Local Testing for Membership in Lattices

TL;DR

This paper develops a theoretical framework for local testing of lattice membership, establishing bounds on query complexity and connecting lattice testing to error-correcting code theory, with applications in cryptography and integer programming.

Contribution

It introduces the first systematic study of local testing for lattices, providing bounds and constructions for code formula lattices, and relates testing complexity to canonical tests.

Findings

Bounds on query complexity for code formula lattices.

Nearly-tight bounds for Reed-Muller based lattices.

Low query complexity achieved via one-sided non-adaptive tests.

Abstract

Motivated by the structural analogies between point lattices and linear error-correcting codes, and by the mature theory on locally testable codes, we initiate a systematic study of local testing for membership in lattices. Testing membership in lattices is also motivated in practice, by applications to integer programming, error detection in lattice-based communication, and cryptography. Apart from establishing the conceptual foundations of lattice testing, our results include the following: 1. We demonstrate upper and lower bounds on the query complexity of local testing for the well-known family of code formula lattices. Furthermore, we instantiate our results with code formula lattices constructed from Reed-Muller codes, and obtain nearly-tight bounds. 2. 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) pp1-21).