# A Metric for Secrecy-Energy Efficiency Tradeoff Evaluation in 3 GPP Cellular Networks
## 3 Metrics for the Evaluation of the Effective Secrecy-Energy Efficiency Tradeoff
### Effective Secret Area  
Suppose that the BS ([PERSON]) has to serve a user ([PERSON]) in the cell and that the requested service has to be provided by means of a secure connection (QoSS--Quality of Service with Security). The BS sets a target secrecy rate \(\overline{R}_{s}\) depending on the QoS of the user. Given the position of the user ([PERSON]), a specific metric is required that can help [PERSON] determine the minimum transmit power that maximizes the secure area of the cell, i.e., the region where [PERSON] can stay without affecting the secrecy capacity \(C_{s}\) of the legitimate link under the target secrecy rate, i.e., \(C_{s}\geq\overline{R}_{s}\). In our analysis, we set a dynamic target secrecy rate, equal to 10% of the capacity of the legitimate link (Alice-Bob), i.e., we initially set \(\overline{R}_{s}=0.1C_{B}\). Nonetheless, in the following, results with different target secrecy rates are also shown.  
Before introducing the metrics, let us show the distribution of the secrecy capacity \(C_{s}\) in the cell that is covered by the central BS (Alice) by assuming a variable distance of the user ([PERSON]) and increasing the transmit power.  
Figure 2 shows the geographical distribution of \(C_{s}\) over the playground of the cellular network. In particular, we focus on the sector of the cell managed by the central BS (Alice) where the user ([PERSON])is present. The map shows the secrecy capacity of each point of the cell calculated, as [PERSON] was in that specific point. While the transmit power of Alice is fixed (20 W), the distance of [PERSON] in the maximum radiation pattern direction is varying from 10 to 100 m. The area where the secrecy capacity is less than the target secrecy rate \(\overline{R}_{s}\) (Unsecure Area) is represented with darker blue and increases as the distance of [PERSON] increases.  
These results are summarized in Table 1.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Bob’s Distance** & **10 m** & **50 m** & **100 m** \\ \hline Unsecure Area & 0.97\% & 16.20\% & 53.79\% \\ Secure Area & 99.03\% & 83.80\% & 46.21\% \\ \hline \end{tabular}
\end{table}
Table 1: Secrecy capacity values when [PERSON]’s distance from Alice is equal to \(\{10,50,100\}\) m and \(P=20\) W. The total area covered by the BS is 9336 m\({}^{2}\).  
Figure 2: Secrecy capacity geographical distribution for \(P=20\) W.  
Figure 3 shows the geographical distribution of \(C_{\text{s}}\) over the playground of the cellular network when the transmit power of Alice is varying from 5 to 40 W while the distance of Bob is fixed (50 m): the area where the secrecy capacity is less than the target secrecy rate \(\overline{R}_{s}\) remains the same, but the overall area of the cell increases. It is important to stress that in this graph the transmit power of all the sectors of other BSs and of the other two sectors which are co-sited with Alice are kept equal to 20 W (In this case, the possibility to change the transmit power of the antenna sector is realized to emulate the behavior of a basic Transmit Power Control (TPC)).  
These results are summarized in the Table 2.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**[PERSON]’s Power** & **10 W** & **20 W** & **40 W** \\ \hline Unsecure Area & 27.87\% & 16.20\% & 11.56\% \\ Secure Area & 72.13\% & 83.80\% & 88.44\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Secrecy capacity values when [PERSON]’s distance from [PERSON] is 50 m and \(P=\{10,20,40\}\) W.  
Figure 3: Secrecy capacity geographical distribution when [PERSON]’s distance from [PERSON] is 50 m.