Lattice 作业2.0

[qijinzing]

请参考单元多项式承诺的evaluation relation，写出多元多项式承诺的evaluation relation。

[hygithub-art]

多元多项式承诺的evaluation relation，就是将下图中“单元多项式承诺的evaluation relation”中的X，换成 vector of X：

[mountaintower]

[wangbard]

[kings2147]

[cc030101]

$\mathcal{N} = \left\{ \begin{array}{l|l} \text{witness: } f_1(\vec{X}) \mid & F = \text{Com}(f_1) \\ \text{public input: } F, \vec{z}, v \mid & v = f_1(\vec{z}) \end{array} \right\}.$

[aoao-zhaozhao]

[lzw65]

$$ \mathfrak{R}_{\text{multi}} = \begin{array}{ll} \text{witness:} & f(X_1, X_2, \dots, X_k) & F = \mathbf{Com}(f(X_1, X_2, \dots, X_k)) \\\ \text{public input:} & F,\ \vec{z} = (z_1, z_2, \dots, z_k),\ v & v = f(z_1, z_2, \dots, z_k) \\\ \end{array} $$

[lingering]

$$R_{mul} : {witness\ f^{\leq d}(x_1,....,x_n), public\ input: F, \vec{Z}=(z_1,....z_n)}.\ We\ have\ c=comm(f), v= f^{\leq d}(z_1,....,z_n) $$

[cmdscz]

[sunjianping-alt]

[loamyve]

[JohnsonLo00]

[oceanqdu]

• 设：
  • $\mathcal{R}_{\text{MPC}}$ 表示多元多项式承诺的评估关系
  • $f_1 \in \mathbb{F}[X_1, X_2, \dots, X_n]$ 为承诺的多变量多项式
  • $\mathbf{z} = (z_1, z_2, \dots, z_n) \in \mathbb{F}^n$ 为评估点
  • $v = f_1(\mathbf{z})$ 为多项式在 $\mathbf{z}$ 处的取值

则关系定义为：

[Kunram]

[lattice-me]

[fuaiphi]

$$ \begin{aligned} &f(\mathbf{x}) - v = \sum_{i=1}^{k} \bigl(x_i - z_i\bigr),h_i(\mathbf{x}), \\ &e!\left(\frac{C}{g_1^{,v}},, g_2\right) = \prod_{i=1}^{k} e!\left(\pi_i,, g_2^{\tau_i - z_i}\right). \end{aligned} $$