# Secrecy Control of Wireless Networks with Finite Encoding Blocklength
## 2 System Models
### Problem Formulation  
In [23], we have proposed a control algorithm according to the secrecy encoding, where the confidential message is encoded into an infinite codewords. Thus, it would involve an infinitely long delay to decode the confidential message. In this paper, we remove the assumption of infinite codewords, i.e., \(N<\infty\). Since the confidential message is encoded into a finite codeword, the perfect secrecy for all message can not be guaranteed. Thus, to embody the security of confidential message, we define the notion of secrecy outage. The secrecy outage occurs if the confidential message is intercepted by eavesdroppers. To evaluate the state of secrecy outage, we assume each source node can collect the knowledge of the confidential message accumulated by eavesdroppers. Such that the source node can identify the occurrence of secrecy outage. Although this assumption is somewhat ideal, it provides a better insight on the performance of secure communication in multi-hop network. In addition, according to the state of secrecy outage, each secrecy codeword \(k\) would encode different confidential message \(R_{n}^{k,cs}\). Thus, an encoding scheme needs to be designed to satisfy the requirement of secrecy outage. Let \(R_{n}^{p,cs}\) be the average confidential message rate transmitted from node \(n\) to \(c\), \(p_{n}^{out}(R_{n}^{k,cs})\) denote the average secrecy outage of codeword \(k\) at node \(n\), \(\gamma_{n}\) be the maximum allowable portion of confidential message intercepted by the eavesdropper. We aim to determine a joint scheduling, secrecy encoding scheme and routing algorithm that maximizes aggregate network utility. Let \(U_{n}^{\epsilon}(x)\) be utility obtained by source \(n\) destined to node \(c\) when the confidential transmission rate is \(x\) bits/channel use. We assume that \(U_{n}^{\epsilon}(\cdot)\) is a continuously differentiable, increasing and strictly concave function. There is a finite backlog at the transport layer, which contains the secrecy-encoded messages. In each slot, source node \(n\) determines the amount of encoded information admitted to its queue at the network level. Let \(\lambda_{n}^{cs}(t)\) be the amount of traffic injected into the queue of source \(n\) and destined to node \(c\) at slot \(t\). Our objective is to support the traffic demand to achieve a long term confidential rate that maximizes the sum of utilities. Then the optimization problem can be formulated as following:  
\[\max_{R_{n}^{p,cs}(t),I_{g}^{p,cs}(t)}:\quad\sum_{n,c}U_{n}^{c}( \bar{r}_{n}^{p,cs}) \tag{3}\] \[\text{s.t.}\quad 0\leq R_{n}^{p,cs}(t)\leq\lambda_{n}^{cs}(t) \ \ \text{for all}\ (n,c) \tag{4}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Definition** \\ \hline \(M\) & Number of legitimate nodes. \\ \(E\) & The set of eavesdroppers. \\ \(N\) & Number of channels. \\ \(T(l),D(l)\) & Transmitter and receiver nodes of link \(l\). \\ \(S_{l}(t)\) & Channel state of link \(l\) in slot \(t\). \\ \(R_{l}(t)\) & Achievable rate on link \(l\) in slot \(t\). \\ \(\bar{R}_{l}^{s}(t)\) & Maximum leakage rate at eavesdropper \(e\). \\ \(\lambda_{n}^{cs}(t)\) & Confidential data arrival rate. \\ \(\lambda_{n}^{cs}(t)\) & The maximum Confidential data arrival rate. \\ \(R_{n}^{cs}(t)\) & Confidential data admitted into the network. \\ \(R_{l}^{s}(t)\) & Secrecy encoding rate. \\ \(\mathbf{Q}(t)\), \(\mathbf{P}(t)\), \(\mathbf{Z}(t)\), \(\mathbf{Y}(t)\) & Queues backlog in the network. \\ \(V\) & Parameter of control algorithm. \\ \(U_{n}^{c}(\cdot)\) & Utility functions. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations. \[\sum_{\{j|(i,j)\in L\}}\tilde{\mu}_{ij}^{n}-\sum_{\{i|(i,j)\in L\}}\tilde{\mu}_{ ij}^{n}\geq 0 \tag{5}\]  
\[\tilde{R}_{n}^{out}\leq\gamma_{n}\tilde{R}_{n}^{cs} \tag{6}\]  
where \(\tilde{R}_{n}^{cs}=\lim_{K\to\infty}\frac{1}{K}\sum_{k=1}^{K}R_{n}^{n,cs}\), \(\tilde{R}_{n}^{out}=\lim_{K\to\infty}\frac{1}{K}\sum_{k=1}^{K}R_{n}^{n,cs}p_{ n}^{out}(R_{n}^{k,cs})\).