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Extractor Lower Bounds, Revisited

Authors Divesh Aggarwal, Siyao Guo, Maciej Obremski, João Ribeiro , Noah Stephens-Davidowitz

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Author Details

Divesh Aggarwal
  • National University of Singapore, Singapore
Siyao Guo
  • New York University Shanghai, China
Maciej Obremski
  • National University of Singapore, Singapore
João Ribeiro
  • Imperial College London, UK
Noah Stephens-Davidowitz
  • Cornell University, Ithaca, NY, USA

Funding

  • Aggarwal, Divesh: This work is supported in part by the Singapore National Research Foundation under NRF RF Award No. NRF-NRFF2013-13, the Ministry of Education, Singapore under the Research Centres of Excellence programme by the Tier-3 grant Grant "Random numbers from quantum processes" No. MOE2012-T3-1-009.
  • Guo, Siyao: Supported by Shanghai Eastern Young Scholar Program. Part of the work was done while visiting the Centre for Quantum Technologies, National University of Singapore.
  • Obremski, Maciej: This author was supported by MOE2019-T2-1-145 Foundations of quantum-safe cryptography.
  • Ribeiro, João: Part of the work was done while visiting the Centre for Quantum Technologies, National University of Singapore.
  • Stephens-Davidowitz, Noah: Parts of this work were done while visiting the Simons Institute for the Theory of Computing, Massachusetts Institute of Technology, and the Centre for Quantum Technologies, National University of Singapore.

Acknowledgements

The authors wish to thank the anonymous reviewers for several comments that improved the paper.

Cite AsGet BibTex

Divesh Aggarwal, Siyao Guo, Maciej Obremski, João Ribeiro, and Noah Stephens-Davidowitz. Extractor Lower Bounds, Revisited. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.1

Abstract

We revisit the fundamental problem of determining seed length lower bounds for strong extractors and natural variants thereof. These variants stem from a "change in quantifiers" over the seeds of the extractor: While a strong extractor requires that the average output bias (over all seeds) is small for all input sources with sufficient min-entropy, a somewhere extractor only requires that there exists a seed whose output bias is small. More generally, we study what we call probable extractors, which on input a source with sufficient min-entropy guarantee that a large enough fraction of seeds have small enough associated output bias. Such extractors have played a key role in many constructions of pseudorandom objects, though they are often defined implicitly and have not been studied extensively. Prior known techniques fail to yield good seed length lower bounds when applied to the variants above. Our novel approach yields significantly improved lower bounds for somewhere and probable extractors. To complement this, we construct a somewhere extractor that implies our lower bound for such functions is tight in the high min-entropy regime. Surprisingly, this means that a random function is far from an optimal somewhere extractor in this regime. The techniques that we develop also yield an alternative, simpler proof of the celebrated optimal lower bound for strong extractors originally due to Radhakrishnan and Ta-Shma (SIAM J. Discrete Math., 2000).

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Expander graphs and randomness extractors
Keywords
  • randomness extractors
  • lower bounds
  • explicit constructions

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References

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