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\title{Notes on Halo}
\author{arnaucube}
\date{July 2022}

\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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