From 0e6992c2537f570d982b53ea3a8ee28408cd2ac1 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Mon, 7 Aug 2023 11:30:13 +0200 Subject: [PATCH] hypernova multifolding slides part 1 (intro) & 2 (multifolding unfolded) --- .gitignore | 1 + slides_hypernova-part1-introduction.pdf | Bin 0 -> 305700 bytes slides_hypernova-part1-introduction.tex | 420 ++++++++++++++++++ ..._hypernova-part2-multifolding-unfolded.pdf | Bin ..._hypernova-part2-multifolding-unfolded.tex | 1 + weil-pairing.pdf | Bin 225307 -> 225315 bytes weil-pairing.tex | 2 +- 7 files changed, 423 insertions(+), 1 deletion(-) create mode 100644 slides_hypernova-part1-introduction.pdf create mode 100644 slides_hypernova-part1-introduction.tex rename hypernova-multifolding-slides.pdf => slides_hypernova-part2-multifolding-unfolded.pdf (100%) rename hypernova-multifolding-slides.tex => slides_hypernova-part2-multifolding-unfolded.tex (99%) diff --git a/.gitignore b/.gitignore index a2d5165..5d7fea2 100644 --- a/.gitignore +++ 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z5>$0Ae1GSwAA5Ve?rqO?0xs6oSGqB{ZmOd7b*Np|QWxeqGu{Kw)JJyjy}B@#WAdf0 zJ74Wx!hA(igQM@SeEh(8md8=okR?Ok9h9{CG!in66ape3#RlJ>G4s-BY{$DgN5f_Zsi3F@RlMkeI>em wRD68TZzbFB?lAVALSreF`G0PH0=t-kTs%OKpFbR|92{JHD6d~jy_H7!KPi`$U;qFB literal 0 HcmV?d00001 diff --git a/slides_hypernova-part1-introduction.tex b/slides_hypernova-part1-introduction.tex new file mode 100644 index 0000000..3db87fe --- /dev/null +++ b/slides_hypernova-part1-introduction.tex @@ -0,0 +1,420 @@ +\documentclass{beamer} +\usefonttheme[onlymath]{serif} + +\mode +{ + \usetheme{Frankfurt} + \usecolortheme{dove} %% grey scale + \useinnertheme{circles} + % \setbeamercovered{transparent} +} + +\hypersetup{ + colorlinks, + citecolor=black, + filecolor=black, + linkcolor=black, + urlcolor=blue +} +\usepackage{graphicx} +\usepackage{listings} % embed code + +\setbeamertemplate{itemize}{$\circ$} +\setbeamertemplate{itemize items}{$\circ$} + +\beamertemplatenavigationsymbolsempty %% no navigation bar + +\setbeamertemplate{footline}{\hspace*{.1cm}\scriptsize{ +\hspace*{50pt} \hfill\insertframenumber/\inserttotalframenumber\hspace*{.1cm}\vspace*{.1cm}}} + +\setbeamertemplate{caption}[numbered] +\setbeamerfont{caption}{size=\tiny} + + + + +\title{HyperNova introduction} +\author{} +\date{\scriptsize{2023-07-25\\\href{https://0xparc.org}{0xPARC}, London}} + +\begin{document} + +\frame{\titlepage} + + +% NOTE: This talk provides an overview, if people is interested we can do another session going more into the technical details of the schemes. + + +\section[Preliminaries]{Preliminaries} + +\begin{frame}{IVC} + +For a function $F$, with initial input $z_0$, an IVC scheme allows a prover to produce a proof $\pi_i$ for the statement $z_i = F^{(i)}(z_0)$, given a proof $\pi_{i-1}$ for the statement $z_{i-1} = F^{(i-1)}(z_0)$ + +TODO add draw +TODO add reference to Valiant paper (2008) + +\end{frame} + +\begin{frame}{Recursion before folding schemes} +We used to use recursive SNARKs to achieve IVC. + + \begin{itemize} + \item Prove verification in circuit: inside a circuit, verify another proof + \begin{itemize} + \item eg. verifying a Groth16 proof inside a Groth16 