# Multi-Antenna Jammer-Assisted Secure Short Packet Communications in IoT Networks
## 3 Secrecy Performance Analysis
### Asymptotic Analysis  
In an effort to understand the relationship between infinite blocklength and finite blocklength and gain more physical insights, in this subsection, we look into the asymptotic secrecy throughput under the infinite blocklength regime, i.e., \(N\rightarrow\infty\). When \(N\rightarrow\infty\), we have \(\epsilon\to 0\) as long as \(\gamma_{b}^{*}>\gamma_{\epsilon}^{*}\) from (9). Therefore, the asymptotic secrecy throughput in the case of infinite blocklength can be mathematically expressed as:  
\[T_{N\rightarrow\infty}^{\star}=\frac{B}{N}\Pr(\gamma_{b}^{ \star}>\gamma_{\epsilon}^{\star}). \tag{14}\]  
**Corollary 1**. : _The asymptotic secrecy throughput of the ZFB scheme in the infinite blocklength regime is given by:_  
\[T_{N\rightarrow\infty}^{\text{ZFB}}=B(\Delta_{1}+\Delta_{2})N^{ -1}, \tag{15}\]  
_where \(\Delta_{1}\) and \(\Delta_{2}\) are given by:_  
\[\Delta_{1}=\frac{\lambda_{a}^{N_{j}-2}\tau_{aa}^{N_{j}-2}}{ \lambda_{j}^{N_{j}-1}\tau_{ja}^{N_{j}-1}}\left(\sum_{k=1}^{N_{j}-2}\frac{(k-1)!\lambda_{j}^{k}\tau_{ja}^{k}(-1)^{N_{j}-2-k}}{(N_{j}-2)!\lambda_{a}^{k}\tau_ {ae}^{k}}\bigg{(}\frac{1}{\lambda_{a}\tau_{ab}}+\frac{1}{\lambda_{a}\tau_{ae}} \bigg{)}^{N_{j}-2-k}\right.\] \[\left.-\frac{e^{\frac{\tau_{ae}}{\gamma_{jb}^{\star}\tau_{ab}}+ \frac{1}{\lambda_{j}^{\star}\tau_{ja}}}}{(N_{j}-2)!}\left(-\frac{1}{\lambda_{ a}\tau_{ab}}-\frac{1}{\lambda_{a}\tau_{ae}}\right)^{N_{j}-2}\text{E}i\bigg{(}- \frac{\tau_{ae}}{\lambda_{j}\tau_{ja}}-\frac{1}{\lambda_{j}\tau_{je}}\bigg{)}\right) \tag{16}\]  
_and_  
\[\Delta_{2}=\frac{\lambda_{a}^{N_{j}-1}\gamma_{ae}^{N_{j}-1}\big{(} N_{j}-1\big{)}}{\lambda_{j}^{N_{j}-1}\tau_{ja}^{N_{j}-1}}\left(\sum_{k=1}^{N_{j}-1} \frac{(k-1)!\lambda_{j}^{k}\tau_{ja}^{k}}{(N_{j}-1)!\lambda_{a}^{k}\tau_{ae} ^{k}}\left(-\frac{1}{\lambda_{a}\tau_{ab}}-\frac{1}{\lambda_{a}\tau_{ae}} \right)^{N_{j}-1-k}\right.\] \[\left.-\frac{e^{\frac{\tau_{ae}}{\gamma_{jb}^{\star}\tau_{ab}}+ \frac{1}{\lambda_{j}^{\star}\tau_{ja}}}}{(N_{j}-1)!}\left(-\frac{1}{\lambda_{ a}\tau_{ab}}-\frac{1}{\lambda_{a}\tau_{ae}}\right)^{N_{j}-1}\text{E}i\bigg{(}- \frac{\tau_{ae}}{\lambda_{j}\tau_{je}\tau_{ab}}-\frac{1}{\lambda_{j}\tau_{je}} \bigg{)}\right). \tag{17}\]  
**Corollary 2**. : _The asymptotic secrecy throughput of the NAN scheme in the infinite blocklength regime is given by:_  
\[T_{N\rightarrow\infty}^{NAN}=B(\Delta_{3}+\Delta_{4})N^{-1}, \tag{18}\]  
\[\max_{N} T^{\star}, \tag{24a}\] \[s.t. \bar{\epsilon}\leq\epsilon_{\max},\] (24b) \[N\leq N_{\max},\] (24c) \[T_{N\rightarrow\infty}^{\text{ZFB}}=B(\Delta_{1}+\Delta_{2})N^{ -1}, \tag{15}\]0  
where (24b) is the reliability constraint of the considered system, (24c) is the transmission latency constraint of the considered system, and \(\mathbb{N}^{+}\) is the non-negative integer set. Next, we will clarify that \(T^{\star}\) is quasi-concave with respect to \(N\). In the first step, by taking the first-order and second-order derivative of \(\epsilon\) on \(N\), we have:  
\[T_{N\rightarrow\infty}^{\text{ZFB}}=B(\Delta_{1}+\Delta_{2})N^{ -1}, \tag{15}\]1  
and  
\[T_{N\rightarrow\infty}^{\text{ZFB}}=B(\Delta_{1}+\Delta_{2})N^{ -1}, \tag{15}\]2  
where \(\phi\!=\!\sqrt{\frac{N}{V_{b}}}\!\left(\ln\frac{1+\gamma_{b}^{\star}}{1+ \gamma_{c}^{\star}}-\sqrt{\frac{V_{c}}{N}}Q^{-1}(\delta)-\frac{B}{N}\ln 2\right)\). Note that \(\epsilon\) is generally much smaller than 0.5 to ensure high reliability [34]; thus, we have \(\phi=Q^{-1}(\epsilon)>0\), \(\frac{\partial\epsilon}{\partial\phi}=-\frac{1}{\sqrt{2\pi}}e^{-\frac{\phi^{ 2}}{2}}<0\), and \(\frac{\partial^{2}\epsilon}{\partial\phi^{2}}=\frac{\phi}{\sqrt{2\pi}}e^{- \frac{\phi^{2}}{2}}>0\). Then,  
_where \(N^{\#}\) is the solution of \(\frac{2\mathcal{T}^{*}}{\delta N}=0\) and \(\lfloor\cdot\rfloor\) is the floor operation._  
Proof. : Firstly, the integer constraint in problem (29) is relaxed.