Asymptotically Ideal Hierarchical Secret Sharing Based on CRT for Integer Ring

TL;DR

This paper introduces two asymptotically ideal and secure hierarchical secret sharing schemes based on the Chinese Remainder Theorem for integer rings, addressing security flaws and efficiency issues in prior CRT-based schemes.

Contribution

It proposes disjunctive and conjunctive hierarchical secret sharing schemes that are asymptotically ideal and secure, improving upon existing CRT-based methods.

Findings

Both schemes are proven to be secure.

They are asymptotically ideal in share size.

Address security flaws of previous CRT-based HSS schemes.

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.