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**2009/071**

**Title:**

Secret sharing on trees: problem solved

**Authors:**
Laszlo Csirmaz, Gabor Tardos

**Abstract:**

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 {\it 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, while the lower bound is achieved by a construction using Stinsonâ€™s decomposition theorem. It is shown that 2-1/c is also the {\it fractional cover number} of the tree where the edges of the tree are covered by stars, and the vertex cover should be minimized. We also give an O(n^2) algorithm which finds an optimal cover on any tree, and thus a perfect secret sharing scheme with optimal rate.

**ePrint:**
https://eprint.iacr.org/2009/071

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