Sliding Secure Symmetric Multilevel Diversity Coding

TL;DR

This paper generalizes SMDC to a sliding secure setting with security priority, introducing the $(L,s)$ framework that governs reconstruction and secrecy as a function of encoder access. It shows that for $s=1$ superposition coding remains optimal, while for $s\ge2$ joint encoding—using prior sources as secret keys—achieves improved rate regions; the authors provide complete rate-region characterizations for the $(3,2)$ MSS and $(3,2)$ sliding SMDC problems and propose a minimum-sum-rate scheme for general $(L,s)$ based on a hybrid of superposition and joint encoding. A key insight is that encoding $X_{\alpha}$ can be secured by correlations with $X_{\alpha-1}$, enabling rate reductions through secret-key chaining. The results establish a nuanced picture where simple superposition is optimal only in the $s=1$ or certain structured cases, while more flexible joint schemes are necessary to approach optimality in broader settings, with practical implications for secure distributed storage and secret sharing configurations. The work also connects sliding secure SMDC to ramp secret sharing and MSS, contributing new coding strategies and provable rate bounds that inform security-priority coding design.

Abstract

Symmetric multilevel diversity coding (SMDC) is a source coding problem where the independent sources are ordered according to their importance. It was shown that separately encoding independent sources (referred to as ``\textit{superposition coding}") is optimal. In this paper, we consider an $(L,s)$ \textit{sliding secure} SMDC problem with security priority, where each source $X_α~(s\leq α\leq L)$ is kept perfectly secure if no more than $α-s$ encoders are accessible. The reconstruction requirements of the $L$ sources are the same as classical SMDC. A special case of an $(L,s)$ sliding secure SMDC problem that the first $s-1$ sources are constants is called the $(L,s)$ \textit{multilevel secret sharing} problem. For $s=1$, the two problems coincide, and we show that superposition coding is optimal. The rate regions for the $(3,2)$ problems are characterized. It is shown that superposition coding is suboptimal for both problems. The main idea that joint encoding can reduce coding rates is that we can use the previous source $X_{α-1}$ as the secret key of $X_α$. Based on this idea, we propose a coding scheme that achieves the minimum sum rate of the general $(L,s)$ multilevel secret sharing problem. Moreover, superposition coding of the $s$ sets of sources $X_1$, $X_2$, $\cdots$, $X_{s-1}$, $(X_s, X_{s+1}, \cdots, X_L)$ achieves the minimum sum rate of the general sliding secure SMDC problem.

Figures (5)

• Figure 1: The sliding secure SMDC model.
• Figure 2: rate region $\mathcal{R}_1^*$: case i ($H_2<\frac{2}{3}H_3$)
• Figure 3: rate region $\mathcal{R}_1^*$: case ii ($\frac{2}{3}H_3\leq H_2<H_3$)
• Figure 4: rate region $\mathcal{R}_1^*$: case iii ($H_3\leq H_2<2H_3$)
• Figure 5: rate region $\mathcal{R}_1^*$: case iv ($2H_3\leq H_2$)

Theorems & Definitions (33)

• Lemma 1
• Theorem 1
• proof
• Remark 1
• Remark 2
• Remark 3
• Theorem 2
• proof
• Lemma 2
• proof