Sufficiently myopic adversaries are blind

In this work we consider the communication setting in which a sender, Alice, wishes to communicate with a receiver, Bob, over a channel controlled by an adversarial entity, Calvin, who is myopic. Roughly speaking, for blocklength n, the codeword Xⁿ transmitted by Alice is corrupted by Calvin who must base his adversarial decisions, on which characters of Xⁿ to corrupt and how to corrupt them, not on the entire view of the codeword Xⁿ but on Zⁿ, the image of Xⁿ through a noisy memoryless channel. More specifically, our communication model may be described by two channels. A memoryless channel p(zx) from Alice to Calvin, and an arbitrarily varying channel from Alice to Bob, p(yx, s) governed by a states Sⁿ determined by Calvin. In standard adversarial channels, the states Sⁿ may depend on the codeword Xⁿ, however in our setting Sⁿ depends only on Calvin's view Zⁿ. The myopic channel captures a broad range of channels and bridges between the standard models of memoryless and adversarial (zero error) channels. In this work we present upper and lower bounds on the capacity of myopic channels. For a number of special cases of interest we show that our bounds are tight. We extend our results to the setting of secure communication in which we require that the transmitted message remain secret from Calvin. For example, we show that if (i) Calvin may flip at most a p fraction of the bits communicated between Alice and Bob, and (ii) Calvin views Xⁿ through a binary symmetric channel with parameter q, then once Calvin is 'sufficiently myopic' (in this case, when q > p), then the optimal communication rate is that of an adversary who is 'blind' (that is, an adversary that does not see Xⁿ at all), which is 1-H(p) for standard communication, and H(q)-H(p) for secure communication. A similar phenomena exists for our general model of communication.