From a8052562e45aadca9459b8ff526ece682b490e59 Mon Sep 17 00:00:00 2001
From: Daira Hopwood <daira@jacaranda.org>
Date: Fri, 16 Mar 2018 15:58:23 +0000
Subject: [PATCH] Cosmetics.

Signed-off-by: Daira Hopwood <daira@jacaranda.org>
---
 protocol/incremental_merkle.png | Bin 34755 -> 18608 bytes
 protocol/key_components.png     | Bin 71625 -> 59061 bytes
 protocol/protocol.tex           | 621 ++++++++++++++++++--------------
 protocol/zcash.bib              |   6 +-
 4 files changed, 360 insertions(+), 267 deletions(-)

diff --git a/protocol/incremental_merkle.png b/protocol/incremental_merkle.png
index bb5180477e036f427a36d33121037ee07770b4dc..5d51c7a2120867024e47c827a1340cad41a901ae 100644
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diff --git a/protocol/key_components.png b/protocol/key_components.png
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aR_jD<*L2e0>K+~nJl;ory{Jd<<o^KfSeX_8

diff --git a/protocol/protocol.tex b/protocol/protocol.tex
index ee1c2ffc..33ac4906 100644
--- a/protocol/protocol.tex
+++ b/protocol/protocol.tex
@@ -1,54 +1,59 @@
 \documentclass{article}
-\RequirePackage[utf8]{inputenc}
-\RequirePackage[T1]{fontenc}
-\RequirePackage{amsmath}
-\RequirePackage{amsthm}
-\RequirePackage{bytefield}
-\RequirePackage{graphicx}
-\RequirePackage{newtxmath}
-\RequirePackage{mathtools}
-\RequirePackage{xspace}
-\RequirePackage{url}
-\RequirePackage{changepage}
-\RequirePackage{enumitem}
-\RequirePackage{tabularx}
-\RequirePackage{hhline}
-\RequirePackage[usestackEOL]{stackengine}
-\RequirePackage{comment}
-\RequirePackage{needspace}
-\RequirePackage[nobottomtitles]{titlesec}
-\RequirePackage[hang]{footmisc}
-\RequirePackage{xstring}
-\RequirePackage[usenames,dvipsnames]{xcolor}
-\RequirePackage[unicode,bookmarksnumbered,bookmarksopen,allbordercolors=MidnightBlue,citebordercolor=Plum,urlbordercolor=BrickRed]{hyperref}
-\RequirePackage{cleveref}
-\RequirePackage{nameref}
-\RequirePackage{etoolbox}
-\RequirePackage{subdepth}
-\RequirePackage{fix-cm}
-\RequirePackage{hyphenat}
-\RequirePackage{tocloft}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{amsmath}
+\usepackage{amsthm}
+\usepackage{bytefield}
+\usepackage{graphicx}
+\usepackage{newtxmath}
+\usepackage{mathtools}
+\usepackage{xspace}
+\usepackage{url}
+\usepackage{changepage}
+\usepackage{enumitem}
+\usepackage{tabularx}
+\usepackage{hhline}
+\usepackage[usestackEOL]{stackengine}
+\usepackage{comment}
+\usepackage{needspace}
+\usepackage[nobottomtitles]{titlesec}
+\usepackage[hang]{footmisc}
+\usepackage{xstring}
+\usepackage[usenames,dvipsnames]{xcolor}
+\usepackage[unicode,bookmarksnumbered,bookmarksopen,allbordercolors=MidnightBlue,
+            citebordercolor=Plum,urlbordercolor=BrickRed]{hyperref}
+\usepackage{cleveref}
+\usepackage{nameref}
+\usepackage{etoolbox}
+\usepackage{subdepth}
+\usepackage{fix-cm}
+\usepackage{hyphenat}
+\usepackage{tocloft}
 
