Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Monotone Circuit Construction for Individually-Secure Multi-Secret Sharing

Cailyn Bass, Alejandro Cohen, Rafael G. L. D'Oliveira, Muriel Médard

TL;DR

This work addresses increasing the information rate in secret sharing with general access structures by embedding multi-secret sharing with individual security into a monotone circuit framework. It introduces a share-replacement technique that substitutes up to $m-1$ shares with linear combinations of messages, while preserving decodability and ensuring each $M_\ell$ remains individually secure against unauthorized subsets. The authors prove that the resulting scheme achieves an information rate of $R_{MS}=\frac{m}{|S|}$ and provide conditions under which replacements are valid; they also show that replacing non-replaceable shares destroys decodability. A concrete four-party example illustrates the method and rate gains, highlighting the practical potential for more efficient, securely distributed secret sharing in general access-structure settings.

Abstract

In this work, we introduce a new technique for taking a single-secret sharing scheme with a general access structure and transforming it into an individually secure multi-secret sharing scheme where every secret has the same general access structure. To increase the information rate, we consider Individual Security which guarantees zero mutual information with each secret individually, for any unauthorized subsets. Our approach involves identifying which shares of the single-secret sharing scheme can be replaced by linear combinations of messages. When $m-1$ shares are replaced, our scheme obtains an information rate of $m/|S|$, where $S$ is the set of shares. This provides an improvement over the information rate of $1/|S|$ in the original single-secret sharing scheme.

A Monotone Circuit Construction for Individually-Secure Multi-Secret Sharing

TL;DR

This work addresses increasing the information rate in secret sharing with general access structures by embedding multi-secret sharing with individual security into a monotone circuit framework. It introduces a share-replacement technique that substitutes up to shares with linear combinations of messages, while preserving decodability and ensuring each remains individually secure against unauthorized subsets. The authors prove that the resulting scheme achieves an information rate of and provide conditions under which replacements are valid; they also show that replacing non-replaceable shares destroys decodability. A concrete four-party example illustrates the method and rate gains, highlighting the practical potential for more efficient, securely distributed secret sharing in general access-structure settings.

Abstract

In this work, we introduce a new technique for taking a single-secret sharing scheme with a general access structure and transforming it into an individually secure multi-secret sharing scheme where every secret has the same general access structure. To increase the information rate, we consider Individual Security which guarantees zero mutual information with each secret individually, for any unauthorized subsets. Our approach involves identifying which shares of the single-secret sharing scheme can be replaced by linear combinations of messages. When shares are replaced, our scheme obtains an information rate of , where is the set of shares. This provides an improvement over the information rate of in the original single-secret sharing scheme.
Paper Structure (13 sections, 10 theorems, 15 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 13 sections, 10 theorems, 15 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries on Single-Secret Sharing
  3. Single-Secret Sharing
  4. Monotone Circuit Construction
  5. Multi-Secret Sharing
  6. An Example of Our Approach
  7. Single-Secret Sharing for a General Access Structure
  8. The Conditions for Replacing a Share
  9. From Single-Secret to Multi-Secret Sharing Schemes
  10. Main Results
  11. Proof of Lemma \ref{['lem: mainSecure']} - Individual Security
  12. Proof of Theorem \ref{['teo: nonRep']} - Non-Replaceable
  13. Preliminaries

Key Result

Proposition 1

$\mathrm{H}(M_1,M_2,M_3,M_4|S_A)=0$ where $S_A$ is the set of shares held by an authorized subset, i.e. every subset $A\in\Gamma_0$ can compute all messages $M_1,M_2,M_3,M_4$.

Figures (1)

  • Figure 1: The monotone circuit for $\Gamma_0$, which can be represented by the following monotone Boolean function: $(P_1\land P_2\land P_4)\lor(P_1\land P_3\land P_4)\lor(P_2\land P_3)$. Each $\land$ gate corresponds to adding the shares on the input wires and thus represents a secret sharing scheme among the participants of $A_i$. These smaller schemes are the key to proving the monotone circuit single-secret sharing scheme is decodable and secure. The replaceable shares for our construction are shown in blue.

Theorems & Definitions (24)

  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Proposition 1: Decodability
  • proof
  • Proposition 2: Individual Security
  • proof
  • Definition 4
  • Theorem 1
  • ...and 14 more