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This may be a stupid question but I've been stuck on parsing these definitions for a while.

I am reading the paper "On Ideal Lattices and Learning with Errors Over Rings" by Lyubashevsky, Peikert, and Regev. I am trying to understand the error distributions they are proposing. In section 3, they define a set $\mathbb T = K_{\mathbb R}/R^V$ where $K$ is any number field and $K_{\mathbb R}$ is $K \otimes_{\mathbb Q} \mathbb R$, $R$ is its ring of integers, and $R^V$ is the dual module of $R$. I am a little unsure on how we are defining $\mathbb T$.

First, can we view $R^V$ as a submodule of $K_{\mathbb R}$ by looking at its image under inclusion? ie, are we really supposed to be looking at $K_{\mathbb R}/\iota(R^V)$ where $\iota: K \to K \otimes_{\mathbb Q} \mathbb R$ sending $x \mapsto x \otimes 1$?

Second, what kind of structure are we viewing $\mathbb T$ as? ie is $\mathbb T$ a $\mathbb Q$ vector space? I know $R^V$ has a natural $\mathbb Z$ module structure as it is a fractional ideal, but it seems weird to view $K_{\mathbb R}$ as a $\mathbb Z$ module.

Finally in definition 3.1 (their definition of the ring-lwe distribution), they state a ring-lwe sample is of the form $(a,b = as/q + e \pmod {modR^V})$? I am struggling to see how to view $b$ as an element of $K_{\mathbb R}$.

I apologize if these are "obvious" questions and I would really appreciate some clarification.

Thanks!

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1 Answer 1

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$R_q=R/qR, R_q^{\lor}=R^{\lor}/qR^{\lor}$.
$R^{\lor}$is R-module, $R_q^{\lor}$ is $R_q$-module(define as: $\forall x\in R,y\in R^{\lor}, (x+qR)(y+qR^{\lor})=xy+qR^{\lor}$)

$a\in R_q,s\in R_q^{\lor}$,So $as\in R_q^{\lor}$,and $as/q \in \frac{1}{q}R^{\lor}/R^{\lor}$.Since $\frac{1}{q}R^{\lor} \subset K_{\mathbb{R}} $,we get $as/q + e\,mod\,R^{\lor}\in K_{\mathbb{R}}/R^{\lor}$

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