From 71af4d6d2b5b19c423905c0b693aeda1b051b8d4 Mon Sep 17 00:00:00 2001 From: Franklin Date: Mon, 23 Dec 2024 16:49:38 +0800 Subject: [PATCH] add gemini notes --- gemini/Gemini-PCS-1.zh.md | 289 ++++++++++++++++++++++++++++++ gemini/Gemini-PCS-2.zh.md | 151 ++++++++++++++++ gemini/image/Gemini-PIOP.png | Bin 0 -> 135533 bytes gemini/image/Gemini-multipoly.png | Bin 0 -> 110897 bytes gemini/image/Gemini-unipoly.png | Bin 0 -> 106254 bytes gemini/image/mle.png | Bin 0 -> 35746 bytes 6 files changed, 440 insertions(+) create mode 100644 gemini/Gemini-PCS-1.zh.md create mode 100644 gemini/Gemini-PCS-2.zh.md create mode 100644 gemini/image/Gemini-PIOP.png create mode 100644 gemini/image/Gemini-multipoly.png create mode 100644 gemini/image/Gemini-unipoly.png create mode 100644 gemini/image/mle.png diff --git a/gemini/Gemini-PCS-1.zh.md b/gemini/Gemini-PCS-1.zh.md new file mode 100644 index 0000000..da08481 --- /dev/null +++ b/gemini/Gemini-PCS-1.zh.md @@ -0,0 +1,289 @@ +# Gemini-PCS (Part I) + +- Tianyu ZHENG + +Gemini [BCH+22] 是一种 elastic SNARK,所谓 elastic 是指证明者可以通过设置参数在证明时间和内存之间权衡,以满足不同使用场景的要求。 + +作为 Gemini 的核心算法,Tensor Product Check 为我们提供了一种证明多元线性多项式(Multilinear Polynomial)求值的方法,如 $\tilde{f}(\vec{\rho}) = u$。换句话说,该方法实现了从多元多项式到一元多项式的转换,从而启发我们构造一种新的多元多项式承诺方案。 + +在具体构造上,Tensor Product Check 采用了与之前的工作(Sumcheck, Bulletproofs, FRI)类似的 split-and-fold 思想,达到了比较高效的通信和验证者复杂度,同时其证明者算法能够实现 elastic 性质。 + +# MLE and Tensor Product + +在 Zeromorph 笔记中我们提到,一个 Multilinear Extension 唯一地对应到一个从 Boolean 向量映射到有限域的函数,形如 $f: \{0,1\}^n \rightarrow \mathbb{F}_q$。下图是一个三维的 MLE 多项式 $\tilde{f}(X_0,X_1,X_2)$ 的示例,这个多项式可以唯一地被 $(a_0, a_1,...,a_7)$ 这个 「点值向量」来表示。 + + + +同样地,一个 MLE 多项式也可以采用「系数式」来表示,例如上图可以写成 + +$$ +\tilde{f}(X_0,X_1,X_2) = f_0+f_1X_0+f_2X_1+f_3X_2+f_4X_0X_1+f_5X_0X_2 + f_6X_1X_2 + f_7X_0X_1X_2 +$$ + +该表达式中单项式的排序基于 Lexicographic Order. + +除了「点值式」和「系数式」之外,接下来我们介绍一种新的表达形式——基于「张量积」(Tensor product)的表达式。 + +简单来说,张量积是两个向量之间的一种特殊”乘法”,记作 $\vec{a} \otimes \vec{b}$。具体来说,我们可以先计算 $a b^T$(假设 $\vec{a}, \vec{b}$ 均为列向量),接着将得到的矩阵按列相接成一个向量,该向量即为张量积的结果。例如 $\vec{a}=(a_1,a_2)$ 和 $\vec{b}=(b_1, b_2, b_3)$: + +$$ +\begin{bmatrix}a_1 \\ a_2\end{bmatrix} \cdot \begin{bmatrix}b_1, b_2,b_3\end{bmatrix} = \begin{bmatrix}a_1b_1, a_1b_2, a_1b_3 \\ a_2b_1, a_2b_2, a_2b_3\end{bmatrix} +$$ + +可得 $\vec{a} \otimes \vec{b} = (a_1b_1, a_2b_1, a_1b_2, a_2b_2, a_1b_3, a_2b_3)$。 + +对比我们之前提到的「系数式」表达的 MLE 多项式, 我们会发现它的所有单项式都可以由一个连续的张量积得到: + +$$ +(1,X_0)\otimes(1,X_1)\otimes(1,X_2) = (1, X_0, X_1, X_0X_1, X_2, X_0X_2, X_1X_2, X_0X_1X_2) +$$ + +我们将左式简记为 $\otimes_{j=0}^2 (1,X_j)$。