Sufficiently Myopic Adversaries are Blind

TL;DR

This paper studies communication over channels with a myopic adversary who can only see a noisy version of the transmitted codeword, providing bounds on capacity and analyzing secure communication scenarios.

Contribution

It introduces a new model of myopic adversarial channels, deriving capacity bounds and demonstrating when the adversary's limited view makes them effectively blind.

Findings

Capacity bounds are tight for certain cases.

When the adversary's view is sufficiently noisy, the capacity matches that of a blind adversary.

Secure capacity is characterized when the adversary's view is noisy and limited.

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 $X^n$ 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) not on the codeword $X^n$ but on $Z^n$, an image of $X^n$ 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 $X^n$ 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 only on James's view $Z^n$. 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 $X^n$ through a binary symmetric channel with parameter $q$, then once James 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 $X^n$ at all), which is $1-H(p)$ for standard communication, and $H(q)-H(p)$ for secure communication. A similar phenomenon exists for our general model of communication.