A Zero-Error Source Coding Solution to the Russian Cards Problem

In the Russian Cards problem, Alice wants to communicate her hand from a deck of cards to Bob through one public announcement such that Cath does not learn her hand. In zero-error source coding, Alice’s goal is to communicate her information to Bob with the least amount of bits. Both problems share the goal of Bob learning Alice’s information in one announcement. However, they have additional objectives: in the Russian Cards case it is security against Cath, for source coding it is minimizing the communication complexity. At a first glance, these two additional properties are not mutually exclusive so we ask whether there is a solution that is both optimal and secure at the same time. This would permit us to model the Russian Cards problem as a zero-error source coding problem to find the optimal secure announcement and it will also enrich source coding by adding the knowledge that it is also secure for an attacker that holds some correlated information. Finally, we deal with the relationship between these two problems. To achieve this, we use results of graph theory and design theory, in particular block graphs and Steiner systems.