eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-01-08
22:1
22:15
10.4230/LIPIcs.ITCS.2019.22
article
Pseudorandom Generators from the Second Fourier Level and Applications to AC0 with Parity Gates
Chattopadhyay, Eshan
1
Hatami, Pooya
2
Lovett, Shachar
3
Tal, Avishay
4
Department of Computer Science, Cornell University, 107 Hoy Rd, Ithaca, NY, USA
Department of Computer Science, University of Texas at Austin, 2317 Speedway, Austin, TX, USA
Department of Computer Science, University of California San Diego, La Jolla, CA, USA
Department of Computer Science, Stanford University, 353 Serra Mall, Stanford, CA, USA
A recent work of Chattopadhyay et al. (CCC 2018) introduced a new framework for the design of pseudorandom generators for Boolean functions. It works under the assumption that the Fourier tails of the Boolean functions are uniformly bounded for all levels by an exponential function. In this work, we design an alternative pseudorandom generator that only requires bounds on the second level of the Fourier tails. It is based on a derandomization of the work of Raz and Tal (ECCC 2018) who used the above framework to obtain an oracle separation between BQP and PH.
As an application, we give a concrete conjecture for bounds on the second level of the Fourier tails for low degree polynomials over the finite field F_2. If true, it would imply an efficient pseudorandom generator for AC^0[oplus], a well-known open problem in complexity theory. As a stepping stone towards resolving this conjecture, we prove such bounds for the first level of the Fourier tails.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol124-itcs2019/LIPIcs.ITCS.2019.22/LIPIcs.ITCS.2019.22.pdf
Derandomization
Pseudorandom generator
Explicit construction
Random walk
Small-depth circuits with parity gates