A Skilled Secret Sharing Scheme for r-Uniform Hypergraph-Based Prohibited Structures

Secret sharing is that a dealer distributes a piece of information (called a share) about a secret to each participant such that authorized subsets of participants can reconstruct the secret but unauthorized subsets of participants cannot determine the secret. If any unauthorized subset of participants can not obtain any more information regarding the secret, then the scheme is called perfect. A hypergraph is a general graph in which each edge may be formed by more than two vertices. A prohibited structure is a family of the subset of participants such that any subset belongs to this set can not reconstruct the secret. In this paper, we propose a new perfect secret sharing schemes for prohibited structure based on r-uniform hypergraph where a vertex denotes a participant and an edge is a set of participants who cannot recover the secret. The information rate of our schemes is max{r/(C(n 1, r 1) d + r + 1), r/(C(n 1, r 1) d + 1)}, where n is the number of participants and d is the minimum degree of the hypergraph. We will show this scheme is perfect and, makes the scheme of Sun to be a special case. Besides, we also show that this scheme is better than TUM scheme when given a same prohibited structure.