### Codes and Game Theory for Key Agreement with Untrusted Participants

**Authors:**

Milosavljevic, Nebojsa

Gastpar, Michael

Ramchandran, Kannan
**Technical Report Identifier:** EECS-2010-179
**December 29, 2010**

EECS-2010-179.pdf

**Abstract:** In this work we study the problem of confidential communication when different resources of randomness are available. We start with the relay channel where source observations are available at each terminal. We study the scenario where the transmitter (Alice) sends a private message to the destination (Bob), which is confidential to the relay (Eve). Alice and Bob also want to agree on a secret key that is protected from Eve. We propose an achievable scheme and show that if the channel is degraded or reversely degraded, the secret message-secret key sum rate is optimal. Then, we consider a scenario where three terminals, Alice, Bob and Charlie, observe memoryless correlated observations. Alice wants to agree with Bob on a key that is secured from Charlie. At the same time Alice wants to agree with Charlie on a key that is secured from Bob. We further assume that Alice has no knowledge on how the distributed sources are correlated, and that she is the one who constructs the codebooks for the key agreement. In order to construct the codebooks which would achieve a desired level of secrecy, Alice request some information from Bob and Charlie about their observations. Since Bob and Charlie act like eavesdroppers to each other, they may not be completely honest about what they observe. Therefore, their reports are based on some objective that is a function of the key rate and the amount of information they can learn about the other user?s key, called the leakage rate. We approach this problem from a game-theoretic point of view. For a class of Bob and Charlie?s objective functions that are linear in the key rate and the leakage rate, we characterize a Nash equilibrium. Then, we propose a strategy that Alice can apply in order to ensure that Bob and Charlie?s honest reporting is always a Nash equilibrium. Finally, for the binary erasure source distributions we extend this concept to the multiple terminal case.