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On Distributed Multi-User Secret Sharing with Multiple Secrets per User

Rasagna Chigullapally, Harshithanjani Athi, Nikhil Karamchandani, V. Lalitha

TL;DR

This work generalizes distributed multi-user secret sharing to allow each user to request multiple secrets, linking weak secrecy to $t$-disjunct storage-structure matrices. It derives a necessary condition on storage sets, constructs a DSSP with optimal storage overhead $SO=1$ using a $t$-disjunct-based storage profile and Shamir decoding, and compares multiple disjunct-matrix constructions (Kautz-Singleton, Porat-Rothschild, sparse-disjunct) as well as Steiner systems. The paper also provides bounds on the optimal communication complexity and characterizes the capacity region for DMUSS under prescribed access structures, extending prior results to multi-secret requests and heterogeneous secret sizes. These results clarify the tradeoffs between storage efficiency, communication, and secrecy in distributed storage environments and offer guidance on selecting matrix-construction paradigms for scalable DMUSS deployments.

Abstract

We consider a distributed multi-user secret sharing (DMUSS) setting in which there is a dealer, $n$ storage nodes, and $m$ secrets. Each user demands a $t$-subset of $m$ secrets. Earlier work in this setting dealt with the case of $t=1$; in this work, we consider general $t$. The user downloads shares from the storage nodes based on the designed access structure and reconstructs its secrets. We identify a necessary condition on the access structures to ensure weak secrecy. We also make a connection between access structures for this problem and $t$-disjunct matrices. We apply various $t$-disjunct matrix constructions in this setting and compare their performance in terms of the number of storage nodes and communication complexity. We also derive bounds on the optimal communication complexity of a distributed secret sharing protocol. Finally, we characterize the capacity region of the DMUSS problem when the access structure is specified.

On Distributed Multi-User Secret Sharing with Multiple Secrets per User

TL;DR

This work generalizes distributed multi-user secret sharing to allow each user to request multiple secrets, linking weak secrecy to -disjunct storage-structure matrices. It derives a necessary condition on storage sets, constructs a DSSP with optimal storage overhead using a -disjunct-based storage profile and Shamir decoding, and compares multiple disjunct-matrix constructions (Kautz-Singleton, Porat-Rothschild, sparse-disjunct) as well as Steiner systems. The paper also provides bounds on the optimal communication complexity and characterizes the capacity region for DMUSS under prescribed access structures, extending prior results to multi-secret requests and heterogeneous secret sizes. These results clarify the tradeoffs between storage efficiency, communication, and secrecy in distributed storage environments and offer guidance on selecting matrix-construction paradigms for scalable DMUSS deployments.

Abstract

We consider a distributed multi-user secret sharing (DMUSS) setting in which there is a dealer, storage nodes, and secrets. Each user demands a -subset of secrets. Earlier work in this setting dealt with the case of ; in this work, we consider general . The user downloads shares from the storage nodes based on the designed access structure and reconstructs its secrets. We identify a necessary condition on the access structures to ensure weak secrecy. We also make a connection between access structures for this problem and -disjunct matrices. We apply various -disjunct matrix constructions in this setting and compare their performance in terms of the number of storage nodes and communication complexity. We also derive bounds on the optimal communication complexity of a distributed secret sharing protocol. Finally, we characterize the capacity region of the DMUSS problem when the access structure is specified.
Paper Structure (14 sections, 6 theorems, 15 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 14 sections, 6 theorems, 15 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. System Model
  4. System Model
  5. Shamir's Secret Sharing Scheme
  6. DSSP with Optimal Storage Overhead
  7. Conditions on storage sets to ensure weak secrecy
  8. DSSP with Optimal Storage Overhead
  9. Comparison between different constructions of disjunct matrices
  10. Kautz-Singleton Construction KS06
  11. Porat-Rothschild Construction PR11
  12. Sparse Disjunct Matrices IKO20
  13. Bounds on Optimal Communication Complexity
  14. Capacity region of Distributed Multi-User Secret Sharing

Key Result

Lemma 1

For any weakly secure DSSP with a storage structure $\mathcal{A}$ defined in eq:accessstructure, we have $A_{j_{t+1}}\nsubseteq \bigcup _{k=1}^t A_{j_k}$, $\forall\ j_1,j_2,\ldots,j_{t+1}\in [m]$ with $j_1\neq j_2\neq \cdots \neq j_{t+1}$.

Figures (2)

  • Figure 1: System Model
  • Figure 2: Comparison of disjunct matrix constructions.

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Definition 4
  • Lemma 2
  • proof
  • Definition 5
  • Conjecture 1
  • ...and 10 more