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\title{Notes on HyperNova}
\author{arnaucube}
\date{May 2023}

\begin{document}

\maketitle

\begin{abstract}
	Notes taken while reading about Spartan \cite{cryptoeprint:2023/573}, \cite{cryptoeprint:2023/552}.

	Usually while reading papers 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.
\end{abstract}

\tableofcontents


\section{CCS}
\subsection{R1CS to CCS overview}

\begin{itemize}
	\item[] R1CS instance: $S_{R1CS} = (m, n, N, l, A, B, C)$
	\item[] CCS instance: $S_{CCS} = (m, n, N, l, t, q, d, M, S, c)$
	\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{itemize}

Then, we can see that the CCS relation:
$$\sum_{i=0}^{q-1} c_i \cdot \bigcirc_{j \in S_i} M_j \cdot z ==0$$

where $z=(w, 1, x) \in \mathbb{F}^n$.

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*}

which is equivalent to the R1CS relation: $Az \circ Bz == Cz$

An example of the conversion from R1CS to CCS implemented in SageMath can be found at\\
\href{https://github.com/arnaucube/math/blob/master/r1cs-ccs.sage}{https://github.com/arnaucube/math/blob/master/r1cs-ccs.sage}.

\subsection{Committed CCS}
$R_{CCCS}$ instance: $(C, \mathsf{x})$, where $C$ is a commitment to a multilinear polynomial in $s'-1$ variables.

Sat if:
\begin{enumerate}[i.]
	\item $\text{Commit}(pp, \widetilde{w}) = C$
	\item $\sum_{i=1}^q c_i \cdot \left( \prod_{j \in S_i} \left( \sum_{y \in \{0,1\}^{\log m}} \widetilde{M}_j(x, y) \cdot \widetilde{z}(y) \right) \right)$\\
		where $\widetilde{z}(y) = \widetilde{(w, 1, \mathsf{x})}(x) ~\forall x \in \{0, 1\}^{s'}$
\end{enumerate}


\subsection{Linearized Committed CCS}
$R_{LCCCS}$ instance: $(C, u, \mathsf{x}, r, v_1, \ldots, v_t)$, where $C$ is a commitment to a multilinear polynomial in $s'-1$ variables, and $u \in \mathbb{F},~ \mathsf{x} \in \mathbb{F}^l,~ r \in \mathbb{F}^s,~ v_i \in \mathbb{F} ~\forall i \in [t]$.

Sat if:
\begin{enumerate}[i.]
	\item $\text{Commit}(pp, \widetilde{w}) = C$
	\item $\forall i \in [t],~ v_i = \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_i(r, y) \cdot \widetilde{z}(y)$\\
		where $\widetilde{z}(y) = \widetilde{(w, u, \mathsf{x})}(x) ~\forall x \in \{0, 1\}^{s'}$
\end{enumerate}


\section{Multifolding Scheme for CCS}
Recall sum-check protocol:\\
\underline{$C \leftarrow <P, V(r)>(g, l, d, T)$}:\\ % 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.

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:\\
		$$c \leftarrow <P, V(r_x')>(g, s, d+1, \sum_{j \in [t]} \gamma^j \cdot v_j)$$
		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)
		\end{align*}
	\item $P \rightarrow V$: $\left( (\sigma_1, \ldots, \sigma_t), (\theta_1, \ldots, \theta_t) \right)$
		where
		$$\sigma_j = \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_1(y)$$
		$$\theta_j = \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_2(y)$$
	\item V: $e_1 \leftarrow \widetilde{eq}(r_x, r_x')$, $e_2 \leftarrow \widetilde{eq}(\beta, r_x')$\\
		check:
		$$c = \left( \sum_{j \in [t]} \gamma^j e_1 \sigma_j + \gamma^{t+1} e_2 \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \sigma \right) \right)$$
	\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*}
			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 $P$: output folded witness: $\widetilde{w}' \leftarrow \widetilde{w}_1 + \rho \cdot \widetilde{w}_2$.
\end{enumerate}


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