Disjunctive Hierarchical Secret Sharing (DHSS)} scheme is a type of secret sharing scheme in which the set of all participants is partitioned into disjoint subsets, and each subset is said to be a level with different degrees of trust and different thresholds. In this work, we focus on the Chinese Remainder Theorem (CRT)-based DHSS schemes due to their ability to accommodate flexible share sizes. We point out that the ideal DHSS scheme of Yang et al. (ISIT, 2024) and the asymptotically ideal DHSS scheme of Tiplea et al. (IET Information Security, 2021) are insecure. Consequently, existing CRT-based DHSS schemes either exhibit security flaws or have an information rate less than $\frac{1}{2}$. To address these limitations, we propose a CRT-based asymptotically perfect DHSS scheme that supports flexible share sizes. Notably, our scheme is asymptotically ideal when all shares are equal in size. Its information rate achieves one and it has computational security.

翻译：析取层次秘密共享（DHSS）方案是一种秘密共享方案，其中所有参与者的集合被划分为互不相交的子集，每个子集被称为一个层级，具有不同的信任度和不同的阈值。在本工作中，我们关注基于中国剩余定理（CRT）的DHSS方案，因为它们能够容纳灵活的份额大小。我们指出，Yang等人（ISIT，2024）的理想DHSS方案和Tiplea等人（IET Information Security，2021）的渐近理想DHSS方案是不安全的。因此，现有的基于CRT的DHSS方案要么存在安全缺陷，要么其信息率小于$\frac{1}{2}$。为了解决这些局限性，我们提出了一种基于CRT的渐近完美DHSS方案，该方案支持灵活的份额大小。值得注意的是，当所有份额大小相等时，我们的方案是渐近理想的。其信息率达到一，并且具有计算安全性。