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Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas

Authors Rocco A. Servedio, Li-Yang Tan

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Author Details

Rocco A. Servedio
  • Department of Computer Science, Columbia University, New York, NY, USA
Li-Yang Tan
  • Department of Computer Science, Stanford University, Palo Alto, CA, USA

Funding

  • Servedio, Rocco A.: Supported by NSF grants CCF-1420349 and CCF-1563155.
  • Tan, Li-Yang: Supported by NSF grant CCF-1563122; part of this research was done during a visit to Columbia University.

Acknowledgements

We thank Prahladh Harsha and Srikanth Srinivasan for helpful discussions.

Cite As Get BibTex

Rocco A. Servedio and Li-Yang Tan. Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 45:1-45:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.45

Abstract

We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: small-depth circuits and sparse F_2 polynomials. Our main results are an epsilon-PRG for the class of size-M depth-d AC^0 circuits with seed length log(M)^{d+O(1)}* log(1/epsilon), and an epsilon-PRG for the class of S-sparse F_2 polynomials with seed length 2^{O(sqrt{log S})}* log(1/epsilon). These results bring the state of the art for unconditional derandomization of these classes into sharp alignment with the state of the art for computational hardness for all parameter settings: improving on the seed lengths of either PRG would require breakthrough progress on longstanding and notorious circuit lower bounds. 
The key enabling ingredient in our approach is a new pseudorandom multi-switching lemma. We derandomize recently-developed multi-switching lemmas, which are powerful generalizations of Håstad’s switching lemma that deal with families of depth-two circuits. Our pseudorandom multi-switching lemma - a randomness-efficient algorithm for sampling restrictions that simultaneously simplify all circuits in a family - achieves the parameters obtained by the (full randomness) multi-switching lemmas of Impagliazzo, Matthews, and Paturi [Impagliazzo et al., 2012] and Håstad [Johan Håstad, 2014]. This optimality of our derandomization translates into the optimality (given current circuit lower bounds) of our PRGs for AC^0 and sparse F_2 polynomials.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • pseudorandom generators
  • switching lemmas
  • circuit complexity
  • unconditional derandomization

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