Guessing secrets efficiently via list decoding

We consider the guessing secrets problem defined by Chung, Graham, and Leighton [CGL01]. This is a variant of the standard 20 questions game where the player has a set of k > 1 secrets from a universe of N possible secrets. The player is asked Boolean questions about the secret for each question, the player picks one of the k secrets adversarially, and answers according to this secret.We present an explicit set of O(log N) questions together with an efficient (i.e., poly(log N) time) algorithm to solve the guessing secrets problem for the case of 2 secrets. This answers the main algorithmic question left unanswered by [CGL01]. The main techniques we use are small ε-biased spaces and the notion of list decoding.We also establish bounds on the number of questions needed to solve the k-secrets game for k > 2, and discuss how list decoding can be used to get partial information about the secrets.

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