circuit. + \end{itemize} + \item Amortized accumulation + \begin{itemize} + \item eg. Halo + \end{itemize} + \end{itemize} +\end{frame} + +\begin{frame}{R1CS refresher} + + R1CS instance: $(\{A, B, C\} \in \mathbb{F}^{m \times n},~ io,~ m,~ n,~ l)$, such that for $z=(io \in \mathbb{F}^l, 1, w \in \mathbb{F}^{m-l-1}) \in \mathbb{F}^m$, + +$$Az \circ Bz = Cz$$ + +Typically we use some scheme to prove that the previous equation is fullfilled by some private $w$ (eg. Groth16, Marlin, Spartan, etc). + +\end{frame} + +% \begin{frame}{R1CS refresher} +% TODO add A, B, C example from Vitalik article +% \end{frame} + +\begin{frame}{Random linear combination} + +Combine 2 instances together through a random linear comibnation, and the outputted instance will still satisfy the relation. + +\begin{itemize} + \item Have 2 values $x_1, x_2$. + \item Set $r \in^R \mathbb{F}$ + \item Compute $x_3 = x_1 + r \cdot x_2$. +\end{itemize} + +\pause + +Combined with homomorphic commitments +\begin{itemize} + \item We can do random linear combinations with the commitments and their witnesses, and the output can still be opened +\end{itemize} + +% TODO check on internet if there is some more standard definition / examples. + +\end{frame} + + +\section[Nova]{Nova} + +\begin{frame}{Folding schemes} +We're not verifying the entire proof +\begin{itemize} + \item Take n instances and 'batch' them together + \begin{itemize} + \item Folds $k$ (eg. 2) instances (eg. R1CS instances) and their respective witnesses into a signle one + \end{itemize} + \item At the end of the chain of folds, we just prove that the last fold is correct through a SNARK + \begin{itemize} + \item Which implies that all the previous folds were correct + \end{itemize} +\end{itemize} + +\pause + +In Nova: folding without a SNARK, we just reduce the satisfiability of the 2 inputted instances to the satisfiability of the single outputted one. + +[TODO image of multiple folding iterations] + +\end{frame} + + +\begin{frame}{Relaxed R1CS} + We work with \emph{relaxed R1CS} + +$$Az \circ Bz = u \cdot Cz + E$$ + +\begin{scriptsize} % TODO use the other simplier font syntax +(= R1CS when $u=1,~ E=0$) +\end{scriptsize} + +\begin{itemize} + \item main idea: allows us to fold, but accumulates \emph{cross terms} + \pause + \item when we do the \emph{relaxed} of higher degree equations (eg. plonkish), the cross terms grow (eg. Sangria with higher degree gates) +\end{itemize} + +\end{frame} + +\begin{frame}{NIFS - setup} +V and P: \emph{committed relaxed R1CS} instances +\begin{align*} + \varphi_1&=(\overline{E}_1, u_1, \overline{w}_1, x_1)\\ + \varphi_2&=(\overline{E}_2, u_2, \overline{w}_2, x_2) +\end{align*} + +P: witnesses +\begin{align*} + (E_1, r_{E_1}, w_1, r_{w_1})\\ + (E_2, r_{E_2}, w_2, r_{w_2}) +\end{align*} + +Let $z_1 = (w_1, x_1, u_1)$ and $z_2 = (w_2, x_2, u_2)$. + +\end{frame} + +\begin{frame}{NIFS} +\begin{footnotesize} +% While Prover works with $w, E$, Verifier works with commitments to them (\emph{Committed Relaxed R1CS}).