# Secrecy Control of Wireless Networks with Finite Encoding Blocklength
## 2 System Models  
### Network Model  
The wireless Ad hoc network is formed by \(M\) legitimate nodes and \(L\) links connecting the nodes. As shown in Figure 1, for a link \(l\in\{1,2,...,L\}\), let \(T(l)\) and \(D(l)\) be the set of transmitter and receiver nodes on link \(l\). The eavesdroppers set is denoted as \(E\). In the network system, each node wishes to transmit its confidential message to the destination via a multi-hops manner against eavesdroppers. We assume the network operates on a time-slotted model and the slot is normalized to integral unit \(t\in\{0,1,2,...\}\). In this work, there exists a reasonable assumption that the system Channel State Information (CSI) is known. As in [10], each node can get full-CSI by utilizing pilot symbols and CSI feedback process. For example, each node reports a received-signal-strength index to PBS in packets such as RSSI reports. Let \(\bar{S}(t)=(S_{1}(t),...,S_{L}(t))\) represent the channel state vector of link set \(L\) in slot \(t\), which is a block fading channel and follows independent and identically distribution (i.i.d) and \(S_{l}(t)\) is the channel state of link \(l\). Note that \(S_{l}(t)\) contains \(N<\infty\) channels which implicates the perfect secrecy [1] cannot satisfy. Let \(R_{l}(t)\) and \(\bar{R}^{e}_{l}(t)\) denote the achievable rate on link \(l\) and the maximum overhearing rate of eavesdropper \(e\), respectively. Since the network is a multi-commodity problem, each flow is identified by its destination node \(c\in\{1,...,M\}\). Let \(\lambda^{cs}_{n}(t)\) be the arrival confidential data at node \(n\) and destined for node \(c\), which is bounded by \(\lambda^{c\max}_{n}\). As shown in Figure 1, there is a control valve at each node to admit \(R^{cs}_{n}(t)\) confidential data into the network. For each flow \(c\), let \(Q^{e}_{n}(t)\) be the queue backlog at node \(n\). If \(c=n\), \(Q^{n}_{n}(t)=0\) for all \(n\) and \(t\).  
Since the data is transmitted by wireless channel, there exists simultaneous transmission interference. Thus, Link set \(L\) can not be fully utilized. Let \(g\) be a link set that can be transmitted simultaneously, and \(G\) denote the collection of all sets of the link set \(g\), where set \(G\) is determined by the network interference model. We also define an indicator variable \(I_{g}(t)\) to represent the decision of the network in slot \(t\), where \(I_{g}(t)=1\) means the link set \(g\) is scheduled in slot \(t\), otherwise it is 0. Similarly, indicator variable \(I^{n,c}_{ij}=1\), if link \((i,j)\) is employed to transmit the flow generated by source \(n\) to node \(c\), otherwise it is 0. Thus, in slot \(t\), the flow rate of source \(n\) to node \(c\) at link \((i,j)\) is  
\[\mu^{n}_{ij}(t)=\begin{cases}R_{ij}(t)&\text{if }(i,j)\in g,I_{g}(t)=1,I^{n,c}_{ ij}=1.\\ 0&\text{else}\end{cases} \tag{1}\]  
and the time average link rate is  
\[\bar{\mu}^{n}_{ij}=\lim_{t\to\infty}\frac{1}{t}\sum_{\tau=1}^{t}\mu^{n}_{ij} (\tau). \tag{2}\]  
Due to the broadcast feature of wireless channel, the confidential message is overheard by eavesdroppers. To guarantee the secure transmission of confidential message, we employ [PERSON]'s encoding scheme [1] to encode the confidential message. Specifically, at each slot \(t\) and link \(l\), using independent randomization encoding strategy, the transmitter, according to \(R_{l}(t)\) and \(\bar{R}^{e}_{l}(t)\), encodes a mount of \(R^{s}_{n}(t)\) (may contain multiple flows) confidential message from its arrival data. Such that  
Figure 1: Network model.  
the maximum output confidential message rate \(R_{l}^{s}(t)\) can be denoted as \(R_{l}(t)-\bar{R}_{l}^{s}(t)\), if \(R_{l}^{s}(t)>R_{l}(t)-\bar{R}_{l}^{s}(t)\), secrecy outage occurs. Using this secrecy encoding strategy, we can guarantee the transmission security in each link. All the parameters that have been defined or would be used are presented in Table 1.