A {\em probabilistic secret sharing scheme} is a joint probability distribution of the shares and the secret together with a collection of {\em secret recovery functions} for qualified subsets. The scheme is measurable if the recovery functions are measurable. Depending on how much information an unqualified subset might have, we define four scheme types: {\em perfect}, {\em almost perfect}, {\em ramp}, and {\em almost ramp}. Our main results characterize the access structures which can be realized by schemes of these types.
We show that every access structure can be realized by a non-measurable perfect probabilistic scheme. The construction is based on a paradoxical pair of independent random variables which determine each other.
For measurable schemes we have the following complete characterization. An access structure can be realized by a (measurable) perfect, or almost perfect scheme if and only if the access structure, as a subset of the Sierpi\'nski space $\{0,1\}^P$, is open, if and only if it can be realized by a span program. The access structure can be realized by a (measurable) ramp or almost ramp scheme if and only if the access structure is a $G_\delta$ set (intersection of countably many open sets) in the Sierpi\'nski topology, if and only if it can be realized by a Hilbert-space program.
Category / Keywords: foundations / secret sharing Date: received 23 Jul 2012 Contact author: csirmaz at degas ceu hu Available formats: PDF | BibTeX Citation Version: 20120725:191659 (All versions of this report) Discussion forum: Show discussion | Start new discussion