那么一个 MLE 多项式可以写成内积形式: + +$$ +\tilde{f}(X_0,X_1,X_2) = \langle \vec{f}, \otimes_{j=0}^2 (1,X_j) \rangle +$$ + +其中左边元素是系数向量 $\vec{f}$ ,右边元素则是一个单项式向量 $\otimes_{j=0}^2 (1,X_j)$。 + +# Split-and-Fold 方法 + +在 Gemini 中,作者给出了一个基于一元多项式承诺方案(例如 KZG10)来检查张量积正确性的协议,基于该协议我们可以进一步构造实现多元到一元多项式转换。我们首先以提到的三维 MLE 多项式为例,解释 Tensor Product Check 的主要思路。 + +假设证明者想要证明实例: $\vec{f} = (f_0,...,f_7)$,满足关系 $\langle\vec{f}, \otimes_{j=0}^{2}(1,\rho_j) \rangle = u$,其中 $\rho_0,\rho_1, \rho_2$ 在 $F$ 有限域上。 + +方便起见,我们将向量 $\vec{f}$ 中元素的下标改写成小端序的二进制表示,即 + +$$ +f_i = f_{i_0i_1i_2}, i = \langle (2^0, 2^1, 2^2), (i_0,i_1,i_2) \rangle +$$ + +其中 $i_0,i_1,i_2 \in \{0,1\}$。 + +将重新编号后的 tensor product 展开后,会得到下边这个等式 + +$$ +\begin{matrix} &&\langle\vec{f}, \otimes_{j=0}^{2}(1,\rho_j) \rangle & \\ & = & f_{000}\rho_0^{0}\rho_1^{0}\rho_2^{0}& + &f_{100}\rho_0^{1}\rho_1^{0}\rho_2^{0}& + &f_{010}\rho_0^{0}\rho_1^{1}\rho_2^{0}& + &f_{110}\rho_0^{1}\rho_1^{1}\rho_2^{0} \\ & + & f_{001}\rho_0^{0}\rho_1^{0}\rho_2^{1}& + &f_{101}\rho_0^{1}\rho_1^{0}\rho_2^{1}& + &f_{011}\rho_0^{0}\rho_1^{1}\rho_2^{1}& + &f_{111}\rho_0^{1}\rho_1^{1}\rho_2^{1} \end{matrix} +$$ + +我们会发现每个系数 $f_{i_0i_1i_2}$ 的下标与相乘的 $\rho_0,\rho_1,\rho_2$ 的指数是一一对应的,即 + +$$ +f_{i_0i_1i_2} \cdot \rho_0^{i_0}\rho_1^{i_1} \rho_2^{i_2}, \text{ for all } i_0,i_1,i_2 \in \{ 0,1 \} +$$ + +因此,我们总能够将 $\vec{f}$ 按 $\rho_j$ 的指数 $i_j$ 分成等长的两部分,且两部分分别满足一个 tensor product 子问题。例如 $\vec{f}$ 根据 $\rho_0$ 划分后,可以得到关于 $\vec{f}_1, \vec{f}_2$ 的两个 tensor product 关系: + +$$ +\langle\vec{f}, \otimes_{j=0}^{2}(1,\rho_j) \rangle = \langle\vec{f}_1, \otimes_{j=1}^{2}(1,\rho_j) \rangle + \rho_0 \langle\vec{f}_2, \otimes_{j=1}^{2}(1,\rho_j) \rangle +$$ + +注意到,这两个子问题中,内积的右边元素相同:均为 $\otimes_{j=0}^1 (1,\rho_j)$,因此它们可以进一步合并成一个 $\langle\vec{f}_1 + \rho_0 \vec{f}_2, \otimes_{j=1}^{2}(1,\rho_j) \rangle$。 + +可以看到,对于一个 $N$ 长度的向量 $\vec{f}$,我们将其分开为两个 $N/2$ 长度的向量,再合并成一个向量。通过这一轮操作,我们把一个 $N$ 大小的 tensor product 问题变成了 $N/2$ 大小的问题。 + +以此类推,该问题可以最终被减少到 $1$ 大小。 + +**【多元多项式 Split-and-fold】** + +之前提到过,我们可以将一个 tensor product 关系看作多元多项式的求值关系,即 + +$$ +\langle\vec{f}, \otimes_{j=0}^{2}(1,\rho_j) \rangle = u \quad \Leftrightarrow \quad \tilde{f}(\rho_0,\rho_1,\rho_2) = u +$$ + +对于多元多项式 $\tilde{f}^{(0)} = \tilde{f}$,其在第 $j\in[1,3]$ 轮时 split-and-fold 过程如下: + +- **split:** 证明者将多元多项式 $\tilde{f}^{(j-1)}$ 分成两部分:第一部分的任意单项式中包含阶数为 0 的 $X_j$(记作 $\tilde{f}_e^{(j-1)}$),第二部分的任意单项式中包含阶数为 1 的 $X_{j-1}$(记作 $X_{j-1} \cdot \tilde{f}_o^{(j-1)}$),三者满足 + +$$ +\tilde{f}^{(j-1)} = \tilde{f}_{e}^{(j-1)} + X_{j-1} \cdot \tilde{f}_{o}^{(j-1)} +$$ + +- **fold:** 证明者将分开的两个多项式 $\tilde{f}_e^{(j-1)}, \tilde{f}_o^{(j-1)}$ 线性组合,组合时使用的权重为 $\rho_{j-1}$, 得到新的多元多项式记作 $\tilde{f}^{(j)}(X) = \tilde{f}_e^{(j-1)}(X) + \rho_{j-1} \cdot \tilde{f}_o^{(j-1)}(X)$。 + +下图我们给出 $j=1$ 时的计算过程: + + + +# Tensor Product 检查协议 + +通过上述递归算法,我们将检查 $N$ 长度的 tensor product 关系的正确性归约到检查 $n = \left \lceil \log N \right \rceil$ 次 split-and-fold 过程的正确性。 + +实际上,这一分治解决问题的思想(split-and-fold)在很多之前的协议中出现过,如Sumcheck,Bulletproofs 和 FRI。不同的是 Gemini 给出了基于 KZG10 来证明 split-and-fold 过程的协议,该协议需要 $n= \log(|\vec{f}|)$ 次交互。 + +我们给出证明 tensor product 关系的 PIOP 协议如下: + +**【Tensor-product 检查协议】** + +目标关系:$\langle\vec{f}, \otimes_{j=0}^{n-1}(1,\rho_j) \rangle = u$ + +证明者输入:公共参数,实例 $x = (\rho_0,...,\rho_{n-1}, u)$, 秘密 $w = \vec{f}$ + +验证者输入:公共参数,实例 $x = (\rho_0,...,\rho_{n-1}, u)$ + +1. 证明者构造一元多项式 $f^{(0)}(X) = f(X)$。 +2. 对 $j \in 1,...,n$,证明者计算 + +$$ +f^{(j)}(X) = f_e^{(j-1)}(X) + \rho_{j-1} \cdot f_o^{(j-1)}(X) +$$ + +其中 $f_e^{(j-1)}, f_o^{(j-1)}$ 分别为 $f^{(j-1)}$ 的偶数阶项和奇数阶项构成的多项式,满足 $f^{(j-1)}(X) = f_e^{(j-1)}(X^2) + X \cdot f_o^{(j-1)}(X^2)$。 + +1. 证明者向验证者发送 $f^{(0)},f^{(1)},...,f^{(n-1)}$ 的 Oracles。 +2. 验证者随机选取挑战值 $\beta \leftarrow \mathbb{F}$ 并对 Oracles 进行以下查询: + +$$ +e^{(j-1)}:= f^{(j-1)}(\beta), \bar{e}^{(j-1)} := f^{(j-1)}(-\beta), \hat{e}^{(j-1)} := f^{(j)}(\beta^2) +$$ + +其中 $j=1,...,n$, 当 $j=n$ 时,忽略查询 $f^{(n)}(\beta^2)$ ,并直接令 $\hat{e}^{(n-1)} := u$。 + +1. 对 $j = 0,...,n-1$,验证者检查 + +$$ +\hat{e}^{(j)} = \frac{e^{(j)} + \bar{e}^{(j)}}{2} + \rho_j \cdot \frac{e^{(j)}-\bar{e}^{(j)}}{2\beta} +$$ + +在每一轮中,证明者会分别为 split 之前,和 fold 之后得到的多项式生成 Oracles,具体来说,一个 split-and-fold 关系可以写成: + +> 给定 $f(X), f_e(X), f_o(X),f'(X)$,权重 $\rho$,它们满足如下关系 +> +> +> $$ +> f(X)=f_e(X^2)+X\cdot f_o(X^2) \qquad \% \ \text{split} \\ f'(X) = f_e(X)+\rho \cdot f_o(X)\qquad \% \ \text{fold} +> $$ +> + +又因为偶次、奇次多项式分别满足(1) $f_e(X^2)=(f(X)+f(-X))/2$,(2)$f_o(X^2)=(f(X)-f(-X))/2X$,我们可以进一步地将上述两个等式写成一个,即 + +$$ +f'(X^2)=\frac{f(X)+f(-X)}{2} + \rho \cdot \frac{f(X)-f(-X)}{2X} +$$ + +要检查该等式成立,验证者只需要在 $F$ 有限域上随机选取一个挑战值 $\beta$,并检查 $X=\beta$ 时 $f,f'$ 的值是否满足关系即可。 + +# 多元到一元转换 + +在介绍多元到一元转换的协议的之前,我们再深入分析一下 tensor product 协议中隐藏的一些原理。虽然 tensor product 协议的目标时证明一个多元多项式的取值,但除了输入多元多项式的系数向量以外,协议中涉及的多项式均为一元的。 + +我们不妨将 Split-and-fold 过程用一元多项式写出来: + +**【一元多项式 Split-and-fold】** 在第 $j$ 轮时: + +- **split:** 证明者将一元多项式 $f^{(j-1)}$ 分成两部分:第一部分的任意单项式中包含阶数为偶数的 $X$(记作 $f_e^{(j-1)}$),第二部分的任意单项式中包含阶数为奇数的 $X$(记作 $X \cdot f_o^{(j-1)}$),三者满足 + +$$ +f^{(j-1)}(X) = f_e^{(j-1)}(X^2) + X \cdot f_o^{(j-1)}(X^2) +$$ + +- **fold:** 证明者将分开的两个多项式 $f_e^{(j-1)}, f_o^{(j-1)}$ 线性组合,组合时使用的权重为 $\rho_{j-1}$, 得到新的一元多项式记作 $f^{(j)}(X) = f_e^{(j-1)}(X) + \rho_{j-1} \cdot f_o^{(j-1)}(X)$。【注】这里我们需要引入一个额外的映射 $X^2 \mapsto X$来得到 $f_e^{(j-1)}(X), f_o^{(j-1)}(X)$。 + +因此,tensor product 协议可以看作是证明者同步地在一元多项式 $f$ 上执行一个递归算法来模拟 $\tilde{f}$ 的计算过程, + +我们以三维多项式为例,描述第 $j=1$ 轮时的计算过程: + +一元多项式中 split-and-fold 计算如下: + + + +与多元多项式 split-and-fold 对比,一元多项式第一轮中的自变量 $X$ 就对应多元中的 $X_0$,在第二轮中则是 $X^2$ 对应 $X_1$。实际上,这两个过程分别是对系数向量 $\vec{f}$ 在不同「基」上的计算。 + +当我们需要在基 $\{X_0,X_1,X_2\}$ 上计算多元多项式的值时,我们只需要对应地在 $\{X^1,X^2,X^4\}$ 上(即一元多项式上)同步地进行相同操作,便可以用一元多项式模拟出多元多项式的求值过程。 + +更正式地说,我们就得到了一个多元基向量空间到一元基向量空间的映射关系: + +$$ +\iota: \{ X_0,X_1,X_2 \} \rightarrow \{ X^1, X^2, X^4\} +$$ + +因此,我们说 tensor product 协议为我们提供了一个从多元到一元的证明方法,即 multi-to-uni IOP。如下图所示,在理想情况下,我们希望证明者能够直接生成一个多元多项式的 Oracle 并发送给验证者。然而在工程中缺少高效的多元多项式承诺方案,证明者只能构造一个对 Tensor Product Check 的证明协议(也就是 Multi-Uni-IOP)来在一元多项式上模拟多元多项式求值的计算过程。 + +证明者需要发送 $n$ 个一元多项式的 Oracles,分别让验证者进行查询并检查。由于这些检查彼此独立,验证者可以一次性对所有 Oracles 在某个点 $\beta$ 上进行查询,而不需要 $O(n)$ 个点。 + + + +# 基于KZG实现 + +对前文给出的 IOP 协议,我们可以部署一元多项式承诺方案(KZG10)将其编译为一个 AoK(Argument of Knowledge)。KZG10 能够支持一个多项式在某个点上的求值证明,其优点是拥有常数大小的证明,且支持批量证明。缺点是需要可信初始化,且证明复杂度相对较高(需要 FFT 运算)。 + +【注】由于我们将 IOP 编译为 Argument of Knowledge,因此 KZG10 需要满足 extractability,该性质的证明在 Marlin [CHM+19] 中给出。 + +我们先简单回顾一下 KZG10 的证明原理:给定公共参数:$\mathbb{G}_1, \mathbb{G}_2, \mathbb{G}_T, G, H, e$。在初始化阶段随机选取 $\tau \in \mathbb{}F$ 并计算 $\tau H \in \mathbb{G}_2$ 和向量 + +$$ +(G,\tau G,\ldots \tau^{D-1}G,\tau^{D}G)\in \mathbb{G}_1^{D+1} +$$ + +我们用中括号记号 $[a]_1$ 来表示一个椭圆曲线群元素的上的标量乘法 $a \cdot G$。KZG 证明过程如下: + +1. 证明者计算 $d$ 阶一元多项式 $f(X)$ 的承诺 $[f(\tau)]_1 = \sum_{j=0}^d f_j \cdot \tau^j G$。 +2. 证明者在 $\rho$ 点上公开多项式的值为 $f(\rho)=u$,并计算商多项式 + +$$ +q(X) = \frac{f(X)-f(\rho)}{X-\rho} +$$ + +生成求值证明 $[q(\tau)]_1$。 + +1. 验证者检查求值证明 $e([f(\tau)]_1-[u]_1, [1]_2) = e([q(\tau)]_1, [\tau-\rho]_2)$。 + +因此,要编译 IOP 协议只需要分别对协议中产生的多项式 $f^{(1)},...,f^{(n-1)}$ 进行承诺,并在指定点 $\beta, -\beta, \beta^2$ 上打开即可。但有两点仍然需要我们注意:(1)这些多项式的阶数并不相同,为了防止证明者作弊使用不满足满足阶数要求的多项式,需要采用 Marlin,Zeromorph [CHM+19, KT23] 中的方法来限制多项式的 Degree Bound。(2)大部分多项式都需要在 $\beta, -\beta, \beta^2$ 这三个点上公开,如果对每个点分别生成求值证明会增加证明大小以及验证复杂度。多点求值证明技巧可以用来优化这个问题。 + +**Degree Bound 证明:** 为了证明 $deg(f)\leq d$ + +- 证明者提供 $[f(\tau)]_1$ 并附加上 $[\tau^{D-d}\cdot f(\tau)]_1$ 发送给验证者 +- 验证者检查等式 $e([f(\tau)]_1, [1]_2) = e([\tau^{D-d}\cdot f(\tau)]_1, [\tau^{D-d}]_2)$ + +**多点求值证明:**为了证明 $f(X)$ 在 $\beta_1, \beta_2, \beta_3$ 公开为 $u_1,u_2,u_3$ + +- 证明者随机生成一个阶数与 $f(X)$ 相同的多项式 $g(X)$,该多项式需要经过点 $(\beta, u_1),(-\beta, u_2),(\beta^2, u_3)$ +- 证明者提供 $[f(\tau)]_1$ 和求值证明 $[q(\tau)]_1 = \frac{f(\tau)-g(\tau)}{(\tau-\beta_1)(\tau-\beta_2)(\tau-\beta_3)}$ +- 验证者检查等式 $e([f(\tau)]_1 - [g(\tau)]_1, [1]_2) = e([q(\tau)]_1, [(\tau-\beta_1)(\tau-\beta_2)(\tau-\beta_3)]_2)$ + +【注】上述两个技巧都需要在 Setup 阶段额外生成 $(H,\tau H,\ldots \tau^{D-1}H,\tau^{D}H)\in \mathbb{G}_2^{D+1}$。 + +**【协议描述】** + +下面我们先给出基于 KZG 编译的 Multi-to-Uni AoK 方案: + +**Instance** + +- 向量 $\vec{f}$ 的一元多项式承诺 $[f(\tau)]_1$,长度 $N$ +- 求值点向量 $\vec{\rho}$ +- 求值结果 $\tilde{f}(\rho) = u$ + +**Witness** + +- 多元多项式的系数向量 $\vec{f}$ + +**交互过程** + +1. 证明者生成多项式 $f^{(1)},...,f^{(n-1)}$ 并计算和发送它们的承诺 $[f^{(1)}(\tau)]_1,\ldots,[f^{(n-1)}(\tau)]_1$ +2. 证明者计算并发送多项式 $f^{(0)},...,f^{(n-1)}$ 的 degree bound 证明 $[\tau^{D-N\cdot 2^{-j} + 1}\cdot f^{(j)}(\tau)]_1, j = 0,\ldots n-1$ +3. 验证者随机选取点 $\beta$ 并发送给证明者 +4. 证明者计算每个多项式的求值证明,其中 +- $[q^{(0)}(\tau)]_1 = \frac{f^{(0)}(\tau)-g^{(0)}(\tau)}{(\tau-\beta)(\tau+\beta)}$ % $f^{(0)}(\beta), f^{(0)}(-\beta)$ +- $[q^{(j)}(\tau)]_1 = \frac{f^{(j)}(\tau)-g^{(j)}(\tau)}{(\tau-\beta)(\tau+\beta)(\tau-\beta^2)}$ % $f^{(j)}(\beta), f^{(j)}(-\beta), f^{(j)}(\beta^2), j=1,...,n-1$ +1. 验证者检查: +- $f^{(0)},...,f^{(n-1)}$ 的 degree bound 证明 $[\tau^{D-N\cdot 2^{-j} + 1}\cdot f^{(j)}(\tau)]_1, j = 0,\ldots n-1$ 的正确性 +- $f^{(0)},...,f^{(n-1)}$ 的多点求值证明 $[q^{(0)}(\tau)]_1,\ldots [q^{(n-1)}(\tau)]_1$ 的正确性 +- split-and-fold 关系的正确性,即对 $j = 0,...,n-1$,下列等式是否成立: + +$$ +\hat{e}^{(j)} = \frac{e^{(j)} + \bar{e}^{(j)}}{2} + \rho_j \cdot \frac{e^{(j)}-\bar{e}^{(j)}}{2\beta} +$$ + +**【性能分析】** + +- 证明大小:$3\log N \ \mathbb{G}_1$元素 +- 验证者计算量: $4 \log N \ \mathsf{Pairing},\ O(\log N) \ \mathsf{EccMul}^{\mathbb{G}_1 \text{ or } \mathbb{G}_2}$ + +# 参考文献 + +[BCH+22] Bootle, Jonathan, Alessandro Chiesa, Yuncong Hu, **et al. "Gemini: Elastic SNARKs for Diverse Environments." *Cryptology ePrint Archive* (2022). [https://eprint.iacr.org/2022/420](https://eprint.iacr.org/2022/420) + +[KT23] Kohrita, Tohru, and Patrick Towa. "Zeromorph: Zero-knowledge multilinear-evaluation proofs from homomorphic univariate commitments." Cryptology ePrint Archive (2023). [https://eprint.iacr.org/2023/917](https://eprint.iacr.org/2023/917) + +[CHM+19] Chiesa, Alessandro, Yuncong