From 472e10e0d80ed80d603c965ac46acb8e88c0e91d Mon Sep 17 00:00:00 2001 From: Yu Guo Date: Thu, 26 Sep 2024 12:34:39 +0800 Subject: [PATCH] feat(binius): add binius-02, subspace polynomial --- fri-binius/binius-02.zh.md | 486 ++++++++++++++++++++++++++++++++++ fri-binius/img/subspace-1.png | Bin 0 -> 207312 bytes 2 files changed, 486 insertions(+) create mode 100644 fri-binius/binius-02.zh.md create mode 100644 fri-binius/img/subspace-1.png diff --git a/fri-binius/binius-02.zh.md b/fri-binius/binius-02.zh.md new file mode 100644 index 0000000..972a027 --- /dev/null +++ b/fri-binius/binius-02.zh.md @@ -0,0 +1,486 @@ +# Notes on Binius (Part II): Subspace Polynomial + +- Yu Guo +- Jade Xie + +FRI-Binus 论文 [DP24] 中讨论了基于 Subspace Polynomial 的 Additive FFT 算法,并给出了用奇偶项分解的视角来理解 [LCH14] 中的基于 Novel Polynomial Basis 的 Additive FFT 算法。本文直接介绍子空间多项式 Subspace Polynomial,然后据此介绍奇偶项分解视角的 Additive FFT 算法。本文省去了 Normalized Subspace Polynomial 的定义,方便读者理解。Normalization 只是影响 FFT 算法的性能,和本文介绍的简化版算法并没有本质的区别。 + +由于 Additive FFT 依赖的代数结构和素数域上的 Multiplicative FFT 非常相似,因此关于 Multiplicative FFT 的知识将有助于理解本文的内容。 + +## 线性子空间多项式 Subspace Polynomial + +我们继续探索基于 $\mathbb{F}_2$ 上的 Extension Field,$\mathbb{F}_{2^m}$ 。不管是用何种方式构造的 $\mathbb{F}_{2^m}$,所有的元素构成了一个向量空间,记为 $V_m$ ,并且存在一组 Basis $(\beta_0, \beta_1, \ldots, \beta_{m-1})$ ,张成这个向量空间,记为 $V_m=\mathsf{Span}(\beta_0, \beta_1, \ldots, \beta_{m-1})$,或者用符号 $\langle \cdots \rangle$ 表示: + +$$ +V_m = \langle \beta_0, \beta_1, \ldots, \beta_{m-1} \rangle +$$ + +这样任意一个元素 $ \theta\in\mathbb{F}_{2^m}$ ,可以写为 Basis 分量的线性组合: + +$$ +\theta = c_0\cdot \beta_0 + c_1\cdot \beta_1 + \ldots + c_{m-1}\cdot \beta_{m-1}, \text{ where $c_i\in \mathbb{F}_2$} +$$ + +同时 $V_m$ 还是一个加法群,单位元为 $V_0=\{0\}$ ,如果 $V_k$ 是 $V_m$ 的一个线性子空间,那么 $V_k$ 也是 $V_m$ 的一个加法子群。对于 $V_k$ ,我们可以用一个多项式来编码其中的所有元素,即该多项式的根集合正好对应 $V_k$ 的所有元素的集合,我们把多项式记为 $s_k(X)$ 。这个多项式也被称为「Subspace Polynomial」子空间多项式: + +$$ +s_k(X) = \prod_{i=0}^{2^k-1}(X-\theta_i), \text{ where $\theta_i\in V_k$ } +$$ + +多项式 $s_k(X)$ 也可以看成是 $V_k$ 这个 Domain 上的 Vanishing Polynomial,因为对于任意的 $\theta\in V_k$ ,都满足: + +$$ +s_k(\theta) = 0 +$$ + +### Linearized Polynomial + +上面介绍的 Subspace 多项式是一种所谓的 Linearized Polynomial,因为它的定义满足下面的形式: + +$$ +L(X) = \sum_{i=0}^{n-1} c_i\cdot X^{q^i}, \quad c_i\in \mathbb{F}_{q} +$$ + +多项式 $L(X)$ 之所以被称为 Linearized Polynomial,因为每一个 $L(X)$ 都对应到 $\mathbb{F}_{q}$ 的扩张域 $K$ 上的一个线性算子(Linear Operator)。