# Exploring ISAC: Information-Theoretic Insights
## 6 Secrecy of ISAC Systems
### Secrecy of Data and Sensing Information  
In [70], not only the message (data) has to be kept secure from an external eavesdropper but also the channel state sequence \(S^{n}\). Depending on the channel, Eve will always learn about the channel state; however, it is required that this knowledge stays beyond a given threshold. In other words, the Tx has to choose transmission strategies in a way that not too much information is leaked about the sensing target. In a practical application thiscould mean that the Tx has to restrict itself to beamforming strategies where it points its beam towards a given direction. To make the problem more interesting from a technical point of view, the authors in [70] focused on the setup in Figure 17 where the Tx learns the state sequence \(S^{n}\) in a non-causal manner (i.e., before transmission starts) and the sensing is performed at the Rx. An external eavesdropper is not allowed to learn any information about the message nor the sequence \(\Xi^{n}\), which is obtained by passing the state sequence \(S^{n}\) through a memoryless channel \(P_{\Xi|S}\) independent of the message and the communication channel. Note that the channel \(P_{\Xi|S}\) needs to be carefully chosen in the model to reflect the desired security constraint. For example, it could select a part of the state if \(S\) is bipartite \(S=(S_{1},S_{2})\) and only one of these states needs to be kept secret, or it could implement a function \(\Xi=\
u(S)\) when only certain characteristics of the target have to be kept secret. In general, the setup in [70] allows the modeling of any stochastic relationship between the state \(S\) and the part that needs to be kept secret \(\Xi\). In the setup of this section, there is only one message \(W\) of rate \(R\) and the Tx produces its channel inputs as \(X_{i}=\phi_{i}(W,S^{n})\), for \(S^{n}\) the i.i.d. state sequence following a given pmf \(P_{S}\). The channel outputs \(Y^{n}\) observed at the legitimate Rx and \(Z^{n}\) observed at the eavesdropper are produced from inputs and states according to a given stationary and memoryless channel law \(P_{YZ|XS}\). Based on the observed outputs \(Y^{n}\), the Rx decodes the message as \(\hat{W}=g(Y^{n})\) and produces an estimate of the state sequence \(\hat{S}^{n}=h(Y^{n})\). The goal of the Tx is to find an encoding strategy for which the Rx can decode with arbitrary small probabilities of error and reconstruct the state sequence with desired distortion \(D\) but such that the eavesdropper does not learn about the related sequence \(\Xi^{n}\) nor the message \(W\). This leads to the following definition of achievability. Definition 6: A rate-distortion pair \((\mathsf{R},\mathsf{D})\) is called securely achievable if there exists a sequence (in \(n\)) of encoding, decoding, and estimation functions such that  
\[rCl\lim_{n\to\infty}\Pr[\hat{W}\
eq W] = 0 \tag{67a}\] \[\lim_{n\to\infty}1(W,\Xi^{n};Z^{n}). = 0\] (67b) \[\varlimsup_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}[d(S_{ i},\hat{S}_{i})] \leq \mathsf{D}. \tag{67c}\]  
The following inner bound was proved in [70]. Theorem 6.1: _For any pmf \(P_{UYX|S}\) so that for the associated tuple \((S,\Xi,U,V,X,YZ)\sim P_{S}P_{\Xi|S}P_{UYX|S}P_{YZ|XS}\), the random variable \(\Xi\) is independent of the pair \((U,Z)\) and any function \(g(\cdot)\) on appropriate domains, all pairs \((R_{M},D)\) satisfying the following inequalities_  
\[rClR \leq I(UV;Y)-I(UV;S) \tag{68}\] \[R \leq I(V;Y\mid\mathsf{U})-I(V;\Xi Z\mid U)+\min\{0,I(U;Y)-I(U;S)\} \tag{69}\]  
Figure 17: ISAC model with state information at the Tx and secrecy constraints on messages and states. _and_  
\[D\ \leq\ E[d(S,g(U,V,Y))] \tag{70}\]  
_are securely achievable._  
The above achievability result is based on the following coding scheme. A two-level superposition code with cloud-center codewords \(U^{n}\) and satellite codewords \(V^{n}\) is considered.