**Secret sharing on infinite graphs**

*Laszlo Csirmaz*

**Abstract: **We extend the notion of perfect secret sharing scheme for access structures with infinitely many participants. In particular we investigate cases when the participants are the vertices of an (infinite) graph, and the minimal qualified sets are the edges. The (worst case) {\it information ratio} of an access structure is the largest lower bound on the amount of information some participant must remember for each bit in the secret -- just the inverse of the information rate. We determine this value for several infinite graphs: infinite path, two-dimensional square and honeycomb lattices; and give upper and lower bounds on the ratio for the triangular lattice.
It is also shown that the information ratio is not necessarily {\em local}, i.e.~all finite spanned subgraphs have strictly smaller ratio than the whole graph. We conclude the paper by posing several open problems.

**Category / Keywords: **foundations / Secret sharing scheme, information

**Publication Info: **Submitted to Tatra Mountains Mathematical Publications

**Date: **received 1 Aug 2007

**Contact author: **csirmaz at renyi hu

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

**Version: **20070807:154231 (All versions of this report)

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