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.\'\'