Local testing for membership in lattices

Karthekeyan Chandrasekaran, Mahdi Cheraghchi, Venkata Gandikota, Elena Grigorescu

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Testing membership in lattices is of practical relevance, with applications to integer programming, error detection in lattice-based communication and cryptography. In this work, we initiate a systematic study of local testing for membership in lattices, complementing and building upon the extensive body of work on locally testable codes. In particular, we formally define the notion of local tests for lattices and present the following: 1. We show that in order to achieve low query complexity, it is sufficient to design one-sided non-adaptive canonical tests. This result is akin to, and based on an analogous result for error-correcting codes due to Ben-Sasson et al. (SIAM J. Computing, 35(1):1-21). 2. We demonstrate upper and lower bounds on the query complexity of local testing for membership in code formula lattices. We instantiate our results for code formula lattices constructed from Reed-Muller codes to obtain nearly-matching upper and lower bounds on the query complexity of testing such lattices. 3. We contrast lattice testing from code testing by showing lower bounds on the query complexity of testing low-dimensional lattices. This illustrates large lower bounds on the query complexity of testing membership in knapsack lattices. On the other hand, we show that knapsack lattices with bounded coefficients have low-query testers if the inputs are promised to lie in the span of the lattice.

Original languageEnglish (US)
Title of host publication36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2016
EditorsAkash Lal, S. Akshay, Saket Saurabh, Sandeep Sen, Saket Saurabh
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages46.1-46.14
ISBN (Electronic)9783959770279
DOIs
StatePublished - Dec 1 2016
Externally publishedYes
Event36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2016 - Chennai, India
Duration: Dec 13 2016Dec 15 2016

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume65
ISSN (Print)1868-8969

Conference

Conference36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2016
CountryIndia
CityChennai
Period12/13/1612/15/16

Keywords

  • Complexity theory
  • Lattices
  • Locally testable codes
  • Property testing

ASJC Scopus subject areas

  • Software

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