# Definition of randomly generate in the parallelepiped

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.

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