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%Information to be included in the title page:
\title{HyperNova's multifolding overview}
\author{}
\date{\scriptsize{2023-06-22\\\href{https://0xparc.org}{0xPARC} Novi team}}

\begin{document}

\frame{\titlepage}

\section[Overview]{Overview}

\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}



\begin{frame}{Multifolding - Overview}
  \begin{tiny}
\begin{center}
	\begin{sequencediagram}
		\newinst[1]{p}{Prover}
		\newinst[3]{v}{Verifier}

		\bloodymess[1]{v}{$\gamma,~\beta,~r_x'$}{p}{L}{
			\shortstack{
				$\gamma \in \mathbb{F},~ \beta \in \mathbb{F}^s$\\
				$r_x' \in \mathbb{F}^s$
			}
		}{}
		\bloodymess[1]{p}{$c,~ \pi_{SC}$}{v}{R}{sum-check prove}{sum-check verify}
		\bloodymess[1]{p}{$\{\sigma_j\},~\{\theta_j\}$}{v}{R}{compute $\{\sigma_j\}, \{\theta_j\}~ \forall j \in [t]$}{verify $c$ with $\{\sigma_j\}, \{\theta_j\}$ relation}
		\bloodymess[1]{v}{$\rho$}{p}{L}{$\rho \in^R \mathbb{F}$}{}
		\callself[0]{p}{fold LCCCS instance}{p}
		\prelevel
		\callself[0]{v}{fold LCCCS instance}{v}
		\callself[0]{p}{fold $\widetilde{w}, r_w$}{p}
	\end{sequencediagram}
\end{center}
  \end{tiny}
\end{frame}


\section[Checks]{Checks}

\begin{tiny}
\begin{frame}{LCCCS checks}
$$
\color{gray}{g(x) :=}
\color{black}{\underbrace{\left( \sum_{j \in [t]} \gamma^j \cdot L_j(x) \right)}_\text{LCCCS} }
\color{gray}{+ \gamma^{t+1} \cdot Q(x)}
$$
$$
L_j(x) := \widetilde{eq}(r_x, x) \cdot \left(
\underbrace{\sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_1(y)}_\text{LCCCS check}
\right)
$$


Notice that, $v_j$ from LCCCS relation check

\begin{align*}
v_j &= \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(r_x, y) \cdot \widetilde{z}_1(y)\\
    &= \sum_{x \in \{0,1\}^s}
	  \widetilde{eq}(r_x, x) \cdot \left( \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_1(y) \right)\\
    &= \sum_{x \in \{0,1\}^s} L_j(x)
\end{align*}
  
\end{frame}
\begin{frame}{CCCS checks}
  $$
  \color{gray}{g(x) := \left( \sum_{j \in [t]} \gamma^j \cdot L_j(x) \right) +}
  \color{black}{\underbrace{\gamma^{t+1} \cdot Q(x)}_\text{CCCS}}
  $$
$$Q(x) := \widetilde{eq}(\beta, x) \cdot \left(
                    \underbrace{ \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \left( \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_2(y) \right) }_\text{CCCS check}
            \right)$$
  
  Recall that Spartan's $\widetilde{F}_{io}(x)$ here is $q(x)$, so we're doing the same Spartan check:
  
$$
0 =G(\beta) = \sum_{x \in \{0,1\}^s} Q(x) = \sum_{x \in \{0,1\}^s} eq(\beta, x) \cdot q(x)$$
$$= \sum_{x \in \{0,1\}^s}
 \underbrace{\widetilde{eq}(\beta , x) \cdot
         \overbrace{
                \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \left( \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_2(y) \right)
        }^{q(x)}
}_{Q(x)}
$$

\end{frame}


\begin{frame}{Verifier checks}
  \textcolor{gray}{
Recall:
$$g(x) := \left( \sum_{j \in [t]} \gamma^j \cdot L_j(x) \right) + \gamma^{t+1} \cdot Q(x)$$
$$c = \left(\sum_{j \in [t]} \gamma^j \cdot e_1 \cdot \sigma_j \right) + \gamma^{t+1} \cdot e_2 \cdot \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \theta_j \right)$$
}

