Covert Communication over Adversarially Jammed Channels

Abstract

We consider a situation in which a transmitter Alice may wish to communicate with a receiver Bob over an adversarial channel. An active adversary James eavesdrops on their communication over a binary symmetric channel (BSC(q)), and may maliciously flip (up to) a certain fraction p of their transmitted bits. The communication should be both covert and reliable. Covertness requires that the adversary James should be unable to estimate whether or not Alice is communicating based on his noisy observations, while reliability requires that the receiver Bob should be able to correctly recover Alice's message with high probability.Unlike the setting with passive adversaries considered thus far in the literature, we show that reliable covert communication in the presence of actively jamming adversaries requires Alice and Bob to have a shared key (unknown to James). The optimal throughput obtainable depends critically on the size of this key:• When Alice and Bob's shared key is less than 1 2 log(n) bits, no communication that is simultaneously covert and reliable is possible. This is true even under reasonable computational assumptions on James, and if his jamming is required to be causal. Conversely, when the shared key is larger than 6 log(n), the optimal throughput scales as O( √ n) -we explicitly characterize even the constant factor (with matching inner and outer bounds) for a wide range of parameters of interest. • When Alice and Bob have a large amount (ω( √ n) bits) of shared key, we again present a tight covert capacity characterization for all parameters of interest, and the capacity is independent of the number of bits in the shared key as long as it is larger than ω( √ n) (again regardless of computational or causality assumptions on James). • When the size of the shared key is "moderate" (belongs to (Ω(log(n)), O( √ n))), we show an achievable coding scheme as well as an outer bound on the information-theoretically optimal throughput. • When Alice and Bob's shared key is O( √ n log(n)) bits, we develop a computationally efficient coding scheme for Alice/Bob whose throughput is only a constant factor smaller than information-theoretically optimal, and it ensures both covertness and reliability even against a computationally unbounded adversary James. I. INTRODUCTIONThe security of our communication schemes is of significant concern -Big Brother is often watching! While much attention focuses on schemes that aim to hide the content of communication, in many scenarios, the fact of communication should also be kept secret. For example, a secret agent being caught communicating with an accomplice is of potentially drastic consequences -merely ensuring secrecy does not guarantee undetectability. This observation has drawn attention to the problem of covert communication. In a canonical information-theoretic setting for this problem, a transmitter Alice may wish to transmit messages to a receiver Bob over a noisy channel, and remains silent otherwise. James eavesdrops on her transmission through another noisy channel...

Undetectable Radios: Covert Communication under Spectral Mask Constraints

“…By combining (10), (13), and the fact that n = T / T * 0 (S W , T ), we have the following Lemma, which establishes a bridge connecting the continuous-time channel (characterized by the time T and the spectral mask S W ) and the discrete-time channel (characterized by the blocklength n).…”

“…Proof: Equations (10) and (13) imply that for every ξ ′ > 0, there exists a T ′ (ξ ′ ) such that for all T > T ′ (ξ ′ ), 1 W min…”

“…Replacing W T min β∈[0,1] c(β) by n in Theorems 1, one can verify that equation (84) implies that BPSK is optimal under the variational distance metric. The proof techniques are leveraged from [5], [13] -in particular, Lemmas 10 and 11 below are analogous to Lemmas 11 and 12 in [5] (which are derived for DMCs). We provide the detailed proofs in the following.…”

“…Covert communication considers the scenario in which a transmitter, Alice, wishes to reliably communicate with a legitimate receiver, Bob, while simultaneously hiding the presence of communication from an eavesdropping adversary, referred to as the warden, Willie. Building upon the formalization of the problem in [1], subsequent studies have progressively refined the information-theoretic analysis of covert communication by considering binary symmetric channels [2], discrete memoryless channels (DMCs) and additive white Gaussian noise (AWGN) channels [3]- [5], multiple-access channels [6], broadcast channels [7]- [9], channels with state [10], [11], compound channels [12], and adversarially jammed channels [13]. The aforementioned works have not only identified a square-root law (SRL) for covert communication -only O( √ n) bits of information can be transmitted reliably and covertly over n channel uses -but also characterized the constant behind the O, which can be interpreted as the covert capacity.…”

Covert Secret Key Generation With an Active Warden

“…Another relevant work is [13], in which the warden's uncertainty about the channel is shown to allows the circumvention of the square root law. Finally, covert communication over adversarial channels, in which the warden flips a certain fraction of the transmitted bits, has been investigated in [14]; covert communication is shown to be possible for all warden's states if the legitimate users have access to enough shared secret key, which again prompts the question of how to covertly generate such a secret key.Our work partially addresses the question by showing that, under conditions that we shall precisely specify, covert and secret key generation is possible even with an active warden. Specifically, the results reported here extend our preliminary results restricted to a passive warden [15] to a model with an active warden who can arbitrarily vary the channel state, except when no information is sent on the main channel.…”