# Exploring ISAC: Information-Theoretic Insights
## 6 Secrecy of ISAC Systems
### Secrecy of the Message: The Memoryless Model  
Based on the memoryless ISAC model in [56], a wiretap equivalent was introduced in [69], see Figure 16. In this model, communication needs to be such that the eavesdropper cannot learn part of the message, which is formalized by the requirement that the equivocation between this specific message part and [PERSON]'s observations should vanish for large blocklengths (This requirement is also known as strong secrecy in the information-theoretic literature).  
Formally, the problem is defined with a Tx, a legitimate Rx, and an eavesdropper ([PERSON]). The Tx aims to communicate a pair of messages \((W_{1},W_{2})\) of rates \(R_{1}\) and \(R_{2}\) to the legitimate receiver, in a way that [PERSON] cannot learn any information about the message \(W_{1}\). (There is no constraint on how much [PERSON] learns about the other message \(W_{2}\). Communication is over a memoryless stationary state-dependent channel \(P_{Y_{1}Y_{2}Z|SX}\) where \(X\) is the channel input, \(S_{1}\) and \(S_{2}\) are the states, \(Y_{1}\) the outputs at the legitimate receiver, \(Y_{2}\) the outputs at the eavesdropper, and \(Z\) The generalized feedback. The state sequences \(\{(S_{1,i},S_{2,i})\}\) are assumed i.i.d. according to the given law \(P_{S_{1}S_{2}}\), and the transmitter creates the time-\(i\) channel inputs as \(X_{i}=\phi_{i}(W_{1},W_{2},Z_{i-1})\) using some appropriate encoding function \(\phi_{i}\). At the end of the communication, the Tx estimates the state sequence as \((\hat{S}_{1},\hat{S}_{2}^{n})=h(X^{n},Z^{n})\). The receiver decodes the two messages as \((\hat{W}_{1},\hat{W}_{2})=g(Y_{1}^{n},S_{1}^{n})\) using an appropriate encoding function. The goal of the communication is that decoding error probability vanish asymptotically, that the reconstructed state sequence matches the correct state up to a given distortion constraints \(D_{1}\) and \(D_{2}\) under given per-symbol distortion measures \(d_{1}(\cdot,\cdot)\) and \(d_{2}(\cdot,\cdot)\), and that [PERSON] learns nothing about message \(W_{2}\) from her observations \(Y_{2}^{n}\) and \(S_{2}^{n}\).  
Definition 5: A secrecy-rate-distortion tuple \((R_{1},R_{2},D_{1},D_{2})\) is achievable if it is possible to find a sequence (in the blocklength \(n\)) of encoding, decoding, and estimation functions satisfying  
\[\lim_{n\rightarrow\infty}\Pr[W_{k}\
eq\hat{W}_{l}]\ =\ 0,\quad k\in\{1,2\}, \tag{57}\]  
\[\lim_{n\rightarrow\infty}I(W_{2};Y_{2}^{n},S_{2}^{n})\ =\ 0, \tag{58}\]  
\[\varlimsup_{n\rightarrow\infty}E[d(S_{k}^{n},\hat{S}_{k}^{n})]\ \leq\ D_{k},\quad k\in\{1,2\}. \tag{59}\]  
The closure of the set of all achievable secrecy-rate-distortion tuples \((R_{1},R_{2},D_{1},D_{2})\) is called the secrecy-capacity-distortion region \(\mathcal{SCD}\).  
Here, we chose to present the slightly restricted model where the Rxs learn the two state sequences, thus not allowing for no or only imperfect state information. A more general model can, however, easily be obtained similar to the models presented in the previous sections. The reason for considering this special case is that in the following, we will limit to the special case where the Tx observes perfect output feedback, i.e., \(Z=(Y_{1},Y_{2})\) without the two states, which does not allow the incorporation of arbitrary channel state information distributions at the Tx and the Rx/Eve.  
Figure 16: ISAC model under partial secrecy, where only \(W_{2}\) should be kept secret from Eve.