### 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|CS388H

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## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Lattice Reduction: Theory

All lattices of dimension $d \geq 2$ admit infinitely many bases, and two bases B, B’ generate (or represent) the same lattice if and only if $\mathbf{B}=\mathbf{B}^{\prime} \cdot \mathbf{U}$ for some unimodular matrix $\mathbf{U} \in \mathrm{GL}_d(\mathbb{Z})$. In other words, the set of (full-rank) lattices can be viewed as the quotient $\mathrm{GL}_d(\mathbb{R}) / \mathrm{GL}_d(\mathbb{Z})$. Lattice reduction is the task of finding a good representative of a lattice, i.e., a basis $\mathbf{B} \in \mathrm{GL}_d(\mathbb{R})$ representing $\Lambda \in \mathrm{GL}_d(\mathbb{R}) / \mathrm{GL}_d(\mathbb{Z})$

While there exists a variety of formal definitions for what is a good representative, the general goal is to make the Gram-Schmidt basis $\mathbf{B}^{\star}$ as small as possible. Using the simple size-reduction algorithm (see [454, Algorithm 3]), it is possible to also enforce the shortness of the basis $\mathbf{B}$ itself.

It should be noted that because we have an invariant $\prod_i\left|\mathbf{b}i^{\star}\right|=\operatorname{vol}(\Lambda)$, we cannot make all GS vectors small at the same time, but the goal becomes to balance their lengths. More pictorially, we consider the log profile of a basis as the graph of $\left(\ell_i-\lg \left|\mathbf{b}_i^{\star}\right|\right){i=0 . . d-1}$ as a function of $i$. By the volume invariant, the area under this graph is fixed, and the goal of reduction is to make this graph flatter.

A very strong ${ }^1$ notion of reduction is the Hermite-Korkine-Zolotarev (HKZ) reduction, which requires each basis vector $\mathbf{b}i$ to be a shortest non-zero vector of the remaining projected lattice $\Lambda{[i: d]}$. The Block-Korkine-Zolotarev (BKZ) reduction relaxes $\mathrm{HKZ}$, only requiring $\mathbf{b}_i$ to be close-to-shortest in a local ‘block’. More formally, we have the following.

Definition $2.3$ ( $\mathrm{HKZ}$ and BKZ [454]). The basis $\mathbf{B}=\left(\mathbf{b}0, \ldots, \mathbf{b}{d-1}\right)$ of a lattice $\Lambda$ is said to be $\mathrm{HKZ}$ reduced if $\left|\mathbf{b}i^{\star}\right|=\lambda_1\left(\Lambda\left(\mathbf{B}{[i: d]}\right)\right)$ for all $i<d$. It is said BKZ reduced with block size $\beta$ and $\epsilon \geq 0$ if $\left|\mathbf{b}i^{\star}\right| \leq(1+\epsilon) \cdot \lambda_1\left(\Lambda\left(\mathbf{B}{[i: \min (i+\beta, d)]}\right)\right)$ for all $i<d$.

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Shape Approximation

The Gaussian heuristic predicts that the number $|\Lambda \cap \mathcal{B}|$ of lattice points inside a measurable body $\mathcal{B} \subset \mathbb{R}^n$ is approximately equal to $\operatorname{vol}(\mathcal{B}) / \operatorname{vol}(\Lambda)$. Applied to Euclidean $d$-balls, it leads to the following prediction of the length of a shortest non-zero vector in a lattice.

