From 5e138717c522c061814e8eef0d89530f790d8e97 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Sat, 15 Apr 2023 20:44:54 +0200 Subject: [PATCH] Add first part Spartan notes, continue Nova notes --- notes_nova.pdf | Bin 192561 -> 228030 bytes notes_nova.tex | 148 +++++++++++++++++++++++++++++++-- notes_spartan.pdf | Bin 0 -> 221265 bytes notes_spartan.tex | 206 ++++++++++++++++++++++++++++++++++++++++++++++ paper-notes.bib | 9 ++ 5 files changed, 357 insertions(+), 6 deletions(-) create mode 100644 notes_spartan.pdf create mode 100644 notes_spartan.tex diff --git a/notes_nova.pdf b/notes_nova.pdf index 367b3d60ecf9c30b0170eaf4914f61be2f955b3e..00278ea7a98f8d2899df2dc52985be869895772c 100644 GIT binary patch delta 138705 zcmce-Wl$hpkR^&V?yilyyEM?aySux)Hc+^`)4034H15zijk~+Mz5Zr)WZ&+eorswq zRZ($oR>X`j9Ll-B@ zzZT4YE&lHRbN_3hWa@11>SS!{OvL_oMF|$3QO4BH+{J>3nVpCzmADm@3YbHhRG^yN z0t1112fczdwKH*caWXZu{paZ)WSoEB`|m`E{~r*!{srRyGeYkFW`tk=10nN2zEy(# 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(mess to) {}; + \draw[->,>=angle 60] (mess from) -- (mess to) node[midway, above] + {#3}; + + \if R#5 + \node (\detokenize{#3} from) at (mess from) {\llap{#6~}}; + \node (\detokenize{#3} to) at (mess to) {\rlap{~#7}}; + \else\if L#5 + \node (\detokenize{#3} from) at (mess from) {\rlap{~#6}}; + \node (\detokenize{#3} to) at (mess to) {\llap{#7~}}; + \else + \node (\detokenize{#3} from) at (mess from) {#6}; + \node (\detokenize{#3} to) at (mess to) {#7}; + \fi + \fi +} + % prevent warnings of underfull \hbox: \usepackage{etoolbox} \apptocmd{\sloppy}{\hbadness 4000\relax}{}{} @@ -41,7 +68,7 @@ \tableofcontents -\section{Folding Scheme for Committed Relaxed R1CS} +\section{NIFS} \subsection{R1CS modification} @@ -95,7 +122,7 @@ $(\overline{E}, u, \overline{W}, x)$, satisfyied by a witness $(E, r_E, W, r_W)$ \end{align*} -\subsection{Folding protocol} +\subsection{Folding scheme for committed relaxed R1CS} V and P take two \emph{committed relaxed R1CS} instances \begin{align*} @@ -133,11 +160,120 @@ Let $Z_1 = (W_1, x_1, u_1)$ and $Z_2 = (W_2, x_2, u_2)$. \end{align*} \end{enumerate} -P uses a zkSNARK showing that knows the valid witness $(E, r_E, W, r_W)$ for the committed relaxed R1CS without revealing its value. -Then, via Fiat-Shamir transform we achieve non-interactivity. +P will proof that knows the valid witness $(E, r_E, W, r_W)$ for the committed relaxed R1CS without revealing its value. + +\begin{center} + \begin{sequencediagram} + \newinst[1]{p}{Prover} + \newinst[3]{v}{Verifier} + + \bloodymess[1]{p}{$\overline{T}$}{v}{R}{ + \shortstack{ + $T=A z_1 \circ B z_1 + A z_2 \circ B z_2 - u_1 C z_2 - u_2 C z_2$\\ + $\overline{T}=Commit(T, r_T)$ + } + }{ + \shortstack{ + $r \in^R \mathbb{F}_p$\\ + $\overline{E} = \overline{E}_1 + r \overline{T} + r^2 \overline{E}_2$\\ + $u= u_1 + r u_2$\\ + $\overline{W} = \overline{W}_1 + r \overline{W}_2$\\ + $\overline{x} = \overline{x}_1 + r \overline{x}_2$\\ + $\varphi=(\overline{E}, u, \overline{W}, x)$ + } + } + \bloodymess[1]{v}{$r$}{p}{L}{}{ + \shortstack{ + $E = E_1 + r T + r^2 E_2$\\ + $u= u_1 + r u_2$\\ + $W = W_1 + r W_2$\\ + $r_{W} = r_{W_1} + r r_{W_2}$\\ + $(E, r_E, W, r_W)$ + } + } + \end{sequencediagram} +\end{center} + + +The previous protocol achieves