Guaranteed Private Communication with Secret Block Structure

TL;DR

The paper addresses physical-layer privacy by exploiting linear inverse problems that are identifiable to a legitimate receiver but intractable for an eavesdropper, achieved by privately sharing a block-structure that induces a block-sparse signal. A transmitter encodes messages as block-sparse vectors, enabling Bob to recover from underdetermined measurements while Eve faces a bilinear inverse problem, with privacy guaranteed under appropriate parameter scaling and block lengths. The authors develop a fourth-order-moment eavesdropping strategy based on spectral clustering, derive asymptotic and finite-snapshot guarantees, and validate the framework through numerical experiments and MIMO applications. This work provides a practical, low-overhead physical-layer privacy mechanism with quantified security lifetimes under key reuse and offers a pathway to extend private communication protocols to broader linear-inverse problems.

Abstract

A novel private communication framework is proposed where privacy is induced by transmitting over a channel instances of linear inverse problems that are identifiable to the legitimate receiver but unidentifiable to an eavesdropper. The gap in identifiability is created in the framework by leveraging secret knowledge between the transmitter and the legitimate receiver. Specifically, the case where the legitimate receiver harnesses a secret block structure to decode a transmitted block-sparse message from underdetermined linear measurements in conditions where classical compressed sensing would provably fail is examined. The applicability of the proposed scheme to practical multiple-access wireless communication systems is discussed. The protocol's privacy is studied under a single transmission, and under multiple transmissions without refreshing the secret block structure. It is shown that, under a specific scaling of the channel dimensions and transmission parameters, the eavesdropper can attempt to overhear the block structure from the fourth-order moments of the channel output. Computation of a statistical lower bound suggests that the proposed fourth-order moment secret block estimation strategy is near optimal. The performance of a spectral clustering algorithm is studied to that end, defining scaling laws on the lifespan of the secret key before the communication is compromised. Finally, numerical experiments corroborating the theoretical findings are conducted.

Figures (9)

• Figure 1: Communication model with secure channel.
• Figure 2: Example of block sparse encoding in dimension $n = 12$, with $r=3$ blocks of length $d=4$.
• Figure 3: Regions of (non)identifiability for Eve and Bob in the single snapshot case for a block-length $d = n \log^{-\delta}(n)$ with $\delta > 0$.
• Figure 4: Success Rate of Bob and Eve to recover $\bx$ for different values of $\beta$ in the absence of noise. The parameters are set to $m = 200$, while $r$ is set to the divisor of $n$ closest to $\log_{10}^2(n)$ (i.e.$r \simeq \log_{10}^2(n)$ and $d \simeq n \log_{10}^{-2}(n)$). The results are averaged over 1000 trials.
• Figure 5: The empirical probabilities of inequality \ref{['eq:coherence-def']} holding for different values of coherence parameters $(\mu,\nu)$. Top row: $\bA$ is a random Gaussian matrix with i.i.d. entries. Bottom row: $\bA$ has columns drawn i.i.d according to a unitary spherical distribution. In blue: $n=50$, in red: $n=200$, in yellow: $n=400$. Herein, we set $\frac{m}{n} = \frac{1}{2}$. Experiments are averaged over 5000 trials.

Theorems & Definitions (13)

• Proposition 1: Success of Bob's decoding
• Proposition 2: Failure of Eve's decoding blanchard2011compressed
• Corollary 3: Single snapshot privacy
• Proposition 4
• Proposition 5: Exact clustering
• proof
• Definition 6: Coherence
• Lemma 7: Concentration of $\bE_{\mathcal{B}}$
• Corollary 8: Asymptotic vulnerability
• Proposition 9: Estimation with Finite Numbers of Snapshots