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Pedersen commitments\\ + Let $g \in \mathbb{G}^n,~ v \in \mathbb{F}_r^n$,\\ + $$Com(v) = \langle g, v \rangle =g_1 \cdot v_1 + g_2 \cdot v_2 + \ldots + g_n \cdot v_n$$ + + % \pause + + RLC\\ + Let $v_1, v_2 \in \mathbb{F}_r^n$, set $cm_1 = Com(v_1),~ cm_2=Com(v_2)$. + \\then, + \begin{align*} + v_3 &= v_1 + r \cdot v_2\\ + cm_3 &=cm_1 + r \cdot cm_2 + \end{align*} + \\so that + $$cm_3 = Com(v_3)$$ + +\end{frame} + +\section[Folding]{Folding} +\begin{frame}{Relaxed R1CS} + R1CS instance: $(\{A, B, C\} \in \mathbb{F}^{n \times n},~ io,~ n,~ l)$, such that for $z=(io \in \mathbb{F}^l, 1, w \in \mathbb{F}^{n-l-1}) \in \mathbb{F}^n$, + +$$Az \circ Bz = Cz$$ + +% \pause + +Relaxed R1CS: + +$$Az \circ Bz = uCz + E$$ + +for $u \in \mathbb{F},~~ E \in \mathbb{F}^n$. + +\vspace{1cm} + +Committed Relaxed R1CS instance: $CI = (\overline{E}, u, \overline{W}, x)$\\ +Witness of the instance: $WI=(E, W)$ + + +\end{frame} + +\begin{frame}{NIFS - Non Interactive Folding Scheme} + \scriptsize{ + \begin{align*} + CI_1 &=(\overline{E}_1, u_1, \overline{W}_1, x_1) ~~~~~~WI_1=(E_1, W_1)\\ + CI_2 &=(\overline{E}_2, u_2, \overline{W}_2, x_2) ~~~~~~WI_2=(E_2, W_2) + \end{align*} + where $\overline{V}=Com(V)$ + + +% \pause + + \begin{align*} + T &= Az_1 \circ Bz_1 + Az_2 \circ Bz_2 - u_1 C z_1 - u_2 C z_2\\ + \overline{T}&=Com(T) + \end{align*} + % \pause + +\begin{minipage}[t]{.45\textwidth} + NIFS.P + \begin{align*} + E &= E_1 + r \cdot T + r^2 \cdot E_2\\ + W &= W_1 + r \cdot W + \end{align*} +\end{minipage} +\hfill\vline\hfill +\begin{minipage}[t]{.45\textwidth} + NIFS.V + \begin{align*} + \overline{E} &= \overline{E}_1 + r \cdot \overline{T} + r^2 \cdot \overline{E}_2\\ + u &= u_1 + r \cdot u_2\\ + \overline{W} &= \overline{W}_1 + r \cdot \overline{W}\\ + x &= x_1 + r \cdot x_2 + \end{align*} +\end{minipage} + +New folded Committed Instance: $(\overline{E}, u, \overline{W}, x)$\\ +New folded witness: $(E, W)$ +} +\end{frame} + +\begin{frame}{IVC} + \small{ + $U_i$: committed instance for the correct execution of invocations $1, \ldots, i-1$ of $F'$\\ + $u_i$: committed instance for the correct execution of invocation $i$ of $F'$ + } + + % draw: sketch of the Augmented F Circuit + % big box for F', inside small box for F. NIFS.V box, how things connect to next iteration + + \vspace{4cm} + + \small{ + F':\\ + i) execute a step of the incremental computation, $z_i+1 = F(z_i)$\\ + ii) invoke the NIFS.V to fold $U_i, u_i$ into $U_{i+1}$\\ + iii) other checks to ensure that the IVC is done properly + } +\end{frame} + +\begin{frame}{Cycle of curves} + \small{ + NIFS.V involves $\mathbb{G}$ point scalar mults, which are not native over $\mathbb{F}_r$. + \\$\longrightarrow$ delegate them into a circuit over a 2nd curve. + + \vspace{0.3cm} + + We 'mirror' the main $F'$ circuit into the 2nd curve\\ + each circuit computes natively the point operations of the other curve + } + + % draw: + % 1st the Nova with duplicated F' circuits over 2 curves + % 2nd the Nova with CycleFold circuits sketch +\end{frame} + + +\begin{frame}{Augmented F Circuit + CycleFold Circuit} + \includegraphics[width=\textwidth]{cyclefold-nova-diagram} +\end{frame} + +\begin{frame}{Other Folding Schemes} + % TODO + % HyperNova + % ProtoGalaxy + % ProtoStar + % LatticeFold + % etc + % mention a bit the different characteristics and folding techniques +\end{frame} + +\section{Decider (Final Proof)} + +\begin{frame}{Decider} + \includegraphics[width=\textwidth]{cyclefold-paper-diagram} + + With Prover knowing the respective witnesses for $U_n, u_n, U_{EC,n}$ + + \vspace{1cm} + + Issue: IVC proof is not succinct +\end{frame} + +\begin{frame}{Decider} + Original Nova: generate a zkSNARK proof with Spartan for $U_n, u_n, U_{EC, n}$\\ + $\longrightarrow$ 2 Spartan proofs, one on each curve (with CycleFold is 1 Spartan proof)\\ + (not EVM-friendly) + + % draw of the 2 circuits over the curves, and how we generate a Spartan proof for each one + +\end{frame} + +\begin{frame}{Decider} + checks (simplified) + \begin{enumerate} + \item $(U_{n+1}, W_{n+1})$ satisfy Relaxed R1CS relation of AugmentedFCircuit + \item verify commitments of $U_{n+1}.\{\overline{E}, \overline{W}\}$ w.r.t. $W_{n+1}.\{E,W\}$ + \item $(U_{EC,n}, W_{EC,n})$ satisfy Relaxed R1CS relation of CycleFoldCircuit + \item verify commitments of $U_{EC,n}.\{\overline{E}, \overline{W}\}$ w.r.t. $W_{EC,n}.\{E,W\}$ + \item $u_n.E==0,~ u_n.u==1$, ie. $u_n$ is a fresh not-relaxed instance + \item $u_n.x_0==H(n, z_0, z_n, U_n)$\\ + $u_n.x_1==H(U_{EC,n})$ + \item $NIFS.V(U_n, u_n)==U_{n+1}$ + \end{enumerate} + + % by draw show which are native and not native + % and that the NIFS.V we do it in Solidity +\end{frame} + +\begin{frame}{Decider} + \includegraphics[width=\textwidth]{decider-onchain-flow-diagram} + % draw of the full flow: from inputting the circuit, to folding to generating the Decider proof to verifying in Ethereum +\end{frame} + +\section{Sonobe} +\begin{frame}{Sonobe} + \footnotesize{ + Experimental folding schemes library implemented jointly by 0xPARC and PSE. + + \vspace{0.3cm} + + Dev flow: + \begin{enumerate} + \item Define a circuit to be folded + \item Set which folding scheme to be used (eg. Nova with CycleFold) + \item Set a final decider to generate the final proof (eg. Spartan over Pasta curves) + \item Generate the the decider verifier + \end{enumerate} + } + + \vspace{1cm} + + \includegraphics[width=\textwidth]{sonobe-lib-pipeline} +\end{frame} + +\begin{frame}{Code example} + +\end{frame} + + +\begin{frame} +\frametitle{Wrappup} +\begin{itemize} + \item \href{https://github.com/privacy-scaling-explorations/sonobe}{https://github.com/privacy-scaling-explorations/sonobe} + \item \href{https://privacy-scaling-explorations.github.io/sonobe-docs/}{https://privacy-scaling-explorations.github.io/sonobe-docs/} +\end{itemize} + + \includegraphics[width=4cm]{qr-sonobe-repo-link} + +\tiny{ + $$\text{2024-04-22}$$ + $$\text{\href{https://0xparc.org}{0xPARC}~\&~\href{https://pse.dev/}{PSE.}}$$ +} +\end{frame} + +\end{document}

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