On the Extremal Source Key Rates for Secure Storage over Graphs

TL;DR

The paper introduces a graph-based model of secure storage where $K$ source symbols and a shared key are stored across $N$ nodes, and edges reveal exactly $M$ of the sources while preserving secrecy from others. It characterizes the extremal source key capacity $C$ under different $M$ and structural conditions, showing $C=1$ for $M=1$ under a no-internal-qualified-edge condition, and $C=1/M$ for general $M$ under a broader no-internal-qualified-edge criterion; it also identifies when secure storage can be achieved with no source key ($L_Z=0$) using a precise union-of-common-sources condition. The proofs rely on alignment techniques (noise alignment and coded-symbol alignment), per-source characteristic graphs, and randomized linear coding with Schwartz–Zippel arguments to ensure decodability, providing both necessity and sufficiency results. These findings reveal how graph topology fundamentally constrains secrecy-storage tradeoffs and offer a complete picture of extremal graph structures supporting secure storage without excessive key material.

Abstract

This paper investigates secure storage codes over graphs, where multiple independent source symbols are encoded and stored at graph nodes subject to edge-wise correctness and security constraints. For each edge, a specified subset of source symbols must be recoverable from its two incident nodes, while no information about the remaining sources is revealed. To meet the security requirement, a shared source key may be employed. The ratio between the source symbol size and the source key size defines the source key rate, and the supremum of all achievable rates is referred to as the source key capacity. We study extremal values of the source key capacity in secure storage systems and provide complete graph characterizations for several fundamental settings. For the case where each edge is associated with a single source symbol, we characterize all graphs whose source key capacity equals one. We then generalize this result to the case where each edge is associated with multiple source symbols and identify a broad class of graphs that achieve the corresponding extremal capacity under a mild structural condition. In addition, we characterize all graphs for which secure storage can be achieved without using any source key.

Figures (7)

• Figure 1: An example of a secure storage problem with $K=3$ source symbols and $N=8$ coded symbols. The source key capacity of this graph is $1/2$ (see Theorem \ref{['thm:d2']}; a corresponding code construction is shown in Fig. \ref{['fig5']}). This model can be interpreted as storing three files $W_1, W_2, W_3$ over eight servers $V_1,\ldots,V_8$, where certain pairs of servers allow secure retrieval of specific files.
• Figure 2: (a) An example graph $G$ with $K=3$ source symbols, $N=8$ coded symbols, and $M=2$, where each edge is associated with a subset of two source symbols; (b) the corresponding characteristic graph $G^{[1]}$ for source symbol $W_1$.
• Figure 3: An example of a graph $G$ with $K=2$, $N=7$, and $M=1$, where $\widetilde{G}=\emptyset$. Here, $W_1=(a_1,a_2)$ and $W_2=(b_1,b_2)$, and no key is required.
• Figure 4: (a) An example graph $G$ with $K=3$, $N=8$, and $M=1$, and (b) the characteristic graph $G^{[1]}$ corresponding to $W_1$. The source key capacity over $G$ is $1$, and a capacity-achieving code construction is illustrated.
• Figure 5: (a) An example graph $G$ with $K=2, N=4, M=1$ and (b) its $G^{[1]}$ of $W_1$. The source key capacity over $G$ cannot be $1$.

Theorems & Definitions (19)

• Definition 1: Characteristic Graph $G^{[k]}$ of $W_k$
• Definition 2: Qualified and Unqualified Edges, Paths, and Components
• Definition 3: Common Sources $\mathcal{C}(V)$
• Definition 4: Non-degenerate Subgraph $\widetilde{G}$ of $G$
• Theorem 1
• Remark 1
• Example 1
• Example 2
• Theorem 2
• Remark 2