TY - GEN

T1 - Log-seed pseudorandom generators via iterated restrictions

AU - Doron, Dean

AU - Hatami, Pooya

AU - Hoza, William M.

N1 - Publisher Copyright:
© Dean Doron, Pooya Hatami, and William M. Hoza; licensed under Creative Commons License CC-BY 35th Computational Complexity Conference (CCC 2020).

PY - 2020/7/1

Y1 - 2020/7/1

N2 - There are only a few known general approaches for constructing explicit pseudorandom generators (PRGs). The “iterated restrictions” approach, pioneered by Ajtai and Wigderson [2], has provided PRGs with seed length polylog n or even Oe(log n) for several restricted models of computation. Can this approach ever achieve the optimal seed length of O(log n)? In this work, we answer this question in the affirmative. Using the iterated restrictions approach, we construct an explicit PRG for read-once depth-2 AC0[⊕] formulas with seed length O(log n) + Oe(log(1/ε)). In particular, we achieve optimal seed length O(log n) with near-optimal error ε = exp(−Ω(log e n)). Even for constant error, the best prior PRG for this model (which includes read-once CNFs and read-once F2-polynomials) has seed length Θ(log n · (log log n)2) [22]. A key step in the analysis of our PRG is a tail bound for subset-wise symmetric polynomials, a generalization of elementary symmetric polynomials. Like elementary symmetric polynomials, subset-wise symmetric polynomials provide a way to organize the expansion of Qmi=1(1 + yi). Elementary symmetric polynomials simply organize the terms by degree, i.e., they keep track of the number of variables participating in each monomial. Subset-wise symmetric polynomials keep track of more data: for a fixed partition of [m], they keep track of the number of variables from each subset participating in each monomial. Our tail bound extends prior work by Gopalan and Yehudayoff [17] on elementary symmetric polynomials.

AB - There are only a few known general approaches for constructing explicit pseudorandom generators (PRGs). The “iterated restrictions” approach, pioneered by Ajtai and Wigderson [2], has provided PRGs with seed length polylog n or even Oe(log n) for several restricted models of computation. Can this approach ever achieve the optimal seed length of O(log n)? In this work, we answer this question in the affirmative. Using the iterated restrictions approach, we construct an explicit PRG for read-once depth-2 AC0[⊕] formulas with seed length O(log n) + Oe(log(1/ε)). In particular, we achieve optimal seed length O(log n) with near-optimal error ε = exp(−Ω(log e n)). Even for constant error, the best prior PRG for this model (which includes read-once CNFs and read-once F2-polynomials) has seed length Θ(log n · (log log n)2) [22]. A key step in the analysis of our PRG is a tail bound for subset-wise symmetric polynomials, a generalization of elementary symmetric polynomials. Like elementary symmetric polynomials, subset-wise symmetric polynomials provide a way to organize the expansion of Qmi=1(1 + yi). Elementary symmetric polynomials simply organize the terms by degree, i.e., they keep track of the number of variables participating in each monomial. Subset-wise symmetric polynomials keep track of more data: for a fixed partition of [m], they keep track of the number of variables from each subset participating in each monomial. Our tail bound extends prior work by Gopalan and Yehudayoff [17] on elementary symmetric polynomials.

KW - Parity gates

KW - Pseudorandom generators

KW - Pseudorandom restrictions

KW - Read-once depth-2 formulas

UR - http://www.scopus.com/inward/record.url?scp=85089376213&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.CCC.2020.6

DO - 10.4230/LIPIcs.CCC.2020.6

M3 - Conference contribution

AN - SCOPUS:85089376213

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 35th Computational Complexity Conference, CCC 2020

A2 - Saraf, Shubhangi

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 35th Computational Complexity Conference, CCC 2020

Y2 - 28 July 2020 through 31 July 2020

ER -