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This seems like a folklore claim but I cannot find any reference to it. If Alice has a bit-string of length $n$ where each entry is independently set to 0 or 1 equiprobably, and Bob's goal is to reconstruct the string with a success probability at least 0.9. What is the simplest way to show that the randomized communication complexity (multiple rounds are allowed) is $\Omega(n)$?

One can use an overkilled proof: such a protocol will lead to a protocol for Disjointness and we know that Disjointness requires $\Omega(n)$ bits of communication.

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Suppose Alice always sends exactly $k$ bits to Bob during the protocol. On average, how many possible candidates for her $n$-bit string are consistent with the communication transcript? What does that tell you about the probability that Bob guesses which one of those candidates is the correct one?

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  • $\begingroup$ Ah, thanks a lot! $\endgroup$
    – polar_bear_cheese
    Jul 16, 2021 at 4:45

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