Cosmetics.

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

protocol/protocol.tex

@@ -147,6 +147,7 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 \DeclareMathSymbol{\binampersand}{\mathbin}{bskadd}{"EE}
 
 \newcommand{\hairspace}{~\!}
+\newcommand{\hparen}{\hphantom{(}}
 
 \newcommand{\hfrac}[2]{\scalebox{0.8}{$\genfrac{}{}{0.5pt}{0}{#1}{#2}$}}
 
@@ -698,6 +699,8 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 \newcommand{\vmacs}{\mathtt{vmacs}}
 \newcommand{\GroupG}[1]{\mathbb{G}_{#1}}
 \newcommand{\PointP}[1]{\mathcal{P}_{#1}}
+\newcommand{\xP}{{x_{\hspace{-0.12em}P}}}
+\newcommand{\yP}{{y_{\hspace{-0.03em}P}}}
 \newcommand{\GF}[1]{\mathbb{F}_{#1}}
 \newcommand{\GFstar}[1]{\mathbb{F}^\ast_{#1}}
 \newcommand{\ECtoOSP}{\mathsf{EC2OSP}}
@@ -1903,27 +1906,25 @@ attempts to add a \nullifier to the \nullifierSet that already exists in the set
 A valid instance of $\JoinSplitProof$ assures that given a \term{primary input}:
 
 \begin{formulae}
-  \item $(\rt \typecolon \MerkleHash,
-            \nfOld{\allOld} \typecolon \typeexp{\PRFOutput}{\NOld},
-            \cmNew{\allNew} \typecolon \typeexp{\CommitOutput}{\NNew},
-            \changed{\vpubOld \typecolon \range{0}{2^{64}-1},}\,
-            \vpubNew \typecolon \range{0}{2^{64}-1},\\
-           \hphantom{(}
-            \hSig \typecolon \hSigType,
-            \h{\allOld} \typecolon \typeexp{\PRFOutput}{\NOld})$,
+  \item $(\rt \typecolon \MerkleHash,\\
+   \hparen\nfOld{\allOld} \typecolon \typeexp{\PRFOutput}{\NOld},\vspace{0.4ex}\\
+   \hparen\cmNew{\allNew} \typecolon \typeexp{\CommitOutput}{\NNew},\vspace{0.8ex}\\
+   \hparen\changed{\vpubOld \typecolon \range{0}{2^{64}-1},}\vspace{0.4ex}\\
+   \hparen\vpubNew \typecolon \range{0}{2^{64}-1},\\
+   \hparen\hSig \typecolon \hSigType,\\
+   \hparen\h{\allOld} \typecolon \typeexp{\PRFOutput}{\NOld})$,
 \end{formulae}
 
 \introlist
 the prover knows an \term{auxiliary input}:
 
 \begin{formulae}
-  \item $(\treepath{\allOld} \typecolon \typeexp{\typeexp{\MerkleHash}{\MerkleDepth}}{\NOld},
-            \nOld{\allOld} \typecolon \typeexp{\NoteType}{\NOld},
-            \AuthPrivateOld{\allOld} \typecolon \typeexp{\bitseq{\AuthPrivateLength}}{\NOld},
-            \nNew{\allNew} \typecolon \typeexp{\NoteType}{\NOld}\changed{,}\\
-           \hphantom{(}
-            \changed{\NoteAddressPreRand \typecolon \bitseq{\NoteAddressPreRandLength},
-            \EnforceMerklePath{\allOld} \typecolon \bitseq{\NOld}})$,
+  \item $(\treepath{\allOld} \typecolon \typeexp{\typeexp{\MerkleHash}{\MerkleDepth}}{\NOld},\\
+   \hparen\nOld{\allOld} \typecolon \typeexp{\NoteType}{\NOld},\\
+   \hparen\AuthPrivateOld{\allOld} \typecolon \typeexp{\bitseq{\AuthPrivateLength}}{\NOld},\\
+   \hparen\nNew{\allNew} \typecolon \typeexp{\NoteType}{\NNew}\changed{,}\vspace{0.8ex}\\
+   \hparen\changed{\NoteAddressPreRand \typecolon \bitseq{\NoteAddressPreRandLength},}\\
+   \hparen\changed{\EnforceMerklePath{\allOld} \typecolon \bitseq{\NOld}})$,
 \end{formulae}
 
 \introlist
@@ -2810,7 +2811,7 @@ Let $r = 21888242871839275222246405745257275088548364400416034343698204186575808
 
 Let $b = 3$.
 
-($q$ and $r$ are prime.)
+(\hairspace $q$ and $r$ are prime.)
 
 \introlist
 The pairing is of type $\GroupG{1} \times \GroupG{2} \rightarrow \GroupG{T}$, where:
@@ -2901,24 +2902,24 @@ Define $\ItoOSP{} \typecolon (k \typecolon \Nat) \times \range{0}{256^k\!-\!1} \
 representing $n$ in big-endian order.
 
 \introlist
-For a point $P \typecolon \GroupG{1} = (x_P, y_P)$:
+For a point $P \typecolon \GroupG{1} = (\xP, \yP)$:
 
 \begin{itemize}
-  \item The field elements $x_P$ and $y_P \typecolon \GF{q}$ are represented as
+  \item The field elements $\xP$ and $\yP \typecolon \GF{q}$ are represented as
         integers $x$ and $y \typecolon \range{0}{q\!-\!1}$.
   \item Let $\tilde{y} = y \bmod 2$.
   \item $P$ is encoded as $\Justthebox{\gonebox}$.
 \end{itemize}
 
 \introlist
-For a point $P \typecolon \GroupG{2} = (x_P, y_P)$:
+For a point $P \typecolon \GroupG{2} = (\xP, \yP)$:
 
 \begin{itemize}
   \item A field element $w \typecolon \GF{q^2}$ is represented as
         a polynomial $a_{w,1} \mult t + a_{w,0} \typecolon \GF{q}[t]$ modulo $t^2 + 1$.
         Define $\FEtoIP \typecolon \GF{q^2} \rightarrow \range{0}{q^2\!-\!1}$ such that
         $\FEtoIP(w) = a_{w,1} \mult q + a_{w,0}$.
-  \item Let $x = \FEtoIP(x_P)$, $y = \FEtoIP(y_P)$, and $y' = \FEtoIP(-y_P)$.
+  \item Let $x = \FEtoIP(\xP)$, $y = \FEtoIP(\yP)$, and $y' = \FEtoIP(-\yP)$.
   \item Let $\tilde{y} = \begin{cases}
           1, &\caseif y > y' \\
           0, &\caseotherwise.