In the paper Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller 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 (RLWE), however they don't give details on that. What would be an equivalent result to this one in the ring setting?

In the paper Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller 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?

In the paper MP12 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: i) $\mathbf{A}$ is indistinguishable from a uniformly chosen matrix; and ii) 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 (RLWE), however they don't give details on that. What would be an equivalent result to this one in the ring setting?

Link of the paper: https://eprint.iacr.org/2011/501

In the paper Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller 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 (RLWE), however they don't give details on that. What would be an equivalent result to this one in the ring setting?