Key agreement from weak bit agreement

Abstract

Assume that Alice and Bob, given an authentic channel, have a protocol where they end up with a bit SA and SB, respectively, such that with probability 1+e/2 these bits are equal. Further assume that conditioned on the event SA =n SB no polynomial time bounded algorithm can predict the bit better than with probability 1-δ/2. Is it possible to obtain key agreement from such a primitive? We show that for constant δ and e the answer is yes if and only if δ > 1-e/1+e, both for uniform and non-uniform adversaries.The main computational technique used in this paper is a strengthening of Impagliazzo's hard-core lemma to the uniform case and to a set size parameter which is tight (i.e., twice the original size). This may be of independent interest.