**Secret sharing on trees: problem solved**

*Laszlo Csirmaz and 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 Stinson'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.

**Category / Keywords: **foundations / Secret sharing scheme; information rate; graph;

**Date: **received 12 Feb 2009

**Contact author: **csirmaz at degas ceu hu

**Available format(s): **PDF | BibTeX Citation

**Version: **20090216:081458 (All versions of this report)

**Short URL: **ia.cr/2009/071

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