Impagliazzo and Levin showed a reduction from average-case hardness of any NP-search problem under any polynomial-time samplable 
distribution to that of another NP-search problem under the uniform distribution in [Impagliazzo and Levin, FOCS '90].
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 [Gutfreund, Shaltiel, and Ta-Shma, CCC '05] 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.