1\documentclass{article}
  2\usepackage[utf8]{inputenc}
  3\usepackage{amsfonts}
  4\usepackage{amsthm}
  5\usepackage{amsmath}
  6\usepackage{mathtools}
  7\usepackage{enumerate}
  8\usepackage{hyperref}
  9\usepackage{xcolor}
 10\usepackage{pgf-umlsd} % diagrams
 11\usepackage{centernot}
 12\usepackage{algorithm}
 13\usepackage{algpseudocode}
 14
 15
 16% prevent warnings of underfull \hbox:
 17\usepackage{etoolbox}
 18\apptocmd{\sloppy}{\hbadness 4000\relax}{}{}
 19
 20\theoremstyle{definition}
 21\newtheorem{definition}{Def}[section]
 22\newtheorem{theorem}[definition]{Thm}
 23
 24% custom lemma environment to set custom numbers
 25\newtheorem{innerlemma}{Lemma}
 26\newenvironment{lemma}[1]
 27{\renewcommand\theinnerlemma{#1}\innerlemma}
 28{\endinnerlemma}
 29
 30
 31\title{Notes on HyperNova}
 32\author{arnaucube}
 33\date{May 2023}
 34
 35\begin{document}
 36
 37\maketitle
 38
 39\begin{abstract}
 40	Notes taken while reading about HyperNova \cite{cryptoeprint:2023/573} and CCS\cite{cryptoeprint:2023/552}.
 41
 42	Usually while reading papers I take handwritten notes, this document contains some of them re-written to $LaTeX$.
 43
 44	The notes are not complete, don't include all the steps neither all the proofs.
 45\end{abstract}
 46
 47\tableofcontents
 48
 49
 50\section{CCS}
 51\subsection{R1CS to CCS overview}
 52
 53\begin{description}
 54	\item[R1CS instance] $S_{R1CS} = (m, n, N, l, A, B, C)$\\
 55		where $m, n$ are such that $A \in \mathbb{F}^{m \times n}$, and $l$ such that the public inputs $x \in \mathbb{F}^l$. Also $z=(w, 1, x) \in \mathbb{F}^n$, thus $w \in \mathbb{F}^{n-l-1}$.
 56	\item[CCS instance] $S_{CCS} = (m, n, N, l, t, q, d, M, S, c)$\\
 57		where we have the same parameters than in $S_{R1CS}$, but additionally:\\
 58		$t=|M|$, $q = |c| = |S|$, $d$= max degree in each variable.
 59	\item[R1CS-to-CCS parameters] $n=n,~ m=m,~ N=N,~ l=l,~ t=3,~ q=2,~ d=2$, $M=\{A,B,C\}$, $S=\{\{0,~1\},~ \{2\}\}$, $c=\{1,-1\}$
 60\end{description}
 61
 62The CCS relation check:
 63$$\sum_{i=0}^{q-1} c_i \cdot \bigcirc_{j \in S_i} M_j \cdot z ==0$$
 64
 65where $z=(w, 1, x) \in \mathbb{F}^n$.
 66
 67In our R1CS-to-CCS parameters is equivalent to
 68
 69\begin{align*}
 70	&c_0 \cdot ( (M_0 z) \circ (M_1 z) ) + c_1 \cdot (M_2 z) ==0\\
 71	\Longrightarrow &1 \cdot ( (A z) \circ (B z) ) + (-1) \cdot (C z) ==0\\
 72	\Longrightarrow &( (A z) \circ (B z) ) - (C z) ==0
 73\end{align*}
 74
 75which is equivalent to the R1CS relation: $Az \circ Bz == Cz$
 76
 77An example of the conversion from R1CS to CCS implemented in SageMath can be found at\\
 78\href{https://github.com/arnaucube/math/blob/master/r1cs-ccs.sage}{https://github.com/arnaucube/math/blob/master/r1cs-ccs.sage}.
 79
 80Similar relations between Plonkish and AIR arithmetizations to CCS are shown in the CCS paper \cite{cryptoeprint:2023/552}, but for now with the R1CS we have enough to see the CCS generalization idea and to use it for the HyperNova scheme.
 81
 82\subsection{Committed CCS}
 83$R_{CCCS}$ instance: $(C, \mathsf{x})$, where $C$ is a commitment to a multilinear polynomial in $s'-1$ variables.
 84
 85Sat if:
 86\begin{enumerate}[i.]
 87	\item $\text{Commit}(pp, \widetilde{w}) = C$
 88	\item $\sum_{i=1}^q c_i \cdot \left( \prod_{j \in S_i} \left( \sum_{y \in \{0,1\}^{\log m}} \widetilde{M}_j(x, y) \cdot \widetilde{z}(y) \right) \right)$\\
 89		where $\widetilde{z}(y) = \widetilde{(w, 1, \mathsf{x})}(x) ~\forall x \in \{0, 1\}^{s'}$
 90\end{enumerate}
 91
 92
 93\subsection{Linearized Committed CCS}
 94$R_{LCCCS}$ 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 $u \in \mathbb{F},~ \mathsf{x} \in \mathbb{F}^l,~ r \in \mathbb{F}^s,~ v_i \in \mathbb{F} ~\forall i \in [t]$.
