  We present the first formal mathematical presentation of the generalized
Russian cards problem, and provide rigorous security definitions that capture
both basic and extended versions of weak and perfect security notions. In the
generalized Russian cards problem, three players, Alice, Bob, and Cathy, are
dealt a deck of $n$ cards, each given $a$, $b$, and $c$ cards, respectively.
The goal is for Alice and Bob to learn each other's hands via public
communication, without Cathy learning the fate of any particular card. The
basic idea is that Alice announces a set of possible hands she might hold, and
Bob, using knowledge of his own hand, should be able to learn Alice's cards
from this announcement, but Cathy should not. Using a combinatorial approach,
we are able to give a nice characterization of informative strategies (i.e.,
strategies allowing Bob to learn Alice's hand), having optimal communication
complexity, namely the set of possible hands Alice announces must be equivalent
to a large set of $t-(n, a, 1)$-designs, where $t=a-c$. We also provide some
interesting necessary conditions for certain types of deals to be
simultaneously informative and secure. That is, for deals satisfying $c = a-d$
for some $d \geq 2$, where $b \geq d-1$ and the strategy is assumed to satisfy
a strong version of security (namely perfect $(d-1)$-security), we show that $a
= d+1$ and hence $c=1$. We also give a precise characterization of informative
and perfectly $(d-1)$-secure deals of the form $(d+1, b, 1)$ satisfying $b \geq
d-1$ involving $d-(n, d+1, 1)$-designs.
