We study deterministic extractors for bit-fixing sources (a.k.a. resilient
functions) and exposure-resilient functions for small min-entropy. That is, of
the n bits given as input to the function, k << n bits are uniformly random and
unknown to the adversary. We show that a random function is a resilient
function with high probability if and only if k is at least roughly log n. In
contrast, we show that a random function is a static (resp. adaptive)
exposure-resilient function with high probability even if k is as small as a
constant (resp. log(log n)).