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\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsfonts}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{enumerate}
\usepackage{hyperref}
\hypersetup{
colorlinks,
citecolor=black,
filecolor=black,
linkcolor=black,
urlcolor=blue
}
\usepackage{xcolor}
% prevent warnings of underfull \hbox:
\usepackage{etoolbox}
\apptocmd{\sloppy}{\hbadness 4000\relax}{}{}
\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
\bibliography{paper-notes.bib}
\bibliographystyle{unsrt}
\end{document}