Parameters for ring-LWE I see on eprint that there are many papers suggesting ways to compute parameters for LWE. How can those be used to compute parameters for ring-LWE (assuming that known algorithms solving LWE are the best for solving ring-LWE)?

Specifically, consider the ring $R_q = \mathbb{Z}_q[X]/(X^n + 1)$. How can an R-LWE instance $(a, as+e)$ in that ring be transformed into an LWE instance $(A, As+e)$? I am missing in particular how to get $m &gt; n$ linear equations from what appear to be only $n$ linear constraints in the original system.
As mentioned by Peikert in [Pei16]:

*) Ring-LWE is parametrized by a ring $R$ of degree $n$ over $\mathbb{Z}$, a positive integer modulus $q$ defining the quotient ring $R_q=R/qR$, and an error distribution $\mathcal{X}$ over $R$.

**) Left-multiplication by any fixed $a\in R_q$ is a $\mathbb{Z}$-linear function from $R$ to $R_q$, so it can be represented by a square matrix $\mathbf{A}_a \in \mathbb{Z}_q^{n \times n}$, which maps $\mathbb{Z}^n$ to $\mathbb{Z}_q^n$.

For example, for $R=\mathbb{Z}[X]/(X^3+1)$ and $R_5=\mathbb{Z}_5[X]/(X^3+1)$, fix $a=3+2X+4X^2 \in R_5$. For $z=1+10X^2 \in R$, we have:
$$
a.z = 4X^2+2X+3 \in R_5
$$
Now, consider the matrix representation of $a$ and $z$:
$$
\mathbf{A}=\begin{bmatrix}
3 &amp; -4 &amp; -2\\
2 &amp; 3 &amp; -4 \\
4 &amp; 2 &amp; 3
\end{bmatrix} = \begin{bmatrix}
3 &amp; 1 &amp; 3\\
2 &amp; 3 &amp; 1 \\
4 &amp; 2 &amp; 3
\end{bmatrix} \in \mathbb{Z}_5^3
\qquad ; \qquad \mathbf{z}=\begin{bmatrix}
1\\
0 \\
10
\end{bmatrix}$$
We have:
$$
\mathbf{A.z}=\begin{bmatrix}
3 &amp; 1 &amp; 3\\
2 &amp; 3 &amp; 1 \\
4 &amp; 2 &amp; 3
\end{bmatrix}\begin{bmatrix}
1\\
0 \\
10
\end{bmatrix}=\begin{bmatrix}
33\\
12 \\
34
\end{bmatrix}=\begin{bmatrix}
3\\
2 \\
4
\end{bmatrix} \in \mathbb{Z}_5^3
$$
which is the matrix representation of $a.z$.

Now from these results, it is easy to see that each random element $a\in R_q$ corresponds to $n$ $\textit{related}$ (non-independent) vectors $\mathbf{a}\in \mathbb{Z}_q^n$, where $n$ is the degree of the ring $R$ over $\mathbb{Z}$.