samples/texts/1897687/page_9.md

We will also need Banaszczyk's celebrated lemma [Ban93, Lemma 1.5].

Lemma 2.6 ([Ban93]). For any lattice $\mathcal{L} \subset \mathbb{R}^n$, shift vector $\mathbf{t} \in \mathbb{R}^n$, and $r \ge 1/\sqrt{2\pi}$,

$$\rho ((\mathcal{L}+\mathbf{t})\setminus \sqrt{n}{B}_2^n)\le (\sqrt{2\pi e{r}^2}{e}^{-\pi r^2}{)}^n\cdot \rho (\mathcal{L})\mathrm{.}\backslash rho((\backslash mathcal\{L\}\; +\; \backslash mathbf\{t\})\; \backslash setminus\; \backslash sqrt\{n\}B\_2^n)\; \backslash le\; (\backslash sqrt\{2\backslash pi\; er^2\}e^\{-\backslash pi\; r^2\})^n\; \backslash cdot\; \backslash rho(\backslash mathcal\{L\}).$$

Micciancio and Regev introduced a lattice parameter called the smoothing parameter. For an $n$-dimensional lattice $\mathcal{L}$ and $\epsilon > 0$, the smoothing parameter $\eta_{\epsilon}(\mathcal{L})$ is defined as the smallest $s$ such that $\rho_{1/s}(\mathcal{L}^*) \le 1 + \epsilon$. The motivation for defining smoothing parameter comes from the following two facts [MR04].

Claim 2.7. For any lattice $\mathcal{L} \subset \mathbb{R}^n$, shift vector $\mathbf{t} \in \mathbb{R}^n$, $\epsilon \in (0, 1)$, and parameter $s \ge \eta_{\epsilon}(\mathcal{L})$,

$$\frac{1-ϵ}{1+ϵ}\cdot {\rho }_s(\mathcal{L})\le {\rho }_s(\mathcal{L}-\mathbf{t})\le {\rho }_s(\mathcal{L})\mathrm{.}\backslash frac\{1\; -\; \backslash epsilon\}\{1\; +\; \backslash epsilon\}\; \backslash cdot\; \backslash rho\_s(\backslash mathcal\{L\})\; \backslash le\; \backslash rho\_s(\backslash mathcal\{L\}\; -\; \backslash mathbf\{t\})\; \backslash le\; \backslash rho\_s(\backslash mathcal\{L\}).$$

Lemma 2.8. For any lattice $\mathcal{L}$, $\mathbf{c} \in \mathbb{R}^n$ and $s \ge \eta_{\epsilon}(\mathcal{L})$,

$$\mathrm{\Delta }((D_{s}B),U({\mathbb{R}}^n\mathrm{/}\mathcal{L}))\le ϵ\mathrm{/}2,\backslash Delta((D\_s\; \backslash bmod\; B),\; U(\backslash mathbb\{R\}^n/\backslash mathcal\{L\}))\; \backslash le\; \backslash epsilon/2,$$

where $D_s$ is the continuous Gaussian distribution with parameter $s$ and $U(\mathbb{R}^n/\mathcal{L})$ denotes the uniform distribution over $\mathbb{R}^n/\mathcal{L}$.

We use the following epsilon-decreasing tool which has been introduced in [CDLP13].

Lemma 2.9 ([CDLP13], Lemma 2.4). For any lattice $\mathcal{L} \subset \mathbb{R}^n$ and any $0 < \epsilon' \le \epsilon < 1$,

$${\eta }_{{ϵ}^{\mathrm{\prime }}}(\mathcal{L})\le \sqrt{\log (1\mathrm{/}{ϵ}^{\mathrm{\prime }})\mathrm{/}\log (1\mathrm{/}ϵ)}\cdot {\eta }_{ϵ}(\mathcal{L})\mathrm{.}\backslash eta\_\{\backslash epsilon\text{'}\}(\backslash mathcal\{L\})\; \backslash le\; \backslash sqrt\{\backslash log(1/\backslash epsilon\text{'})/\backslash log(1/\backslash epsilon)\}\; \backslash cdot\; \backslash eta\_\{\backslash epsilon\}(\backslash mathcal\{L\}).$$

Proof. We may assume without loss of generality that $\eta_{\epsilon}(\mathcal{L}) = 1$. Notice that this implies that $\lambda_1(\mathcal{L}^*) \ge \sqrt{\log(1/\epsilon)/\pi}$. Then, for any $s \ge 1$,

$${\rho }_{1\mathrm{/}s}({\mathcal{L}}^{\ast }\setminus \{\mathbf{0}\})=\sum \exp (-\pi (s^2-1)\mathrm{\parallel }\mathbf{w}{\mathrm{\parallel }}^2)\cdot \rho (\mathbf{w})\le \exp (-\pi (s^2-1){\lambda }_1(\mathcal{L}{)}^2)\rho ({\mathcal{L}}^{\ast }\setminus \{\mathbf{0}\})\le {ϵ}^{s^2}\mathrm{.}\backslash rho\_\{1/s\}(\backslash mathcal\{L\}^*\; \backslash setminus\; \backslash \{\backslash mathbf\{0\}\backslash \})\; =\; \backslash sum\; \backslash exp(-\backslash pi(s^2-1)\backslash |\backslash mathbf\{w\}\backslash |^2)\; \backslash cdot\; \backslash rho(\backslash mathbf\{w\})\; \backslash le\; \backslash exp(-\backslash pi(s^2-1)\backslash lambda\_1(\backslash mathcal\{L\})^2)\backslash rho(\backslash mathcal\{L\}^*\; \backslash setminus\; \backslash \{\backslash mathbf\{0\}\backslash \})\; \backslash le\; \backslash epsilon^\{s^2\}.$$

Setting $s := \sqrt{\log(1/\epsilon')/\log(1/\epsilon)}$ gives the result. $\square$

Lemma 2.10. For any lattice $\mathcal{L} \subset \mathbb{Q}^n$ with basis $\mathbf{B}$ whose bit length is $\beta$ and any $\epsilon \in (0, 1/2)$, we have $\eta_{\epsilon}(\mathcal{L}(\mathbf{B})) \le 2^{\text{poly}(\beta)}\sqrt{\log(1/\epsilon)}$, and $\lambda_n(\mathcal{L}) \le 2\mu(\mathcal{L}) \le 2^{\text{poly}(\beta)}$.

2.4 Sampling from the Discrete Gaussian

For any $\mathbf{B} = (\mathbf{b}_1, \dots, \mathbf{b}_n) \in \mathbb{R}^{n \times n}$, let

$$\mathrm{\parallel }\tilde{\mathbf{B}}\mathrm{\parallel }:=\underset{i}{\max }\mathrm{\parallel }{\pi }_{\{{\mathbf{b}}_1,\dots ,{\mathbf{b}}_{i-1}\}}^{\perp }({\mathbf{b}}_i)\mathrm{\parallel },\backslash |\; \backslash tilde\{\backslash mathbf\{B\}\}\; \backslash |\; :=\; \backslash max\_i\; \backslash |\; \backslash pi\_\{\backslash \{\backslash mathbf\{b\}\_1,\; \backslash dots,\; \backslash mathbf\{b\}\_\{i-1\}\backslash \}\}^\backslash perp\; (\backslash mathbf\{b\}\_i)\; \backslash |,$$

i.e., $|\tilde{\mathbf{B}}|$ is the length of the longest Gram-Schmidt vector of $\mathbf{B}$.

We recall the following result from a sequence of works due to Klein [Kle00]; Gentry, Peikert, and Vaikuntanathan [GPV08]; and Brakerski et al. [BLP$^{+}$13].