Covert communication, also known as communication with low probability of detection (LPD), aims to reliably transmit a message to an intended receiver while simultaneously preventing the detection of the transmission by an adversary. Unlike other security approaches, such as cryptography or information-theoretic secrecy that aim at denying the adversary access to the information content of the transmitted signals, covertness aims at hiding the presence of the transmitted signal themselves. While practical radio-frequency covert communication systems have been around since the advent of spread-spectrum, exploration of their fundamental limits is a fairly new direction in information theory. Although related ideas can be found in the steganography literature, the initial mathematical analysis of covert communications was presented at ISIT 2012 and revealed that, while the Shannon capacity of a covert communication channel is zero, it allows transmission of a large volume of covert data. This is because covert communication is governed by the square root law: no more than L \sqrt(n)+o(\sqrt(n)) covert bits can be transmitted reliably in n channel uses. Subsequent works, have characterized the covert capacity L of discrete memoryless channels (DMCs) and additive white Gaussian noise (AWGN) channels, as well as classical-quantum channels. These papers seeded a large body of follow-on work resulting in “Covert Communications” sessions at recent ISITs and numerous presentations at ITWs, ICCs, etc.

The goal of this tutorial is to introduce the audience to the mathematical foundations of covert communications. We will present both the classical and quantum information theory perspectives. For the classical treatment we will assume familiarity with basic information theory. While the knowledge of the material from the recent ISIT tutorials on quantum information theory and quantum limits of optical communication will certainly benefit the listener, we will not assume any prior quantum background and will introduce the necessary concepts and results. Although we will present several proofs, we will focus on key insights rather than mathematical details.