Deciding Orthogonality in Construction-A Lattices

TL;DR

This paper characterizes when Construction-A lattices derived from binary and ternary codes have orthogonal bases, providing an efficient algorithm to decide orthogonality and find such bases, advancing understanding of lattice orthogonality complexity.

Contribution

It offers a complete characterization and an efficient algorithm for the orthogonality decision problem specifically for Construction-A lattices from binary and ternary codes.

Findings

Characterization of orthogonal bases for Construction-A lattices from binary and ternary codes

An efficient algorithm to decide orthogonality and find orthogonal bases for these lattices

Potential insights into the complexity of the orthogonality decision problem for general lattices

Abstract

Lattices are discrete mathematical objects with widespread applications to integer programs as well as modern cryptography. A fundamental problem in both domains is the Closest Vector Problem (popularly known as CVP). It is well-known that CVP can be easily solved in lattices that have an orthogonal basis \emph{if} the orthogonal basis is specified. This motivates the orthogonality decision problem: verify whether a given lattice has an orthogonal basis. Surprisingly, the orthogonality decision problem is not known to be either NP-complete or in P. In this paper, we focus on the orthogonality decision problem for a well-known family of lattices, namely Construction-A lattices. These are lattices of the form $C+q\mathbb{Z}^n$, where $C$ is an error-correcting $q$-ary code, and are studied in communication settings. We provide a complete characterization of lattices obtained from binary and ternary codes using Construction-A that have an orthogonal basis. We use this characterization to give an efficient algorithm to solve the orthogonality decision problem. Our algorithm also finds an orthogonal basis if one exists for this family of lattices. We believe that these results could provide a better understanding of the complexity of the orthogonality decision problem for general lattices.