Reduction for NP-search Problems from Samplable to Uniform Distributions: Hard Distribution Case

Impagliazzo and Levin showed a reduction from average-case hardness of any NP-search problem under any polynomialtime samplable distribution to that of another NP-search problem under the uniform distribution in [12]. Their target was the hardness of positive instances occurring with probability 1/poly(n) under the distributions. In this paper, we focus on hardness of a larger fraction of instances. We reduce the hardness of positive instances for any NP-search problem occurring with probability 1−1/poly(n) under any polynomial-time samplable distribution over positive instances to that for another NP-search problem with similar hardness under the uniform distribution. In order to illustrate the usage/importance of this technique, we show a simple way to modify the technique of Gutfreund, Shaltiel and Ta-Shma in [8] to construct an NP-search problem hard on average under the uniform distribution based on the assumption that NP , RP and some worst-case mild derandomization holds.