Gaussian distribution over ideal lattices

Question

I am reading Lyubashevsky's paper [LPR10] about ideal lattices and R-LWE problem and I did not understand the following part:

"Using the canonical embedding also allows us to think of the distribution $D_r$ (for $r \in (R^+)^n$) over $H$ as a distribution over $K$. Strictly speaking, the distribution $D_r$ is not quite over $K$, but rather over the field $K_{\mathbb{R}} = K ⊗_{\mathbb{Q}}\mathbb{R}$, which, roughly speaking, is to $K$ as $\mathbb{R}$ is to $\mathbb{Q}$."

Can someone please explain this?

Answer 1

The space $H$ is defined such that it is isomorphic to $\mathbb{R}^n$ for some $n$. Therefore, we can choose a basis $\{h_1,\ldots,h_n\}$ such that for every $h\in H$ there exist $x_1,\ldots,x_n\in\mathbb{R}$ such that $$h=x_1h_1+\cdots+x_nh_n.$$ So we think of elements of $H$ as linear combinations of a basis with coefficients in $\mathbb{R}$.

On the other hand, we defined a number field $K=\mathbb{Q}(\zeta)$. As a vector space, this is isomorphic to $\mathbb{Q}^n$ (this $n$ is the same $n$ from above, we simply construct $H$ that way). Choosing a basis $\{k_1,\ldots,k_n\}$, for every $k\in K$ we can find $y_1,\ldots,y_n\in\mathbb{Q}$ such that $$k=y_1k_1+\cdots+y_nk_n.$$ So we think of elements of $K$ as linear combinations of a basis with coefficients in $\mathbb{Q}$.

The distribution $D_r$ is defined on the space $H$, which has coefficients in $\mathbb{R}$. We could try and apply this to $K$, but the coefficients of $K$ are only in $\mathbb{Q}$. The statement you cite says that we apply $D_r$ to the space which has the same basis as $K$, but with coefficients in $\mathbb{R}$. This space is denoted $K\otimes_\mathbb{Q}\mathbb{R}$.