Secret key establishment over a pair of independent broadcast channels

We consider the problem of secret key establishment (SKE) in the presence of a passive adversary, Eve, when Alice and Bob are connected by a pair of independent discrete memoryless broadcast channels in opposite directions. We refer to this setup as 2DMBC. We define the secret-key capacity in this setup and provide a lower bound on the capacity by proposing a two-round SKE protocol. We also prove an upper bound and show that the two bounds coincide in the case of degraded 2DMBCs.