How the condition $s \geq 8$ is determined in Lindner-Peikert cryptosystem? In Lindner &amp; Peikert paper, the authors propose that to set the cryptosystem's parameters, one should choose $q$ to be large enough to allow for a Gaussian parameter $s \geq 8$.

My question is, how this condition ($s \geq 8$) is obtained? I guess this lower-bound on $s$ relates to the smoothing parameter, but I can't figure out how it is determined?
Note: I edited a lot of my speculation and calculations after comments from Chris Peikert (see comments below). 

The smoothing parameter on a lattice, $\eta_\epsilon(\Lambda)$, is roughly the width of a discrete Gaussian $\mathcal{D}_{\Lambda,s}$ above which it is well-approximated by a continuous Gaussian; see the Regev quote on page 17 here.

In particular, for $\mathbb{Z}^n$, there is an upper bound for the smoothing parameter (see page 8):

$\eta_\epsilon(\mathbb{Z}^n) \leq \sqrt{\ln(2n(1 + 1/\epsilon))/\pi}$

Again, on page 17, if $s \geq \sqrt{2} \eta_\epsilon(\mathbb{Z}^n)$, then the sum of the discrete Gaussian $\mathcal{D}_{\Lambda,s}$ and a continuous Gaussian with width $s$ is basically equivalent ("within statistical distance $4\epsilon$") to a continous Gaussian of width $\sqrt{2} s$, and you want this to hide the discrete structure in the continuous noise.

So I thought at first that the critical quantity was $\sqrt{2} \eta_\epsilon(\mathbb{Z}^n)$. However! After a comment from one of the authors of the paper (see comments below!), it's enough to just consider $s \geq \eta_\epsilon$ with $\epsilon \ll 1/2$. Using the formula above with $\epsilon = 2^{-256}$:


$n=128 \rightarrow s \geq 7.63$
$n=1024 \rightarrow s \geq 7.68$
$n=2048 \rightarrow s \geq 7.89$ 


So $\eta_\epsilon$ doesn't fluctuate that much and you can take $s \geq 8$ for a nice round number.

Of course you can vary $\epsilon$; I used Lemma 4.2 in this paper to justify constraining it $0 &lt; \epsilon &lt; 1$, but it should be a negligible function, like $\epsilon = 2^{-n}$.

Thanks to Chris Peikert for discussion and clarification in the comments!