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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

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    $\begingroup$ Am I missing something? Surely $\pi_W(b)$ is the zero vector? $\endgroup$
    – Daniel S
    May 15, 2023 at 14:55
  • $\begingroup$ 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. $\endgroup$
    – aussy
    May 16, 2023 at 11:20

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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.

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  • $\begingroup$ Thank you @Mark. I think i have an answer.) $\endgroup$
    – aussy
    May 16, 2023 at 11:24

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