假如 $L(X)$ 的所有根都在扩张域 $K=\mathbb{F}_{q^s}$ 中,那么对于所有的 $\theta\in K$,都有 $L(\theta)\in K$。而且,如果 $\theta\neq\theta'$,那么 $L(\theta)\neq L(\theta')$。每一个 $L(X)$ 都可以被视为一个矩阵 $B\in \mathbb{F}_q^{s\times s}$ ,完成 向量空间 $\mathbb{F}^s_q$ 上的线性变换,使得: + +$$ +(c_0, c_1, \ldots, c_{s-1}) B = (d_0, d_1, \ldots, d_{s-1}) +$$ + +对于 Subspace Polynomial 而言,每一个 $s_k(X)$ 都是一个 Linearized Polynomial,反过来,任何一个 Linearized polynomial $L(X)\in \mathbb{F}_{q^m}[X]$,它的所有根都构成某个线性子空间 $V_n\subset V_m$。详细的证明过程请参考 [LN97]。 + +### 线性性质 + +由于 $s_k(X)$ 的每一项都是 $a_i\cdot X^{2^i}$ 的形式,因此它具有加法的同态性: + +$$ +\begin{align*} +s_k(x + y) &= s_k(x) + s_k(y), &\quad \forall x, y \in \mathbb{F}_{2^m}\\ +s_k(c\cdot x) &= c\cdot s_k(x), &\quad \forall x\in \mathbb{F}_{2^m}, \forall c\in \mathbb{F}_2 +\end{align*} +$$ + +我们来尝试简单证明第一个等式,根据有限域理论的一个常见定理(Freshman's dream): + +$$ +(x + y)^{2} = x^2 + 2xy + y^2 = x^{2} + y^{2}, \quad\text{where $x, y\in \mathbb{F}_{2^m}$} +$$ + +显然,$2xy=0$,因为在二进制域中,$2=0$。所以下面的等式也同理成立: + +$$ +(x + y)^{2^i} = x^{2^i} + y^{2^i} +$$ + +接下来验证下 $s_k(X)$ 的加法同态性: + +$$ +s_k(x + y) = \sum_{i=0}^{k} a_i\cdot (x+y)^{2^i} = \sum_{i=0}^{k} a_i\cdot \big(x^{2^i} + y^{2^i}\big) = s_k(x) + s_k(y) +$$ + +### 子空间多项式的递推式 + +对于子空间 $V_k$,它可以被拆分为两个不相交的集合: + +$$ +V_k = V_{k-1} \cup (\beta_{k-1}+V_{k-1}) +$$ + +这里 $V_k=\langle \beta_0, \beta_1, \ldots, \beta_{k-1} \rangle$ , $V_{k-1}=\langle \beta_0, \beta_1, \ldots, \beta_{k-2} \rangle $ , +那么 $V_k$, $V_{k-1}$, $\beta_{k-1}+V_{k-1}$ 所对应的子空间多项式满足下面的关系: + +$$ +s_k(X) = s_{k-1}(X) \cdot s_{k-1}(X+\beta_{k-1}) +$$ + +举个简单例子,假设 $k=3$,$V_3=\langle \beta_0,\beta_1,\beta_2\rangle $ 由两部分构成,一部分是 $V_2=\langle \beta_0,\beta_1\rangle $,另一部分是 $V_2$ 中的每一个元素加上 ${\color{blue}\beta_2}$。因此,$V_3$ 的元素个数为 $2^2 + 2^2 = 8$,下面列出 $V_3$ 的全部元素: + +$$ +V_3 = \{0, \beta_0, \beta_1, \beta_0+\beta_1\} \cup \{{\color{blue}\beta_2}, \beta_0+{\color{blue}\beta_2}, \beta_1+{\color{blue}\beta_2}, (\beta_0+\beta_1)+{\color{blue}\beta_2}\} +$$ + +我们容易验证: $s_3(X) = s_{2}(X) \cdot s_{2}(X+\beta_{k-1})$ 。