Equation (13) which is equal to the channel capacity. \[I(X;Y)=1+P_{ey}\log_{2}\Bigl{(}P_{ey}\Bigr{)}+\Bigl{(}1-P_{ey}\Bigr{)}\log_{ 2}\Bigl{(}1-P_{ey}\Bigr{)}=C_{y} \tag{13}\]  
Similarly, \(Z\) corresponds to the received information corrected by the shortened RA code, \(P(z_{1})=P(z_{2})=1/2\), and the error is a random error with an error rate of \(P_{ez}\), the mutual information between the sender and the cybererrorist is expressed by Equation (14).