Definition 2.6 (Gaussian heuristic). We denote by gh( $\Lambda$ ) the expected first minimum of a lattice $\Lambda$ according to the Gaussian heuristic. For a full-rank lattice $\Lambda \subset \mathbb{R}^d$, it is given by
$$\operatorname{gh}(\Lambda)=\left(\frac{\operatorname{vol}(\Lambda)}{\operatorname{vol}(\mathfrak{B})}\right)^{1 / d}=\frac{\Gamma\left(1+\frac{d}{2}\right)^{1 / d}}{\sqrt{\pi}} \cdot \operatorname{vol}(\Lambda)^{1 / d} \approx \sqrt{\frac{d}{2 \pi e}} \cdot \operatorname{vol}(\Lambda)^{1 / d},$$
where $\mathfrak{B}$ denotes the $d$-dimensional Euclidean ball. We also denote by $\operatorname{gh}(d)$ the quantity $\operatorname{gh}(\Lambda)$ of any $d$-dimensional lattice $\Lambda$ of volume 1: $\operatorname{gh}(d) \approx$ $\sqrt{d / 2 \pi e}$. For convenience we also denote $\operatorname{lgh}(x)$ for $\lg (\operatorname{gh}(x))$.

Combining the Gaussian heuristic with the definition of a BKZ reduced basis, after BKZ- $\beta$ reduction we expect
\begin{aligned} \ell_i=\lg \left(\lambda_1\left(\Lambda\left(\mathbf{B}{[i: \min (i+\beta, d)]}\right)\right)\right) & \approx \operatorname{lgh}(\min (\beta, d-i))+\frac{\lg \left(\operatorname{vol}\left(\Lambda\left(\mathbf{B}{[i \min (i+\beta, d)]}\right)\right)\right)}{\min (\beta, d-i)} \ &=\operatorname{lgh}(\min (\beta, d-i))+\frac{\sum_{j=i}^{\min (i+\beta, d)-1} \ell_j}{\min (\beta, d-i)} . \end{aligned}
If $d \gg \beta$ this linear recurrence implies a geometric series for the $\left|\mathbf{b}i^{\star}\right|$. Considering one block of dimension $\beta$ and unit volume, we expect $\ell_i=(\beta-$ $i-1) \cdot \lg \left(\alpha\beta\right)$ for $i=0, \ldots, \beta-1$ and some $\alpha_\beta$. We obtain
\begin{aligned} \ell_0=(\beta-1) \cdot \lg \left(\alpha_\beta\right) & \approx \operatorname{lgh}(\beta)+\frac{1}{\beta} \sum_{j=0}^\beta j \cdot \lg \left(\alpha_\beta\right) \ &=\operatorname{lgh}(\beta)+(\beta-1) / 2 \cdot \lg \left(\alpha_\beta\right) . \end{aligned}

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Lattice Reduction: Theory

$\left|\mathbf{b} i^{\star}\right|=\lambda_1(\Lambda(\mathbf{B}[i: d]))$ 对所有人 $i<d$. 据说 BKZ 随着区块大小的增加而减少 $\beta$ 和 $\epsilon \geq 0$ 如果
$\left|\mathbf{b} i^{\star}\right| \leq(1+\epsilon) \cdot \lambda_1(\Lambda(\mathbf{B}[i: \min (i+\beta, d)]))$ 对所有人 $i<d$.

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Shape Approximation

$$\operatorname{gh}(\Lambda)=\left(\frac{\operatorname{vol}(\Lambda)}{\operatorname{vol}(\mathfrak{B})}\right)^{1 / d}=\frac{\Gamma\left(1+\frac{d}{2}\right)^{1 / d}}{\sqrt{\pi}} \cdot \operatorname{vol}(\Lambda)^{1 / d} \approx \sqrt{\frac{d}{2 \pi e}} \cdot \operatorname{vol}(\Lambda)^{1 / d}$$

$$\ell_i=\lg \left(\lambda_1(\Lambda(\mathbf{B}[i: \min (i+\beta, d)]))\right) \approx \operatorname{lgh}(\min (\beta, d-i))+\frac{\lg (\operatorname{vol}(\Lambda(\mathbf{B}[i \min (i+\beta, d)])))}{\min (\beta, d-i)}$$

$$\ell_0=(\beta-1) \cdot \lg \left(\alpha_\beta\right) \approx \operatorname{lgh}(\beta)+\frac{1}{\beta} \sum_{j=0}^\beta j \cdot \lg \left(\alpha_\beta\right) \quad=\operatorname{lgh}(\beta)+(\beta-1) / 2 \cdot \lg \left(\alpha_\beta\right)$$

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