From 47ed478835fe6798e38d197bc516d4bcebcfd47f Mon Sep 17 00:00:00 2001 From: arnaucube Date: Thu, 18 May 2023 00:08:03 +0200 Subject: [PATCH] hypernova: add sparse multilinear extension from matrix details --- notes_hypernova.pdf | Bin 227741 -> 243944 bytes notes_hypernova.tex | 120 ++++++++++++++++++++++++++++++++++++-------- 2 files changed, 98 insertions(+), 22 deletions(-) diff --git a/notes_hypernova.pdf b/notes_hypernova.pdf index c558ff361b4a2f06a5db33088327059f71913055..f3708998965b9393d27a07f6544d1786e5f51a11 100644 GIT binary patch delta 143978 zcmV)RK(oJ{^9|_n4zMx?7&bW|Fd%PYY6?6&3NK7$ZfA68ATc;LIFn%nD1X&hNpIUm z6u#$I@Ky?_CbKUIdQgnG3t}H)wJ6}`KueUxRuUD7vYq_;zFA0-^vIIo6i6=`a`fgc z-&=<`xlM5L^DBI0eK&sf`kb1CC`5_i$#|Af#*h{&kwhS7XfmEA*Xda4u`kdi}5M+d)mLnI4N|_9)M%ob1Fx#xM-DGzI zEL57?Qwc$mYM^3>5$35bvH|@41q6ZsiYc>%kBoSI&P}3`k-dlmiZr9}Adn=o3jk+nl?_Px&w%PQU-_MT*q9fK+<(7q@@!hHZvBg; zwMLq>+*N+-YMpJW%pIlaHTm?T-@c}wki!9kWDMXLFZuWFE?X=t1_UGpF@(3o!2UY3 z&H#-zTmSE`e!t2qTDW78a%ejMWd} 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ziAL3V*_jk(8S1L!MU}?HPGkge8sXl}9kRN2ugmD*g$ZPmD&G z7tz%u2ie{0qp6uugFMq3>Uu`|@felW4lCt0@9KJGZZv17ErKX!aZ7`2D}}A-RfK7+ z`7+BxaF+*G)s&#D43Vt4XL&S`Tx!8NL@-YcztBKt7E01CufMhZbO9MqO-NCA%QVjr z_lOw30Z?#qMA)3uoYVBvR1QtuhJcL#x3gel&@|v;x>D(3gmQV(N#O3=vh+xJfZ`TZ zyCLIwVBa(_)Sye|gLUf^3_s}GaS+3C2p@iEfQ<6$78gMB>v^@L~{dd ziETP`L-ffW?95QdR6lqJ6be->vyLnBq??63eYLcI+m8+ zgLeTb^?qqeCEN!sBlQ*_@m-a99G>yn^h_wN>)2YTsJ+B@6{F-P}PdpZ@WaWPvp-&H&+Gn jdtaM?$A8|TqZn+lQo34z1PCk~+^igM(g, l, d, T)$}:\\ % TODO use proper <, > -$T=\sum_{x_1 \in \{0,1\}} \sum_{x_2 \in \{0,1\}} \cdots \sum_{x_l \in \{0,1\}} g(x_1, x_2, \ldots, x_l)$ -$l$-variate polynomial g, degree $\leq d$ in each variable. +Recall sum-check protocol notation: \underline{$C \leftarrow \langle P, V(r) \rangle (g, l, d, T)$}: +$$T=\sum_{x_1 \in \{0,1\}} \sum_{x_2 \in \{0,1\}} \cdots \sum_{x_l \in \{0,1\}} g(x_1, x_2, \ldots, x_l)$$ +where $g$ is a $l$-variate polynomial, with degree at most $d$ in each variable, and $T$ is the claimed value. -let $s= \log m,~ s'= \log n$. +\vspace{1cm} + +Let $s= \log m,~ s'= \log n$. \begin{enumerate} \item $V \rightarrow P: \gamma \in^R \mathbb{F},~ \beta \in^R \mathbb{F}^s$ \item $V: r_x' \in^R \mathbb{F}^s$ - \item $V \leftrightarrow P$: sum-check protocol:\\ - $$c \leftarrow (g, s, d+1, \sum_{j \in [t]} \gamma^j \cdot v_j)$$ - where:\\ + \item $V \leftrightarrow P$: sum-check protocol: + $$c \leftarrow \langle P, V(r_x') \rangle (g, s, d+1, \overbrace{\sum_{j \in [t]} \gamma^j \cdot v_j}^\text{T})$$ + where: \begin{align*} g(x) &:= \left( \sum_{j \in [t]} \gamma^j \cdot L_j(x) \right) + \gamma^{t+1} \cdot Q(x)\\ - L_j(x) &:= \widetilde{eq}(r_x, x) \cdot \left( \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_1(y) \right)\\ - Q(x) &:= \widetilde{eq}(\beta, x) \cdot \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \left( \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_2(y) \right) \right) + \text{for LCCCS:}~ L_j(x) &:= \widetilde{eq}(r_x, x) \cdot \left( + \underbrace{\sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_1(y)}_\text{this is the check from LCCCS} + \right)\\ + \text{for CCCS:}~ Q(x) := &\widetilde{eq}(\beta, x) \cdot \left( + \underbrace{ \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \left( \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_2(y) \right) }_\text{this is the check from CommittedCCS} + \right) \end{align*} - \item $P \rightarrow V$: $\left( (\sigma_1, \ldots, \sigma_t), (\theta_1, \ldots, \theta_t) \right)$ - where - $$\sigma_j = \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_1(y)$$ - $$\theta_j = \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_2(y)$$ + Notice that $v_j= \sum_{y\in \{0,1\}^{s'}} \widetilde{M}_j(r, y) \cdot \widetilde{z}(y) = \sum_{x\in \{0,1\}^s} L_j(x)$. + \item $P \rightarrow V$: $\left( (\sigma_1, \ldots, \sigma_t), (\theta_1, \ldots, \theta_t) \right)$, where $\forall j \in [t]$, + $$\sigma_j = \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \widetilde{z}_1(y)$$ + $$\theta_j = \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \widetilde{z}_2(y)$$ + where $\sigma_j,~\theta_j$ are the checks from LCCCS and CCCS respectively with $x=r_x'$. \item V: $e_1 \leftarrow \widetilde{eq}(r_x, r_x')$, $e_2 \leftarrow \widetilde{eq}(\beta, r_x')$\\ check: $$c = \left( \sum_{j \in [t]} \gamma^j e_1 \sigma_j + \gamma^{t+1} e_2 \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \sigma \right) \right)$$ + which should be equivalent to the $g(x)$ computed by $V,P$ in the sum-check protocol. \item $V \rightarrow P: \rho \in^R \mathbb{F}$ \item $V, P$: output the folded LCCCS instance $(C', u', \mathsf{x}', r_x', v_1', \ldots, v_t')$, where $\forall i \in [t]$: \begin{align*} @@ -134,6 +144,72 @@ let $s= \log m,~ s'= \log n$. \end{enumerate} + +%%%%%% APPENDIX +\appendix +\section{Appendix: Some details} +This appendix contains some notes on things that don't specifically appear in the paper, but that would be needed in a practical implementation of the scheme. + +\subsection{Matrix and Vector to Sparse Multilinear Extension} + +Let $M \in \mathbb{F}^{m \times n}$ be a matrix. We want to compute its MLE +$$\widetilde{M}(x_1, \ldots, x_l) = \sum_{e \in \{0, 1 \}^l} M(e) \cdot \widetilde{eq}(x, e)$$ + +We can view the matrix $M \in \mathbb{F}^{m \times n}$ as a function with the following signature: +$$M(\cdot): \{0,1\}^s \times \{0,1\}^{s'} \rightarrow \mathbb{F}$$ +where $s = \lceil \log m \rceil,~ s' = \lceil \log n \rceil$. + +An entry in $M$ can be accessed with a $(s+s')$-bit identifier. + +eg.: +$$ +M = \begin{pmatrix} +1 & 2 & 3\\ +4 & 5 & 6\\ +\end{pmatrix} +\in \mathbb{F}^{3 \times 2} +$$ + +$m = 3,~ n = 2,~~~ s = \lceil \log 3 \rceil = 2,~ s' = \lceil \log 2 \rceil = 1$ + +So, $M(s_0, s_1) = x$, where $s_0 \in \{0,1\}^s,~ s_1 \in \{0,1\}^{s'},~ x \in \mathbb{F}$ + +$$ +M = \begin{pmatrix} +M(00,0) & M(01,0) & M(10,0)\\ +M(00,1) & M(01,1) & M(10,1)\\ +\end{pmatrix} +\in \mathbb{F}^{3 \times 2} +$$ + +This logic can be defined as follows: + +\begin{algorithm}[H] +\caption{Generating a Sparse Multilinear Polynomial from a matrix} +\begin{algorithmic} + \State set empty vector $v \in (\text{index:}~ \mathbb{Z}, x: \mathbb{F})^{s \times s'}$ + \For {$i$ to $n$} + \For {$j$ to $m$} + \If {$M_{i,j} \neq 0$} + \State $v.\text{append}( \{ \text{index}: i \cdot m + j,~ x: M_{i,j} \} )$ + \EndIf + \EndFor + \EndFor + \State return $v$ \Comment {$v$ represents the evaluations of the polynomial} +\end{algorithmic} +\end{algorithm} + +Once we have the polynomial, its MLE comes from +$$\widetilde{M}(x_1, \ldots, x_{s+s'}) = \sum_{e \in \{0,1\}^{s+s'}} M(e) \cdot \widetilde{eq}(x, e)$$ + +$$M(X) \in \mathbb{F}[X_1, \ldots, X_s]$$ + +\paragraph{Multilinear extensions of vectors} +Given a vector $u \in \mathbb{F}^m$, the polynomial $\widetilde{u}$ is the MLE of $u$, and is obtained by viewing $u$ as a function mapping ($s=\log m$) +$$u(x): \{0,1\}^s \rightarrow \mathbb{F}$$ +$\widetilde{u}(x, e)$ is the multilinear extension of the function $u(x)$ +$$\widetilde{u}(x_1, \ldots, x_s) = \sum_{e \in \{0,1\}^s} u(e) \cdot \widetilde{eq}(x, e)$$ + \bibliography{paper-notes.bib} \bibliographystyle{unsrt}