Deciding whether a Lattice has an Orthonormal Basis is in co-NP

TL;DR

This paper proves that determining whether a Euclidean lattice has an orthonormal basis is in both NP and co-NP, linking it to the lattice isomorphism problem and its complexity class SZK.

Contribution

It establishes the complexity classification of the orthonormal basis decision problem within NP and co-NP, relating it to lattice isomorphism.

Findings

The problem is in NP and co-NP.

Deciding if a lattice has an orthonormal basis is equivalent to lattice isomorphism.

Connects the problem to the complexity class SZK.

Abstract

We show that the problem of deciding whether a given Euclidean lattice L has an orthonormal basis is in NP and co-NP. Since this is equivalent to saying that L is isomorphic to the standard integer lattice, this problem is a special form of the Lattice Isomorphism Problem, which is known to be in the complexity class SZK.