Hu, Mary Maller, et al. "Marlin: Preprocessing zkSNARKs with Universal and Updatable SRS." *Cryptology ePrint Archive* (2019). [https://eprint.iacr.org/2019/1047](https://eprint.iacr.org/2019/1047) \ No newline at end of file diff --git a/gemini/Gemini-PCS-2.zh.md b/gemini/Gemini-PCS-2.zh.md new file mode 100644 index 0000000..b9e1a28 --- /dev/null +++ b/gemini/Gemini-PCS-2.zh.md @@ -0,0 +1,151 @@ +# Gemini-PCS (Part II) + +- Tianyu ZHENG + +在第一部分中, 我们介绍了 Gemini [BCH+22] 中实现多元多项式求值证明的 Tensor product 检查协议,并简要说明了如何将它应用到实际证明系统中来实现多元多项式到一元多项式的转换。 这一部分,我们主要关注 Tensor product 检查协议安全性,并在 Gemini 的基础上提出了一些优化。 + +# 回顾 + +方便阅读,我们先回顾一下 tensor product 检查协议的流程: + +**【Tensor-product 检查协议】** + +目标关系:$\langle\vec{f}, \otimes_{j=0}^{n-1}(1,\rho_j) \rangle = u$ + +证明者输入:公共参数,实例 $x = (\rho_0,...,\rho_{n-1}, u)$, 秘密 $w = \vec{f}$ + +验证者输入:公共参数,实例 $x = (\rho_0,...,\rho_{n-1}, u)$ + +1. 证明者构造一元多项式 $f^{(0)}(X) = f(X)$。 +2. 对 $j \in 1,...,n$,证明者计算 + +$$ +f^{(j)}(X) = f_e^{(j-1)}(X) + \rho_{j-1} \cdot f_o^{(j-1)}(X) +$$ + +其中 $f_e^{(j-1)}, f_o^{(j-1)}$ 分别为 $f^{(j-1)}$ 的偶数阶项和奇数阶项构成的多项式,满足 $f^{(j-1)}(X) = f_e^{(j-1)}(X^2) + X \cdot f_o^{(j-1)}(X^2)$。 + +1. 证明者向验证者发送 $f^{(0)},f^{(1)},...,f^{(n-1)}$ 的 Oracles。 +2. 验证者随机选取挑战值 $\beta \leftarrow \mathbb{F}$ 并对 Oracles 进行以下查询: + +$$ +e^{(j-1)}:= f^{(j-1)}(\beta), \bar{e}^{(j-1)} := f^{(j-1)}(-\beta), \hat{e}^{(j-1)} := f^{(j)}(\beta^2) +$$ + +其中 $j=1,...,n$, 当 $j=n$ 时,忽略查询 $f^{(n)}(\beta^2)$ ,并直接令 $\hat{e}^{(n-1)} := u$。 + +1. 对 $j = 0,...,n-1$,验证者检查 + +$$ +\hat{e}^{(j)} = \frac{e^{(j)} + \bar{e}^{(j)}}{2} + \rho_j \cdot \frac{e^{(j)}-\bar{e}^{(j)}}{2\beta} +$$ + +# 安全性分析 + +**【Completeness】** + + 给定一个多项式系数向量满足 $\langle\vec{f}, \otimes_{j=0}^{2}(1,\rho_j) \rangle = u$, 一个诚实执行上述协议的证明者一定能够通过验证。根据我们在【Split-and-fold 检查协议】中的讨论,当证明者正确地执行 split 得到 $f_e^{(j-1)}(X), f_o^{(j-1)}(X)$ 后,其 fold 得到的 $f^{(j)}(X)$ 一定能够通过验证,并且 $f^{(j)}(X)$ 满足如下表达式 + +$$ +f^{(j)}(X) = \sum_{i_0,...,i_{n-1} \in \{ 0,1 \}} f_{i_0\cdots i_{n-1}} \cdot \rho_0^{i_0} \cdots \rho_{j-1}^{i_{j-1}} \cdot X^{\langle \vec{i}_{[j:n-1]}, \vec{2}^{n-1-j}\rangle} +$$ + + 其中 $\vec{2}^{n-1-j} = (2^0,...,2^{n-1-j})$。显然当 $j=n$ 时,$f^{(n)}(X) = \tilde{f}(\vec{\rho})= u$。 + +**【Soundness】** + +这里我们采用和 Gemini 原文不同,但更加容易理解的证明方法。假设 $\langle\vec{f}, \otimes_{j=0}^{2}(1,\rho_j) \rangle \neq u$,对于一个恶意的证明者,如果它能够输出一系列的交互数据,其中包括多项式 $f^{(j)}, j=0,...,n-1$ 的 Oracles,并通过验证。那么一定至少存在一对 Oracles,它们所包含的多项式不满足 split-and-fold 关系。 