Necessary and sufficient numbers of cards for sharing secret keys on hierarchical groups

Suppose that there are players in two hierarchical groups and a computationally unlimited eavesdropper. Using a random deal of cards, a player in the higher group wishes to send a one-bit message information-theoretically securely either to all the players in her group or to all the players in the two groups. This can be done by the so-called 2-level key set protocol. In this paper we give a necessary and sufficient condition for the 2-level key set protocol to succeed.