# Realizing Efficient Security and Privacy in IoT Networks
## 2 Privacy Capacity Model in Wireless Networks  
In this section, the privacy capacity model of the transmitting network is expressed. In the privacy capacity model of the network, it is assumed that the malicious node may eavesdrop the source as well as well as the receiver. However, in order to get a full use of the signals transmitted from the transmitter,the eavesdropper must be fully synchronized in the network. According to traditional privacy definitions in [8], the channel of communication can be modeled as a channel of broadcast in line with the wiretap channel as illustrated in Figure 1.  
Considering the wiretap channel, the transmitting channel's message is represented as \(a^{n}\in\mathrm{A}^{n}\) and is encoded and broadcasted as a codeword \(c^{m}\in C^{m}\). The receiver \(\left(R_{x}\right)\) and eavesdropper ([PERSON]) receive \(b^{m}\in\mathbf{B}^{m}\) and \(e^{m}\in\mathrm{E}^{m}\), respectively. [PERSON]'s received information via her receiving signal is then modeled and described as in Equation (1),  
\[I\left(e^{m};a^{n}\right)=g\left(a^{n}\right)-g\left(a^{n}\left|e^{m}\right.\right) \tag{1}\]  
where \(I(e^{m};a^{n})\) represents the mutual information shared by the transmitter and the legitimate receivers, while \(g\) is considered as the entropy. As long as [PERSON] cannot decode any bit of the transmitted information, then perfect privacy is achieved. Thus, Equation (2),  
\[I\left(e^{m};a^{n}\right)=0\leftrightarrow g\left(a^{n}\right)=g\left(a^{n} \left|e^{m}\right.\right) \tag{2}\]  
This implies that the quantity of uncertainty about [PERSON]'s private information is not altered after \(e^{m}\) is received.  
By definition, the probability of experiencing an error \(P_{i}\) in the message estimation of message \(a^{n}\), and \(\hat{a}^{n}\) is defined as the estimate of \(a^{n}\); therefore,  
\[P_{i}=P\left\{a^{n}\
eq\hat{a}^{n}\right\}. \tag{3}\]  
[PERSON]'s rate of uncertainty about message \(a^{n}\) is term as the rate of equivocation and can be described as  
\[Q_{i}=\frac{1}{m}g\left(a^{n}\left|e^{m}\right.\right) \tag{4}\]  
where  
\[0\leq Q_{i}\leq g(a^{n})/m \tag{5}\]  
Figure 1: A Wiretap Channel for Privacy Capacity model  
Evidently, if \(Q_{i}=g(a^{n})/m\), then perfect privacy, which is related to perfect privacy rate \(Q_{s}\), is realized. For each \(\varepsilon>0\), a particular \(Q_{s}\) is assumed to be realizable, and there is a sequence of \((2^{mQ_{s}},m)\) codes such that for any \(m>m(\varepsilon)\) the following states are obtained.  
\[\left\{\begin{array}{l}P_{i}<\varepsilon\\ \\ Q_{s}-\varepsilon<Q_{i}\end{array}\right. \tag{6}\]  
The first state is the constraint for the realizable rate, whereas the second is the equivocation rate constraint which guarantees prefect privacy. In summary, privacy capacity \(S_{c}\) is the optimal realizable rate of privacy in a network transmission. Thus, in [8], it is established that the difference between the main channel capacity \(C_{mc}\) and the wiretap channel capacity \(C_{wc}\) is the privacy capacity \(S_{c}\), in other words,  
\[S_{c}=(C_{mc}-C_{wc})^{+}, \tag{7}\]  
as \((.)^{+}\triangleq\max(0,.)\), where negative rate is meaningless. Although [PERSON] demonstrated this for the distinct no-memory channel, the principle that the capacity of privacy is the difference of the capacity of the legitimate channel and that of the eavesdropper's is established to be accurate for several systems like in multiple-input, multiple-output (MIMO) set-ups [12; 13].