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hypernova: add multifolding diagram

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also add some more notes to spartan
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arnaucube
2023-05-21 23:37:09 +02:00
parent dc443a8c6a
commit 3ea610705a
7 changed files with 80 additions and 12 deletions
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57
notes_hypernova.tex
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@@ -7,11 +7,38 @@
\usepackage{enumerate}
\usepackage{hyperref}
\usepackage{xcolor}
\usepackage{pgf-umlsd} % diagrams
\usepackage{centernot}
\usepackage{algorithm}
\usepackage{algpseudocode}
\usepackage{pgf-umlsd} % diagrams
% message between threads. From https://tex.stackexchange.com/a/174765
% Example:
% \bloodymess[delay]{sender}{message content}{receiver}{DIR}{start note}{end note}
\newcommand{\bloodymess}[7][0]{
\stepcounter{seqlevel}
\path
(#2)+(0,-\theseqlevel*\unitfactor-0.7*\unitfactor) node (mess from) {};
\addtocounter{seqlevel}{#1}
\path
(#4)+(0,-\theseqlevel*\unitfactor-0.7*\unitfactor) node (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}
@@ -147,6 +174,30 @@ Let $s= \log m,~ s'= \log n$.
\item $P$: output folded witness: $\widetilde{w}' \leftarrow \widetilde{w}_1 + \rho \cdot \widetilde{w}_2$.
\end{enumerate}
\vspace{1cm}
Multifolding flow:
\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 $\widetilde{w}$}{p}
\end{sequencediagram}
\end{center}
\vspace{1cm}
Now, to see the verifier check from step 5, observe that in LCCCS, since $\widetilde{w}$ satisfies,
@@ -178,7 +229,7 @@ Then we can see that
\begin{align*}
c &= g(r_x')\\
&= \left( \sum_{j \in [t]} \gamma^j \cdot L_j(r_x') \right) + \gamma^{t+1} \cdot Q(r_x')\\
&= \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
&= \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)}
\end{align*}
where $e_1 = \widetilde{eq}(r_x, r_x')$ and $e_2=\widetilde{eq}(\beta, r_x')$.
@@ -229,7 +280,7 @@ This logic can be defined as follows:
\begin{algorithm}[H]
\caption{Generating a Sparse Multilinear Polynomial from a matrix}
\begin{algorithmic}
\State set empty vector $v \in (\text{index:}~ \mathbb{Z}, x: \mathbb{F})^{s \times s'}$
\State set empty vector $v \in (\text{index:}~ \mathbb{Z}, x: \mathbb{F}^{s \times s'})$
\For {$i$ to $m$}
\For {$j$ to $n$}
\If {$M_{i,j} \neq 0$}