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Title: Local Testing for Membership in Lattices
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 {\em 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: \begin{enumerate} \item We show that in order to achieve low query complexity, it is sufficient to design one-sided non-adaptive {\em canonical} tests. This result is akin to, and based on an analogous result for error-correcting codes due to Ben-Sasson \etal\ (SIAM J. Computing 35(1) pp1--21). \item We demonstrate upper and lower bounds on the query complexity of local testing for membership in {\em 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. \item 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 {\em 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. \end{enumerate}  more » « less
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