Gaussian distribution propoprties

pr flag

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 $\pi_{W}(\,c)\,$?


Daniel S avatar
ru flag
Am I missing something? Surely $\pi_W(b)$ is the zero vector?
aussy avatar
pr flag
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.
ng flag

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.

aussy avatar
pr flag
Thank you @Mark. I think i have an answer.)
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