Proving LWE inversion in Micciancio-Peikert-2012 lattice trapdoors
We are given some $b^t = s^t A + e^t$ for $A$ of the above form and short $e$, and wish to recover $s$ (which will immediately give us $e$ as well). If $e$ is short enough, then $\hat{b}^t = (s^t H) G + \hat{e}$ for some sufficiently short $\hat{e}$. (This is where we use the bound on the expansion, or top singular value, of $R$.) Therefore, the LWE inverter for $G$ will recover $s^t H$ (and $\hat{e}$), from which the original $s$ can be recovered via $H^{-1}$.
It’s very important that $e$ is short enough to ensure that $\hat{e}$ is short enough. If the latter is too long, then the LWE inverter for $G$ might return the wrong answer for $s^t H$, leading to a wrong $s$ and thereby a wrong, probably very long $e$.