Abstract:
Through transforming the access structures on seven participants to a total of 94 connected hypergraphs on seven vertices, the optimal information rate and the construction of perfect secret sharing schemes corresponding to these access structures are given in terms of the relationship between access structures and connected hypergraphs. The exact values for the optimal information rate of the 80 access structures are computed by using hypergraph theory and method, and the relevant construction of perfect secret sharing schemes is discussed. The upper and lower bounds on the information rate of other 14 access structures based on hypergraphs are given by using λ-Decomposition method and so on. At the same time, it is shown that the hypergraph with n vertices and r rank has at least (n-r)/(r-1)+1 edges, and at most Crn edges, and that the optimal information rates of the non-ideal hypergraphs with n(4≤n≤9) vertices and r ranks are all equal to 2/3 if they meets certain conditions.
