# First and Second Order Asymptotics in Covert Communication

## TL;DR

This paper analyzes the fundamental limits of covert communication over binary-input channels, providing precise asymptotic characterizations under various covertness metrics and demonstrating the optimality of PPM schemes.

## Contribution

It offers the first and second-order asymptotic characterizations for covert communication under multiple covertness metrics, clarifying the impact of metric choice on limits.

## Key findings

- PPM achieves optimal first-order asymptotics across all metrics.
- Exact second-order asymptotics are derived for relative entropy covertness.
- The choice of covertness metric significantly influences the communication limits.

## Abstract

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.