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L. Csirmaz:

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Secret sharing on infinite graphs

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) *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 *local*,
i.e. all finite spanned subgraphs have strictly smaller ratio than the
whole graph.

We conclude the paper with open problems concerning secret sharing on
infinite sets.