We can see now that V's check in step 5,

\begin{align*}
  c &= 
\left( \sum_{j \in [t]} \gamma^j \cdot \overbrace{e_1 \cdot \sigma_j}^{L_j(r_x')} \right) + \gamma^{t+1} \cdot \overbrace{e_2 \cdot \sum_{i \in [q]} c_i \prod_{j \in S_i} \theta_j}^{Q(x)}\\
&= \left( \sum_{j \in [t]} \gamma^j \cdot L_j(r_x') \right) + \gamma^{t+1} \cdot Q(r_x')\\
&= g(r_x')
\end{align*}

where $e_1 = \widetilde{eq}(r_x, r_x')$, $e_2=\widetilde{eq}(\beta, r_x')$.
\end{frame}
\end{tiny}

\section[Multiple instances]{Multiple instances}

\begin{footnotesize}
\begin{frame}{Multifolding multiple instances}
Hypernova paper: $\mu=1, \nu=1$ \emph{(ie. 1 LCCCS instance and 1 CCCS instance)}

\vspace{1cm}
In next slides
\begin{itemize}
  \item example with: $\color{orange}{LCCCS: \mu = 2},~ \color{blue}{CCCS: \nu = 2}$
  \item generalized equations for $\mu,~\nu$
\end{itemize}

Let $z_1,~ \color{orange}{z_2}$ be the two LCCCS instances, and $z_3,~ \color{blue}{z_4}$ be the two CCCS instances

\end{frame}
\end{footnotesize}

\begin{tiny}
\begin{frame}
  In \emph{step 3},

\begin{align*}
	g(x) &:= \left( \sum_{j \in [t]} \gamma^j \cdot L_{1,j}(x) + \textcolor{orange}{\gamma^{t+j} \cdot L_{2,j}(x)} \right)
+ \gamma^{2t+1} \cdot Q_1(x) + \textcolor{cyan}{\gamma^{2t+2} \cdot Q_2(x)} \\
	     &L_{1,j}(x) := \widetilde{eq}(r_{1,x}, x) \cdot \left(
	\sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_1(y)
\right)\\
	     &\textcolor{orange}{L_{2,j}(x)} := \widetilde{eq}(\textcolor{orange}{r_{2,x}}, x) \cdot \left(
	\sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(x, y) \cdot \textcolor{orange}{\widetilde{z}_2(y)}
\right)\\
	     &Q_1(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}_3(y) \right)\right)\\
	     &\textcolor{cyan}{Q_2(x)} := \widetilde{eq}(\textcolor{cyan}{\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 \textcolor{cyan}{\widetilde{z}_4(y)} \right)\right)
\end{align*}

\framebox{\begin{minipage}{4.3 in}
A generic definition of $g(x)$ for $\mu>1~\nu>1$, would be

$$
g(x) := \left( \sum_{i \in [\mu]} \left( \sum_{j \in [t]} \gamma^{i \cdot t+j} \cdot L_{i,j}(x) \right) \right)
+ \left( \sum_{i \in [\nu]} \gamma^{\mu \cdot t + i} \cdot Q_i(x) \right)
$$
\end{minipage}}

Recall, the original $g(x)$ definition was
$$\textcolor{gray}{g(x) := \left( \sum_{j \in [t]} \gamma^j \cdot L_j(x) \right) + \gamma^{t+1} \cdot Q(x)}$$


\end{frame}

\begin{frame}
  In \emph{step 4}, $P \rightarrow V$: $(\{\sigma_{1,j}\}, \textcolor{orange}{\{\sigma_{2,j}\}}, \{\theta_{1,j}\}, \textcolor{cyan}{\{\theta_{2,j}\}}),~ \text{where} ~\forall j \in [t]$,

$$\sigma_{1,j} = \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \widetilde{z}_1(y)$$
$$\textcolor{orange}{\sigma_{2,j}} = \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \textcolor{orange}{\widetilde{z}_2(y)}$$
$$\theta_{1,j} = \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \widetilde{z}_3(y)$$
$$\textcolor{cyan}{\theta_{2,j}} = \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \textcolor{cyan}{\widetilde{z}_4(y)}$$

