Private broadcasting with probing constraint

We consider a three-receiver wiretap channel in which the transmitter aims to send a common message to legitimate receivers 1 and 2 while keeping it secret from the receiver 3 (eavesdropper). The transmitter takes cost constrained actions to probe the transmitter-to-receiver 1 and transmitter-to-receiver 2 channels. Under the probing cost constraint, we provide a lower bound to the secrecy capacity of this setting. The achievability strategy employs a block Markov coding strategy in which the channel state sequence of the previous block is mapped to a key which secures the part of the confidential message to be transmitted in the current block. The challenge in using the state sequence as a source of the key is the fact that the legitimate receivers are not aware of each other's channel state. To that end, we employ a novel strategy in which the transmitter takes XOR of individual keys generated from the state sequences of transmitter-to receiver 1 and transmitter-to-receiver 2 channels and send the XOR-ed key along with the confidential message. We furthermore analyze the trade-off between the achievable secrecy rate and the probing cost constraint. We conclude that there is a linear dependency between the key rate and probing cost constraint.