Join Log in
Score:2
a196884 avatar Crypto

Volume of an NTRU lattice

cn flag
a196884

Let $K$ be a number field of degree $n$ and $\Lambda^q_h=\{(f,g)\in\mathcal{O}_K\text{ : }fh-g = 0\bmod q\mathcal{O}_K\}$, where $h$ is an NTRU public key. Then $\{(1,h),(0,q)\}$ generates a lattice. I've found it stated in the literature that $Vol(\Lambda^q_h) = Vol(\mathcal{O}_K)^2q^n$ (e.g. here), but how does the proof of this statement run? Or where can I find a proof?

66
1 + 0
Score:3
Mark avatar Crypto
ng flag
Mark

This is a standard computation in number theory. The idea behind it is that the matrix you have written down is a basis of the lattice as an $\mathcal{O}_K$-module, but to find the volume you first find a $\mathbb{Z}$-basis for the lattice, and then do "standard" computations with this. If $B$ is a $\mathbb{Z}$-basis of $\mathcal{O}_K$, then one has that:

$$B' = \begin{pmatrix}B & hB\\ 0 & qB\end{pmatrix} = \begin{pmatrix}1 & h\\ 0 & q\end{pmatrix}\otimes B$$

is a $\mathbb{Z}$-basis for your lattice. You can then compute the volume of this in the "standard" way, e.g. taking determinants, to get that:

$$\det B' = q^{\deg \mathcal{O}_K}(\det B)^2$$

which is precisely the expression you have written.

You can probably find this in many (if not all) textbooks on algebraic number theory. For example, I believe this is a corollary of lemma 2.23 of Milne's notes, but there are a number of abstractions one would have to unwind to verify this.

0 + 0
ธงชาติไทย
Elon Musk
I sit in a Tesla and translated this thread with Ai:

mangohost

Post an answer

Most people don’t grasp that asking a lot of questions unlocks learning and improves interpersonal bonding. In Alison’s studies, for example, though people could accurately recall how many questions had been asked in their conversations, they didn’t intuit the link between questions and liking. Across four studies, in which participants were engaged in conversations themselves or read transcripts of others’ conversations, people tended not to realize that question asking would influence—or had influenced—the level of amity between the conversationalists.