-\RequirePackage[style=alphabetic,maxbibnames=99,dateabbrev=false,urldate=iso8601,backref=true,backrefstyle=none,backend=biber]{biblatex}
+\usepackage[style=alphabetic,maxbibnames=99,dateabbrev=false,urldate=iso8601,
+            backref=true,backrefstyle=none,backend=biber]{biblatex}
 \addbibresource{zcash.bib}
 
 % Fonts
-\RequirePackage{lmodern}
-\RequirePackage{quattrocento}
-\RequirePackage[bb=ams]{mathalfa}
-\RequirePackage[scr]{rsfso}
-%\RequirePackage{txfonts}
+\usepackage{lmodern}
+\usepackage{quattrocento}
+\usepackage[bb=ams]{mathalfa}
+\usepackage[scr]{rsfso}
+%\usepackage{txfonts}
 
 % Quattrocento is beautiful but doesn't have an italic face. So we scale
 % New Century Schoolbook italic to fit in with slanted Quattrocento and
 % match its x height.
+% (This has the side effect of making all italic text into boxes, so
+% it won't linebreak, which has pluses and minuses.)
 \renewcommand{\emph}[1]{\hspace{0.15em}{\fontfamily{pnc}\selectfont\scalebox{1.02}[0.999]{\textit{#1}}}\hspace{0.02em}}
 
-% While we're at it, let's match the tt x height to Quattrocento as well.
+% While we're at it, let's match the tt x height to Quattrocento as well,
+% and compress it a little to space space in tables.
 \let\oldtexttt\texttt
 \let\oldmathtt\mathtt
-\renewcommand{\texttt}[1]{\scalebox{1.02}[1.07]{\oldtexttt{#1}}}
-\renewcommand{\mathtt}[1]{\scalebox{1.02}[1.07]{$\oldmathtt{#1}$}}
+\renewcommand{\texttt}[1]{\scalebox{0.97}[1.07]{\oldtexttt{#1}}}
+\renewcommand{\mathtt}[1]{\scalebox{0.97}[1.07]{$\oldmathtt{#1}$}}
 
 % bold but not extended
 \newcommand{\textbnx}[1]{{\fontseries{b}\selectfont #1}}
@@ -72,21 +77,23 @@
 \setlength{\topmargin}{-0.75in}
 \setlength{\textheight}{9.2in}
 \setlength{\parindent}{0ex}
-\renewcommand{\arraystretch}{1.4}
+
+\newcommand{\defaultarraystretch}{1.4}
+\renewcommand{\arraystretch}{\defaultarraystretch}
 
 % <https://tex.stackexchange.com/a/49898/78411>
 \makeatletter
 \renewcommand{\@pnumwidth}{2em}
 \makeatother
 
-\newcommand{\pagenumfont}{\fontfamily{pnc}\selectfont\rule[-.2\baselineskip]{0pt}{1.4\baselineskip}}
+\newcommand{\pagenumfont}{\fontfamily{pnc}\selectfont\rule[-.2\baselineskip]{0pt}{1.35\baselineskip}}
 \renewcommand{\cftsecpagefont}{\pagenumfont}
 \renewcommand{\cftsubsecpagefont}{\pagenumfont}
 \renewcommand{\cftsubsubsecpagefont}{\pagenumfont}
 \renewcommand{\cftparapagefont}{\pagenumfont}
 
 \hfuzz=1pt
-%\overfullrule=2cm
+\overfullrule=2cm
 
 \setlength{\footnotemargin}{0.6em}
 \setlength{\footnotesep}{2ex}
@@ -100,14 +107,17 @@
 
 \setlength{\bibitemsep}{1.2ex}  % default is too cramped!
 
