Construction of secret key exchange spanning trees by random deals of cards on hierarchical structures

We propose the problem of how to transmit an information-theoretically secu re bit using random deals of cards among players in hierarchical groups and a computationally unlimited eavesdropper. A player in the highest group wa nts to send players in lower groups a secret bit which is secure from the e avesdropper and some other players. We formalize this problem and design pr otocols for constructing secret key exchange spanning trees on hierarchical groups. For each protocol we give sufficient conditions to successfully co nstruct a secret key exchange spanning tree for the hand sizes of the playe rs and the eavesdropper.