Hardness Self-Amplification from Feasible Hard-Core Sets

We consider the question of hardness self-amplification: Given a Boolean function f that is hard to compute on an o (1)-fraction of inputs drawn from some distribution, can we prove that f is hard to compute on a $(\displaystyle \frac{1}{2}-o(1))$-fraction of inputs drawn from the same distribution? We prove hardness self-amplification results for natural distributional problems studied in fine-grained average-case complexity, such as the problem of counting the number of the triangles modulo 2 in a random tripartite graph and the online vector-matrix-vector multiplication problem over $\mathbb{F}_{2}$. More generally, we show that any problem that can be decomposed into "computationally disjoint" subsets of inputs admits hardness self-amplification. This is proved by generalizing the security proof of the NisanWigderson pseudorandom generator, in which case nearly disjoint subsets of inputs are considered. At the core of our proof techniques is a new notion of feasible hard-core set, which generalizes Impagliazzo’s hard-core set [Impagliazzo, FOCS’95]. We show that any weak average-case hard function f has a feasible hard-core set H: any small H-oracle circuit (that is allowed to make queries q to H if $f(q)$ can be computed without the oracle) fails to compute f on a $(\displaystyle \frac{1}{2}-o(1))$-fraction of inputs in H.