Suppose that a transmitter Alice potentially wishes to communicate with a receiver Bob over an adversarially jammed binary 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 based on his observation. We consider a setting where the communication must be simultaneously covert as well as reliable, i.e., James should be unable to accurately distinguish whether or not Alice is communicating, while Bob should be able to correctly recover Alice's message with high probability regardless of the adversarial jamming strategy. We show that, unlike the setting with passive adversaries, reliable covert communication against active adversaries requires Alice and Bob to have a shared key (of length at least Ω(log n)) even when Bob has a better channel than James. We present inner and outer bounds on the informationtheoretically optimal throughputs as a function of the channel parameters, the desired level of covertness, and the amount of shared key available. Further, these bounds match for a wide range of parameters of interest. Full version [1]: https:// arxiv.org/ pdf/ 1805.02426.pdf unable to detect if transmission has indeed taken place. To model this scenario, we take in this work a somewhat codingtheoretic view -we let the channel from Alice to James be probabilistic, but we allow James to try to jam the channel to Bob adversarially, as a function of his noisy observation of Alice's potential transmissions. Semi-formally, in our setting, Alice's channel input is an nlength binary vector X. When Alice is silent, X must be the all-zeros vector 0; when Alice is active, the X she transmits may depend on the message she wishes to send. The channel from Alice to James is BSC(q). James uses his observation Z in two ways -to detect if communication is being attempted, and to choose a binary jamming vector S of Hamming weight at most pn -Bob receives the vector Y = X ⊕ S. We denote the channel from Alice to Bob as ADVC(p|q). The covertness is measured via a hypothesis-testing metric -we say the communication is (1 − d )-covert if regardless of James' detection strategy Φ, his probability of false alarm plus his probability of missed detection is always lower-bounded by 1 − d . Secondly, reliability requires Bob should be able to reconstruct Alice's transmission correctly w.h.p. regardless of James' jamming strategy. Unfortunately, in our setting this turns out to be impossible without shared key (as shown in Theorem 1) -James can cleverly choose the jamming vector S to ensure that any such communication protocol must be either non-covert or unreliable. This is in stark contrast to the probabilistic channel setting wherein covert and reliable communication is possible for a wide range of parameters without shared key. Hence, we mildly relax our problem -prior to transmission, Alice and Bob secretly share a ∆(n)-bit randomly generated shared key that is unknown to James. The optimal throughput obtainable depends critically on the size of this key: 1) When the shared key is less than 1 2 log(n) bits 1 , 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 [6], [7] . Conversely, when the shared key is larger than 6 log(n), the optimal throughput scales as O( √ n) -we explicitly characterize the constant factor (with matching inner and outer bounds) for a wide range of parameters of interest. 2) When Alice and Bob have a large amount (ω( √ n) bits) of shared key, we 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). 3) When the size of the shared key is "moderate" (belongs to 1 All logarithms in this paper are binary.
We defer the detailed proofs to the full version REF .
Covert Communication over Adversarially Jammed Channels
2018 IEEE Information Theory Workshop (ITW)
2018 IEEE Information Theory Workshop (ITW)
Computer Science Mathematics