I'm trying to understand CKKS (non bootstrappable) and I'm struggling with the encoding part. Particularly with to questions. I'm using the original paper , "Homomorphic Encryption for Arithmetic of Approximate Numbers ", also this notes "Lattices, Homomorphic Encryption, and CKKS" and this blog.

Given a vector $z\in \mathbb{C}^{N/2}$ they use the canonical embedding $\sigma^{-1}$ to map it to $\mathcal{R} =\mathbb{Z}[X]/(X^N+1)$. So you want $\sigma^{-1}:z\to\mathcal{R}$. First they expand (or projects) z adding the complex conjugates so $\pi^{-1}(z)\in\mathbb{H}\subset \mathbb{C}^{N}$.

First question.

  1. They use this notation for the space of this expansion that I don't understand: $\mathbb{H}=\{(z_j)_{j\in \mathbb{Z}^{*}_M}:z_{-j}=\overline{z_j},\forall j\in\mathbb{Z}_M^*\}\subseteq \mathbb{C}^{\Phi(M)}$

Where $T$ is a subgroup of the multiplicative group $\mathbb{Z}^*_M$ satisfying $\mathbb{Z}^*_M/T=\{\pm1\}$ and $\Phi(M)$ is the M-th Cyclotomic polynomial (I think... here it change a little bit the notation of this, the "original" was $\Phi_M(X)$).

1.1) So what it means that something is a subset of $\mathbb{C}^{\Phi(M)}$? This is really $\mathbb{C}^{N}$?

1.2) What this formula means in english? I know what the expansion do. Example: $\pi_{-1}((1+2i, 3-4i))\to (1+2i, 3-4i, 3+4i, 1-2i)$, but can't figure out the notation.

Second question.

  1. Before they apply $\sigma^{-1}$ to $\pi^{-1}(z)$ they need to use a round-off algorithm to discretize so that the output becomes an integral polynomial. Because they say that an element of $\mathbb{H}$ is not necessarily in $\sigma(\mathcal{R})$. Why?

2.1) This is because an particular element of $\mathbb{H}$ may be imposible to create with the base of $\sigma(\mathcal{R})$?


1 Answer 1

  1. You don't mention $T$ in your definition of $\mathbb{H}$. Still, the idea is that it's a mathematical fact that the embeddings $\sigma_i$ (for $i\in(\mathbb{Z}/N\mathbb{Z})^*$, i.e. there are $\varphi(N)$ total, which for $N = 2^k$ is $2^{k-1} = N/2$) come in two kinds

    a. "Real embeddings" --- meaning the image of $\sigma_i$ is a subset of $\mathbb{R}\subseteq \mathbb{R}$

    b. "Conjugate Pairs" --- meaning there are a pair of embeddings $(\sigma_i, \sigma_j)$ such that $\sigma_i(x) = \overline{\sigma_j(x)}$ are related by complex conjugation.

    I believe this notation $\mathbb{H}$ is an (explicit) way of describing these second type of embeddings (I don't think cyclotomics have real embeddings). Conceptually, one can view $\mathbb{Y}$ as being made entirely of these "conjugate pairs", so (as you say) you add conjugate pairs if they are missing.

1.1 No. In general, for a number field $K$, it embeds into $\mathbb{C}^M$ for $M$ the order of the "Galois group" (= number of embeddings into $\mathbb{C}$). For a cyclotomic of degree $N$, then $M = |\mathsf{Gal}(K/\mathbb{Q})| = \varphi(N)\neq N$. For $N = 2^k$ a power of two, $M = \varphi(N) = 2^{k-1} = N/2\neq N$. Note that for power-of-two cyclotomics, we do have that it embeds into $\mathbb{C}^M = \mathbb{C}^{N/2}\cong\mathbb{R}^N$, but this is really a coincidence.

1.2. In english, given a polynomial $a(x) = \sum_i a_ix^i$, $\sigma_j(a)$ is given by evaluating it at $\zeta_N^j = \exp(2\pi ij/N)$ for $j\in(\mathbb{Z}/N\mathbb{Z})^*$. As mentioned, these come in conjugate pairs, as $\overline{\zeta_N} = \zeta_N^{-1}$ (so $\overline{\sigma_i} = \sigma_{-i\bmod N}$).

  1. $\mathbb{H}$ is the image of $\mathbb{R}^N$ under all of the relevant embeddings. Instead, we (roughly) want the image of $\mathbb{Z}^N$ under all of the relevant embeddings. So morally, $\mathbb{H}\neq \sigma(\mathcal{R})$ for similar reasons why $\mathbb{Z}^N\neq \mathbb{R}^N$.

2.1 Yes, at least with integer coefficients. If you use arbitrary real coefficients you can do it (I'm pretty sure), but this isn't useful for cryptography.


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