Conjunctive Hierarchical Secret Sharing (CHSS) is a type of secret sharing that divides participants into multiple distinct hierarchical levels, with each level having a specific threshold. An authorized subset must simultaneously meet the threshold of all levels. Existing Chinese Remainder Theorem (CRT)-based CHSS schemes either have security vulnerabilities or have an information rate lower than $\frac{1}{2}$. In this work, we utilize the CRT for polynomial ring and one-way functions to construct an asymptotically perfect CHSS scheme. It has computational security, and permits flexible share sizes. Notably, when all shares are of equal size, our scheme is an asymptotically ideal CHSS scheme with an information rate one.

翻译：合取分层秘密共享（CHSS）是一种将参与者划分为多个不同层级、每个层级具有特定阈值的秘密共享方式。任何授权子集必须同时满足所有层级的阈值要求。现有基于中国剩余定理（CRT）的CHSS方案要么存在安全漏洞，要么信息率低于$\frac{1}{2}$。本文利用多项式环上的中国剩余定理与单向函数，构造了一个渐近完美的CHSS方案。该方案具有计算安全性，且允许灵活的份额大小。值得注意的是，当所有份额大小相等时，本方案即为信息率为1的渐近理想CHSS方案。