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small update to IPA notes

master
arnaucube 1 year ago
parent
commit
d509900028
3 changed files with 85 additions and 16 deletions
  1. +3
    -2
      ipa.sage
  2. BIN
      notes_halo.pdf
  3. +82
    -14
      notes_halo.tex

+ 3
- 2
ipa.sage

@ -186,6 +186,9 @@ class IPA_halo:
def verify(self, P, a, v, x_powers, r, u, U, lj, rj, L, R): def verify(self, P, a, v, x_powers, r, u, U, lj, rj, L, R):
print("methid verify()") print("methid verify()")
# compute P' = P + [v] U
P = P + int(v) * U
s = build_s_from_us(u, self.d) s = build_s_from_us(u, self.d)
b = inner_product_field(s, x_powers) b = inner_product_field(s, x_powers)
G = inner_product_point(s, self.gs) G = inner_product_point(s, self.gs)
@ -445,8 +448,6 @@ class TestIPA_halo(unittest.TestCase):
while (u[j] == 0): # prevent u[j] from being 0 while (u[j] == 0): # prevent u[j] from being 0
u[j] = ipa.F.random_element() u[j] = ipa.F.random_element()
P = P + int(v) * U
# prover # prover
a_ipa, lj, rj, L, R = ipa.ipa(a, x_powers, u, U) a_ipa, lj, rj, L, R = ipa.ipa(a, x_powers, u, U)

