  We study the first- and second-order asymptotics of covert communication over
binary-input DMC for three different covertness metrics and under maximum
probability of error constraint. When covertness is measured in terms of the
relative entropy between the channel output distributions induced with and
without communication, we characterize the exact first- and second-order
asymptotics of the number of bits that can be reliably transmitted with a
maximum probability of error less than $\epsilon$ and a relative entropy less
than $\delta$. When covertness is measured in terms of the variational distance
between the channel output distributions or in terms of the probability of
missed detection for fixed probability of false alarm, we establish the exact
first-order asymptotics and bound the second-order asymptotics. PPM achieves
the optimal first-order asymptotics for all three metrics, as well as the
optimal second-order asymptotics for relative entropy. The main conceptual
contribution of this paper is to clarify how the choice of a covertness metric
impacts the information-theoretic limits of covert communications. The main
technical contribution underlying our results is a detailed expurgation
argument to show the existence of a code satisfying the reliability and
covertness criteria.
