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Let $R$ be the ring of integers of a cyclotomic field $\mathbb{Q}(\zeta_n)$, where $n$ is a power of two, and $\boldsymbol{a} \in R_{q}^{m}$, for $m\in\mathbb{Z}^+$, $q\in\mathbb{Z}_{\geq2}$ prime. Define the following $R$-modules, where $I$ is an ideal of $R_{q} = R/qR$: $$ \begin{gathered} \boldsymbol{a}^{\perp}(I):=\left\{\left(t_{1}, \ldots, t_{m}\right) \in R^{m}: \forall i,\left(t_{i} \bmod q\right) \in I \text { and } \sum_{i} t_{i} a_{i}=0 \bmod q\right\}, \\ L(\boldsymbol{a}, I):=\left\{\left(t_{1}, \ldots, t_{m}\right) \in R^{m}: \exists s \in R_{q}, \forall i,\left(t_{i} \bmod q\right)=a_{i} \cdot s \bmod I\right\}. \end{gathered} $$ Ideals of $R_{q}$ can be written in the form $I_{S}:=\prod_{i \in S}\left(x-\zeta_n^{i}\right) \cdot R_{q}=\left\{a \in R_{q}: \forall i \in S, a\left(\zeta_n^{i}\right)=0\right\}$, where $S$ is any subset of $\{1, \ldots, n\}$ (the $\zeta_n^{i}$'s are the roots of $\Phi_n$ modulo $q$ ). Define $I_{S}^{\times}=\prod_{i \in S}\left(x-{\zeta_n^{i}}^{-1}\right) \cdot R_{q}$.

The authors of this paper then prove (Lemma 7): let $S \subseteq\{1, \ldots, n\}$ and $\boldsymbol{a} \in R_{q}^{m}$. Let $\bar{S}=\{1, \ldots, n\} \backslash S$ and $\boldsymbol{a}^{\times} \in$ $R_{q}^{m}$ be defined by $a_{i}^{\times}=a_{i}\left(x^{-1}\right)$. Then, with $\widehat{\cdot}$ denoting the dual of a lattice: $$ \widehat{\boldsymbol{a}^{\perp}\left(I_{S}\right)}=\frac{1}{q} L\left(\boldsymbol{a}^{\times}, I_{\bar{S}}^{\times}\right). $$ My question is: while the containment $\frac{1}{q} L\left(\boldsymbol{a}^{\times}, I_{\bar{S}}^{\times}\right)\subset \widehat{\boldsymbol{a}^{\perp}\left(I_{S}\right)}$ is clear to me, I cannot prove the reverse direction $\widehat{\boldsymbol{a}^{\perp}\left(I_{S}\right)}\subset\frac{1}{q} L\left(\boldsymbol{a}^{\times}, I_{\bar{S}}^{\times}\right)$. How is the result obtained?

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Their paper contains a proof of this, they "just" first appeal to lattice duality. In short, to prove that lattices

$$A = B,$$

it suffices (as you say) to prove that $A\subseteq B$ and $B\subseteq A$. What they do is use that

$$B\subseteq A\iff A^*\subseteq B^*,$$

and instead prove that $A\subseteq B$ and $A^*\subseteq B^*$. You can verify that their proof does precisely this, but with $A = L(\cdot)$, and $B = \widehat{\alpha^\perp(\cdot)}$ your lattices. Concretely, the containment you are missing is $\widehat{L(\cdot)}\subseteq \frac{1}{q}\alpha^\perp(\cdot)$. Regarding this, they state

This can be seen by considering elements of $L(\cdot)$ that correspond to $s = 1$.

I haven't checked, this but I imagine they mean that $\widehat{L(\cdot)} = \{\vec t\in R^m :\forall \ell \in L(\cdot): \langle \ell, t\rangle\equiv 0\bmod q\}$. If we replace $L(\cdot)$ in this with some subset $S\subseteq L(\cdot)$, we get a superset of $\widehat{L(\cdot)}$. It seems they in particular state you should replace $L(\cdot)$ with the subset corresponding to the choice of $s = 1$. Concretely, this gives us the containment.

$$\widehat{L(I_{\alpha^\times, \overline{S}}^\times)} \subseteq \{\vec t\in R^m : \forall i : (t_i\bmod q) = \alpha_i^\times\bmod I_{\overline{S}}^\times\}.$$

I don't know if this is precisely $\frac{1}{q}\alpha^\perp(\cdot)$, but their hint makes it sound like the right thing to look at.

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