即存在 $j$,使得 + +$$ +p_j(X) = f^{(j)}(X^2) - \frac{f^{(j-1)}(X)+f^{(j-1)}(-X)}{2} - \rho_{j-1}\frac{f^{(j-1)}(X)-f^{(j-1)}(-X)}{2X} +$$ + +是非零多项式。注意到 $p_j(X)$ 的最高阶数为 $2^{n-(j-1)}-1$。令事件 $E_j$ 表示 $p_j(X)$ 为非零多项式且 $p_j(\beta) = 0$,根据 Schwartz-Zippel 引理, $Pr(E_j) \leq deg(p_j)/|F|$。那么,对于所有 $p_1(X),...,p_{n}(X)$,存在非零多项式且在 $\beta$ 点上刚好为 0 的概率可以用 union bound 来约束 + +$$ +Pr(E_1 \vee \cdots \vee E_n) \leq Pr(E_1) + \cdots + Pr(E_n) = \sum_{j=1}^n \frac{deg(p_j)}{|F|} = \frac{2N}{|F|} +$$ + +**【关于 degree bound】** + +注意在 Gemini 原论文中,验证者需要检查每个 $f^{(j)}$ 的阶数都严格小于等于 $2^{n-j}-1$。这一点在上述证明中也有体现:即假设 $p_j(X)$ 的最高阶数为 $2^{n-(j-1)}-1$。 + +但经过研究我们会发现 degree check 并不是必需的:即使恶意的证明者被允许在每一轮构造一个非法的 $f^{(j')}$,其阶数大于合法 $f^{(j)}$ 但小于等于 $N$,我们仍然能够得到一个可以忽略的 soundness error(尽管比原来的略大一些)。具体来说,对于任意 $E_j$,有 $Pr(E_j) \leq N/|F|$,可得 $Pr(E_1 \vee \cdots \vee E_n) \leq N \log N/|F|$ (当 $N < D$ 时,为 $D \log N / |F|$)。 + +因此,第一部分中基于 KZG10 实现的 Tensor-product 检查协议中的 degree bound 检查可以忽略,以减少证明中的 $\log N$ 个 $\mathbb{G}_1$ 元素。 + +# 实现 Zero-Knowledge + +Gemini 当中并没有讨论如何实现 tensor product check 的 ZK 性质,这里我们给出两种可行的方案。 + +**【方案一】** + +采用与论文 [CAS17] 中实现 zk sumcheck 类似的思想,我们可以直接在原多项式 $f(X)$ 上加一个相同大小的盲化多项式 $g(X)$,即对于 $f(X)$ 中每一个系数不为零的单项式,$g(X)$ 都包含对应的一个单项式,且系数为随机值。令 $\langle\vec{g}, \otimes_{j=0}^{n-1}(1,\rho_j) \rangle = v$。 + +接下来,证明者只需要额外对 $g(X)$ 进行承诺,并将承诺值 $cm(g(X))$ 和 $v$ 发送给验证者。验证者则随机选取一个挑战者 $c$ 将 $f,g$ 的 tensor product 关系合并为 + +$$ +u+c\cdot v = \langle\vec{f}+c\cdot \vec{g}, \otimes_{j=0}^{n-1}(1,\rho_j) \rangle +$$ + +接下来,证明者和验证者共同完成对上述关系式的 Tensor product check 协议即可。我们不过多赘述这个方案的具体构造。 + +**【方案二】** + +上述方法十分简单直接,但缺点是证明者需要额外增加一个和 $f(X)$ 一样大($N$ 长度)的随机多项式。参考Libra [XZZPS19] 中对 [CFS17] 中 zk sumcheck 的优化方案,我们同样可以提出一个实现 zk tensor product 协议的优化方案,能够显著减少盲化多项式的大小。 + +该优化方案的思路是:由于证明者在 tensor product check 中一共只发送了 $3n$ 个点值,因此盲化多项式至少需要包含 $3n$ 个随机系数才能够保证协议的零知识性。 + +具体地,令盲化多项式为: + +$$ +g(X) = a_{0,1}X + a_{0,2}X^2 + \sum_{i=1}^{n-1} \left( a_{i,1} X^{3i} + a_{i,2} X^{3i+1} + a_{i,3} X^{3i+2} \right) + a_n +$$ + +**【Zero-Knowledge Tensor Product 检查协议】** + +为了证明 $\langle\vec{f}, \otimes_{j=0}^{n-1}(1,\rho_j) \rangle = u$ + +1. 证明者构造盲化多项式 $g(X)$ 并将其系数向量用零补全长度为 $N$ +2. 证明者计算并发送下面数据给验证者 + +$$ +v = \langle\vec{g}, \otimes_{j=0}^{n-1}(1,\rho_j) \rangle \\ C_g = cm(g(X)) +$$ + +1. 验证者随机选取 $c \in F^*$ 发送给证明者,并计算 $u + c\cdot v, C_f + c\cdot C_g$。 +2. 证明者和验证者运行 tensor product 检查协议证明如下关系 + +$$ +u+c\cdot v = \langle\vec{f}+c\cdot \vec{g}, \otimes_{j=0}^{n-1}(1,\rho_j) \rangle +$$ + +方便起见,我们令 $h(X) = f(X) + c\cdot g(X)$。