**Probabilistic Infinite Secret Sharing**

*Laszlo Csirmaz*

**Abstract: **The study of probabilistic secret sharing schemes using arbitrary
probability spaces and possibly infinite number of participants lets us
investigate abstract properties of such schemes. It highlights important
properties, explains why certain definitions work better than others,
connects this topic to other branches of mathematics, and might yield new
design paradigms.

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 format(s): **PDF | BibTeX Citation

**Version: **20120725:191659 (All versions of this report)

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