non-interactivity via Fiat-Shamir transform, obtaining a \emph{Non-Interactive Folding Scheme for Committed Relaxed R1CS}. + +Note: the paper later uses $\mathsf{u}_i,~ \mathsf{U}_i$ for the two inputed $\varphi_1,~ \varphi_2$, and later $\mathsf{u}_{i+1}$ for the outputed $\varphi$. Also, the paper later uses $\mathsf{w},~ \mathsf{W}$ to refer to the witnesses of two folded instances (eg. $\mathsf{w}=(E, r_E, W, r_W)$). + +\section{Nova} +IVC (Incremental Verifiable Computation) scheme for a non-interactive folding scheme. + +\subsection{IVC proofs} + +Allows prover to show $z_n = F^{(n)}(z_0)$, for some count $n$, initial input $z_0$, and output $z_n$.\\ +$F$: program function (polynomial-time computable)\\ +$F'$: augmented function, invokes $F$ and additionally performs fold-related stuff. + +\vspace{0.5cm} +Two committed relaxed R1CS instances:\\ +$\mathsf{U}_i$: represents the correct execution of invocations $1, \ldots, i-1$ of $F'$\\ +$\mathsf{u}_i$: represents the correct execution of invocations $i$ of $F'$ + +\paragraph{Simplified version of $F'$ for intuition} +\vspace{0.5cm} +$F'$ performs two tasks: +\begin{enumerate}[i.] + \item execute a step of the incremental computation: + instance $\mathsf{u}_i$ contains $z_i$, used to output $z_{i+1}=F(z_i)$ + \item invokes the verifier of the non-interactive folding scheme to fold the task of checking $\mathsf{u}_i$ and $\mathsf{U}_i$ into the task of checking a single instance $\mathsf{U}_{i+1}$ +\end{enumerate} + +\vspace{0.5cm} +$F'$ proves that: +\begin{enumerate} + \item $\exists ( (i, z_0, z_i, \mathsf{u}_i, \mathsf{U}_i), \mathsf{U}_{i+1}, \overline{T})$ such that + \begin{enumerate}[i.] + \item $\mathsf{u}_i.x = H(vk, i, z_0, z_i, \mathsf{U}_i)$ + \item $h_{i+1} = H(vk, i+1, z_0, F(z_i), \mathsf{U}_{i+1})$ + \item $\mathsf{U}_{i+1} = NIFS.V(vk, \mathsf{U}_i, \mathsf{u}_i, \overline{T})$ + \end{enumerate} +\item $F'$ outputs $h_{i+1}$ +\end{enumerate} + + +$F'$ is described as follows:\\ +$F'(vk, \mathsf{U}_i, \mathsf{u}_i, (i, z_0, z_i), w_i, \overline{T}) \rightarrow x$:\\ +if $i=0$, output $H(vk, 1, z_0, F(z_0, w_i), \mathsf{u}_{\bot})$\\ +otherwise +\begin{enumerate} + \item check $\mathsf{u}_i.x = H(vk, i, z_0, z_i, \mathsf{U}_i)$ + \item check $(\mathsf{u}_i.\overline{E}, \mathsf{u}_i.u) = (\mathsf{u}_{\bot}.\overline{E}, 1)$ + \item compute $\mathsf{U}_{i+1} \leftarrow NIFS.V(vk, U, u, \overline{T})$ + \item output $H(vk, i+1, z_0, F(z_i, w_i), \mathsf{U}_{i+1})$ +\end{enumerate} + +% TODO add diagram + +\paragraph{IVC Proof} +iteration $i+1$: prover runs $F'$ and computes $\mathsf{u}_{i+1},~ \mathsf{U}_{i+1}$, with corresponding witnesses $\mathsf{w}_{i+1},~ \mathsf{W}_{i+1}$. +$(\mathsf{u}_{i+1},~ \mathsf{U}_{i+1})$ attest correctness of $i+1$ invocations of $F'$, the IVC proof is $\pi_{i+1} = ( (\mathsf{U}_{i+1}, \mathsf{W}_{i+1}), (\mathsf{u}_{i+1}, \mathsf{w}_{i+1}))$. + + +\vspace{0.5cm} + +\underline{$P(pk, (i, z_0, z_i), \mathsf{w}_i, \pi_i) \rightarrow \pi_{i+1}$}:\\ +Parse $\pi_i = ( (\mathsf{U}_i, \mathsf{W}_i), (\mathsf{u}_i, \mathsf{w}_i))$, then +\begin{enumerate} + \item if $i=0$: $(\mathsf{U}_{i+1}, \mathsf{W}_{i+1}, \overline{T}) \leftarrow (\mathsf{u}_{\perp}, \mathsf{w}_{\perp}, \mathsf{u}_{\perp}.