Abstract

What is the value of one bit of side information to a guesser? We study this problem in a setup where Alice wishes to guess a uniform binary random vector, and can obtain a single bit of information from Bob, who observes this vector through a binary symmetric channel. Our goal is to charaterize the guessing efficiency, namely the maximal reduction factor in Alice's guessing-time moments obtainable by observing Bob's bit. We provide two lower bounds on the guessing efficiency by analyzing the performance of the Dictator and Majority functions, and two upper bounds via maximum entropy and Fourier-analytic/hypercontractivity arguments.

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