Quadratically Constrained Myopic Adversarial Channels

Abstract

We study communication in the presence of a jamming adversary where quadratic power constraints are imposed on the transmitter and the jammer. The jamming signal is allowed to be a function of the codebook, and a noncausal but noisy observation of the transmitted codeword. For a certain range of the noise-to-signal ratios (NSRs) of the transmitter and the jammer, we are able to characterize the capacity of this channel under deterministic encoding or stochastic encoding, i.e., with no common randomness between the encoder/decoder pair. For the remaining NSR regimes, we determine the capacity under the assumption of a small amount of common randomness (at most 2 log(n) bits in one sub-regime, and at most Ω(n) bits in the other sub-regime) available to the encoder-decoder pair. Our proof techniques involve a novel myopic list-decoding result for achievability, and a Plotkin-type push attack for the converse in a subregion of the NSRs, both of which may be of independent interest. We also give bounds on the strong secrecy capacity of this channel assuming that the jammer is simultaneously eavesdropping.

List Decoding Random Euclidean Codes and Infinite Constellations

“…‚ If we restrict ourselves to structured codes, e.g., nested lattice codes [11], then what list sizes are achievable? It was shown in [8] that O `1 δ log 1 δ ˘list sizes are achievable using random spherical codes. If we define S n´1 p0, ?…”

“…For instance, Langberg [6] showed that if there exists a coding scheme that achieves a list size that is at most polynomial in the blocklength n, then even a small amount of shared secret key (just about Θplog nq bits kept secret from the adversary) between the sender-receiver pair suffices to ensure that the true message can be uniquely decoded by the receiver. List decoding can also serve as a useful proof technique for obtaining bounds on the capacities of other adversarial channels [7], [8], [9].…”

“…For an adversarial channel with L 2 power constraints, it is known (see e.g. [8,Appendix D]) that if the transmission rate R is greater than 1 2 log P N , then no coding scheme can achieve subexponential (in n) list sizes. On the other hand, it is also known ([8,Appendix D]) that random spherical codes of rate R ă 1 2 log P N can achieve constant (in n) list sizes.…”

“…[8,Appendix D]) that if the transmission rate R is greater than 1 2 log P N , then no coding scheme can achieve subexponential (in n) list sizes. On the other hand, it is also known ([8,Appendix D]) that random spherical codes of rate R ă 1 2 log P N can achieve constant (in n) list sizes. We can therefore call 1 2 log P N to be the list decoding capacity of this channel.…”

“…In this model, we assume that a myopic adversary sees a noncausal noisy version of the transmitted codeword. This problem was studied in [8].…”