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Pseudorandom Generators for Unbounded-Width Permutation Branching Programs

Authors William M. Hoza , Edward Pyne, Salil Vadhan

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ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Pseudorandom generators
  • permutation branching programs

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Abstract

We prove that the Impagliazzo-Nisan-Wigderson [Impagliazzo et al., 1994] pseudorandom generator (PRG) fools ordered (read-once) permutation branching programs of unbounded width with a seed length of Õ(log d + log n ⋅ log(1/ε)), assuming the program has only one accepting vertex in the final layer. Here, n is the length of the program, d is the degree (equivalently, the alphabet size), and ε is the error of the PRG. In contrast, we show that a randomly chosen generator requires seed length Ω(n log d) to fool such unbounded-width programs. Thus, this is an unusual case where an explicit construction is "better than random."
Except when the program’s width w is very small, this is an improvement over prior work. For example, when w = poly(n) and d = 2, the best prior PRG for permutation branching programs was simply Nisan’s PRG [Nisan, 1992], which fools general ordered branching programs with seed length O(log(wn/ε) log n). We prove a seed length lower bound of Ω̃(log d + log n ⋅ log(1/ε)) for fooling these unbounded-width programs, showing that our seed length is near-optimal. In fact, when ε ≤ 1/log n, our seed length is within a constant factor of optimal. Our analysis of the INW generator uses the connection between the PRG and the derandomized square of Rozenman and Vadhan [Rozenman and Vadhan, 2005] and the recent analysis of the latter in terms of unit-circle approximation by Ahmadinejad et al. [Ahmadinejad et al., 2020].

Cite As Get BibTex

William M. Hoza, Edward Pyne, and Salil Vadhan. Pseudorandom Generators for Unbounded-Width Permutation Branching Programs. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ITCS.2021.7

Author Details

William M. Hoza
  • University of Texas at Austin, TX, USA
Edward Pyne
  • Harvard University, Cambridge, MA, USA
Salil Vadhan
  • Harvard University, Cambridge, MA, USA

Funding

  • Hoza, William M.: Supported by the NSF GRFP under grant DGE-1610403 and a Harrington Fellowship from UT Austin.
  • Pyne, Edward: Supported by NSF grant CCF-1763299.
  • Vadhan, Salil: Supported by NSF grant CCF-1763299 and a Simons Investigator Award.

Acknowledgements

We thank Jack Murtagh for collaboration at the start of this research. The first author thanks Dean Doron for insightful and relevant discussions about the works by Murtagh et al. [Murtagh et al., 2017; Murtagh et al., 2019]. We thank Shyam Narayanan, Dean Doron and David Zuckerman for valuable comments on a draft of this paper.

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