**Infinite Secret Sharing -- Examples**

*Alexander Dibert and Laszlo Csirmaz*

**Abstract: **The motivation for extending secret sharing schemes to cases when either the
set of players is infinite or the domain from which the secret and/or the
shares are drawn is infinite or both, is similar to the case when switching
to abstract probability spaces from classical combinatorial probability. It
might shed new light on old problems, could connect seemingly unrelated
problems, and unify diverse phenomena.

Definitions equivalent in the finitary case could be very much different when switching to infinity, signifying their difference. The standard requirement that qualified subsets should be able to determine the secret has different interpretations in spite of the fact that, by assumption, all participants have infinite computing power. The requirement that unqualified subsets should have no, or limited information on the secret suggests that we also need some probability distribution. In the infinite case events with zero probability are not necessarily impossible, and we should decide whether bad events with zero probability are allowed or not.

In this paper, rather than giving precise definitions, we enlist an abundance of hopefully interesting infinite secret sharing schemes. These schemes touch quite diverse areas of mathematics such as projective geometry, stochastic processes and Hilbert spaces. Nevertheless our main tools are from probability theory. The examples discussed here serve as foundation and illustration to the more theory oriented companion paper ``Probabilistic Infinite Secret Sharing.''

**Category / Keywords: **foundations / secret sharing, information theory

**Date: **received 23 Jul 2012

**Contact author: **csirmaz at degas ceu hu

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

**Version: **20120725:191522 (All versions of this report)

**Short URL: **ia.cr/2012/411

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