Second-Order Asymptotics in Covert Communication

We study the firstand second-order asymptotics of covert communication with Pulse-Position Modulation (PPM) over binary-input Discrete Memoryless Channels (DMCs) for three different metrics of covertness. When covertness is measured in terms of the relative entropy between the channel output distributions induced with and without communication, we characterize the exact PPM second-order asymptotics of the number of bits that can be reliably transmitted with a probability of error less than and a relative entropy less than δ. 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 bounds on the PPM second-order asymptotics. For all three metrics, we show that PPM achieves the optimal first-order asymptotics among all codes; for relative entropy, we show that PPM also achieves the optimal second-order asymptotics among all constant composition codes. The main conceptual contribution of this paper is to clarify how the choice of a covertness metric impacts the information-theoretic limits of the number of covert and reliable bits. The main technical contribution underlying our results is a detailed analysis of the probability of existence of a code satisfying the reliability and covertness criteria.