International Association for Cryptologic Research

International Association
for Cryptologic Research

CryptoDB

Secret sharing on trees: problem solved

Authors:
Laszlo Csirmaz
Gabor Tardos
Download:
URL: http://eprint.iacr.org/2009/071
Search ePrint
Search Google
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.
BibTeX
@misc{eprint-2009-18222,
  title={Secret sharing on trees: problem solved},
  booktitle={IACR Eprint archive},
  keywords={foundations / Secret sharing scheme; information rate; graph;},
  url={http://eprint.iacr.org/2009/071},
  note={ csirmaz@degas.ceu.hu 14287 received 12 Feb 2009},
  author={Laszlo Csirmaz and Gabor Tardos},
  year=2009
}