The role of game theory in key agreement over a public channel

In this work we study the problem of key agreement over a public noiseless channel when Alice, Bob and Charlie observe discrete memoryless sources of an unknown distribution. Alice and Bob want to agree on a key K_AB that is protected from Charlie. At the same time, Alice and Charlie want to agree on a key K_AC that is protected from Bob. In order to construct codebooks for the key agreement, Alice has to know how the sources are distributed. Therefore, she requests Bob and Charlie to send her sufficient information about their observations. We also assume that Bob and Charlie, besides agreeing with Alice on the keys, want to learn as much as possible about the other user´s key: we call this quantity the leakage. We model these reports by having Bob and Charlie select discrete memoryless channels and passing their true observations through them. We approach this problem from a game-theoretic point of view. For a class of Bob and Charlie´s objective functions which are linear in the key rate and the leakage rate, we characterize a Nash equilibrium. Also, we propose a strategy that Alice can apply in order to ensure that Bob and Charlie´s honest reporting is a Nash equilibrium.