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Simplified derandomization of BPP using a hitting set generator. (English) Zbl 1343.68303
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, 59-67 (2011).
Summary: A hitting-set generator is a deterministic algorithm that generates a set of strings such that this set intersects every dense set that is recognizable by a small circuit. A polynomial time hitting-set generator readily implies \(\mathcal{RP}=\mathcal{P}\), but it is not apparent what this implies for \(\mathcal{BPP}\). Nevertheless, A. E. Andreev et al. [Lect. Notes Comput. Sci. 1099, 357–368 (1996; Zbl 1046.68536); J. ACM 45, No. 1, 179–213 (1998; Zbl 0903.68089)] showed that a polynomial-time hitting-set generator implies the seemingly stronger conclusion \(\mathcal{BPP=P}\). We simplify and improve their (and later) constructions.
For the entire collection see [Zbl 1220.68005].

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