## 4 Formulation and Policy  
### Problem Formulation  
_Without loss of generality, we consider the wireless communication, as shown in Figure_ 2b _for the problem formulation, where_ UAV\({}_{L}\) _aims to eavesdrop data packets from_ UAV\({}_{ST}\) _via jamming_ UAV\({}_{SR}\)_. Note that our algorithm is common in the other three cases because channel gains for eavesdropping links are associated with_ \(D(x)\) _according to Equation (11)._ \(D(x)\) _is the only parameter that influences eavesdropped data packets. Based on the notations in the system model, we formulate the optimization problem to maximize the eavesdropped data packets via optimizing jamming power. Assume that each suspicious data packet has_ \(b\) _bytes and then successfully eavesdropped data (in bytes) can be calculated as_ \(\sum_{x=1}^{m}b\cdot R(x)\) _in_ \(m\) _time slots. To prevent legitimate jamming and eavesdropping being detected by suspicious UAVs, SINR of the suspicious link has to be maintained at a certain threshold_ \(\delta\)_, which presents_ \(\gamma_{s}(x)=\delta\)_. Specifically, the modulation of_ UAV\({}_{ST}\) _that is used to transmit data to_ UAV\({}_{SR}\) _is_ \(2^{\rho(x)}\) _Quadrature Amplitude Modulation (QAM), where_ \(\rho(x)=\{1,\ldots,\rho_{max}\}\)_._ \(\rho_{max}\) _indicates the number of modulation levels available for rate adaptation. Constraint_ \(0\leq\sum_{x=1}^{m}P_{L}(x)\leq P_{L}^{total}\) _specifies that the total consuming power (eavesdropping plus jamming) of_ UAV\({}_{L}\) _during the eavesdropping period is required to be less than the total obtained power of the_ UAV\({}_{L}\)_,_ \(P_{L}^{total}\)_. Constraint_ \(P_{L}(x)\leq P_{L}^{max}\ (\forall x,\ x=1,2,\ldots,m)\) _specifies that, in each eavesdropping period,_ UAV\({}_{L}\) _consumes no more than_ \(P_{L}^{max}\) _power. Then, the formulation of the problem is presented as follows._  
\[\max_{P_{L}(x),\rho(x)}\sum\limits_{x=1}^{m}b\cdot R(x) \tag{12}\]  
_Subject to:_  
\[\gamma_{s}(x)=\delta \tag{13}\]  
\[0\leq\sum\limits_{x=1}^{m}P_{L}(x)\leq P_{L}^{total} \tag{14}\]  
\[P_{L}(x)\leq P_{L}^{max}\ (\forall x,\ x=1,2,\ldots,m) \tag{15}\]  
\[1\leq\rho(x)\leq\rho_{max} \tag{16}\]  
_Furthermore, in terms of Equation (13), we have_  
\[\rho(x)=\log_{2}\Biggl{(}\frac{\delta^{2}(N_{0}+P_{L}(x))}{H_{s}(x)\cdot K_{2} ^{-1}\ln\frac{K_{1}}{\epsilon}}+1\Biggr{)} \tag{17}\]  
_which indicates that the modulation level is adapted by_ UAV\({}_{ST}\) _in terms of the consuming power_ \(P_{L}(x)\) _of_ UAV\({}_{L}\)_. Specifically,_ UAV\({}_{ST}\) _increases_ \(\rho(x)\) _to transmit data with an increasing_ \(P_{L}(x)\) _so thatSINR of the suspicious link at time slot \(x\) is maintained at \(\delta\). Moreover, considering Equation (5) and Equation (13), the upper bound and the lower bound of the consuming power \(P_{L}(x)\) can be obtained by  
\[P_{L}(x)=\left\{\begin{array}{c}\frac{H_{s}(x)\cdot K_{2}^{-1}\ln\frac{K_{1}} {c}}{\delta^{2}}-N_{0}\;if\;\rho(x)=1\\ \frac{(2^{max}-1)H_{s}(x)\cdot K_{2}^{-1}\ln\frac{K_{1}}{c}}{\delta^{2}}-N_{0} \;if\;\rho(x)=\rho_{max}\end{array}\right. \tag{18}\]  
Consequently, by substituting Equations (6), (7), (8), (9), (10), and (11) into (13), (14), (15), and (16) the optimization problem is reformulated as follows:  
**Optimal Eavesdropping and Jamming Problem:**  
\[\underset{P_{L}(x)}{\text{max}}b\cdot\sum_{x=1}^{m}\left(1-\frac{1}{2}exp^{ \beta_{1}-\beta_{0}\delta\sqrt{\frac{H_{e}(x)+H_{j}(x)}{H_{s}(x)}\cdot(1+\frac {P_{l}(x)}{N_{0}})}}\right)^{8(2f-l)}\]  
Subject to:  
\[\begin{array}{c}0\leq\sum_{x=1}^{m}P_{L}(x)\leq P_{L}^{total}\\ P_{L}(x)\leq P_{L}^{max}\;(\forall x,\;x=1,2,\ldots,m)\\ P_{L}(x)\geq\frac{H_{s}(x)\cdot K_{2}^{-1}\ln\frac{K_{1}}{c}}{\delta^{2}}-N_{0} \\ P_{L}(x)\leq\frac{(2^{\beta_{max}}-1)H_{s}(x)\cdot K_{2}^{-1}\ln\frac{K_{1}}{ c}}{\delta^{2}}-N_{0}\end{array}\]