Distortion of the spherical gaussian error through the canonical embedding in Ring - LWE

Question

In the paper, "On Error Distributions in Ring-based LWE" by Castryck, Iliashenko and Vercauteren, page 3, It is shown that the distrotion to the spherical gaussian in Ring - LWE is caused by the inverse of the vandermonde matrix and the matrix of multiplication by f'(x). It is explicitly stated that the inverse of Vandermonde matrix B is converted into a real matrix by a "easy unitary transformation". Can anyone help me out in figuring how get such a unitary transformation that takes a complex matrix to a real one?

Answer 1

The section you quote is from the introduction. The paper goes into more technical detail later. For example, equation 3.1 states (roughly) the same equation, and then states that:

$$B = U\Sigma$$

Where $\Sigma$ is the matrix of the canonical embedding, expressed relative to the basis $\{\alpha_1,\dots, \alpha_n\}$, and $U$ is the matrix (defined on page 5):

$$ U = \begin{pmatrix} I_{s\times s} & 0 & 0 \\ 0 & \frac{1}{\sqrt 2}I_{t\times t} & \frac{i}{\sqrt 2}I_{t\times t}\\ 0 & \frac{1}{\sqrt 2}I_{t\times t} & \frac{-i}{\sqrt 2}I_{t\times t} \end{pmatrix} $$

$U$ is unitary, and known to be an isomorphism $(H, \langle\cdot, \cdot\rangle_H)\cong (\mathbb{R}^n, \langle\cdot, \cdot\rangle)$, where $\langle x,y\cdot\rangle_H = \sum_i x_i\overline{y_i}$ is the Hermitian inner product, and $H$ is a certain conjugation-invariant subspace of $\mathbb{C}^n$ that contains the image of the canonical embedding. This is all discussed on page 5.