For a positive integer $m$, the $m$-th cyclotomic ring is $R = \mathbb{Z}[\zeta_m]$, the ring extension of the integers $\mathbb{Z}$ obtained by adjoining an element $\zeta_m$ having multiplicative order $m$. The ring $R$ is contained in $m$-th cyclotomic number field $K = \mathbb{Q}(\zeta_m)$, where $\zeta_m = e^{2\pi i / m}$.
Let $\mathcal{D}(R)$ be a discrete Gaussian distribution over $R$ of parameter $\sigma$. For $\ell \in [1,k]$ sample random ideals $g_1^\ell, g_2^\ell \gets \mathcal{D}(R)$. Define the lattices $\Lambda_1^\ell \overset{\Delta}{=} g_1^\ell \cdot R$ and $\Lambda_2^\ell \overset{\Delta}{=} g_2^\ell \cdot R$. For $i \in [1,r]$ sample random values, $g_2^\ell\xi_{1,i}^\ell \gets \mathcal{D}(\Lambda_2^\ell)$ and $g_1^\ell \xi_{2,i}^\ell \gets \mathcal{D}(\Lambda_1^\ell)$, where $r, k$ are positive integers. Consider the following term $$ \sum_\ell \left(\sum_{i,j} v_{i,j} \xi_{1,i}^\ell \xi_{2,j}^\ell \right)g_2^\ell g_1^\ell, $$ where $(v_{i,j})_{i,j} \in R^{r \times r}$.
For what value of $\ell$ is the above term well-spread over the ring $R$, i.e., lives in $R$ and not in the ideal $g_2g_1$? It feels to me that even for $\ell = 2$ the term would live in $R$, since 2 random elements are likely to be coprime. But I suppose this depends on the Dedekind zeta function $\zeta_K(\ell)$ of the number field $K$.
- How can we compute $\zeta_K(\ell)$ over the cyclotomic number fields?
- How large $\ell$ needs to be to ensure that the probability of $\ell$ elements not being coprime is negligible, i.e., $1 - 1/\zeta_K(\ell) \approx 2^{-\kappa}$ (for some integer $\kappa \geq 128$)?