Asymptotically Ideal Hierarchical Secret Sharing Based on CRT for Integer Ring

2026-03-23

AI Summary

This work addresses the limitations of traditional Shamir’s secret sharing, which lacks support for differentiated access privileges, and improves upon existing Chinese Remainder Theorem (CRT)-based hierarchical secret sharing schemes that either suffer from security flaws or achieve an information rate below 1/2. The paper proposes two novel hierarchical secret sharing schemes—disjunctive and conjunctive—leveraging CRT over integer rings combined with one-way functions to enable flexible authorization through carefully designed hierarchical access structures. Notably, the proposed schemes are the first to achieve asymptotically ideal security, with an information rate approaching 1 as the number of participants grows large, thereby offering a significant improvement over current CRT-based hierarchical approaches.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Machine Learning: Multi-instance/Multi-view Learning

Application Category

• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

In Shamir's secret sharing scheme, all participants possess equal privileges. However, in many practical scenarios, it is often necessary to assign different levels of authority to different participants. To address this requirement, Hierarchical Secret Sharing (HSS) schemes were developed, which partitioned all participants into multiple subsets and assigned a distinct privilege level to each. Existing Chinese Remainder Theorem (CRT)-based HSS schemes benefit from flexible share sizes, but either exhibit security flaws or have an information rate less than $\frac{1}{2}$. In this work, we propose a disjunctive HSS scheme and a conjunctive HSS scheme by using the CRT for integer ring and one-way functions. Both schemes are asymptotically ideal and are proven to be secure.

Problem

• Hierarchical Secret Sharing
• Chinese Remainder Theorem
• information rate
• security
• asymptotically ideal

Innovation

• Hierarchical Secret Sharing
• Chinese Remainder Theorem
• Asymptotically Ideal
• One-way Functions
• Information Rate