  We consider the problem of covert communication over a state-dependent
channel, where the transmitter has causal or noncausal knowledge of the channel
states. Here, "covert" means that a warden on the channel should observe
similar statistics when the transmitter is sending a message and when it is
not. When a sufficiently long secret key is shared between the transmitter and
the receiver, we derive closed-form formulas for the maximum achievable covert
communication rate ("covert capacity") for discrete memoryless channels and,
when the transmitter's channel-state information (CSI) is noncausal, for
additive white Gaussian noise (AWGN) channels. For certain channel models,
including the AWGN channel, we show that the covert capacity is positive with
CSI at the transmitter, but is zero without CSI. We also derive lower bounds on
the rate of the secret key that is needed for the transmitter and the receiver
to achieve the covert capacity.