当然 $s_{2}(X)$ 也可以拆成关于 $s_1(X)$ 和 $s_1(X+\beta_1)$ 的乘积,我们不妨试着拆解到底: + +$$ +\begin{split} +s_3(X) & = s_2(X)\cdot s_2(X + \beta_2) \\ +& = s_2(X)^2 + \beta_2\cdot s_2(X) \\ +& = s_1(X)\cdot s_1(X + \beta_1)\cdot s_1(X)\cdot s_1(X + \beta_1) + \beta_2\cdot s_1(X)\cdot s_1(X + \beta_1) \\ +& = (s_1(X)^2 + \beta_1\cdot s_1(X))^2 + \beta_2\cdot s_1(X)^2 + \beta_1\beta_2\cdot s_1(X) \\ +& = s_1(X)^4 + \beta_1^2\cdot s_1(X)^2 + \beta_2\cdot s_1(X)^2 + \beta_1\beta_2\cdot s_1(X) \\ +& = s_1(X)^4 + (\beta_1^2+\beta_2)\cdot s_1(X)^2 + \beta_1\beta_2\cdot s_1(X) \\ +& = (X\cdot(X+\beta_0))^4 + (\beta_1^2+\beta_2)\cdot (X\cdot(X + \beta_0))^2 + \beta_1\beta_2\cdot (X\cdot(X + \beta_0)) \\ +& = (X^2 + \beta_0\cdot X)^4 + (\beta_1^2+\beta_2)\cdot (X^2 + \beta_0\cdot X)^2 + \beta_1\beta_2\cdot (X^2 + \beta_0\cdot X) \\ +& = X^8 + \beta_0^4X^4 + (\beta_1^2+\beta_2)X^4 + \beta_0^2(\beta_1^2+\beta_2)X^2 + \beta_1\beta_2X^2 + \beta_0\beta_1\beta_2X \\ +\end{split} +$$ + +最后 $s_3(X)$ 的展开式满足 $\sum_{i=0}^{k} a_i\cdot X^{2^i}$ 这样的模式,也符合了我们上面的结论。 + +## 子空间上的同态映射 + +因为 Subspace Polynomial 实际上是一种 Vanishing Polynomial,并且它还具有加法同态,所以我们可以利用 Subspace Polynomial 来定义子空间之间的同态映射。 + +例如对于 $V_3=\langle \beta_0, \beta_1, \beta_2\rangle$ ,我们定义 $V_3$ 的子空间 $V_1=\{0, \beta_0\}$ 及其 Subspace Polynomial $s_1(X)$ + +$$ +s_1(X) = X\cdot (X+\beta_0) +$$ + +很显然,$s_1(V_1)=\{0, 0\}$,如果将 $s_1(X)$ 作用到 $V_3$ ,我们会得到下面的结果: + +$$ +\begin{split} +s_1(0) &= 0\\ +s_1(\beta_0) & = 0 \\ +s_1(\beta_1) & = \beta_0\beta_1 + \beta_1^2\\ +s_1(\beta_0+\beta_1) & = \beta_0\beta_1 + \beta_1^2\\ +s_1(\beta_2) &= \beta_0\beta_2 + \beta_2^2\\ +s_1(\beta_0+\beta_2) &= \beta_0\beta_2 + \beta_2^2\\ +s_1(\beta_1+\beta_2) &= \beta_0\beta_1 + \beta_1^2 + \beta_0\beta_2 + \beta_2^2\\ +s_1(\beta_0+\beta_1+\beta_2) &= \beta_0\beta_1 + \beta_1^2 + \beta_0\beta_2 + \beta_2^2\\ +\end{split} +$$ + +上面的等式显示 $s_1(V_3)$ 被映射到了一个大小只有 $V_3$ 一半的集合,记为 $V_2$。该集合也是一个子空间,$V_2= \langle \beta'_0, \beta'_1\rangle = \langle \beta_0\beta_1 + \beta_1^2, \beta_0\beta_2 + \beta_2^2 \rangle $,维度为 $2$ 。 + +这不是巧合,根据群同构定理,同态映射 $\phi: H \to G$ 的 Image $G$ 满足 $G\cong H/Ker(\phi)$ ,其中 $G$ 是一个商群,并且 $|G| = |H|/|Ker(\phi)|$ 。在上面这个例子里,$s_1: V_3\to V_2$ 是同态映射,$V_1=Ker(s_1)$ 。 + +### 映射构成的链 + +对于 $V_2 = s_1(V_3)$,我们仍然可以继续构造一个 Degree 为 $2$ 的 Subspace Polynomial, + +$$ +s_1'(X) = X\cdot (X+\beta_0\beta_1 + \beta_1^2) +$$ + +可以继续将 $V_2$ 映射到一个一维的子空间 $V_1=\langle \beta''\rangle$。我们只需要计算 $s_1(\beta'_0)$ 与 $s_1(\beta'_1)$ 即可,这些 Basis 分量构成了 $V_2$: + +$$ +\begin{split} +s_1'(\beta_0\beta_1 + \beta_1^2) &= 0 \\ +s_1'(\beta_0\beta_1 + \beta_1^2 + \beta_0\beta_2 + \beta_2^2)& = \beta_0^2\beta_1\beta_2 + \beta_0\beta_1^2\beta_2 + \beta_0^2\beta_2^2 + \beta_0\beta_1\beta_2^2 + \beta_1^2\beta_2^2 + \beta_2^4 \\ + & = \beta'' +\end{split} +$$ + +其中 $V_2$ 的 Basis $(\beta'_0, \beta'_1)$ 中第一个分量会被映射到 $0$,第二个分量映射到 $\beta''$ 。 + +至此,我们得到一个映射的链: + +$$ +V_3 \overset{s_1}{\longrightarrow} V_2 \overset{s'_1}{\longrightarrow} V_1 +$$ + +或者也可以写为: + +$$ +\langle \beta_0, \beta_1, \beta_2 \rangle \overset{s_1}{\longrightarrow} \langle s_1(\beta_1), s_1(\beta_2) \rangle +\overset{s'_1}{\longrightarrow} \langle s'_1(s_1(\beta_2)) \rangle +$$ + +并且每次映射到的线性子空间的维度都减一,即集合大小减半。这个代数结构是我们后续构造 FFT 与 FRI 协议的关键。 + +不难证明,对于任意的线性子空间,只要我们选定一个 Basis 之后,就可以依次构造 Degree 为 2 的 Subspace Polynomial 作为映射函数,然后通过映射得到一个维度减 1 的子空间,然后不断重复,直到子空间降至 1 维。当然 Basis 选择不同,以及 Subspace Polynomial 的选择不同都会导致不同的映射链。选择恰当的映射链可以显著提高计算的效率。 + +## ${s}_1$ 映射的复合 + +我们定义映射链的初始子空间为 $S^{(0)}$,映射后的子空间为 $S^{(1)}$,$i$ 次映射之后的子空间为 $S^{(i)}$ : + +$$ +S^{(0)} \overset{s_1}{\longrightarrow} S^{(1)} \overset{s^{(1)}_1}{\longrightarrow} \cdots \overset{s^{(n-1)}_1}{\longrightarrow} S^{(n)} +$$ + +给定 $S^{(i)}$ 的一组 Basis 后,假设为 $B^{(i)}=(\beta^{(i)}_0, \beta^{(i)}_1,\ldots, \beta^{(i)}_s)$,在 Basis 上定义 Subspace Polynomial ${s}^{(i)}_1$,并用其作为群同态映射函数,把 $S^{(i)}$ 降维到 $S^{(i+1)}$。降维后的线性子空间 $S^{(i+1)}$ 的 Basis 需要把 $S^{(i)}$ 的 Basis 同步跟着 ${s}^{(i)}_1$ 转换到一个新的 Basis。切换到新 Basis 之后,我们又可以定义一组新的 Subspace Polynomial ${s}^{(i+1)}_i(X)$。 + +我们假设 $S^{(0)}=\langle \beta_0, \beta_1, \ldots, \beta_{k-1}\rangle $ 开始,给定一组 Basis $B_k$,经过 ${s}_1$ 的映射之后,我们得到了 $S^{(1)}$,以及其 Basis $B^{(1)}$: + +$$ +B^{(1)} = \langle {\color{blue}{s}_1(\beta_1)}, {\color{blue}{s}_1(\beta_2)}, \ldots, {s}_1(\beta_{k-1}) \rangle +$$ + +在 $S^{(1)}$ 再定义 ${s}^{(1)}_1(X)$: + +$$ +{s}^{(1)}_1(X) = {X(X+{s}_1(\beta_1))} +$$ + +那么,$S^{(1)}$ 映射后产生的 $S^{(2)}$ 和 $S^{(0)}$ 之间的关系是什么?