From 996a3e8f6a16896a39af1859a45564289865a431 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Tue, 16 Aug 2022 19:22:18 +0200 Subject: [PATCH] split paper notes files --- notes_bls-sig.pdf | Bin 0 -> 155189 bytes notes_bls-sig.tex | 87 ++++++++++++ notes_halo.pdf | Bin 0 -> 172753 bytes notes_halo.tex | 145 ++++++++++++++++++++ notes_sonic.pdf | Bin 0 -> 212412 bytes notes_sonic.tex | 181 ++++++++++++++++++++++++ paper-notes.pdf | Bin 239908 -> 0 bytes paper-notes.tex | 340 ---------------------------------------------- 8 files changed, 413 insertions(+), 340 deletions(-) create mode 100644 notes_bls-sig.pdf create mode 100644 notes_bls-sig.tex create mode 100644 notes_halo.pdf create mode 100644 notes_halo.tex create mode 100644 notes_sonic.pdf create mode 100644 notes_sonic.tex delete mode 100644 paper-notes.pdf delete mode 100644 paper-notes.tex diff --git a/notes_bls-sig.pdf b/notes_bls-sig.pdf new file mode 100644 index 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+\theoremstyle{definition} +\newtheorem{definition}{Def}[section] +\newtheorem{theorem}[definition]{Thm} + + +\title{Notes on Halo} +\author{arnaucube} +\date{} + +\begin{document} + +\maketitle + +\begin{abstract} + Notes taken while reading Halo paper \cite{cryptoeprint:2019/1021}. 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{modified IPA (from Halo paper)} +Notes taken while reading about the modified Inner Product Argument (IPA) from the Halo paper \cite{cryptoeprint:2019/1021}. + +\subsection{Notation} +\begin{description} + \item[Scalar mul] $[a]G$, where $a$ is a scalar and $G \in \mathbb{G}$ + \item[Inner product] $<\overrightarrow{a}, \overrightarrow{b}> = a_0 b_0 + a_1 b_1 + \ldots + a_{n-1} b_{n-1}$ + \item[Multiscalar mul] $<\overrightarrow{a}, \overrightarrow{b}> = [a_0] G_0 + [a_1] G_1 + \ldots [a_{n-1}] G_{n-1}$ +\end{description} + + +\subsection{Transparent setup} +$\overrightarrow{G} \in^r \mathbb{G}^d$, $H \in^r \mathbb{G}$ + +Prover wants to commit to $p(x)=a_0$ +\subsection{Protocol} +Prover: +$$P=<\overrightarrow{a}, \overrightarrow{G}> + [r]H$$ +$$v=<\overrightarrow{a}, \{1, x, x^2, \ldots, x^{d-1} \} >$$ + +where $\{1, x, x^2, \ldots, x^{d-1} \} = \overrightarrow{b}$. + +We can see that computing $v$ is the equivalent to evaluating $p(x)$ at $x$ ($p(x)=v$). + +We will prove: +\begin{enumerate}[i.] + \item polynomial $p(X) = \sum a_i X^i$\\ + $p(x) = v$ (that $p(X)$ evaluates $x$ to $v$). + \item $deg(p(X)) \leq d-1$ +\end{enumerate} + + +Both parties know $P$, point $x$ and claimed evaluation $v$. For $U \in^r \mathbb{G}$, + +$$P' = P + [v] U = <\overrightarrow{a}, G> + [r]H + [v] U$$ + +Now, for $k$ rounds ($d=2^k$, from $j=k$ to $j=1$): +\begin{itemize} + \item random blinding factors: $l_j, r_j \in \mathbb{F}_p$ + \item + $$L_j = < \overrightarrow{a}_{lo}, \overrightarrow{G}_{hi}> + [l_j] H + [< \overrightarrow{a}_{lo}, \overrightarrow{b}_{hi}>] U$$ + $$L_j = < \overrightarrow{a}_{lo}, \overrightarrow{G}_{hi}> + [l_j] H + [< \overrightarrow{a}_{lo}, \overrightarrow{b}_{hi}>] U$$ + \item Verifier sends random challenge $u_j \in \mathbb{I}$ + \item Prover computes the halved vectors for next round: + $$\overrightarrow{a} \leftarrow \overrightarrow{a}_{hi} \cdot u_j^{-1} + \overrightarrow{a}_{lo} \cdot u_j$$ + $$\overrightarrow{b} \leftarrow \overrightarrow{b}_{lo} \cdot u_j^{-1} + \overrightarrow{b}_{hi} \cdot u_j$$ + $$\overrightarrow{G} \leftarrow \overrightarrow{G}_{lo} \cdot u_j^{-1} + \overrightarrow{G}_{hi} \cdot u_j$$ +\end{itemize} + +After final round, $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{G}$ are each of length 1. + +Verifier can compute +$$G = \overrightarrow{G}_0 = < \overrightarrow{s}, \overrightarrow{G} >$$ +and $$b = \overrightarrow{b}_0 = < \overrightarrow{s}, \overrightarrow{b} >$$ +where $\overrightarrow{s}$ is the binary counting structure: + +\begin{align*} + &s = (u_1^{-1} ~ u_2^{-1} \cdots ~u_k^{-1},\\ + &~~~~~~u_1 ~~~ u_2^{-1} ~\cdots ~u_k^{-1},\\ + &~~~~~~u_1^{-1} ~~ u_2 ~~\cdots ~u_k^{-1},\\ + &~~~~~~~~~~~~~~\vdots\\ + &~~~~~~u_1 ~~~~ u_2 ~~\cdots ~u_k) +\end{align*} + + +And verifier checks: +$$[a]G + [r'] H + [ab] U == P' + \sum_{j=1}^k ( [u_j^2] L_j + [u_j^{-2}] R_j)$$ + +where the synthetic blinding factor $r'$ is $r' = r + \sum_{j=1}^k (l_j u_j^2 + r_j u_j^{-2})$. + +\vspace{1cm} + +Unfold: + +$$ +\textcolor{brown}{[a]G} + \textcolor{cyan}{[r'] H} + \textcolor{magenta}{[ab] U} +== +\textcolor{blue}{P'} + \sum_{j=1}^k ( \textcolor{violet}{[u_j^2] L_j} + \textcolor{orange}{[u_j^{-2}] R_j}) +$$ + +\begin{align*} +&Right~side = \textcolor{blue}{P'} + \sum_{j=1}^k ( \textcolor{violet}{[u_j^2] L_j} + \textcolor{orange}{[u_j^{-2}] R_j})\\ +&= \textcolor{blue}{< \overrightarrow{a}, \overrightarrow{G}> + [r] H + [v] U}\\ +&+ \sum_{j=1}^k (\\ +&\textcolor{violet}{[u_j^2] \cdot <\overrightarrow{a}_{lo}, \overrightarrow{G}_{hi}> + [l_j] H + [<\overrightarrow{a}_{lo}, \overrightarrow{b}_{hi}>] U}\\ +&\textcolor{orange}{+ [u_j^{-2}] \cdot <\overrightarrow{a}_{hi}, \overrightarrow{G}_{lo}> + [r_j] H + [<\overrightarrow{a}_{hi}, \overrightarrow{b}_{lo}>] U} +) +\end{align*} + +\begin{align*} +&Left~side = \textcolor{brown}{[a]G} + \textcolor{cyan}{[r'] H} + \textcolor{magenta}{[ab] U}\\ +& = \textcolor{brown}{< \overrightarrow{a}, \overrightarrow{G} >}\\ +&+ \textcolor{cyan}{[r + \sum_{j=1}^k (l_j \cdot u_j^2 + r_j u_j^{-2})] \cdot H}\\ +&+ \textcolor{magenta}{< \overrightarrow{a}, \overrightarrow{b} > U} +\end{align*} + + +\section{Amortization Strategy} +TODO + 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