Absolutely Secure Message Transmission using a Key Sharing Graph

Assume that there are players and an eavesdropper Eve of unlimited computational power and that several pairs of players have shared secret keys beforehand. A key sharing graph is one in which each vertex corresponds to a player, and each edge corresponds to a secret key shared by the two players corresponding to the ends of the edge. Given a key sharing graph, a player wishes to send a message to another player so that the eavesdropper Eve and any other player can get no information on the message. In this paper, we give a necessary and sufficient condition on a key sharing graph for the existence of a protocol for such a purpose. We also give a sufficient condition for the existence of a protocol for the case where a player sends a message to a multiple number of designated players.