Disjunctive Hierarchical Secret Sharing (DHSS) scheme is a secret sharing scheme in which the set of all participants is partitioned into disjoint subsets. Each disjoint subset is said to be a level, and different levels have different degrees of trust and different thresholds. If the number of cooperating participants from a given level falls to meet its threshold, the shortfall can be compensated by participants from higher levels. Many ideal DHSS schemes have been proposed, but they often suffer from big share sizes. Conversely, existing non-ideal DHSS schemes achieve small share sizes, yet they fail to be both secure and asymptotically ideal simultaneously. In this work, we present an explicit construct of an asymptotically ideal DHSS scheme by using a polynomial, multiple linear homogeneous recurrence relations and one-way functions. Although our scheme has computational security and many public values, it has a small share size and the dealer is required polynomial time.

翻译：析取层次秘密共享（DHSS）方案是一种将全体参与者划分为互不相交子集的秘密共享方案。每个互不相交子集称为一个层级，不同层级具有不同的信任度与不同的阈值。若某层级中参与协作的参与者数量未达到其阈值，可由更高层级的参与者进行差额补偿。现有许多理想DHSS方案被提出，但常面临份额尺寸过大的问题。反之，现有非理想DHSS方案虽能实现较小的份额尺寸，却无法同时满足安全性与渐进理想性。本研究通过运用多项式、多重线性齐次递推关系及单向函数，提出了一种渐进理想DHSS方案的显式构造。尽管该方案具有计算安全性且包含大量公开值，但其份额尺寸较小，且分发者仅需多项式时间即可完成操作。