samples/texts/1897687/page_23.md

We will also need the following result from [MP12, Lemma 2.8], which shows that the discrete Gaussian distribution is subgaussian.

Lemma A.4. For any lattice $\mathcal{L} \subset \mathbb{R}^n$ with $\eta_{1/2}(\mathcal{L}) \le 1$, shift vector $\mathbf{t} \in \mathbb{R}^n$, unit vector $\mathbf{v} \in \mathbb{R}^n$, and any $r > 0$,

$$\underset{\mathbf{X}\sim D_{\mathcal{L}-\mathbf{t}}}{\mathrm{Pr}}[\mathrm{\mid }⟨\mathbf{v},\mathbf{X}⟩\mathrm{\mid }\ge r]\le 10\exp (-\pi r^2)\mathrm{.}\backslash Pr\_\{\backslash mathbf\{X\}\; \backslash sim\; D\_\{\backslash mathcal\{L\}-\backslash mathbf\{t\}\}\}\; [|\backslash langle\; \backslash mathbf\{v\},\; \backslash mathbf\{X\}\; \backslash rangle|\; \backslash ge\; r]\; \backslash le\; 10\; \backslash exp(-\backslash pi\; r^2).$$

Proof of Lemma 3.3. Let ${\mathbf{v}_1, \dots, \mathbf{v}_N}$ be a (1/10)-net of the unit sphere with $N \le 25^n$, as guaranteed by Lemma A.2. By Lemma A.4, we have that for any $\mathbf{e}_i$ in the proof, any $\mathbf{v}_j$, and any $r \ge 0$, $\Pr[|\langle \mathbf{v}_j, \mathbf{e}_i \rangle| \ge r] \le 10 \exp(-\pi r^2)$. Therefore, by Lemma 2.20

$$\mathrm{Pr}[\sum _i\mathrm{\mid }⟨{\mathbf{v}}_j,{\mathbf{e}}_i⟩{\mathrm{\mid }}^2\ge r]\le 2^m{e}^{-\pi r\mathrm{/}2}\mathrm{.}\backslash Pr\; \backslash left[\; \backslash sum\_i\; |\backslash langle\; \backslash mathbf\{v\}\_j,\; \backslash mathbf\{e\}\_i\; \backslash rangle|^2\; \backslash geq\; r\; \backslash right]\; \backslash leq\; 2^m\; e^\{-\backslash pi\; r/2\}.$$

Applying the union bound, we have

$$\mathrm{Pr}[\mathrm{\exists }j,\sum _i\mathrm{\mid }⟨{\mathbf{v}}_j,{\mathbf{e}}_i⟩{\mathrm{\mid }}^2\ge r]\le N{2}^m{e}^{-\pi r\mathrm{/}2}\mathrm{.}\backslash Pr[\backslash exists\; j,\; \backslash sum\_i\; |\backslash langle\; \backslash mathbf\{v\}\_j,\; \backslash mathbf\{e\}\_i\; \backslash rangle|^2\; \backslash geq\; r]\; \backslash leq\; N\; 2^m\; e^\{-\backslash pi\; r/2\}.$$

Taking $r := 2m$, we see that this probability is negligible. Applying Lemma A.3 shows that

except with negligible probability, as needed. $\square$