上述构造满足零知识性的证明如下: + +**【证明】** + +首先,模拟器 $S$ 可以按照下面步骤构造: + +1. $S$ 首先输入一个随机挑战值 $c \neq 0$ +2. $S$ 均匀随机生成向量 $\vec{h}$ 并计算多项式 $h(X)$,以及 $C_h = \mathsf{Commit}(h(X)), w = \langle\vec{h}, \otimes_{j=0}^{n-1}(1,\rho_j) \rangle$ +3. $S$ 计算 $v = (w-u)/c$ 和承诺 $C_g = (C_h/C_f)^{1/c}$ +4. $S$ 与 $V^*$ 运行一个 tensor product 检查协议,其证明关系为 $w = \langle\vec{h}, \otimes_{j=0}^{n-1}(1,\rho_j) \rangle$. + +显然,第1,3步中的消息和一个诚实证明者 $P$ 发送的消息是概率上不可区分的。 + +接下来,我们只需要说明第4步中运行的 $S$ 与 $V^*$ 运行 tensor product 检查协议同样满足这一性质即可。具体来说,因为协议满足 soundness,对于每一个 oracle $h^{(j)}, j=0,...,n-1$,其满足 + +$$ +h^{(j)}(X^2)=\frac{h^{(j-1)}(X)+f^{(j-1)}(-X)}{2} + \rho \cdot \frac{h^{(j-1)}(X)-h^{(j-1)}(-X)}{2X} +$$ + +注意到,对于等式右边的 $h^{(j-1)}(X)$ 它对应的 oracle 同样满足一个与 $h^{(j-2)}(X)$ 相关的等式。因此我们总能够将任意 $h^{(j)}(X^2)$ 所满足的右式展开为一个只包含 $h^{(0)}(X), h^{(0)}(-X), h^{(0)}(X^2)$ 形式。因此 $V^*$ 对 oracle $h^{(j)}$ 在任意点 $\beta$ 上进行问询所得到的回复,一定是一个关于 $\vec{h}$ 的一个线性独立的约束。 + +总的来说,在进行 tensor product 检查协议后,$V^*$ 会得到 $3\cdot (n-1)$ ,$2$ 个 $h^{(0)}(X)$ 上的取值,和一个 $h^{(n)}(X)$ 的值,即 $u+c\cdot v$。因为 $h$ 中包含了一个大小为 $3n$ 的盲化多项式,验证者无法通过插值求出盲化多项式的所有系数,所以该协议与诚实验证者执行的协议是不可区分的。 + +# 参考文献 +[BCH+22] Bootle, Jonathan, Alessandro Chiesa, Yuncong Hu, **et al. "Gemini: Elastic SNARKs for Diverse Environments." *Cryptology ePrint Archive* (2022). [https://eprint.iacr.org/2022/420](https://eprint.iacr.org/2022/420) + +[CFS17**]** Chiesa, Alessandro, Michael A. Forbes, and Nicholas Spooner. "A zero knowledge sumcheck and its applications." [*arXiv preprint arXiv:1704.02086* (2017)](https://eprint.iacr.org/2017/305). + +[XZZPS19] Xie, T., Zhang, J., Zhang, Y., Papamanthou, C., & Song, D. “Libra: Succinct zero-knowledge proofs with optimal prover computation.” [https://eprint.iacr.org/2019/317](https://eprint.iacr.org/2019/317) \ No newline at end of file diff --git a/gemini/image/Gemini-PIOP.png b/gemini/image/Gemini-PIOP.png new file mode 100644 index 0000000000000000000000000000000000000000..6ba9b01b8072abdac55207e2ab8156225301b029 GIT binary patch literal 135533 zcmeEv1z1$u7q21;h)61mNW%b1*B~jK15%=tbPe4tozh5yfJmr_pdc-cN*Ev@-Q5Dx z?+nZ+D&BkF|J~60uHSqzb87FiSO3=9d;7~mL@|yN96xmE5Qeyzkldj|$O(rIA?cwW z1CE?sjf(~T9JZ1Z6*!dBOgM4qP}rcgu(Gv@y`CXd_YftE;NCY%W<~=uD{D#?AxdUu zVW_UQwWXOU@D(_2Vy0uLXQ->QcaE8nnU#)-3HT^5a!|5dW&*?h~?PO!7?eYIz2r2&py zHaCarUe(o-G_;1@&%!~^!VVm!mK0NvmZD@91inKJjdg)PqPp6~X0TgyEH&-u4b6bd znVIPs>0zB0G_W)?0WK0@Vqygjv2n1l(KCZNfiquTQiPd}9!$>)9Fy17)3h|)*BtB- 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