Sufficiently Myopic Adversaries are Blind

Abstract: In this work we consider a communication problem in which a sender, Alice, wishes to communicate with a receiver, Bob, over a channel controlled by an adversarial jammer, James, who is {\em myopic}. Roughly speaking, for blocklength $n$, the codeword $X<sup>n$ transmitted by Alice is corrupted by James who must base his adversarial decisions (of which locations of $X<sup>n$ to corrupt and how to corrupt them) not on the codeword $X<sup>n$ but on $Z<sup>n$, an image of $X<sup>n$ through a noisy memoryless channel. More specifically, our communication model may be described by two channels. A memoryless channel $p(z|x)$ from Alice to James, and an {\it Arbitrarily Varying Channel} from Alice to Bob, $p(y|x,s)$ governed by a state $X<sup>n$ determined by James. In standard adversarial channels the states $S<sup>n$ may depend on the codeword $X<sup>n$, but in our setting $S<sup>n$ depends only on James's view $Z<sup>n$. 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 {\em secure} communication in which we require that the transmitted message remain secret from James. For example, we show that if (i) James may flip at most a $p$ fraction of the bits communicated between Alice and Bob, and (ii) James views $X<sup>n$ through a binary symmetric channel with parameter $q$, then once James is "sufficiently myopic" (in this case, when $q&gt;p$), then the optimal communication rate is that of an adversary who is "blind" (that is, an adversary that does not see $X<sup>n$ at all), which is $1-H(p)$ for standard communication, and $H(q)-H(p)$ for secure communication. A similar phenomenon exists for our general model of communication.