Channel Capacity for Adversaries with Computationally Bounded Observations
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We study reliable communication over point-to-point adversarial channels in which the adversary can observe the codeword via some function that takes the n-bit codeword as input and computes an rn-bit output. 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 channel error. For some number r^* ∈ [0,1] and for r ∈ [0,r^*], 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.
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