Deciding whether a lattice has an orthonormal basis is in co-NP

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. We achieve this by deploying a result on characteristic vectors by Elkies that gained attention in the context of 4-manifolds and Seiberg-Witten equations, but seems rather unnoticed in the algorithmic lattice community. On the way, we also show that for a given Gram matrix \(G \in \mathbb {Q}^{n \times n}\), we can efficiently find a rational lattice that is embedded in at most four times the initial dimension n, i.e. a rational matrix \(B \in \mathbb {Q}^{4n \times n}\) such that \(B^\intercal B = G\).