\\ +% To keep the relations working with the random linear combinations, we use homomorphic commitments. + +\begin{itemize} + \item V, P: folded instance $\varphi = (\overline{E}, u, \overline{w}, x)$ + \begin{align*} + &\overline{E}=\overline{E}_1 + r \overline{T} + r^2 \overline{E}_2\\ + &u = u_1 + r u_2\\ + &\overline{w} = \overline{w}_1 + r \overline{w}_2\\ + &x = x_1 + r x_2 + \end{align*} + \item P: folded witness $(E, r_E, w, r_W)$ + \begin{align*} + &E = E_1 + r T + r^2 E_2\\ + &r_E = r_{E_1} + r \cdot r_T + r^2 r_{E_2}\\ + &w=w_1 + r w_2\\ + &r_W = r_{w_1} + r \cdot r_{w_2} + \end{align*} +\end{itemize} +\end{footnotesize} +\pause +\begin{scriptsize} +Note: $T$ are the cross-terms comming from combining the two R1CS instances from +\begin{align*} + Az \circ Bz &=A(z_1 + r \cdot z_2) \circ B(z_1 + r z_2)\\ + &=A z_1 \circ B z_1 + r(A z_1 \circ B z_2 + A z_2 \circ B z_1) + r^2 (A z_2 \circ B z_2) = \ldots +\end{align*} +\end{scriptsize} + +\end{frame} + +\begin{frame}{NIFS} + +\begin{small} +$$E=E_1 + r \underbrace{ (A z_1 \circ B z_2 + A z_2 \circ B z_1 - u_1 C z_2 - u_2 C z_1) }_\text{cross-terms} + r^2 E_2$$ +\end{small} + +$Az \circ Bz = uCz + E$ will hold for valid $z$ (which comes from valid $z_1,~ z_2$). + +[TODO add image of function F' with F inside with extra checks] + +\end{frame} + + + +\begin{frame}{NIFS} + +Each fold: $2~EC_{Add} + 1~EC_{Mul} + 1~hash$ + +20k R1CS constraints (using curve cycles) + +{\footnotesize +(so folding makes sense when we have a circuit with more than $2 \cdot 20k$ constraints) +} + +\pause +After all the folding iterations, Nova generates a SNARK proving the last folding instance. + +In Nova implementation, they use Spartan. +\end{frame} + +\begin{frame}{Benchmarks} + +% TODO: review names, and add links to profiles. +Benchmarks that Oskar, Carlos, et al did during the Vietnam residency in April +\href{https://hackmd.io/u3qM9s_YR1emHZSg3jteQA?view}{https://hackmd.io/u3qM9s\_YR1emHZSg3jteQA} + +\begin{center} +\begin{tabular}{ |c|c|c| } + \hline + Size & Constraints & Time\\ + \hline + 2KB & 883k & 320ms\\ + 4KB & 1.7m & 521ms\\ + 8KB & 3.4m & 1s\\ + 16KB & 6.8m & 1.9s\\ + 32KB & 13.7m & 4.1s \\ + \hline +\end{tabular}\\ +{\footnotesize eg. for 8kb, x100 Halo2 and Plonky2} +\end{center} + +(this is for the folding, without the last snark) + +\end{frame} + +\begin{frame}{SuperNova} +\begin{itemize} + \item iteration on Nova, combining \emph{different circuits} in a single one with \emph{selectors} + \item so we can work with a big circuit with \emph{subcircuits} without paying the whole size cost on each iteration + \item in IVC terms: fold multiple $F_i$ in a single $F'$ (in Nova was a single $F$ in $F'$) +\end{itemize} + +This is useful for example for a VM, doing one $F_i$ for each opcode + +\end{frame} + +\section[HyperNova]{HyperNova} + +% \begin{frame}{CCS} +% \begin{itemize} +% \item kind of a generalization of constraint systems +% \item can translate R1CS,Plonk,AIR to CCS +% \end{itemize} +% $$\sum_{i=0}^{q-1} c_i \cdot \bigcirc_{j \in S_i} M_j \cdot z ==0$$ +% \end{frame} + +\begin{frame}{R1CS to CCS example} + +\begin{scriptsize} +\begin{itemize} + \item Kind of a generalization of constraint systems + \item Can translate R1CS,Plonk,AIR to CCS +\end{itemize} +\pause +\begin{description} + \item[CCS instance] $S_{CCS} = (m, n, N, l, t, q, d, M, S, c)$\\ + where we have the same parameters than in $S_{R1CS}$, but additionally:\\ + $t=|M|$, $q = |c| = |S|$, $d$= max degree in each variable. + \item[R1CS-to-CCS parameters] $n=n,~ m=m,~ N=N,~ l=l,~ t=3,~ q=2,~ d=2$, $M=\{A,B,C\}$, $S=\{\{0,~1\},~ \{2\}\}$, $c=\{1,-1\}$ +\end{description} +\pause + +The CCS relation check: +\end{scriptsize} + +$$\sum_{i=0}^{q-1} c_i \cdot \bigcirc_{j \in S_i} M_j \cdot z ==0$$ + +\begin{scriptsize} +In our R1CS-to-CCS parameters is equivalent to +\begin{align*} + &c_0 \cdot ( (M_0 z) \circ (M_1 z) ) + c_1 \cdot (M_2 z) ==0\\ + \Longrightarrow &1 \cdot ( (A z) \circ (B z) ) + (-1) \cdot (C z) ==0\\ + \Longrightarrow &( (A z) \circ (B z) ) - (C z) ==0 +\end{align*} +\end{scriptsize} + +\end{frame} + + +\begin{frame}{Multifolding} +\begin{itemize} + \item Nova: 2-to-1 folding + \item HyperNova: multifolding, k-to-1 folding + \item We fold while through a SumCheck proving the correctness of the fold +\end{itemize} + +SumCheck's polynomial work is trivial, most of the cost comes from Poseidon hash in the transcript + +[TODO WIP section] + +\end{frame} + +\begin{frame}{Multifolding - Overview} + + \begin{tiny} + \begin{enumerate} + \item[1.] $V \rightarrow P: \gamma \in^R \mathbb{F},~ \beta \in^R \mathbb{F}^s$ + \item[2.] $V: r_x' \in^R \mathbb{F}^s$ + \item[3.] $V \leftrightarrow P$: sum-check protocol: + $c \leftarrow \langle P, V(r_x') \rangle (g, s, d+1, \underbrace{\sum_{j \in [t]} \gamma^j \cdot v_j}_\text{T})$, where: + \begin{align*} + g(x) &:= \underbrace{\left( \sum_{j \in [t]} \gamma^j \cdot L_j(x) \right)}_\text{LCCCS check} + \underbrace{\gamma^{t+1} \cdot Q(x)}_\text{CCCS check}\\ + 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{LCCCS check} + \right)\\ + 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{CCCS check} + \right) + \end{align*} + \end{enumerate} + \end{tiny} + +\end{frame} + +\begin{frame}{Multifolding - Overview} + + \begin{tiny} + \begin{enumerate} + \item[4.] $P \rightarrow V$: $\left( (\sigma_1, \ldots, \sigma_t), (\theta_1, \ldots, \theta_t) \right)$, where $\forall j \in [t]$, + $$\sigma_j = \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \widetilde{z}_1(y)$$ + $$\theta_j = \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \widetilde{z}_2(y)$$ + \item[5.] V: $e_1 \leftarrow \widetilde{eq}(r_x, r_x')$, $e_2 \leftarrow \widetilde{eq}(\beta, r_x')$\\ + check: + $$c = \left(\sum_{j \in [t]} \gamma^j \cdot e_1 \cdot \sigma_j \right) + \gamma^{t+1} \cdot e_2 \cdot \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \theta_j \right)$$ + \item[6.] $V \rightarrow P: \rho \in^R \mathbb{F}$ + \item[7.] $V, P$: output the folded LCCCS instance $(C', u', \mathsf{x}', r_x', v_1', \ldots, v_t')$, where $\forall i \in [t]$: + \begin{align*} + C' &\leftarrow C_1 + \rho \cdot C_2\\ + u' &\leftarrow u + \rho \cdot 1\\ + \mathsf{x}' &\leftarrow \mathsf{x}_1 + \rho \cdot \mathsf{x}_2\\ + v_i' &\leftarrow \sigma_i + \rho \cdot \theta_i + \end{align*} + \item[8.] $P$: output folded witness and the folded $r_w'$: + \begin{align*} + \widetilde{w}' &\leftarrow \widetilde{w}_1 + \rho \cdot \widetilde{w}_2\\ + r_w' &\leftarrow r_{w_1} + \rho \cdot