[Paper Review] How to Collaborate between Threshold Schemes

Wang, Daoshun, Zi‐Wei Ye+1 more|arXiv (Cornell University)|May 6, 2013

Cryptography and Data Security2 references3 citations

TL;DR

This paper proposes a novel method to enable efficient collaboration between multiple threshold secret sharing schemes by allowing shared participants to hold a single combined share instead of multiple individual shares. By constructing polynomials with common crossover points, the scheme reduces storage overhead and improves scalability in multi-secret threshold environments, with applications extending to more than two schemes.

ABSTRACT

Threshold schemes have been used to protect secrets by distributing shares to participants. To protect two secrets, we can use two separate traditional schemes, say, a (t1, n1) scheme and a (t2, n2) scheme. If there are u (<=min(t1, t2)) participants involved in both schemes, each of these u participants must keep two different shares. This paper proposes a method that allows each common participant to keep only one share. Our method constructs two polynomials with u common crossover points. We give theoretical details and two demonstrative examples. This algorithm can also handle the collaboration between more than two schemes.

Motivation & Objective

• To address the inefficiency of managing multiple secret shares when participants are involved in more than one threshold scheme.
• To reduce the number of shares each common participant must store from multiple to a single share.
• To design a unified framework that supports collaboration between two or more threshold schemes.
• To maintain security and threshold properties while minimizing storage and communication overhead.

Proposed method

• Construct two or more polynomials, each corresponding to a different secret, such that they share u common points.
• Use interpolation to ensure that the u common participants can reconstruct their respective secrets using only one shared value per participant.
• Apply Lagrange interpolation to derive the shared values at crossover points, ensuring correctness and security.
• Ensure that the degree of each polynomial matches the required threshold (t1, t2, etc.) to preserve the threshold property.
• Generalize the method to handle more than two threshold schemes by extending the polynomial construction to include multiple shared roots.
• Verify that the scheme maintains information-theoretic security and prevents unauthorized reconstruction of secrets.

Experimental results

Research questions

• RQ1 How can participants involved in multiple threshold schemes be reduced to storing only one share instead of multiple?
• RQ2 What mathematical construction allows multiple secret sharing schemes to share common participants with minimal overhead?
• RQ3 Can the proposed method be extended to more than two threshold schemes while preserving security and correctness?
• RQ4 What conditions must be met to ensure that the shared values at crossover points do not compromise the threshold property?
• RQ5 How does the scheme compare in efficiency and security to using separate, independent threshold schemes?

Key findings

• The proposed method reduces the number of shares per common participant from multiple to a single share, significantly improving storage efficiency.
• The scheme maintains information-theoretic security as long as the threshold conditions are satisfied for each individual secret.
• The use of polynomials with u common crossover points enables joint reconstruction of multiple secrets using only one value per participant.
• The method is generalizable to more than two threshold schemes, enabling scalable multi-secret collaboration.
• Two demonstrative examples in the paper confirm the correctness and feasibility of the approach under standard threshold conditions.
• The construction ensures that no participant can learn more than their authorized share, preserving the security guarantees of individual schemes.