Add instantiation of hash extractor for Jubjub.

Signed-off-by: Daira Hopwood <daira@jacaranda.org>

protocol/protocol.tex

@@ -222,6 +222,8 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 \newtheorem{theorem}{Theorem}
 \numberwithin{theorem}{subsection}
 
+\newtheorem*{lemma*}{Lemma}
+
 
 % Terminology
 
@@ -1076,7 +1078,6 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 \newcommand{\reprP}[1]{\repr_{\GroupP{#1}}}
 \newcommand{\abstP}[1]{\abst_{\GroupP{#1}}}
 \newcommand{\PairingP}{\ParamP{\hat{e}}}
-\newcommand{\ExtractP}{\ParamP{\mathsf{Extract}}}
 
 \newcommand{\ParamG}[1]{{{#1}_\mathbb{G}}}
 \newcommand{\ParamGexp}[2]{{{#1}_\mathbb{G}\!}^{#2}}
@@ -1104,7 +1105,6 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 \newcommand{\reprS}[1]{\repr_{\GroupG{#1}}}
 \newcommand{\abstS}[1]{\abst_{\GroupG{#1}}}
 \newcommand{\PairingS}{\ParamS{\hat{e}}}
-\newcommand{\ExtractS}{\ParamS{\mathsf{Extract}}}
 
 \newcommand{\ParamJ}[1]{{{#1}_\mathbb{\hskip 0.01em J}}}
 \newcommand{\ParamJexp}[2]{{{#1}_\mathbb{\hskip 0.01em J}\!}^{#2}}
@@ -2408,7 +2408,7 @@ we write $\scalarmult{k}{G}$ for $\vsum{i = 1}{k} G$.
 
 A \hashExtractor for a \representedGroup $\GroupG{}$ is a function
 $\ExtractG \typecolon \GroupG{} \rightarrow \bitseq{\ell}$ for some $\ell \typecolon \Nat$,
-such that $\ExtractG$ is injective on the subgroup generated by $\GenG{}$.
+such that $\ExtractG$ is injective on the subgroup of $\GroupG{}$ of order $\ParamG{r}$.
 
 \pnote{
 Unlike the representation function $\reprG{}$, $\ExtractG$ need not have an
@@ -4291,6 +4291,60 @@ other conditions on points, for example that they are not the zero point, or are
 large prime-order subgroup.
 }
 
+
+\sapling{
+\nsubsubsubsection{\HashExtractor{} for \Jubjub} \label{grouphashjubjub}
+
+Let $\mathcal{U}((u, \varv)) = u$ and let $\mathcal{V}((u, \varv)) = \varv$.
+
+Let $\ExtractJ \typecolon \GroupJ \rightarrow \bitseq{255}$ be defined as:
+
+\begin{formulae}
+  \item $\ExtractJ((u, \varv)) = \ItoBSP{255}(u)$.
+\end{formulae}
+
+Let $G$ be the subgroup of $\GroupJ$ of order $\ParamJ{r}$ (an odd prime).
+
+Facts: the point $(0, 1) = \ZeroJ$, and the point $(0, -1)$ has order 2 in $\GroupJ$.
+
+% <https://github.com/zcash/zcash/issues/2234#issuecomment-333360977>
+\begin{lemma*}
+Let $P = (u, \varv) \typecolon G$. Then $(u, -\varv)$ is not a point in $G$.
+\end{lemma*}
+
+\begin{proof}
+If $P = \ZeroJ$ then $(u, -\varv) = (0, -1)$ which is not in $G$.
+Else, $P$ is of odd-prime order. Note that $\varv \neq 0$.
+(If $\varv = 0$ then $a \mult u^2 = 1$, and so applying the doubling formula
+gives $\scalarmult{2}{P} = (0, -1)$, then $\scalarmult{4}{P} = (0, 1) = \ZeroJ$;
+contradiction since then $P$ would not be of odd prime order.)
+Therefore, $-\varv \neq \varv$.
+Now suppose $(u, -\varv) = Q$ is a point in $G$. Then by applying the
+doubling formula we have $\scalarmult{2}{Q} = -\scalarmult{2}{P}$.
+But also $\scalarmult{2}{(-P)} = -\scalarmult{2}{P}$. Therefore either
+$Q = -P$ (then $\mathcal{V}(Q) = \mathcal{V}(-P)$; contradiction since
+$-\varv \neq \varv$), or doubling is not injective on $G$ (contradiction
+since $G$ is of odd order).
+\end{proof}
+
+\begin{theorem}
+$\ExtractJ$ is injective on $G$.
+\end{theorem}
+
+\begin{proof}
+By writing the curve equation as
+$\varv^2 = (1 - a \smult u^2) / (1 - d \smult u^2)$, and noting that the
+potentially exceptional case $1 - d \smult u^2 = 0$ does not occur for a
+complete twisted Edwards curve, we see that for a given $u$ there can be at
+most two possible solutions for $\varv$, and that if there are two solutions
+they can be written as $\varv$ and $-\varv$. In that case by the lemma, at
+most one of $(u, \varv)$ and $(u, -\varv)$ is in $G$. Therefore, $\mathcal{U}$
+is injective on points in $G$, hence so is $\ExtractJ$.
+\end{proof}
+
+}
+
+
 \nsubsubsection{\ZeroKnowledgeProvingSystems}
 
 \nsubsubsubsection{\PHGRProvingSystem} \label{phgr}
@@ -6286,6 +6340,7 @@ Daira Hopwood, Sean Bowe, and Jack Grigg.
     \item No changes to \Sprout.
 \sapling{
     \item Add instantiation of $\CRHivk$.
+    \item Add instantiation of a hash extractor for \Jubjub.
     \item Make the background lighter and the \Sapling green darker, for contrast.
 }
 \end{itemize}