Sufficiently Myopic Adversaries Are Blind

Abstract

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 myopic . Specifically, for blocklength $n$ , the codeword ${X^ {n}}$ transmitted by Alice is corrupted by James who must base his adversarial decisions (of which locations of ${X^ {n}}$ to corrupt and how to corrupt them) on the non-causal observation ${Z^ {n}}$ of ${X^ {n}}$ obtained 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 arbitrarily varying channel from Alice to Bob, $p_{Y|XS}$ governed by a state ${S^ {n}}$ determined by James. In standard adversarial channels, the states ${S^ {n}}$ may depend on the codeword ${X^ {n}}$ , but in our setting ${S^ {n}}$ depends non-causally only on James’s view ${Z^ {n}}$ . 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 then extend our results to the setting of secure communication, in which we require that the transmitted message remains secret from James. For example, we show that if 1i) James may flip at most a $p$ fraction of the bits communicated between Alice and Bob and 2) James views ${X^ {n}}$ through a binary symmetric channel with crossover probability $q$ , then once James is “sufficiently myopic” (in this case, when $p and $H(q)>H(p)$ ), then the optimal communication rate is that of an adversary who is “blind” (that is, an adversary that does not have any knowledge of ${X^ {n}}$ at all), which is $1-H(p)$ for standard communication, and $H(q)-H(p)$ for secure communication. A similar phenomenon exists for more general models of communication.