Guessing more secrets via list decoding

Abstract

We consider the following game introduced by Chung, Graham, and Leighton in [Chung et al. 01]. One player, A, picks k > 1 secrets from a universe of N possible secrets, and another player, B, tries to gain as much information about this set as possible by asking binary questions f: [N] → {0, 1}. Upon receiving a question f, A adversarially chooses one of her k secrets, and answers f according to it. In this paper we present an explicit set of 2O(k)(log N) questions, along with a(Formula presented.) recovery algorithm that achieves B’s goal in this game. This, in particular, completely solves the problem for any constant number of secrets k. Our strategy is based on the list decoding of Reed-Solomon codes, and it extends and generalizes ideas introduced by Alon, Guruswami, Kaufman, and Sudan in [Alon et al. 02].