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In a world of \(\mathrm{P}=\mathrm{BPP}\). (English) Zbl 1343.68084
Goldreich, Oded (ed.), Studies in complexity and cryptography. Miscellanea on the interplay between randomness and computation. In collaboration with Lidor Avigad, Mihir Bellare, Zvika Brakerski, Shafi Goldwasser, Shai Halevi, Tali Kaufman, Leonid Levin, Noam Nisan, Dana Ron, Madhu Sudan, Luca Trevisan, Salil Vadhan, Avi Wigderson, David Zuckerman. Berlin: Springer (ISBN 978-3-642-22669-4/pbk). Lecture Notes in Computer Science 6650, 191-232 (2011).
Summary: We show that proving results such as \(\mathcal{BPP}=\mathcal{P}\) essentially necessitate the construction of suitable pseudorandom generators (i.e., generators that suffice for such derandomization results). In particular, the main incarnation of this equivalence refers to the standard notion of uniform derandomization and to the corresponding pseudorandom generators (i.e., the standard uniform notion of “canonical derandomizers”). This equivalence bypasses the question of which hardness assumptions are required for establishing such derandomization results, which has received considerable attention in the last decade or so (starting with [R. Impagliazzo and A. Wigderson, J. Comput. Syst. Sci. 63, No. 4, 672–688 (2001; Zbl 1052.68034)]).
We also identify a natural class of search problems that can be solved by deterministic polynomial-time reductions to \(\mathcal{BPP}\). This result is instrumental to the construction of the aforementioned pseudorandom generators (based on the assumption \(\mathcal{BPP}=\mathcal{P}\)), which is actually a reduction of the “construction problem” to \(\mathcal{BPP}\).
Caveat: Throughout the text, we abuse standard notation by letting \(\mathcal{BPP}\), \(\mathcal{P}\) etc denote classes of promise problems. We are aware of the possibility that this choice may annoy some readers, but believe that promise problem actually provide the most adequate formulation of natural decisional problems.
For the entire collection see [Zbl 1220.68005].

68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
68W20 Randomized algorithms
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