Fundamental Limits of Covert Bit Insertion in Packets

TL;DR

This paper investigates the fundamental limits of covertly inserting bits into packet streams without detection, establishing the maximum covert throughput under various packet size dependency scenarios.

Contribution

It characterizes the maximum covert bit insertion capacity in packet streams with different size distributions and dependencies, extending previous work on packet insertion and timing.

Findings

Maximum covert bits for i.i.d. packet sizes is proportional to √n.

Insertion of more than √n bits can be detected with high probability.

Dependent packet sizes allow covert insertion proportional to c(n)/√n.

Abstract

Covert communication is necessary when revealing the mere existence of a message leaks sensitive information to an attacker. Consider a network link where an authorized transmitter Jack sends packets to an authorized receiver Steve, and the packets visit Alice, Willie, and Bob, respectively, before they reach Steve. Covert transmitter Alice wishes to alter the packet stream in some way to send information to covert receiver Bob without watchful and capable adversary Willie being able to detect the presence of the message. In our previous works, we addressed two techniques for such covert transmission from Alice to Bob: packet insertion and packet timing. In this paper, we consider covert communication via bit insertion in packets with available space (e.g., with size less than the maximum transmission unit). We consider three scenarios: 1) packet sizes are independent and identically distributed (i.i.d.) with a probability mass function (pmf) whose support is a set of one bit spaced values; 2) packet sizes are i.i.d. with a pmf whose support is arbitrary; 3) packet sizes may be dependent. For the first and second assumptions, we show that Alice can covertly insert $\mathcal{O}(\sqrt{n})$ bits of information in a flow of $n$ packets; conversely, if she inserts $\omega(\sqrt{n})$ bits of information, Willie can detect her with arbitrarily small error probability. For the third assumption, we prove Alice can covertly insert on average $\mathcal{O}(c(n)/\sqrt{n})$ bits in a sequence of $n$ packets, where $c(n)$ is the average number of conditional pmf of packet sizes given the history, with a support of at least size two.