samples/texts/1897687/page_15.md

volume roughly one. And, by definition, if $\eta_\epsilon(\mathcal{L}) \ge \Omega(C_\eta(n)\sqrt{\log(1/\epsilon)})$, then there exists a projection $\pi$ such that, say, $\text{vol}(\pi(\mathbb{R}^n/\mathcal{L})) = \det(\pi(\mathcal{L})) \ge 100$. Therefore, the projected Gaussian lies in a small fraction of $\pi(\mathbb{R}^n/\mathcal{L})$ with high probability.

To make this precise, we must discretize $\mathbb{R}^n/\mathcal{L}$ appropriately to, say, $(\mathcal{L}/q)/\mathcal{L}$ for some large integer $q > 1$ and sample from a discretized version of the continuous Gaussian. Naturally, we choose $D_{\mathcal{L}/q}$. The following theorem shows that $D_{\mathcal{L}/q} \bmod \mathcal{L}$ lies in a small subset of $(\mathcal{L}/q)/\mathcal{L}$ when $\eta_{1/2}(\mathcal{L})$ is large.

Theorem 3.8. For any lattice $\mathcal{L} \subset \mathbb{R}^n$ with sufficiently large $n$ and integer $q \ge 2^n (\eta_{2-n}(\mathcal{L}) + \mu(\mathcal{L}))$, if $\eta_{1/2}(\mathcal{L}) \ge 1000C_\eta(n)$ (and in particular if $\eta_{1/2}(\mathcal{L}) \ge 10^4(\log(n)+2)$), then there is a subset $S \subset (\mathcal{L}/q)/\mathcal{L}$ with $|S| \le q^n/200$ such that

$$\mathbf{x}\sim D_{\mathcal{L}\mathrm{/}q}\mathcal{L}[\mathbf{X}\in S]\ge \frac{9}{10}\mathrm{.}\backslash mathbf\{x\}\; \backslash sim\; D\_\{\backslash mathcal\{L\}/q\}\; \backslash bmod\; \backslash mathcal\{L\}\; [\backslash mathbf\{X\}\; \backslash in\; S]\; \backslash ge\; \backslash frac\{9\}\{10\}.$$

Proof. It is easy to see that $D_{\mathcal{L}/q}$ is statistically close to the distribution obtained by sampling from a continuous Gaussian with parameter one and rounding to the closest vector in $\mathcal{L}/q$. (One must simply recall from Lemma 2.6 that nearly all of the mass of $D_{\mathcal{L}/q}$ lies in a ball of radius $\sqrt{n}$ and notice that for such short points, shifts of size $\mu(\mathcal{L}/q) < 2^{-n}$ have little effect on the Gaussian mass.) It therefore suffices to show that the above probability is at least 19/20 when $\mathbf{X}$ is sampled from this new distribution. We write $CVP(\mathbf{t})$ for the closest vector in $\mathcal{L}/q$ to $\mathbf{t}$.

By assumption, there is a lattice projection $\pi$ onto a $k$-dimensional subspace such that $\det(\pi(\mathcal{L})) \ge 1000^k$. Notice that $|\pi(CVP(\mathbf{t}))| \le |\pi(\mathbf{t})| + \mu(\mathcal{L})/q \le |\mathbf{t}| + 2^{-n}$ for any $\mathbf{t} \in \mathbb{R}^n$. In particular, if $\mathbf{X}$ is sampled from a continuous Gaussian with parameter one,

$$\mathrm{Pr}[\mathrm{\parallel }\pi (CVP(\mathbf{X}))\mathrm{\parallel }\ge \sqrt{k}]\le \mathrm{Pr}[\mathrm{\parallel }\pi (\mathbf{X})\mathrm{\parallel }\ge \sqrt{k}-2^{-n}]\le \frac{1}{20},\backslash Pr\; \backslash left[\; \backslash |\; \backslash pi(CVP(\backslash mathbf\{X\}))\; \backslash |\; \backslash ge\; \backslash sqrt\{k\}\; \backslash right]\; \backslash le\; \backslash Pr\; \backslash left[\; \backslash |\; \backslash pi(\backslash mathbf\{X\})\; \backslash |\; \backslash ge\; \backslash sqrt\{k\}\; -\; 2^\{-n\}\; \backslash right]\; \backslash le\; \backslash frac\{1\}\{20\},$$

where we have applied Lemma 2.20. But, by Lemma 2.2, there are at most $(q/200)^k$ points $y \in (\pi(\mathcal{L})/q)/\pi(\mathcal{L}) \cap \sqrt{k}B_2^k$. Therefore, there are at most $q^n/200^k \le q^n/200$ points $y \in (\mathcal{L}/q)/\mathcal{L}$ with 19/20 of the mass, as needed. $\square$

Corollary 3.9. For any lattice $\mathcal{L} \subset \mathbb{R}^n$ with $n \ge 2$, $\varepsilon \in (0, 1/2)$, and integer $q \ge 2$, let $\mathbf{X} \sim D_{\mathcal{L}/q} \bmod \mathcal{L}$. Then,

1. if $\eta_\varepsilon(\mathcal{L}) \le 1$, then $H(\mathbf{X}) > n \log_2 q - 2$; but

2. if $\eta_\varepsilon(\mathcal{L}) \ge 1000C_\eta(n) \cdot \sqrt{\log(1/\varepsilon)}$ (and in particular if $\eta_\varepsilon(\mathcal{L}) \ge 10^4 \log(n)\sqrt{\log(1/\varepsilon)})$ and $q \ge 2^n(\eta_{2-n}(\mathcal{L}) + \mu(\mathcal{L}))$, then $H(\mathbf{X}) < n \log_2 q - 6$.

Proof. Suppose that $\eta_\varepsilon(\mathcal{L}) \le 1$. Then, by Claim 2.7, for any $y \in (\mathcal{L}/q)/\mathcal{L}$,

$$\underset{\mathbf{x}\sim D_{\mathcal{L}\mathrm{/}q}\mathcal{L}}{\mathrm{Pr}}[\mathbf{X}=y]=\frac{\rho (\mathcal{L}+y)}{\rho (\mathcal{L}\mathrm{/}q)}\le \frac{1+\epsilon }{1-\epsilon }\cdot \frac{1}{q^n}\mathrm{.}\backslash Pr\_\{\backslash mathbf\{x\}\; \backslash sim\; D\_\{\backslash mathcal\{L\}/q\}\; \backslash bmod\; \backslash mathcal\{L\}\}\; [\backslash mathbf\{X\}\; =\; y]\; =\; \backslash frac\{\backslash rho(\backslash mathcal\{L\}\; +\; y)\}\{\backslash rho(\backslash mathcal\{L\}/q)\}\; \backslash le\; \backslash frac\{1+\backslash varepsilon\}\{1-\backslash varepsilon\}\; \backslash cdot\; \backslash frac\{1\}\{q^n\}.$$

It follows that

$$H(D_{\mathcal{L}\mathrm{/}q}\mathcal{L})\ge n{\log }_{2}q+{\log }_2(1-\epsilon )-{\log }_2(1+\epsilon )>n{\log }_{2}q-2,H(D\_\{\backslash mathcal\{L\}/q\}\; \backslash bmod\; \backslash mathcal\{L\})\; \backslash ge\; n\; \backslash log\_2\; q\; +\; \backslash log\_2(1-\backslash varepsilon)\; -\; \backslash log\_2(1+\backslash varepsilon)\; >\; n\; \backslash log\_2\; q\; -\; 2,$$

as needed.