 95
 96Sat if:
 97\begin{enumerate}[i.]
 98	\item $\text{Commit}(pp, \widetilde{w}) = C$
 99	\item $\forall i \in [t],~ v_i = \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_i(r, y) \cdot \widetilde{z}(y)$\\
100		where $\widetilde{z}(y) = \widetilde{(w, u, \mathsf{x})}(x) ~\forall x \in \{0, 1\}^{s'}$
101\end{enumerate}
102
103
104\section{Multifolding Scheme for CCS}
105Recall sum-check protocol notation: \underline{$C \leftarrow \langle P, V(r) \rangle (g, l, d, T)$} means
106$$T=\sum_{x_1 \in \{0,1\}} \sum_{x_2 \in \{0,1\}} \cdots \sum_{x_l \in \{0,1\}} g(x_1, x_2, \ldots, x_l)$$
107where $g$ is a $l$-variate polynomial, with degree at most $d$ in each variable, and $T$ is the claimed value.
108
109\vspace{1cm}
110
111Let $s= \log m,~ s'= \log n$.
112
113\begin{enumerate}
114	\item $V \rightarrow P: \gamma \in^R \mathbb{F},~ \beta \in^R \mathbb{F}^s$
115	\item $V: r_x' \in^R \mathbb{F}^s$
116	\item $V \leftrightarrow P$: sum-check protocol:
117		$$c \leftarrow \langle P, V(r_x') \rangle (g, s, d+1, \underbrace{\sum_{j \in [t]} \gamma^j \cdot v_j}_\text{T})$$
118		(in fact, $T=(\sum_{j \in [t]} \gamma^j \cdot v_j) \underbrace{+ \gamma^{t+1} \cdot Q(x)}_{=0}) = \sum_{j \in [t]} \gamma^j \cdot v_j$)\\
119		where:
120		\begin{align*}
121			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}\\
122			\text{for LCCCS:}~ L_j(x) &:= \widetilde{eq}(r_x, x) \cdot \left(
123				\underbrace{\sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_1(y)}_\text{this is the check from LCCCS}
124			\right)\\
125				\text{for CCCS:}~ Q(x) := &\widetilde{eq}(\beta, x) \cdot \left(
126				\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{this is the check from CCCS}
127			\right)
128		\end{align*}
129		Notice that
130		$$v_j= \sum_{y\in \{0,1\}^{s'}} \widetilde{M}_j(r, y) \cdot \widetilde{z}(y) = \sum_{x\in \{0,1\}^s} L_j(x)$$
131	\item $P \rightarrow V$: $\left( (\sigma_1, \ldots, \sigma_t), (\theta_1, \ldots, \theta_t) \right)$, where $\forall j \in [t]$,
132		$$\sigma_j = \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \widetilde{z}_1(y)$$
133		$$\theta_j = \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \widetilde{z}_2(y)$$
134		where $\sigma_j,~\theta_j$ are the checks from LCCCS and CCCS respectively with $x=r_x'$.
135	\item V: $e_1 \leftarrow \widetilde{eq}(r_x, r_x')$, $e_2 \leftarrow \widetilde{eq}(\beta, r_x')$\\
136		check:
137		$$c = \left( \sum_{j \in [t]} \gamma^j e_1 \sigma_j + \gamma^{t+1} e_2 \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \sigma \right) \right)$$
138		which should be equivalent to the $g(x)$ computed by $V,P$ in the sum-check protocol.
139	\item $V \rightarrow P: \rho \in^R \mathbb{F}$
140	\item $V, P$: output the folded LCCCS instance $(C', u', \mathsf{x}', r_x', v_1', \ldots, v_t')$, where $\forall i \in [t]$:
141		\begin{align*}
142			C' &\leftarrow C_1 + \rho \cdot C_2\\
143			u' &\leftarrow u + \rho \cdot 1\\
144			\mathsf{x}' &\leftarrow \mathsf{x}_1 + \rho \cdot \mathsf{x}_2\\
145			v_i' &\leftarrow \sigma_i + \rho \cdot \theta_i
146		\end{align*}
147	\item $P$: output folded witness: $\widetilde{w}' \leftarrow \widetilde{w}_1 + \rho \cdot \widetilde{w}_2$.
