Trapdoor Quality for Lattice Crypto

In these two papers the authors mention the "quality" of a trapdoor

• [GPV] https://eprint.iacr.org/2007/432
• [MP] https://eprint.iacr.org/2011/501

But the best detail on this matter I could find was "The quality of a trapdoor S roughly corresponds to the Euclidean lengths of its vectors — shorter is better."

I wonder where could I find a more formal treatment on this matter. Thanks!

From definition 5.2 of the second paper:

Let $\mathbf A\in\mathbb Z^{n\times m}_q$ and $\mathbf G \in\mathbb Z^{n\times w}_q$ be matrices with $m \ge w \ge n$. A $\mathbf G$-trapdoor for $\mathbf A$ is a matrix $\mathbf R\in \mathbb Z^{(m−w)\times w}$ such that $\mathbf A \begin{bmatrix} \mathbf R\\ \mathbf I\end{bmatrix}= \mathbf H\mathbf G$ for some invertible matrix $\mathbf H\in\mathbb Z^{n\times n}_q.$ We refer to $\mathbf H$ as the tag or label of the trapdoor. The quality of the trapdoor is measured by its largest singular value $s_1(\mathbf R)$.