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.

翻译：在Shamir的秘密共享方案中，所有参与者拥有相同的权限。然而，在许多实际场景中，常常需要为不同参与者分配不同级别的权限。为满足这一需求，层次秘密共享（HSS）方案应运而生，该方案将所有参与者划分为多个子集，并为每个子集分配独特的权限级别。现有基于中国剩余定理（CRT）的HSS方案虽具有份额大小灵活的优势，但要么存在安全缺陷，要么信息率低于$\frac{1}{2}$。本文利用整数环上的中国剩余定理与单向函数，分别提出了析取型HSS方案和合取型HSS方案。两种方案均为渐近理想方案，并已被证明具有安全性。