r_{w_2} + \end{align*} + \end{enumerate} + \end{tiny} + +\end{frame} + +\section[Wrappup]{Wrappup} + +\begin{frame}{Mysteries \& unsolved things} +\begin{itemize} + \item how HyperNova compares to Protostar + \item prover knows the full witness [TODO update/rm this] +\end{itemize} + +[TODO WIP section] +\end{frame} + + +\begin{frame} +\frametitle{Wrappup} +\begin{itemize} + \item HyperNova: \href{https://eprint.iacr.org/2023/573}{https://eprint.iacr.org/2023/573} + \item multifolding PoC on arkworks: \href{https://github.com/privacy-scaling-explorations/multifolding-poc}{github.com/privacy-scaling-explorations/multifolding-poc} + \item PSE hypernova WIP \href{https://github.com/privacy-scaling-explorations/Nova}{github.com/privacy-scaling-explorations/Nova} +\end{itemize} + +\vspace{2cm} +\tiny{ + $$\text{2023-07-25}$$ + $$\text{\href{https://0xparc.org}{0xPARC}}$$ +} +\end{frame} + + +% from Michael +% - Why Nova? +% - Nova's limitations +% - Why Hypernova +% - Hypernova concepts explained to General Technologist (minimal ZK understanding) +% - Final output + + + +%%%%% +% - We used recursive SNARKs to achieve IVC +% - get a proof and prove that it's verification passes, inside another proof +% - Folding: we're not verifying the entire proof +% - we take n proofs and 'batch' them together +% - at the end of the chain of folds, we just prove that the last fold is correct +% - which implies that all the previous folds were correct +% - Random Linear Combination: combine 2 instances together through a random linear comibnation, and the outputted instance will still satisfy the relation +% - Multifolding SumCheck: SumCheck's polynomial work is trivial, most of the cost comes from Poseidon hash in the transcript + +\end{document} diff --git a/hypernova-multifolding-slides.pdf b/slides_hypernova-part2-multifolding-unfolded.pdf similarity index 100% rename from hypernova-multifolding-slides.pdf rename to slides_hypernova-part2-multifolding-unfolded.pdf diff --git a/hypernova-multifolding-slides.tex b/slides_hypernova-part2-multifolding-unfolded.tex similarity index 99% rename from hypernova-multifolding-slides.tex rename to slides_hypernova-part2-multifolding-unfolded.tex index 126ba87..4b8ecaf 100644 --- a/hypernova-multifolding-slides.tex +++ b/slides_hypernova-part2-multifolding-unfolded.tex @@ -1,4 +1,5 @@ \documentclass{beamer} +% \usefonttheme[onlymath]{serif} \mode { diff --git a/weil-pairing.pdf b/weil-pairing.pdf index 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\href{https://sites.google.com/site/matanprasma/artifact}{Matan Prsma} math seminars and also while reading about Bilinear Pairings. Usually while reading papers and books I take handwritten notes, this document contains some of them re-written to $LaTeX$. + Notes taken from \href{https://sites.google.com/site/matanprasma/artifact}{Matan Prasma} math seminars and also while reading about Bilinear Pairings. Usually while reading papers and books I take handwritten notes, this document contains some of them re-written to $LaTeX$. The notes are not complete, don't include all the steps neither all the proofs. I use these notes to revisit the concepts after some time of reading the topic. \end{abstract}