# Reduction modulo lattice parallelepipeds

I was checking the following lecture notes, and on page 5, it says that $$m$$ vectors $$x_1,\ldots,x_m$$ are independently sampled from the Gaussian distribution, and then they are reduced modulo the parallelepiped $$\mathcal{P}(B)$$ of the lattice $$\Lambda(B)$$, where $$B$$ is the basis, in order to obtain new vectors $$y_1,\ldots,y_m$$. Given that the parallelepiped is defined as $$\mathcal{P}(B) = \{ \sum a_ib_i \mid a_i \in \mathbb{R}, a_i \in [0,1) \}$$, what exactly it means to reduce a vector modulo the parallelepiped, i.e., compute $$y_i = x_i \mod \mathcal{P}(B)$$?

• Decompose $x_{i}$ into its coordinate representation in $B$ and take all the coordinates $\pmod{1}$. Commented Jul 4, 2023 at 20:26
• @JoshuaWang What exactly you mean by coordinate representation in 𝐵? Something like $Bz = x$, so $z$ is the coordinate representation? What if $x$ is not a lattice point? Commented Jul 5, 2023 at 7:54

The equation $$y_i = x_i \bmod \mathcal{P}(B)$$ denotes that $$y_i$$ represents the unique vector in $$\mathcal{P}(B)$$ such that $$y_i - x_i \in \mathcal{L}(B)$$. To derive this $$y_i$$, we can express $$x_i = \sum_{i=1}^{n} c_i b_i$$ where $$c_i \in \mathbb{R}$$. Then, it is easy to see that $$y_i = \sum_{i=1}^{n} (c_i - \lfloor c_i \rfloor) \cdot b_i$$ satisfies the above conditions.