Secret sharing on trees: problem solved

We determine the worst case information rate for all secret sharing schemes based on trees. It is the inverse of 2 − 1/c, where c is the size of the maximal core in the tree. A core is a connected subset of the vertices so that every vertex in the core has a neighbor outside the core. The upper bound comes from an application of the entropy method [2, 3], while the lower bound is achieved by a construction using Stinson’s decomposition theorem [7]. It is shown that 2 − 1/c is also the fractional cover number of the tree where the edges of the tree are covered by stars, and the vertex cover should be minimized, cf [5]. We also give an O(n) algorithm which finds an optimal cover on any tree, and thus a perfect secret sharing scheme with optimal rate.