Journal Extension Discussion

Secure ISAC

Sidong Guo

2026-07-03

Previously on the Converse Attempts

  • Type extraction via pigeonhole principle
    • For fixed length converse, polynomial n-type and exponential codewords
    • Extract a constant composition codebook

Problem: This will not work with feedback, since each message corresponds to a tree of codewords, there is no single type to pigeonhole over

  • Wagner Style Proof [1]
    • If a policy achieves capacity, show that its induced distribution has to be \(\gamma\)-close to ideal distribution
    • Divide the meta-converse distribution into two sets and use a modified policy and change of measure argument to evaluate the \(\gamma\)-close distribution

Problem: This does not conclude a real subcode since there is no reliability and secrecy guarantee, it is a neat way to analyze a mathematical quantity but insufficient for our application

[1] A. Wagner et al., “A New Method for Employing Feedback to Improve Coding Performance,” IEEE Trans. Info. Theory., 2020.

New Converse Proof Attempt–Combining both ideas

Partition messages, not codeword paths — and attach a deterministic profile to each message root.

  1. Reduce the rate constraint to a fixed prefix message. Let \(W\) be the first \(k_n=\lfloor nR\rfloor\) message bits. The event that fewer than \(k_n\) bits are sent is absorbed into decoding error, so the prefix code has rate \(R-o(1)\) and error \(\le \delta_1+\delta_4\).

Note that this first step will require a doubled sided concentration of stopping time around \(n\), which is true in our achievability but not an assumption in converse

  1. Attach a deterministic distribution to each message. For each prefix message \(w\) and state \(s\), \[ Q_{w,s}^{(n)}(x)=\frac{\mathbb{E}_{s}\!\left[\sum_{t=1}^{\tau}\mathbb{1}\{X_t=x\}\,\middle|\,W=w\right]}{\mathbb{E}_{s}[\tau\mid W=w]}. \] This averages over the full feedback tree — deterministic once \(w,s\) are fixed.

New Converse Proof Attempt (cont.)

  1. Quantize the statewise vector \(\bigl(Q_{w,s}^{(n)}\bigr)_{s\in \mathcal{S}}\) with a \(\gamma\)-net, \(\gamma\to 0\), polynomially many cells.

A \(\gamma\)-net \(\mathcal{N}_{\gamma}\) of \(\mathcal{P}_{\mathcal X}^{|\mathcal{S}|}\) is a finite set such that, for every \(\left(P_s\right)_{s\in\mathcal{S}}\in \mathcal{P}_{\mathcal X}^{|\mathcal{S}|}\), there exists \(\left(P_{X,s}^{(j)}\right)_{s\in\mathcal{S}}\in\mathcal{N}_{\gamma}\) satisfying \[ \max_{s\in\mathcal{S}} \mathbb{V}\left(P_s,P_{X,s}^{(j)}\right) \leq \gamma. \]

  1. Select a large secure shell cell. With cell index \(J\), \[ \mathbb{I}(W;Y_2^\tau)\ge \sum_j \Prob(J=j)\,\mathbb{I}(W;Y_2^\tau\mid J=j), \] so almost all cell mass is secure. One cell has exponentially many messages, leakage \(\to 0\), and a common shell center \(P_{X,s}^{(n)}\): \[ \V\!\left(Q_{X,s}^{\star(n)},\,P_{X,s}^{(n)}\right)\le \gamma,\qquad \forall s\in\mathcal S. \]

The extracted subcode is not constant composition code

New Converse Proof Attempt (cont.)

  1. Rate. Fano on the retained prefix message, then the feedback chain rule and concavity of mutual information, gives the main-channel bound directly in \(Q_{X,s}^{\star(n)}\). Similarly get the secrecy-rate bound. Finally \(\V(Q_{X,s}^{\star(n)},P_{X,s}^{(n)})\le\gamma\) and TV continuity bound replace \(Q_{X,s}^{\star(n)}\) by \(P_{X,s}^{(n)}\). \(E_1\) is linear in input type, so we can probably get the same bound transfer with \(E_1\).

  2. What probably does not work. My current \(E_2\) is a problem

\(R\) — Likely valid

Distribution extraction by message partition \(\to\) direct feedback converse \(\to\) continuity.

\(E_1\) — Probably valid

continuity.

\(E_2\) — probably not valid. Eve’s hypothesis testing exponent depends on the full feedback-induced output mixture (stated below), therefore not linear in input. The shell center \(P_{X,s}^{(n)}\) does not by itself certify Eve’s exponent.

\[ E_2 \geq \underset{s'\neq s}{\min} \min_{\varrho \in [0,1)}\frac{\varrho^2\mathbb{D}^2(W_{Y_1|X,s}||W_{Y_1|X,s'}|P_{X,s})}{2(1-\varrho) \phi^2_{s,s'}} +(1-\varrho)\mathbb{C}(P_{X,s}\circ W_{Y_2|X,s}||P_{X,s'}\circ W_{Y_2|X,s'}) \]

  • A potential solution involves a looser converse that does not involve Chernoff Information between mixture distributions.