Check if two bases form the same lattice

Question

Hi I am currently working on a problem regarding different bases and their respective lattices. I am a physicist, so maybe the question might be trivial (I don't know), but I was not able to find anything regarding this problem yet.

The Question

Basically my question is, whether there is a clever way to determine if two bases form the same lattice.

Some Context and Definitions

Definition: Let $B=\{b_1,\dots,b_n\}$ be linearly independent vectors in $\mathbb{R}^n$. The lattice $\mathcal{L}(B)$ generated by $B$ is the set
$$\mathcal{L}(B) = \left\{\sum_{i=1}^n x_i b_i : x_i\in \mathbb{Z} \right\}$$
of all the integer linear combinations of the vectors in $B$, and the set $B$ is called a basis for $\mathcal{L}(B)$.

It is now fairly easy to find statements of the form if $B$ and $B^\prime$ are a basis of the same lattice e.g. $\mathcal{L}(B)=\mathcal{L}(B^\prime)$, then … .

An example would be the determinants of the bases which are the same.

It is now of interest to me, if there is a simple way/algorithm to check if two bases for the same lattice.

Answer 1

From chapter 1 of Micciancio & Goldwasser¹:

Equivalent bases (i.e., bases that generate the same lattice) can be algebraically characterized as follows. Two bases ${\bf B}, {\bf B}' \in \Bbb R^{m \times n}$ are equivalent if and only if there exists a unimodular matrix ${\bf U} \in \Bbb Z^{n \times n}$ (i.e., an integral matrix with determinant $\det({\bf U}) = \pm 1$) such that ${\bf B}' = {\bf B} {\bf U}$.

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From chapter 2 of Peikert's survey²:

A lattice basis $\bf B$ is not unique: for any unimodular matrix ${\bf U} \in \Bbb Z^{n \times n}$ (i.e., one having determinant $\pm 1$), ${\bf B} \cdot {\bf U}$ is also a basis of $\mathcal L({\bf B})$, because ${\bf U} \cdot \Bbb Z^n = \Bbb Z^n$.

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1. Daniele Micciancio, Shafi Goldwasser, Complexity of Lattice Problems — a Cryptographic Perspective, Springer, 2002.

2. Chris Peikert, A Decade of Lattice Cryptography [PDF], February 17, 2016.