Multilevel-Coded Pulse-Position Modulation for Covert Communications

Abstract

We develop a low-complexity coding scheme to achieve covert communications over binary-input discrete memoryless channels (BI-DMCs). We circumvent the impossibility of covert communication with linear codes by introducing non-linearity through the use of pulse position modulation (PPM) and multilevel coding (MLC). We show that the MLC-PPM scheme exhibits many appealing properties; in particular, the channel at a given index level remains stationary as the number of level increases, which allows one to use families of channel capacity-and channel resolvability-achieving codes to concretely instantiate the covert communication scheme.A small misconception: One might think that linear codes might not be covert because the linearity constraint requires the sum of codewords to be a codeword; consequently, linear combinations of low-weight codewords would result in a codeword with high weight. This is, however, insufficient to argue that linear codes cannot be covert. In fact, one could consider a systematic code with unit-weight codewords in the generator matrix, i.e, G = ( I k 0 k×n−k ), where I k is the identity matrix of size k. All codewords of this code have weight at most k. Of course, this code would hardly be covert, mainly because the structure of the generator matrix allows an attacker to dismiss the last n − k codeword components. The next proposition formalizes this.Proposition 1. Consider an (n, k) binary linear code with (n, k) ∈ (N * ) 2 , n k. Assume that the code is used for communication over a BI-DMC (X , W Z|X , Z) with uniformly distributed messages. There exists a binary hypothesis test with false alarm probability α and missed-detection probability β such that

First- and Second-Order Asymptotics in Covert Communication

“…All our achievability results are established using Pulse-Position Modulation (PPM), which optimality was previously only established for the first-order asymptotics with relative entropy [16]. The operational relevance of codes used in conjunction with PPM, which may be viewed as a highly structured subset of constant composition codes, is justified by recent work towards practical code design [17], in which PPM plays a crucial role. We also emphasize from the outset that the focus on maximal probability of error is essential to our analysis.…”

“…towards practical code design [17], in which PPM plays a crucial role. We also emphasize from the outset that the focus on maximal probability of error is essential to our analysis.…”

An IR-HARQ scheme for covert communications

“…It has been shown that in many relevant scenarios when the Warden channel is worse than the Alice-Bob channel the maximum number of bits that can be reliably transmitted while keeping the transmission covert from the Warden grows asymptotically as O( √ n), being n the codeword length, even if Alice and Bob do not share any secret key. Furthermore, a scheme based on pulse position modulation (PPM) that is asymptotically optimal has been proposed for binary-input discrete memoryless channels and the Gaussian channel ( [5], [6], [7]). In all these works the focus has been on the amount of information bits that could be sent when the codeword length increased unbounded.…”

Correcting Insertions/Deletions in DPPM Using Hidden Markov Model

“…The chips in symbols (0) and (3) flip and recombine, resulting in a two-symbol substitution (03) → (21). The chip in symbol (0) flips from '1' to '0', and these '0' chips merge with the successive symbol (2), resulting in a deletion, i.e., (02) → (3). Therefore, the DPPM modulation, the AWGN channel and the DPPM detection can be considered as a general insertion/deletion/substitution (IDS) channel [28].…”