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# Gaussian distribution propoprties

Good day,

I've a question regarding Gaussian distribution properties over lattices :

Let $$\mathcal{L}$$ := $$\mathcal{L}(\,b_{1}$$,..., $$b_{m})$$ be a lattice over $$\mathbb{R}^{n}$$, and $$W$$ = span($$b_{1}$$,..,$$b_{m}$$)$$^{\perp}$$. define $$\pi_{W}$$ to be the orthogonal projection onto $$W$$.

If i sample a vector b from a Gaussian distribution of support $$\mathcal{L}$$, standard deviation parameter $$s$$ and center parameter $$c \in$$ span($$\mathcal{L}$$).

Can i pretend that $$\pi_{W}(\,b)\,$$ can be sampled from a Gaussian distribution of support $$\pi_{W}(\,\mathcal{L})\,$$, standard deviation parameter $$s$$ and center parameter $$\pi_{W}(\,c)\,$$?

Thanks

Am I missing something? Surely $\pi_W(b)$ is the zero vector?
Certainly it is. but, what i was trying to say that i didn't expressed very well- and I'm really sorry about it-; is how the Gaussian distribution handle linear transformation such as orthogonal projections.
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As Daniel's comment mentions, the answer to your question appears to be "no, for trivial reasons" (plausibly due to some typo in your question).

Still, any question of this form has a relatively straightforward answer --- namely to look into Improved Discrete Gaussian and Subgaussian Analysis for Lattice Cryptography. In particular, section 3 discusses how Gaussians on lattices transform under linear transformations (including projections), and seems to be what you are interested in.

Thank you @Mark. I think i have an answer.)
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