Deterministic extractors for bit-fixing sources and exposure-resilient cryptography

We give an efficient deterministic algorithm which extracts Ω(n^2γ) almost-random bits from sources where n¹2 + γ/ of the n bits are uniformly random and the rest are fixed in advance. This improves on previous constructions which required that at least n/2 of the bits be random. Our construction also gives 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 which 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.