Local Testing of Lattices

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).In this work we initiate the study of local testability for membership in point lattices, a class of infinite algebraic objects that form subgroups of Z n . Lattices are well-studied in mathematics, physics and computer science due to their rich algebraic structure [9]. Algorithms for various lattice problems have directly influenced the ability to solve integer programs [10,23,17]. Recently, lattices have found applications in modern cryptography due to attractive properties that enable efficient computations and security guarantees [28,26,31,32]. Lattices are also used in practical communication settings to encode data in a redundant manner in order to protect it from channel noise during transmission [12].A point lattice L ⊂ R n of rank k and dimension n is specified by a set of linearly independent vectors b 1 , . . . , b k ∈ Z n known as a basis, for some k ≤ n. If k = n the lattice is said to have full rank. The set L is defined to be the set of all vectors in R n that are integer linear combinations of the basis vectors, i.e., L :Lattices are the analogues over Z of linear error-correcting codes over a finite field F, which are generated as F-linear combinations of a linearly independent set of basis vectors b 1 , . . . , b k ∈ F n .Given a basis for a lattice L, we are interested in testing if a given input t ∈ R n belongs to L, or is far from all points in L by querying a small number of coordinates of t. We emphasize that this setting does not limit the computational space or time in pre-processing the lattice as well as the queried coordinates. The main goal is to design a tester that queries only a small number of coordinates of the input. MotivationInteger Programming. Lattices are the fundamental structures underlying integer programming problems. An integer programming problem (IP) is specified by a constraint matrix A ∈ Z n×m , a vector b ∈ R n . The goal is to verify if there exists an integer solution to the system Ax = b, x ≥ 0. Although IP is NP-complete [18], its instances are solv...

On Lattice Constructions D and D' from q-ary Linear Codes

“…Several works in lattice-based cryptography, for instance, analyze the complexity of certain computational problems related to lattices in the L P -norms, such as the closed and the shortest vector problems (CVP and SVP) [47] and the bounded decoding distance (BDD) [5]. Under a cryptography perspective and aiming at possible applications to error-detection in lattice-based communications, in [14] it has been proposed the study of a computational problem called local testability for membership in lattices, for L P -distances. It should be noted that, in order to obtain nearly matching bounds on the complexity, the authors of [14] focus on families of lattices constructed by Code Formula from a chain of binary Reed-Muller codes closed under the Schur product.…”

“…Under a cryptography perspective and aiming at possible applications to error-detection in lattice-based communications, in [14] it has been proposed the study of a computational problem called local testability for membership in lattices, for L P -distances. It should be noted that, in order to obtain nearly matching bounds on the complexity, the authors of [14] focus on families of lattices constructed by Code Formula from a chain of binary Reed-Muller codes closed under the Schur product.…”