Abstract:
We show that a mild derandomization assumption together with the worst-case hardness of NP implies the average-case hardness of a language in non-deterministic quasi-polynomial time. Previously such connections were only known for high classes such as EXP and PSPACE. There has been a long line of research trying to explain our failure in proving worst-case to average-case reductions within \NP \cite{FF93,V03,BT03,AGGM06}. The bottom line of this research is essentially that (under plausible assumptions) black-box techniques cannot prove such results. Indeed, our proof is not black-box, as it uses a non-black-box reduction of Gutfreund, Shaltiel and Ta-Shma \cite{GST05}. Furthermore, we prove using the same arguments as the above mentioned negative results, that this reduction cannot be done in a black-box way (again, under a plausible assumption). Thus our techniques show a way to bypass black-box impossibility arguments regarding worst-case to average-case reductions.