Sufficiently Myopic Adversaries Are Blind

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 <italic>myopic</italic>. Specifically, for blocklength <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, the codeword <inline-formula> <tex-math notation="LaTeX">${X^ {n}}$ </tex-math></inline-formula> transmitted by Alice is corrupted by James who must base his adversarial decisions (of which locations of <inline-formula> <tex-math notation="LaTeX">${X^ {n}}$ </tex-math></inline-formula> to corrupt and how to corrupt them) on the non-causal observation <inline-formula> <tex-math notation="LaTeX">${Z^ {n}}$ </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">${X^ {n}}$ </tex-math></inline-formula> obtained through a noisy memoryless channel. More specifically, our communication model may be described by two channels. A memoryless channel <inline-formula> <tex-math notation="LaTeX">$p_{Z|X}$ </tex-math></inline-formula> from Alice to James, and an <italic>arbitrarily varying channel</italic> from Alice to Bob, <inline-formula> <tex-math notation="LaTeX">$p_{Y|XS}$ </tex-math></inline-formula> governed by a state <inline-formula> <tex-math notation="LaTeX">${S^ {n}}$ </tex-math></inline-formula> determined by James. In standard adversarial channels, the states <inline-formula> <tex-math notation="LaTeX">${S^ {n}}$ </tex-math></inline-formula> may depend on the codeword <inline-formula> <tex-math notation="LaTeX">${X^ {n}}$ </tex-math></inline-formula>, but in our setting <inline-formula> <tex-math notation="LaTeX">${S^ {n}}$ </tex-math></inline-formula> depends non-causally only on James’s view <inline-formula> <tex-math notation="LaTeX">${Z^ {n}}$ </tex-math></inline-formula>. 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 <italic>secure</italic> 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 <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> fraction of the bits communicated between Alice and Bob and 2) James views <inline-formula> <tex-math notation="LaTeX">${X^ {n}}$ </tex-math></inline-formula> through a binary symmetric channel with crossover probability <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>, then once James is “sufficiently myopic” (in this case, when <inline-formula> <tex-math notation="LaTeX">$p< 0.5$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$H(q)>H(p)$ </tex-math></inline-formula>), then the optimal communication rate is that of an adversary who is “blind” (that is, an adversary that does not have any knowledge of <inline-formula> <tex-math notation="LaTeX">${X^ {n}}$ </tex-math></inline-formula> at all), which is <inline-formula> <tex-math notation="LaTeX">$1-H(p)$ </tex-math></inline-formula> for standard communication, and <inline-formula> <tex-math notation="LaTeX">$H(q)-H(p)$ </tex-math></inline-formula> for secure communication. A similar phenomenon exists for more general models of communication.