{\overline{E}})$\\ + otherwise: $(\mathsf{U}_{i+1}, \mathsf{W}_{i+1}, \overline{T}) \leftarrow NIFS.P(pk, (\mathsf{U}_i, \mathsf{W}_i), (\mathsf{u}_i, \mathsf{w}_i))$ + \item compute $(\mathsf{u}_{i+1}, \mathsf{w}_{i+1}) \leftarrow trace(F', (vk, \mathsf{U}_i, \mathsf{u}_i, (i, z_0, z_i), \mathsf{w}_i, \overline{T}))$ + \item output $\pi_{i+1} \leftarrow ((\mathsf{U}_{i+1}, \mathsf{W}_{i+1}), (\mathsf{u}_{i+1}, \mathsf{w}_{i+1}))$ +\end{enumerate} + +\underline{$V(vk, (i, z_0, z_i), \pi_i) \rightarrow \{0,1\}$}: +if $i=0$: check that $z_i=z_0$\\ +otherwise, parse $\pi_i = ( (\mathsf{U}_i, \mathsf{W}_i), (\mathsf{u}_i, \mathsf{w}_i))$, then +\begin{enumerate} + \item check $\mathsf{u}_i.x = H(vk, i, z_0, z_i, \mathsf{U}_i)$ + \item check $(\mathsf{u}_i.{\overline{E}}, \mathsf{u}_i.u) = (\mathsf{u}_{\perp}.{\overline{E}}, 1)$ + \item check that $\mathsf{W}_i,~ \mathsf{w}_i$ are satisfying witnesses to $\mathsf{U}_i,~ \mathsf{u}_i$ respectively +\end{enumerate} -\section{IVC proofs} -\textbf{WIP} +\paragraph{A zkSNARK of a Valid IVC Proof} \bibliography{paper-notes.bib} diff --git a/notes_spartan.pdf b/notes_spartan.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fe689f95b0a54a2d9401bdd67cee10cf26001940 GIT binary patch literal 221265 zcmce6V~}pml5X3!ZQHhcw{7=sd$(=d-fi1<@3w8*?)QG*%sCNr@12O5Kj*B7s;E_! 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\mathbb{F}^{log~m}$ to P + \item let $T_1=0,~ \mu_1=log~m,~ l_1=3$ + \item V: set $r_x \in^R \mathbb{F}^{\mu_1}$ + \item Sum-check 1. $e_x \leftarrow (\mu_1, l_1, T_1)$ + \item P: compute $v_A=\overline{A}(r_x),~ v_B=\overline{B}(r_x),~ v_C=\overline{C}(r_x)$, send $(v_A, v_B, v_C)$ to V + \item V: abort with $b=0$ if $e_x \neq (v_A \cdot v_B - v_C)\cdot \widetilde{eq}(r_x, \tau)$ + \item V: send $r_A, r_B, r_C \in^R \mathbb{F}$ to P + \item let $T_2 = r_A \cdot v_A + r_B \cdot v_B + r_C \cdot v_C,~ \mu_2=log~m,~ l_2=2$ + \item V: set $r_y \in^R \mathbb{F}^{\mu_2}$ + \item Sum-check 2. $e_y \leftarrow (\mu_2, l_2, T_2)$ + \item P: $v \leftarrow \widetilde{w}(r_y[1..])$, send $v$ to V + \item $b_e \leftarrow (pp, C, r_y, v, \mu_2)$ + \item V: abourt with $b=0$ if $b_e==0$ + \item V: $v_z \leftarrow (1 - r_y[0]) \cdot \widetilde{w}(r_y [1..]) + r_y[0] \widetilde{(io, 1)} (r_y[1..])$ + \item V: $v_1 \leftarrow \widetilde{A}(r_x, r_y),~ v_2 \leftarrow \widetilde{B}(r_x, r_y),~ v_3 \leftarrow \widetilde{C}(r_x, r_y)$ + \item V: abort with $b=0$ if $e_y \neq (r_A v_1 + r_B v_2 + r_C v_3) \cdot v_z$ + \item V: output $b=1$ + \end{enumerate} +\end{itemize} + +\vspace{2cm} +\framebox{WIP: covered until sec.6} + + + +\bibliography{paper-notes.bib} +\bibliographystyle{unsrt} + +\end{document} diff --git a/paper-notes.bib b/paper-notes.bib index 6827f61..0090122 100644 --- a/paper-notes.bib +++ b/paper-notes.bib @@ -100,3 +100,12 @@ note = {\url{https://github.com/arnaucube/fri-commitment}}, url = {https://github.com/arnaucube/fri-commitment} } + +@misc{cryptoeprint:2019/550, + author = {Srinath Setty}, + title = {Spartan: Efficient and general-purpose zkSNARKs without trusted setup}, + howpublished = {Cryptology ePrint Archive, Paper 2019/550}, + year = {2019}, + note = {\url{https://eprint.iacr.org/2019/550}}, + url = {https://eprint.iacr.org/2019/550} +}