NDLI: Pseudorandom Bits for Polynomials

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Viola, Emanuele Bogdanov, Andrej
Copyright Year	2010
Abstract	We present a new approach to constructing pseudorandom generators that fool low-degree polynomials over finite fields, based on the Gowers norm. Using this approach, we obtain the following main constructions of explicitly computable generators $G:\mathbb{F}^s\to\mathbb{F}^n$ that fool polynomials over a finite field $\mathbb{F}$: We stress that the results in (1) and (2) are unconditional, i.e., do not rely on any unproven assumption. Moreover, the results in (3) rely on a special case of the conjecture which may be easier to prove. Our generator for degree-d polynomials is the componentwise sum of d generators for degree-1 polynomials (on independent seeds). Prior to our work, generators with logarithmic seed length were only known for degree-1 (i.e., linear) polynomials [J. Naor and M. Naor, SIAM J. Comput., 22 (1993), pp. 838856]. In fact, over small fields such as $\mathbb{F}_2=\{0,1\}$, our results constitute the first progress on these problems since the long-standing generator by Luby, Velikovi, and Wigderson [Deterministic approximate counting of depth-2 circuits, in Proceedings of the 2nd Israeli Symposium on Theoretical Computer Science (ISTCS), 1993, pp. 1824], whose seed length is much bigger: $s=\exp\left(\Omega\left(\sqrt{\log n}\right)\right)$, even for the case of degree-2 polynomials over $\mathbb{F}_2$.
Starting Page	2464
Ending Page	2486
Page Count	23
File Format	PDF
ISSN	00975397
DOI	10.1137/070712109
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	6 (Special Section On Foundations of Computer Science)
Volume Number	39
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2010-04-30
Access Restriction	Subscribed
Subject Keyword	Computational difficulty of problems Modes of computation pseudorandom generator Gowers norm polynomial bias degree General Complexity classes
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Pseudorandom Bits for ConstantDepth Circuits with Few Arbitrary Symmetric Gates

Pseudorandom bits for polynomials (2007)

Pseudorandom bits for polynomials (2007)

Gowers Uniformity, Influence of Variables, and PCPs

Pseudorandom bits for polynomials

Arithmetic Circuits and Polynomial Replacement Systems

Holographic Algorithms

The Sum of D Small-Bias Generators Fools Polynomials of Degree D

Holographic Proofs and Derandomization

Pseudorandom Bits for Polynomials

Similar Documents

Pseudorandom Bits for ConstantDepth Circuits with Few Arbitrary Symmetric Gates

Pseudorandom bits for polynomials (2007)

Pseudorandom bits for polynomials (2007)

Gowers Uniformity, Influence of Variables, and PCPs

Pseudorandom bits for polynomials

Arithmetic Circuits and Polynomial Replacement Systems

Holographic Algorithms

The Sum of D Small-Bias Generators Fools Polynomials of Degree D

Holographic Proofs and Derandomization

Pseudorandom Bits for Polynomials