148\end{enumerate}
149
150\vspace{1cm}
151
152Now, to see the verifier check from step 5, observe that in LCCCS, since $\widetilde{w}$ satisfies,
153\begin{align*}
154	v_j &= \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(r_x, y) \cdot \widetilde{z}_1(y)\\
155	    &= \sum_{x \in \{0,1\}^s}
156		\underbrace{
157			\widetilde{eq}(r_x, x) \cdot \left( \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_1(y) \right)
158		}_{L_j(x)}\\
159	    &= \sum_{x \in \{0,1\}^s} L_j(x)
160\end{align*}
161
162Observe also that in CCCS, since $\widetilde{w}$ satisfies,
163$$
1640=\sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \left( \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_2(y) \right)
165$$
166for $\beta$,
167\begin{align*}
168	0&=\sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \left( \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(\beta, y) \cdot \widetilde{z}_2(y) \right)\\
169	 &= \sum_{x \in \{0,1\}^s}
170	 \underbrace{\widetilde{eq}(\beta , x) \cdot
171		\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)
172	}_{Q(x)}\\
173	 &= \sum_{x \in \{0,1\}^s} Q(x)
174\end{align*}
175
176Then we can see that
177
178\begin{align*}
179	c &= g(r_x')\\
180	  &= \left( \sum_{j \in [t]} \gamma^j \cdot L_j(r_x') \right) + \gamma^{t+1} \cdot Q(r_x')\\
181	  &= \left( \sum_{j \in [t]} \gamma^j \cdot e_q \cdot \sigma_j \right) + \gamma^{t+1} \cdot e_2 \cdot \sum_{i \in [q]} c_i \prod_{j \in S_i} \theta_j
182\end{align*}
183
184where $e_1 = \widetilde{eq}(r_x, r_x')$ and $e_2=\widetilde{eq}(\beta, r_x')$.
185
186Which is the check that $V$ performs at step $5$.
187
188
189
190%%%%%% APPENDIX
191\appendix
192\section{Appendix: Some details}
193This appendix contains some notes on things that don't specifically appear in the paper, but that would be needed in a practical implementation of the scheme.
194
195\subsection{Matrix and Vector to Sparse Multilinear Extension}
196
197Let $M \in \mathbb{F}^{m \times n}$ be a matrix. We want to compute its MLE
198$$\widetilde{M}(x_1, \ldots, x_l) = \sum_{e \in \{0, 1 \}^l} M(e) \cdot \widetilde{eq}(x, e)$$
199
200We can view the matrix $M \in \mathbb{F}^{m \times n}$ as a function with the following signature:
201$$M(\cdot): \{0,1\}^s \times \{0,1\}^{s'} \rightarrow \mathbb{F}$$
202where $s = \lceil \log m \rceil,~ s' = \lceil \log n \rceil$.
203
204An entry in $M$ can be accessed with a $(s+s')$-bit identifier.
205
206eg.:
207$$
208M = \begin{pmatrix}
2091 & 2 & 3\\
2104 & 5 & 6\\
211\end{pmatrix}
212\in \mathbb{F}^{3 \times 2}
213$$
214
215$m = 3,~ n = 2,~~~ s = \lceil \log 3 \rceil = 2,~ s' = \lceil \log 2 \rceil = 1$
216
217So, $M(x, y) = x$, where $x \in \{0,1\}^s,~ y \in \{0,1\}^{s'},~ x \in \mathbb{F}$
218
219$$
220M = \begin{pmatrix}
221M(00,0) & M(01,0) & M(10,0)\\
222M(00,1) & M(01,1) & M(10,1)\\
223\end{pmatrix}
224\in \mathbb{F}^{3 \times 2}
225$$
226
227This logic can be defined as follows:
228
229\begin{algorithm}[H]
230\caption{Generating a Sparse Multilinear Polynomial from a matrix}
231\begin{algorithmic}
232	\State set empty vector $v \in (\text{index:}~ \mathbb{Z}, x: \mathbb{F})^{s \times s'}$
233	\For {$i$ to $m$}
234	\For {$j$ to $n$}
235		\If {$M_{i,j} \neq 0$}
236			\State $v.\text{append}( \{ \text{index}: i \cdot n + j,~ x: M_{i,j} \} )$
237		\EndIf
238	\EndFor
239	\EndFor
240	\State return $v$    \Comment {$v$ represents the evaluations of the polynomial}
241\end{algorithmic}
242\end{algorithm}
243
244Once we have the polynomial, its MLE comes from
245$$\widetilde{M}(x_1, \ldots, x_{s+s'}) = \sum_{e \in \{0,1\}^{s+s'}} M(e) \cdot \widetilde{eq}(x, e)$$
246
247$$M(X) \in \mathbb{F}[X_1, \ldots, X_s]$$
248
249\paragraph{Multilinear extensions of vectors}
250Given a vector $u \in \mathbb{F}^m$, the polynomial $\widetilde{u}$ is the MLE of $u$, and is obtained by viewing $u$ as a function mapping ($s=\log m$)
251$$u(x): \{0,1\}^s \rightarrow \mathbb{F}$$
252$\widetilde{u}(x, e)$ is the multilinear extension of the function $u(x)$
253$$\widetilde{u}(x_1, \ldots, x_s) = \sum_{e \in \{0,1\}^s} u(e) \cdot \widetilde{eq}(x, e)$$
254
255\bibliography{paper-notes.bib}
256\bibliographystyle{unsrt}
257
258\end{document}