skip to main content
research-article
Public Access

Quadratic Maps Are Hard to Sample

Published:14 June 2016Publication History
Skip Abstract Section

Abstract

This note proves the existence of a quadratic GF(2) map p: {0, 1}n → {0, 1} such that no constant-depth circuit of size poly(n) can sample the distribution (u, p(u)) for uniform u.

References

  1. László Babai. 1987. Random oracles separate PSPACE from the polynomial-time hierarchy. Inform. Process. Lett. 26, 1 (1987), 51--53. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Chris Beck, Russell Impagliazzo, and Shachar Lovett. 2012. Large deviation bounds for decision trees and sampling lower bounds for AC0-circuits. Electron. Colloquium Computat. Complexity (ECCC) 19 (2012), 42.Google ScholarGoogle Scholar
  3. Itai Benjamini, Gil Cohen, and Igor Shinkar. 2014. Bi-Lipschitz bijection between the boolean cube and the hamming ball. In IEEE Symp. on Foundations of Computer Science (FOCS’14). Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. Anindya De and Thomas Watson. 2012. Extractors and lower bounds for locally samplable sources. ACM Trans. Computat. Theory 4, 1 (2012), 3. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Russell Impagliazzo and Moni Naor. 1996. Efficient cryptographic schemes provably as secure as subset sum. J. Cryptol. 9, 4 (1996), 199--216. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. Shachar Lovett and Emanuele Viola. 2012. Bounded-depth circuits cannot sample good codes. Computat. Complexity 21, 2 (2012), 245--266. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Emanuele Viola. 2012a. The complexity of distributions. SIAM J. Comput. 41, 1 (2012), 191--218. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Emanuele Viola. 2012b. Extractors for turing-machine sources. In Workshop on Randomization and Computation (RANDOM’12).Google ScholarGoogle ScholarCross RefCross Ref
  9. Emanuele Viola. 2014a. Is Nature a low complexity sampler? http://emanueleviola.wordpress.com/2014/11/09/is-nature-a-low-comple xity-sampler.Google ScholarGoogle Scholar
  10. Emanuele Viola. 2014b. Extractors for circuit sources. SIAM J. Comput. 43, 2 (2014), 355--972.Google ScholarGoogle ScholarCross RefCross Ref

Index Terms

  1. Quadratic Maps Are Hard to Sample

      Recommendations

      Reviews

      Chris Heunen

      A quadratic map is a function of the form f ( x ) = ax 2 + bx + c , or more generally, The only simpler polynomials are those of degree 1, omitting the terms a ijx ix j , and the constants f ( x ) = c . This delightful short note proves that quadratic maps, simple as they may seem, are hard to sample. That means there is no constant-depth circuit of size polynomial in n and complexity AC 0 that outputs the probability distribution ( x , f ( x )), where x is uniformly distributed. This is in contrast to degree 1 maps, which can be sampled by an AC 0 circuit by a theorem of Babai. A quadratic map that cannot be sampled is explicitly constructed, where x ranges over the field 0,1 with two elements. Namely, the counterexample is the composition of the inner product quadratic map ( x 1, x n ) x 1 x 2 + x 2 x 3 + + x n - 1 x n with a random linear transformation. This is remarkable because the inner product quadratic map on its own can be sampled with an AC 0 circuit. This example advances the understanding of which maps can be samples in AC 0. Online Computing Reviews Service

      Access critical reviews of Computing literature here

      Become a reviewer for Computing Reviews.

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in

      Full Access

      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 8, Issue 4
        July 2016
        97 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/2956681
        Issue’s Table of Contents

        Copyright © 2016 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 14 June 2016
        • Revised: 1 January 2016
        • Accepted: 1 January 2016
        • Received: 1 August 2015
        Published in toct Volume 8, Issue 4

        Permissions

        Request permissions about this article.

        Request Permissions

        Check for updates

        Qualifiers

        • research-article
        • Research
        • Refereed

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader
      About Cookies On This Site

      We use cookies to ensure that we give you the best experience on our website.

      Learn more

      Got it!