I am reading the [CLT13] paper regarding multilinear maps over the integers and I have some troubles in understanding what the following random generation means:

The vectors of the matrix $r_i \in \mathbb{Z}^n$ are randomly and independently chosen from the half-open parallelepiped spanned by some vectors $\pi_1,...,\pi_n \in \mathbb{Z}^n$.

What I've understood: Those $r_i$ can be written as $r_i = a_1\pi_1+...+a_n\pi_n$ where each $a_i$ is random in $\mathbb{Z}$. Is it correct?

You might want to search for the word span to see exactly what the authors meant in the paper.

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    $\begingroup$ Does Appendix E (Uniform Sampling of a Parallelepiped) of the same paper answers your question in sufficient details ? $\endgroup$ – LeoDucas Apr 15 '17 at 5:26
  • $\begingroup$ Great @LeoDucas, I do not know how I missed that. The actually answer is in Appendix E of the original paper. $\endgroup$ – penguina Apr 15 '17 at 7:13

As @LeoDucas already commented, the answer is contained in Appendix E (page 22) of the same paper:

screenshot of Appendix E

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