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Revision #1 to TR11-167 | 12th December 2012 09:30
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#### Improving on Gutfreund, Shaltiel, and Ta-Shma's paper "If NP Languages are Hard on the Worst-Case, Then it is Easy to Find Their Hard Instances''

**Abstract:**
Assume that NP is not included in BPP. Gutfreund, Shaltiel, and Ta-Shma in [Computational Complexity 16(4):412-441 (2007)] have proved that for every randomized polynomial time decision algorithm D for SAT there is a polynomial time samplable distribution such that D errs with probability at least 1/6-epsilon

on a random formula chosen with respect to that distribution. In this paper, we show how to increase the error probability to 1/3-epsilon.

**Changes to previous version:**
Removed the result on generating hard instances for NP search problems, as it has been published in a paper of Gutfreund (2006) of which I was unaware.

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TR11-167 | 6th December 2011 16:06
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#### Improving on Gutfreund, Shaltiel, and Ta-Shma's paper "If NP Languages are Hard on the Worst-Case, Then it is Easy to Find Their Hard Instances''

**Abstract:**
Assume that $NP\ne RP$. Gutfreund, Shaltiel, and Ta-Shma in [Computational Complexity 16(4):412-441 (2007)] have proved that for every randomized polynomial time decision algorithm $D$ for SAT there is a polynomial time samplable distribution such that $D$ errs with probability at least $1/6-\epsilon$ on a random formula chosen with respect to that distribution. In this paper, we show how to increase the error probability to $1/3-\epsilon$. We also generalize this result to the search version of SAT: we prove that for every randomized polynomial time algorithm $S$ searching for a satisfying assignment of a given formula, there is a polynomial time samplable distribution such that $S$ errs with probability at least $1-\epsilon$ on a random formula chosen with respect to that distribution.