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 ∈ [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 ∈ [0,1 − H(p)] where H(•) 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.