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In the generalized Russian cards problem, we have a card deck $X$ of $n$cards and three participants, Alice, Bob, and Cathy, dealt $a$, $b$, and $c$cards, respectively. Once the cards are dealt, Alice and Bob wish to privatelycommunicate their hands to each other via public announcements, without theadvantage of a shared secret or public key infrastructure. Cathy should remainignorant of all but her own cards after Alice and Bob have made theirannouncements. Notions for Cathy's ignorance in the literature range from Cathynot learning the fate of any individual card with certainty (weak $1$-security)to not gaining any probabilistic advantage in guessing the fate of some set of$\delta$ cards (perfect $\delta$-security). As we demonstrate, the generalizedRussian cards problem has close ties to the field of combinatorial designs, onwhich we rely heavily, particularly for perfect security notions. Our mainresult establishes an equivalence between perfectly $\delta$-secure strategiesand $(c+\delta)$-designs on $n$ points with block size $a$, when announcementsare chosen uniformly at random from the set of possible announcements. We alsoprovide construction methods and example solutions, including a constructionthat yields perfect $1$-security against Cathy when $c=2$. We leverage a knowncombinatorial design to construct a strategy with $a=8$, $b=13$, and $c=3$ thatis perfectly $2$-secure. Finally, we consider a variant of the problem thatyields solutions that are easy to construct and optimal with respect to boththe number of announcements and level of security achieved. Moreover, this isthe first method obtaining weak $\delta$-security that allows Alice to hold anarbitrary number of cards and Cathy to hold a set of $c = \lfloor\frac{a-\delta}{2} \rfloor$ cards. Alternatively, the construction yieldssolutions for arbitrary $\delta$, $c$ and any $a \geq \delta + 2c$.

Additional Constructions to Solve the Generalized Russian Cards Problem using Combinatorial Designs