Local testing of lattices

Karthekeyan Chandrasekaran, Mahdi Cheraghchi, Venkata Gandikota, Elena Grigorescu

Research output: Contribution to journalArticlepeer-review

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 1-sided nonadaptive canonical tests. This result is akin to, and based on, an analogous result for error-correcting codes due to [E. Ben-Sasson, P. Harsha, and S. Raskhodnikova, SIAM J. Comput., 35 (2005), pp. 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 to 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 the well-known 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)
Pages (from-to)1265-1295
Number of pages31
JournalSIAM Journal on Discrete Mathematics
Volume32
Issue number2
DOIs
StatePublished - 2018
Externally publishedYes

Keywords

  • Construction-D
  • Lattices
  • Linear test
  • Locally testable codes
  • Property testing

ASJC Scopus subject areas

  • Mathematics(all)

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