In the paper [Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller][1] by Micciancio and Peikert, they present the following theorem about the existence of trapdoor for LWE.

*Theorem 5.1:* There is an algorithm $\mathsf{GenTrap}(1^n,1^m,q)$ that, given any $n\geq 1, q\geq 2$ and $m=\mathcal{O}(n\log q)$, outputs a matrix $\mathbf{A}\in\mathbb{Z}^{n\times m}_q$ and a trapdoor $\mathbf{R}$ such that: 
 
 1. $\mathbf{A}$ is indistinguishable from a uniformly chosen matrix; and 
 2. there is an algorithm $\mathsf{Invert}$ that, given $\mathbf{b}=\mathbf{A}\mathbf{s}+\mathbf{e}$ (with $\mathbf{s}\in\mathbb{Z}^{n}_q$, $\mathbf{e}\in\mathbb{Z}^m$ and $||\mathbf{e}||<q/\mathcal{O}(\sqrt{n\log q})$) and a trapdoor $\mathbf{R}$, outputs $\mathbf{s}$ and $\mathbf{e}.$

They mention that the results in the paper can be straightforwardly adapted to the ring setting (Ring-LWE), however, they don't give details on that. What would be an equivalent result to this one in the ring setting?


  [1]: https://eprint.iacr.org/2011/501.pdf