Channel Capacity for Adversaries with Computationally Bounded Observations

TL;DR

This paper investigates the capacity of point-to-point channels with adversaries who have computationally bounded, observation-based access to transmitted codewords, revealing that capacity remains unchanged for certain observation rates.

Contribution

It characterizes the channel capacity when the adversary's observations are computationally bounded and depend on the codeword, extending known results on myopic adversaries.

Findings

Capacity equals the trivial case for observation rates up to 1-H(p).

The result generalizes known models like myopic adversaries.

Provides a capacity characterization for a new class of adversarial channels.

Abstract

We study reliable communication over point-to-point adversarial channels in which the adversary can observe the transmitted codeword via some function that takes the $n$-bit codeword as input and computes an $rn$-bit output for some given $r \in [0,1]$. We consider the scenario where the $rn$-bit observation is computationally bounded -- the adversary is free to choose an arbitrary observation function as long as the function can be computed using a polynomial amount of computational resources. This observation-based restriction differs from conventional channel-based computational limitations, where in the later case, the resource limitation applies to the computation of the (adversarial) channel error. For all $r \in [0,1-H(p)]$ where $H(\cdot)$ is the binary entropy function and $p$ is the adversary's error budget, we characterize the capacity of the above channel. For this range of $r$, we find that the capacity is identical to the completely obvious setting ($r=0$). This result can be viewed as a generalization of known results on myopic adversaries and channels with active eavesdroppers for which the observation process depends on a fixed distribution and fixed-linear structure, respectively, that cannot be chosen arbitrarily by the adversary.