对于任何一个元素 $a \in S^{(0)}$,它先经过 ${s}_1$ 映射到 $S^{(1)}$,然后再经过 ${s}^{(1)}_1$ 映射到 $S^{(2)}$ 中的某个元素,因此经过两次映射后的值可以被写为两次映射函数的复合, ${s}^{(1)}_1({s}_1(X))$,我们化简下这个复合函数: + +$$ +\begin{align*} +{s}^{(1)}_1({s}_1(X)) &= {s}_1(X)({s}_1(X)+{s}_1(\beta_1)) +& \\[3ex] +&= {s}_1(X)({s}_1(X+\beta_1)) & (\text{additive homomorphism })\\[3ex] +&= {s}_2(X) & (\text{recurrency } ) +\end{align*} +$$ + +于是我们推导出了 ${s}^{(1)}_1({s}_1(X))={s}_2(X)$。这意味着经过两次 2-to-1 的映射,等价于做一次 4-to-1 的映射,并且对应的同态映射函数为 ${s}_2$: + +$$ +\begin{split} +{s}_2: &S^{(0)} \to S^{(2)} \\[3ex] +& X \mapsto {X(X+\beta_0)(X+\beta_1)(X+\beta_1+\beta_0)} \\ +\end{split} +$$ + +如下图所示,左右两种映射方式都会得到 $S^{(2)}$: + + + +同理,我们可以得到下面的结论,对于折叠 $j$ 次之后的线性子空间 $S^{(j)}$ 为 + +$$ +S^{(j)} = \langle {s}_{j}(\beta_j), {s}_{j}(\beta_{j+1}), \ldots, {s}_{j}(\beta_{k-1}) \rangle +$$ + +并且 ${s}_{j}$ 满足下面的复合等式: + +$$ +s_{j}(X) = s^{(1)}_{j-1}(s_1(X)) = s^{(1)}_{j-1}\circ s_1 +$$ + +这个复合映射的等式可以解读为:先做一次 $s_1$ 映射得到 $S^{(1)}$,然后再做一次 $j-1$ 维的映射 $s^{(1)}_{j-1}$,等价于直接一次 $j$ 维映射 $s_{j}$,两者都映射到同一个子空间 $S^{(j)}$。 + +同样,我们还能证明:如果先做一次 $j-1$ 维的映射 $s_{j-1}$,再在映射后的子空间 $S^{(j-1)}$ 上做一次一维映射,同样可以得到子空间 $S^{(j)}$ : + +$$ +s_{j}(X) = s^{(1)}_{1}(s_{j-1}(X)) = s^{(1)}_{1}\circ s_{j-1} +$$ + +更一般地,我们可以证明得到下面的重要性质,即在任意子空间 $S^{(i)}$ 上做一次 $j$ 维映射,等价于在其上连续做 $j$ 次 1 维映射: + +$$ +s^{(i)}_{j}(X) = s^{(i+j-1)}_1\circ s^{(i+j-2)}_1\circ \cdots \circ s^{(i)}_1 +$$ + + +## 多项式的 Polynomial Basis + + +对于一个次数小于 $N=2^n$ 的一元多项式 $f(X) \in \mathbb{F}[X]^{ z?Tu%2j9*_)F>+j^`}%U==DOD7n`ebS@G_{kMoWrS`4m5AHRF0%G$e7=+Wz4xqWIIN zyoeLETUSna{W5HT;2?5T3cd%`m!eX62H+^FDEv?|hwVmM(G7mic}<;r+MiuTxP9Z;j@IlKLv)F;>dj)Eisch0pe*9Ex>wuAwi#H)qf z=>WH`!4=AHTxu<%2g{scpS}Gk2zO8HsQVK=;v%TT3D!%lwsU{oCYI z@{h?Eea>w$-UN&+u?ON8v=-GA zC~1F4O^l|RZFyYQfJ#{Dcb;Wxzk@!r?IUjQk##Qj23GVzO!!M9&RHORT92UhhxeCH zMJHe~PGW<&d(IC8kH2f*1cr_SW^Y)z-un1Lr=Oj%&r{2F^Q&*`(mLI$DZ z7h$3bkFDeiu10XlJQay=xibd|Q+y?LW-E5ZCiMJl6K^QO!Tfk2pcrgS@AHx#bhYCR zOFW**Y0l&6W^=h^!of|uqql4@*fr;8VATfH^B|%T(1I6~J8&Q*wh{5JzxCP|?A1^; z*RFon_gmlDRq#A*VvDn)P9rsk2Ae@K14BF@v*#VN@w3xKi*JbtlqCU_Wz!fpVM9R<~ka-s43`l=+Ft9Cc6qL6LHlCD8TfntfHdd zyb8|{eG?M0zYjs+^UiM!e?HSxeWl=*=~cX&gT{0CDf#`c%XA`^C-Y_kQl5)%Fg6GQ 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