A Characterization of the Capacity of Online (causal) Binary Channels

Chen, Zitan; Jaggi, Sidharth; Langberg, Michael

doi:10.1145/2746539.2746591

Cited by 34 publications

(40 citation statements)

References 22 publications

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“…• Achievability: Motivated by the stochastic encoder designed in [12], we construct an ensemble of concatenated codes (with independent stochasticity in each chunk) in Section 6.…”

Section: Classes Of Adversariesmentioning

confidence: 99%

“…Hence, regardless of the list he chooses to impose on Bob via the prefix, with high probability over the specific suffix that Alice transmits, there will be no "reasonably close" suffix for any u = w in this list. Proving such a fact requires one to prove a somewhat subtle "code goodness" property, analogous to the one in [12], which may be viewed as a generalization of a Gilbert-Varsahmov-type about the specific stochastic codeword transmitted in those chunks. (iii) Another relatively straightforward issue pertains to the fact that Alice and James may use non-uniform power allocations.…”

Section: Intuition Behind the Formalismmentioning

confidence: 99%

“…Later in Section 6, we shall show that any rate less than C γ K is achievable. To prove this, we use the same construction of codes in [12]. First, an encode transmits a concatenation of K chunks of θ-length codewords.…”

Section: Optimization (P3) For Achievabilitymentioning

confidence: 99%

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Quadratically Constrained Channels with Causal Adversaries

Dey

Jaggi

et al. 2018

2018 IEEE International Symposium on Information Theory (ISIT)

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We consider the problem of communication over a channel with a causal jamming adversary subject to quadratic constraints. A sender Alice wishes to communicate a message to a receiver Bob by transmitting a real-valued length-n codeword x = (x 1 , . . . , x n ) through a communication channel. Alice and Bob do not share common randomness. Knowing Alice's encoding strategy, a jammer James chooses a real-valued length-n adversarial noise sequence s = (s 1 , . . . , s n ) in a causal manner: each s t (1 ≤ t ≤ n) can only depend on (x 1 , . . . , x t ).Bob receives y, the sum (over R) of Alice's transmission x and James' jamming vector s, and is required to reliably estimate Alice's message from this sum.In this work we characterize the channel capacity for such a channel as the limit superior of the optimal values C n P N of a series of optimizations. Upper and lower bounds on C n P N are provided both analytically and numerically. Interestingly, unlike many communication problems, in this causal setting Alice's optimal codebook may not have a uniform power allocation -for certain SNR a codebook with a two-level uniform power allocation results in a strictly higher rate than a codebook with a uniform power allocation would.Kabatiansky and Levenshtein in [7] derived the tightest known outer bounds 2 . In Figure 2, we plot the GV-type bound in [5] and the LP bound in [7] , together with the omniscient capacity C ob , as references for our results in this work.Causal The primary focus of this work is on causal adversaries. Channels with causal adversaries can be considered as a special case of arbitrarily varying channels (AVCs) [8,9]. The causality assumption is physically reasonable in many engineering situations. In [10] (c.f., in page 224), an arbitrarily "star" varying channel is introduced such that at each time step t, the state s t can only depend on previously transmitted symbols x 1 , . . . , x t−1 . Yet, as mentioned in [10], the general problem has not been fully tackled. It turns out that techniques from previous works on AVCs cannot be applied directly to channels with causal adversaries. The capacities of channels with causal adversaries are known in some special cases. For example, recent papers by Dey et al. [11] and Chen et al. [12] characterize the capacity region of binary bit-flip channels with causal (online) adversaries. The results in [13] extend these techniques to characterize the capacities of q-ary additive-error/erasure channels. Also, the capacities of binary erasure channels with causal adversaries are known by [12, 14]. Dey et al. [15, 16] considered a "delayed" causal adversary such that the state s t is only decided by x 1 , . . . , x t−∆ where ∆ can be an arbitrarily small (but constant) fraction of the code block-length n.At the risk of missing much of the relevant literature, we summarize below the known results, parametrized by the delay parameter ∆ going from −n to n.In general, adversaries with causality constraints may be weaker than omniscient adversaries, and stronger than obliv...

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“…• Achievability: Motivated by the stochastic encoder designed in [12], we construct an ensemble of concatenated codes (with independent stochasticity in each chunk) in Section 6.…”

Section: Classes Of Adversariesmentioning

confidence: 99%

Section: Intuition Behind the Formalismmentioning

confidence: 99%

See 1 more Smart Citation

Quadratically Constrained Channels with Causal Adversaries

Dey

Jaggi

et al. 2018

2018 IEEE International Symposium on Information Theory (ISIT)

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“…[4]) implies James can ensure any such communication protocol must be either non-covert or unreliable. This is true even if James has computational restrictions, or is required to behave causally [11]. This is in stark contrast to the probabilistic channel setting wherein covert and reliable communication is possible for a wide range of parameters.…”

Section: Introductionmentioning

confidence: 99%

“…regardless of James' jamming strategy.Unfortunately, in our setting this turns out to be impossible -it turns out (as we show in our first main result) that the noise S on Bob's channel is adversarially chosen (rather than randomly as in the classical setting, e.g.[4]) implies James can ensure any such communication protocol must be either non-covert or unreliable. This is true even if James has computational restrictions, or is required to behave causally [11]. This is in stark contrast to the probabilistic channel setting wherein covert and reliable communication is possible for a wide range of parameters.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.…”

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confidence: 99%

Covert Communication over Adversarially Jammed Channels

Zhang

Bakshi

Jaggi

2018

2018 IEEE Information Theory Workshop (ITW)

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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...

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Error Correcting Codes that Achieve BSC Capacity Against Channels that are Poly-Size Circuits

Shaltiel

Silbak

2022

2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)

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A Characterization of the Capacity of Online (causal) Binary Channels

Cited by 34 publications

References 22 publications

Quadratically Constrained Channels with Causal Adversaries

Quadratically Constrained Channels with Causal Adversaries

Covert Communication over Adversarially Jammed Channels

Error Correcting Codes that Achieve BSC Capacity Against Channels that are Poly-Size Circuits

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