I have a question about the definition of the (Closest Vector Problem) CVP. In the literature you can find for example this definition of the approximation variant of CVP:

$CVP_\gamma$, Search: Given a basis $B \in \mathbb{Z}^{m \times n}$ and a point $t \in \mathbb{Z}^m$, find a point $x \in \mathcal{L}(B)$ such that $\forall y \in \mathcal{L}(B)$,$||x-t|| \leq \gamma|| y-t ||$


My question is now and unfortunately this is a bit misleading, since some authors define $t$ as an element of $\mathbb{R}^m$. What are the arguments that speak for using $t \in \mathbb{Z}^m$ in the definition for the CSV instead of $t \in \mathbb{R}^m$?

In the hope that I have not formulated my question too unclearly, I would be interested in what you think of it?

  • 1
    $\begingroup$ This is perhaps a pedantic point, but one can efficiently represent $t\in\mathbb{Z}^m$ with $\approx m\log_2 \lVert t\rVert_\infty$ bits, while it is not clear you can represent an arbitrary point $t\in\mathbb{R}^m$. Of course, one can always simply choose a good-enough fixed-point approximation of $t\approx \tilde{t}\in\frac{1}{2^k}\mathbb{Z}^m$, hence why this mostly pedantic. $\endgroup$
    – Mark
    Oct 24, 2022 at 5:10


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