# 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
### New Metrics  
Let us introduce an auxiliary parameter that is defined as _effective secrecy area ratio_ and is equal to:  
\[A_{s}^{[\text{eff}]}=\frac{A_{s}}{A}, \tag{5}\]  
where \(A\) is the area of the cell sector that is managed by the BS ([PERSON]) and \(A_{s}\) is the secure area, i.e., the set of points of the total cell sector surface where the attacker ([PERSON]) can stay without decreasing the secrecy rate of the legitimate link under the target rate; therefore, the parameter \(A_{s}^{[\text{eff}]}\) defines the percentage of area (related to the overall area of the cell) where the attacker ([PERSON]) can stay without decreasing the secrecy rate of the legitimate link under the target rate.  
The effective secrecy area ratio is computed by supposing that the eavesdropper could be located in any point \((x,y)\) of the area managed by the base station. The position of the legitimate receiver ([PERSON]) in the cell is supposed to be fixed as well as the transmit power. Results are shown with different transmit powers and distances Alice-Bob, while [PERSON] could be located in any point of the cell of the BS. The extension of the cell depends on the transmit power. All of this information gives us a new parameter for evaluating how the area is extended where the legitimate link has a minimum target secrecy rate. The algorithm for the BS could be the following:  
1. Decide a target secrecy rate that [PERSON] wants to keep with [PERSON];
2. Given the position of [PERSON] and the transmit power of [PERSON], the extension of the cell is known;
3. Compute the secrecy capacity (\(C_{B}-C_{E}\)) of each point \((x,y)\) of the cell area, as [PERSON] was located at that point; in other words, the surface managed by the BS is divided into infinitesimal squares whose surface is equal to \(dx\cdot dy\) and the eavesdropper is supposed to be there for the computing of the secrecy capacity;
4. Count each point of the cell area that gives a secrecy capacity equal to or greater than the target rate;
5. Compute the effective secrecy area as the ratio between the set of points that give a secrecy capacity equal to or greater than the target rate and the transmit power. We propose a first metric that is called _effective secrecy area per power unit_ [W\({}^{-1}\)] and defined as: \[\rho_{1}=\frac{A_{s}^{[\text{eff}]}}{P},\] (6)  
where \(P\) is the power transmitted by the BS (Alice). Given the distance of the user ([PERSON]), this metric can identify the transmit power to be used by the BS ([PERSON]) in order to maximize the secure area in the sector. Thus, the metric allows the BS to maximize the area of security while minimizing the transmit power, i.e., saving energy at the same time.  
Since the main goal of this paper is the maximization of the effective secrecy area for the affordable target date rate with the BS minimum power consumption, we propose another metric that is called _Effective Secrecy-Energy Efficiency_ [Bit/Joule] and is defined as:  
\[\rho_{2}=A_{s}^{[\text{eff}]}\cdot\frac{\overline{R}_{S}}{P_{in}}, \tag{7}\]  
where \(P_{in}\) is the BS power consumption as defined in Equation (2) and \(\overline{R}_{s}\) is the target secrecy rate. Given the distance of the user ([PERSON]), this metric helps to identify the power to be used by BS (Alice) to maximize the secure area as well as the cost in terms of energy requested to send a secret bit stream to the legitimate receiver. Hence, this metric is a toll that helps to maximize the area of security, and, at the same time, minimize the BS power consumption. It is important to note that the secrecy area is intended as the area where the eavesdropper can stay without driving the secrecy capacity of the legitimate link under the target secrecy rate. Note that the power consumption of the BS (Alice)has been calculated by using a complete model (2), which takes into account any source of energy consumption in the BS equipment, even if the width of the area coverage is determined only by the transmit power \(P\). The results are shown in the following section.