# Exploring ISAC: Information-Theoretic Insights
## 6 Secrecy of ISAC Systems
### Secrecy of Data and Sensing Information
The Tx uses the \(U^{n}\)-codewords to describe information about the state sequence \(S^{n}\) to the receiver, where this cloud-center codeword can also be decoded by the eavesdropper. It further uses the \(V^{n}\)-codewords to send more refined information about \(S^{n}\) as well as the message \(W\) to the Rx. The Rx decodes both the \(U^{n}\) and \(V^{n}\) codewords to recover the transmitted message \(W\). It also reconstructs the state sequence based on the two decoded codewords and its own observed sequence of channel outputs. Security of the scheme against the external eavesdropper is obtained by choosing the \(U^{n}\)-codewords so that the decoded does not reveal information about the \(\Xi^{n}\)-sequence (because the \(U^{n}\)-codeword is also decoded by the eavesdropper). In fact, in the construction, only the \(V^{n}\)-codeword can contain information about \(\Xi^{n}\) and \(W\), and they are chosen of sufficiently high rate so that the eavesdropper cannot decode them. The theorem includes several interesting special cases. When \(Z\) is independent of the input-state pair \((X,S)\), the setup reduces to the setup without secrecy constraint studied in [89], in which case Theorem 13 can be simplified by choosing \(U=\)const. On a different note, when the entire state \(S\) has to be kept secret, \(\Xi=S\), then \(U\) has to be chosen independently of \(S\) and thus \(I(U;S)=0\) and the minimum in the right-hand side of (69) evaluates to \(0\). Moreover, for \(\Xi=S\) the right-hand side of (68) is larger than the right-hand side of (69) because \(I(V;SZ|U)\geq I(V;S|U)\). Thus, for \(\Xi=S\), Constraint (68) is less stringent than Constraint (69) where \(U\) only plays the role of a convexification random variable. Comparing the results where both message and state have to be kept secret with the results with no secrecy constraint is applied, the price for the double state and message secrecy in the proposed scheme seems to be independence of \(S\) with \(Z\) and the rate reduction of \(I(V;Z|S)=I(SV;Z)\), see ([70], Corollaries 1-3). **Example 3**. : _From [70]. Consider the interesting examples with Gaussian channels_  
\[Y_{i}\ =\ X_{i}+S_{i}+N_{i}, \tag{71}\]  
\[Z_{i}\ =\ aX_{i}+bS_{i}+N_{e,i}, \tag{72}\]  
_for some given parameters \(a,b\) and \(\{N_{i}\}\) and \(\{N_{e,i}\}\) memoryless standard Gaussian noise sequences. Let further_  
\[\Xi=S+A, \tag{73}\]  
_for \(A\sim\mathcal{N}(0,\sigma_{A}^{2}\geq 0)\) independent of all other r.vs. This setup covers the scenario where the entire state sequence has to be kept secret, with the choice \(\sigma_{A}^{2}=0\), and (with a slight abuse of notation) the scenario where the state does not have to be kept secret at all, with the choice \(\sigma_{A}^{2}\to\infty\). The following Figure 18 shows an achievable set of rate-distortion pairs according to above Theorem 13 for \(a=0.7\) and \(b=0.3\) and \(S\sim\mathcal{N}(0,3)\) and with a input block power constraint of \(P=30\). The set of achievable rate-distortion pairs is provided without any secrecy constraints neither on messages nor state, with full secrecy constraints on both (\(\Xi=S\)), and with security constraints only on the message but not on the state (\(\Xi\) const). As shown in this example, the additional secrecy constraint on the state is clearly harmful and significantly reduces the set of achievable rates and distortions._