Triangle Network Secrecy

We characterize the secret message capacity of the triangle network, that consists of a source, a relay and a destination connected through orthogonal erasure channels. A passive eavesdropper, Eve, wiretaps any one of the three channels. The source and the relay can each generate unlimited private randomness; the relay and the destination can publicly provide strictly causal channel state information. Our achievable scheme is expressed through a linear program (LP) with 11 inequalities that captures a minimal set of secret key generation methods and the use of them for message encryption. Our outer bound is expressed also through a linear program, in this case with 41 constraints, constructed from general information inequalities. We prove that the optimal value of the outer bound LP is no larger than that of the scheme LP, which implies that the solution of the achievable scheme LP is the capacity. We find that equipping the relay with private randomness increases the secrecy rate by more than 40% in some cases and that cut-set bounds, directly applied in the network, are not always tight. Because the derivation of the inner and outer bound are both lengthy, we describe in this paper the achievability scheme, outline the outer bound, and provide the full derivations online. We also make available Matlab functions that take as input the erasure probabilities and evaluate the inner and outer bounds.