Two-source condensers with low error and small entropy gap via entropy-resilient functions

In their seminal work, Chattopadhyay and Zuckerman (STOC’16) constructed a two-source extractor with error ε for n-bit sources having min-entropy polylog(n/ε). Unfortunately, the construction’s running-time is poly(n/ε), which means that with polynomial-time constructions, only polynomially-small errors are possible. Our main result is a poly(n, log(1/ε))-time computable two-source condenser. For any k ≥ polylog(n/ε), our condenser transforms two independent (n, k)-sources to a distribution over m = k − O(log(1/ε)) bits that is ε-close to having min-entropy m − o(log(1/ε)). Hence, achieving entropy gap of o(log(1/ε)). The bottleneck for obtaining low error in recent constructions of two-source extractors lies in the use of resilient functions. Informally, this is a function that receives input bits from r players with the property that the function’s output has small bias even if a bounded number of corrupted players feed adversarial inputs after seeing the inputs of the other players. The drawback of using resilient functions is that the error cannot be smaller than ln r/r. This, in return, forces the running time of the construction to be polynomial in 1/ε. A key component in our construction is a variant of resilient functions which we call entropy-resilient functions. This variant can be seen as playing the above game for several rounds, each round outputting one bit. The goal of the corrupted players is to reduce, with as high probability as they can, the min-entropy accumulated throughout the rounds. We show that while the bias decreases only polynomially with the number of players in a one-round game, their success probability decreases exponentially in the entropy gap they are attempting to incur in a repeated game.