Channel Capacity for Adversaries With Computationally Bounded Observations

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/corruption. 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 and find that the capacity is identical to the completely oblivious setting (
$r=0$
). This result can be viewed as a generalization of known results on myopic adversaries and on 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.