Secure Source Coding Resilient Against Compromised Users via an Access Structure

TL;DR

This work addresses secure source coding with access structures under a lossy distortion constraint for Gaussian sources observed by multiple users. It derives a closed-form rate–leakage region, revealing how the best and worst side-information among authorized and unauthorized sets (captured by tr(Σ_A^{−1}) and tr(Σ_B^{−1})) govern the trade-off, including linear leakage growth when authorized side information is sufficiently informative. The analysis exploits sufficient statistics to reduce vector Gaussian problems to scalar forms and introduces a saddle-point argument to resolve inner–outer region mismatches, enabling exact capacity results. For threshold access structures, the paper provides conditions under which the leakage-rate bound is monotone in the threshold and presents concrete numerical illustrations. These results extend secure secret-sharing-like settings to lossy reconstructions and multiple user sets, with implications for storage systems using public databases and correlated side information.

Abstract

Consider a source and multiple users who observe the independent and identically distributed (i.i.d.) copies of correlated Gaussian random variables. The source wishes to compress its observations and store the result in a public database such that (i) authorized sets of users are able to reconstruct the source with a certain distortion level, and (ii) information leakage to non-authorized sets of colluding users is minimized. In other words, the recovery of the source is restricted to a predefined access structure. The main result of this paper is a closed-form characterization of the fundamental trade-off between the source coding rate and the information leakage rate. As an example, threshold access structures are studied, i.e., the case where any set of at least $t$ users is able to reconstruct the source with some predefined distortion level and the information leakage at any set of users with a size smaller than $t$ is minimized.

Figures (3)

• Figure 1: Secure source coding with three users, i.e., $\mathcal{L}=\{1,2,3\}$, when any single user must not learn more than $n\Delta$ bits of information about the source $X^n$, i.e., we set $\mathbb{A} = \{\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$, and $\mathbb{B} = \{\{1\},\{2\},\{3\}\}$. $\hat{X}^n(\{i,j\})\mathop\lesssim \limits^{D}X^n$, for $i,j\in\{1,2,3\}$ and $i\ne j$, means that the distortion between the reconstructed source by the users $i$ and $j$ together and the source sequence $X^n$ must be less than $D$.
• Figure 2: $(R^\star,\Delta^\star)$ represents the corner points of the rate-leakage region $\mathcal{R}(D,\mathbb{A})$ characterized in Theorem \ref{['thm:Capacity']}, for fixed noise variances, when $\sigma_X^2=2$, $D=0.1$, and ${\mathop{\mathrm{tr}}\nolimits(\boldsymbol{\Sigma}_{\mathcal{B}^\star}^{-1})}=3.5$.
• Figure 3: The rate-leakage region for threshold access structures when $D=0.1$, $\sigma_X^2=2$, and $\boldsymbol{\Sigma}_\mathcal{L}=[10.80.90.70.6]^\intercal$.

Theorems & Definitions (26)

• Definition 1
• Definition 2
• Theorem 1
• Remark 1: Comparison with EkremUlukus13_Lossy and VillardPianta
• Theorem 2
• Theorem 3
• Example 1
• Theorem 4
• proof
• Remark 2: Leakage Measure