\framebox{\begin{minipage}{4.3 in}
	so in a generic way,\\
$P \rightarrow V$:
$(\{\sigma_{i,j}\}, \{\theta_{k,j}\}),~ \text{where} ~\forall~ j \in [t],~ \forall~ i \in [\mu],~ \forall~ k \in [\nu]$
where
$$\sigma_{i,j} = \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \widetilde{z}_i(y)$$
$$\theta_{k,j} = \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \widetilde{z}_{\mu+k}(y)$$
\end{minipage}}

\end{frame}

\begin{frame}
  And in \emph{step 5}, $V$ checks

\begin{align*}
	c &= \left(\sum_{j \in [t]} \gamma^j \cdot e_1 \cdot \sigma_{1,j}
~\textcolor{orange}{+ \gamma^{t+j} \cdot e_2 \cdot \sigma_{2,j}}\right)
	  + \gamma^{2t+1} \cdot e_3 \cdot \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \theta_j \right)
	  + \textcolor{cyan}{\gamma^{2t+2} \cdot e_4 \cdot \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \theta_j \right)} 
\end{align*}

where
$e_1 \leftarrow \widetilde{eq}(r_{1,x}, r_x'),~ e_2 \leftarrow \widetilde{eq}(r_{2,x}, r_x')$, $e_3, e_4 \leftarrow \widetilde{eq}(\beta, r_x')$.

\vspace{0.5cm}

\framebox{\begin{minipage}{4.3 in}
A generic definition of the check would be
$$
c = \sum_{i \in [\mu]} \left(\sum_{j \in [t]} \gamma^{i \cdot t + j} \cdot e_i \cdot \sigma_{i,j} \right) \\
+ \sum_{k \in [\nu]} \gamma^{\mu \cdot t+k} \cdot e_k \cdot \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \theta_{k,j} \right)
$$
\end{minipage}}

where the original check was\\
$\textcolor{gray}{c = \left(\sum_{j \in [t]} \gamma^j \cdot e_1 \cdot \sigma_j \right) + \gamma^{t+1} \cdot e_2 \cdot \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \theta_j \right)}$

\end{frame}

\begin{frame}
And for the \emph{step 7},
\begin{align*}
	C' &\leftarrow C_1 + \rho \cdot C_2 + \rho^2 C_3 + \rho^3 C_4 + \ldots = \sum_{i \in [\mu + \nu]} \rho^i \cdot C_i \\
	u' &\leftarrow \sum_{i \in [\mu]} \rho^i \cdot u_i + \sum_{i \in [\nu]} \rho^{\mu + i-1} \cdot 1\\
	\mathsf{x}' &\leftarrow \sum_{i \in [\mu+\nu]} \rho^i \cdot \mathsf{x}_i\\
	v_i' &\leftarrow \sum_{i \in [\mu]} \rho^i \cdot \sigma_i + \sum_{i \in [\nu]} \rho^{\mu + i-1} \cdot \theta_i\\
\end{align*}

and \emph{step 8},
\begin{align*}
	\widetilde{w}' &\leftarrow \sum_{i \in [\mu+\nu]} \rho^i\cdot \widetilde{w}_i\\
	r_w' &\leftarrow \sum_{i \in [\mu+\nu]} \rho^i \cdot r_{w_i}\\
\end{align*}

\end{frame}

\end{tiny}

\section[Wrappup]{Wrappup}

\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}
\end{itemize}

\vspace{2cm}
\tiny{
  $$\text{2023-06-22}$$
  $$\text{\href{https://0xparc.org}{0xPARC} Novi team}$$
}
\end{frame}

\end{document}