NDLI: Local Testing of Lattices

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Gandikota, Venkata Chandrasekaran, Karthekeyan Cheraghchi, Mahdi Grigorescu, Elena
Copyright Year	2018
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.
Sponsorship	Purdue Research Foundation. National Science Foundation
Starting Page	1265
Ending Page	1295
Page Count	31
File Format	PDF
ISSN	08954801
DOI	10.1137/17M1110353
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	2
Volume Number	32
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2018-06-07
Access Restriction	Subscribed
Subject Keyword	Construction-D Lattice packing and covering locally testable codes Linear codes, general linear test property testing lattices
Content Type	Text
Resource Type	Article
Subject	Mathematics

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