Deterministic Extractors for Bit-Fixing Sources and Exposure-Resilient Cryptography

We give an efficient deterministic algorithm that extracts Ω(n2γ) almost-random bits from sources where n 1 2 +γ of the n bits are uniformly random and the rest are fixed in advance. This improves upon previous constructions, which required that at least n/2 of the bits be random in order to extract many bits. Our construction also has applications in exposure-resilient cryptography, giving explicit adaptive exposure-resilient functions and, in turn, adaptive all-or-nothing transforms. For sources where instead of bits the values are chosen from [d], for d > 2, we give an algorithm that extracts a constant fraction of the randomness. We also give bounds on extracting randomness for sources where the fixed bits can depend on the random bits.