BIN
notes_halo.pdf


+ 82
- 14
notes_halo.tex

@ -14,6 +14,33 @@
} }
\usepackage{xcolor} \usepackage{xcolor}
\usepackage{pgf-umlsd} % diagrams
% message between threads
% 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 (#3 from) at (mess from) {\llap{#6~}};
\node (#3 to) at (mess to) {\rlap{~#7}};
\else\if L#5
\node (#3 from) at (mess from) {\rlap{~#6}};
\node (#3 to) at (mess to) {\llap{#7~}};
\else
\node (#3 from) at (mess from) {#6};
\node (#3 to) at (mess to) {#7};
\fi
\fi
}
% prevent warnings of underfull \hbox: % prevent warnings of underfull \hbox:
\usepackage{etoolbox} \usepackage{etoolbox}
\apptocmd{\sloppy}{\hbadness 4000\relax}{}{} \apptocmd{\sloppy}{\hbadness 4000\relax}{}{}
@ -42,11 +69,14 @@
\section{modified IPA (from Halo paper)} \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}. Notes taken while reading about the modified Inner Product Argument (IPA) from the Halo paper \cite{cryptoeprint:2019/1021}.
\paragraph{Objective:}
Prover wants to prove that the polynomial $p(X)$ from the commitment $P$ evaluates to $v$ at $x$, and that $deg(p(X)) \leq d-1$.
\subsection{Notation} \subsection{Notation}
\begin{description} \begin{description}
\item[Scalar mul] $[a]G$, where $a$ is a scalar and $G \in \mathbb{G}$ \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[Inner product] $<\overrightarrow{a}, \overrightarrow{b}> = a_0 b_0 + a_1 b_1 + \ldots + a_{n-1} b_{n-1}$
\item[Multiscalar mul] $<\overrightarrow{a}, \overrightarrow{b}> = [a_0] G_0 + [a_1] G_1 + \ldots [a_{n-1}] G_{n-1}$
\item[Multiscalar mul] $<\overrightarrow{a}, \overrightarrow{G}> = [a_0] G_0 + [a_1] G_1 + \ldots + [a_{n-1}] G_{n-1}$
\end{description} \end{description}
@ -61,7 +91,7 @@ $$v=<\overrightarrow{a}, \{1, x, x^2, \ldots, x^{d-1} \} >$$
where $\{1, x, x^2, \ldots, x^{d-1} \} = \overrightarrow{b}$. 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 can see that computing $v$ is the equivalent to evaluating $p(X)$ at $x$ ($p(x)=v$).
We will prove: We will prove:
\begin{enumerate}[i.] \begin{enumerate}[i.]
@ -71,16 +101,18 @@ We will prove:
\end{enumerate} \end{enumerate}
Both parties know $P$, point $x$ and claimed evaluation $v$. For $U \in^r \mathbb{G}$,
Both parties know $P$, point $x$ and claimed evaluation $v$. For $U \in^r \mathbb{G}$.
Prover computes $P'$:
$$P' = P + [v] U = <\overrightarrow{a}, G> + [r]H + [v] U$$ $$P' = P + [v] U = <\overrightarrow{a}, G> + [r]H + [v] U$$
Now, for $k$ rounds ($d=2^k$, from $j=k$ to $j=1$): Now, for $k$ rounds ($d=2^k$, from $j=k$ to $j=1$):
\begin{itemize} \begin{itemize}
\item random blinding factors: $l_j, r_j \in \mathbb{F}_p$
\item
$$L_j = < \overrightarrow{a}_{lo}, \overrightarrow{G}_{hi}> + [l_j] H + [< \overrightarrow{a}_{lo}, \overrightarrow{b}_{hi}>] U$$
\item Prover sets random blinding factors: $l_j, r_j \in \mathbb{F}_p$
\item Prover computes
$$L_j = < \overrightarrow{a}_{lo}, \overrightarrow{G}_{hi}> + [l_j] H + [< \overrightarrow{a}_{lo}, \overrightarrow{b}_{hi}>] U$$ $$L_j = < \overrightarrow{a}_{lo}, \overrightarrow{G}_{hi}> + [l_j] H + [< \overrightarrow{a}_{lo}, \overrightarrow{b}_{hi}>] U$$
$$R_j = < \overrightarrow{a}_{lo}, \overrightarrow{G}_{hi}> + [l_j] H + [< \overrightarrow{a}_{lo}, \overrightarrow{b}_{hi}>] U$$
\item Verifier sends random challenge $u_j \in \mathbb{I}$ \item Verifier sends random challenge $u_j \in \mathbb{I}$
\item Prover computes the halved vectors for next round: \item Prover computes the halved vectors for next round:
$$\overrightarrow{a} \leftarrow \overrightarrow{a}_{hi} \cdot u_j^{-1} + \overrightarrow{a}_{lo} \cdot u_j$$ $$\overrightarrow{a} \leftarrow \overrightarrow{a}_{hi} \cdot u_j^{-1} + \overrightarrow{a}_{lo} \cdot u_j$$
@ -119,22 +151,58 @@ $$
\textcolor{blue}{P'} + \sum_{j=1}^k ( \textcolor{violet}{[u_j^2] L_j} + \textcolor{orange}{[u_j^{-2}] R_j}) \textcolor{blue}{P'} + \sum_{j=1}^k ( \textcolor{violet}{[u_j^2] L_j} + \textcolor{orange}{[u_j^{-2}] R_j})
$$ $$
\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*}
\begin{align*} \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})\\ &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}\\ &= \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}\\
&+ \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} &\textcolor{orange}{+ [u_j^{-2}] \cdot <\overrightarrow{a}_{hi}, \overrightarrow{G}_{lo}> + [r_j] H + [<\overrightarrow{a}_{hi}, \overrightarrow{b}_{lo}>] U}
) )
\end{align*} \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*}
\vspace{1.5cm}
The following diagram ilustrates the main steps in the scheme:
\begin{center}
\begin{sequencediagram}
\newinst[1]{p}{Prover}
\newinst[3]{v}{Verifier}
\bloodymess[1]{p}{P}{v}{R}{knows $p(X)\in \mathbb{F[X]}$, commits to $p(X)$, $P$}{rand $x \in \mathbb{F},~U\in \mathbb{G},~\overrightarrow{u} \in \mathbb{F}^d$}
\bloodymess[1]{v}{$x, U, u$}{p}{R}{}{}
\bloodymess[1]{p}{$proof, a, L_j, R_j, v$}{v}{R}{gen proof}{$verify(proof, P, a, x, L_j, R_j)$}
% \begin{callself}{p}{knows $p(X) \in \mathbb{F}[X]$}{}
% \end{callself}
% \begin{callself}{p}{commit to $p(X),~P$}{}
% \end{callself}
%
% \mess[0]{p}{$P$}{v}
% \begin{callself}{v}{rand $x \in \mathbb{F},~U\in \mathbb{G},~\overrightarrow{u} \in \mathbb{F}^d$}{}
% \end{callself}
%
% \mess[0]{v}{$x,U,u$}{p}
% \node[anchor=west] (p2) at (mess to) {gen proof2}
% \begin{callself}{p}{gen proof $\pi$}{}
% \end{callself}
%
% \mess[0]{p}{$a, L_j, R_j, v$}{v}
%
% \begin{callself}{v}{$verify(P, a, x, v, L_j, R_k$)}{}
% \end{callself}
\end{sequencediagram}
\end{center}
\section{Amortization Strategy} \section{Amortization Strategy}
TODO TODO

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