### 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.

Available format(s)
Category
Foundations
Publication info
Published elsewhere. Unknown where it was published
Keywords
Secret sharing schemeinformation rategraph
Contact author(s)
csirmaz @ degas ceu hu
History
Short URL
https://ia.cr/2009/071

CC BY

BibTeX

@misc{cryptoeprint:2009/071,
author = {Laszlo Csirmaz and Gabor Tardos},
title = {Secret sharing on trees: problem solved},
howpublished = {Cryptology ePrint Archive, Paper 2009/071},
year = {2009},
note = {\url{https://eprint.iacr.org/2009/071}},
url = {https://eprint.iacr.org/2009/071}
}

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