Hard-Core Distributions for Somewhat Hard Problems

Consider a decision problem that cannot be 1 ? approximated by circuits of a given size in the sense that any such circuit fails to give the correct answer on at least a fraction of instances. We show that for any such problem there is a speciic \hard-core" set of inputs which is at least a fraction of all inputs and on which no circuit of a slightly smaller size can get even a small advantage over a random guess. More generally , our argument holds for any non-uniform model of computation closed under majorities. We apply this result to get a new proof of the Yao XOR lemma Y], and to get a related XOR lemma for inputs that are only k-wise independent.