-\setlist[enumerate]{before=\vspace{-1ex}}
-\setlist[itemize]{itemsep=0.5ex,topsep=0.2ex,before=\vspace{-1ex},after=\vspace{1.5ex}}
+\setlist[enumerate]{before=\vspace{-0.8ex}}
+\setlist[itemize]{itemsep=0.5ex,topsep=0.2ex,before=\vspace{-0.8ex},after=\vspace{1.5ex}}
+
+\newlist{compactitemize}{itemize}{3}
+\setlist[compactitemize]{itemsep=-1ex,topsep=0ex,before=\vspace{-0.2ex},leftmargin=1.2em,label=$\cdot$,after=\vspace{-3.3ex}}
 
 \newlist{formulae}{itemize}{3}
 \setlist[formulae]{itemsep=0.2ex,topsep=0ex,leftmargin=1.5em,label=,after=\vspace{1.5ex}}
 
 \newlist{lines}{itemize}{3}
-\setlist[lines]{itemsep=-0.5ex,topsep=0ex,before=\vspace{1ex},leftmargin=0.6em,label=,after=\vspace{1ex}}
+\setlist[lines]{itemsep=-0.5ex,topsep=0ex,before=\vspace{1ex},leftmargin=1.6em,label=,after=\vspace{1ex}}
 
 \newcommand{\docversion}{Version unavailable (check protocol.ver)}
 \newcommand{\SaplingSpec}{Overwinter+Sapling}
@@ -134,27 +144,47 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 }
 
 \makeatletter
-\renewcommand*{\@fnsymbol}[1]{\ensuremath{\ifcase#1\or \dagger\or \ddagger\or
-    \mathsection\or \mathparagraph\else\@ctrerr\fi}}
-\newcommand\slightlylarge{\fontsize{10.5}{10.5}\selectfont}
-\newcommand\notsolarge{\fontsize{11}{11}\selectfont}
-\newcommand\largeish{\fontsize{12}{12}\selectfont}
-\newcommand\larger{\fontsize{13}{13}\selectfont}
-\newcommand\Larger{\fontsize{16}{16}\selectfont}
+\renewcommand*{\@fnsymbol}[1]{\ensuremath{\ifcase#1\or \dagger\or \ddagger\or \mathsection\or \mathparagraph\else\@ctrerr\fi}}
 \makeatother
 
+\newcommand{\slightlylarge}{\fontsize{10.5}{10.5}\selectfont}
+\newcommand{\notsolarge}{\fontsize{11}{11}\selectfont}
+\newcommand{\largeish}{\fontsize{12}{12}\selectfont}
+\newcommand{\larger}{\fontsize{13}{13}\selectfont}
+\newcommand{\Larger}{\fontsize{16}{16}\selectfont}
+
 \titleformat*{\subsection}{\larger\bfseries}
 \titleformat*{\subsubsection}{\largeish\bfseries}
 \titleformat*{\paragraph}{\notsolarge\bfseries}
 \titleformat*{\subparagraph}{\slightlylarge\bfseries}
 
-\renewcommand{\sectionautorefname}{\S\!}
-\renewcommand{\subsectionautorefname}{\S\!}
-\renewcommand{\subsubsectionautorefname}{\S\!}
-\renewcommand{\paragraphautorefname}{\S\!}
-\renewcommand{\subparagraphautorefname}{\S\!}
-\newcommand{\crossref}[1]{\autoref{#1}\, \emph{`\nameref*{#1}\kern -0.05em'} on p.\,\pageref*{#1}}
-\newcommand{\theoremref}[1]{\autoref{#1} on p.\,\pageref*{#1}}
+% Fix the height of citation link underlines.
+% Also, biblatex really doesn't want to support Unicode citation labels, but I will not be beaten.
+\newcommand{\linkstrut}{\rule[-0.4ex]{0ex}{\fontcharht\font`X}}
+\DeclareFieldFormat{labelalpha}{\linkstrut\smash{\StrSubstitute{#1}{MAEA2010}{MAEÁ2010}}}
+\DeclareFieldFormat{postnote}{\linkstrut\smash{#1}}
+\let\oldcite\cite
+\renewcommand{\cite}[2][]{\raisebox{0ex}{\oldcite[{#1}]{#2}}}
+
+\let\oldfootnote\footnote
+\renewcommand{\footnote}[1]{\hairspace{\oldfootnote{#1}}}
+\newcommand{\footnotewithlabel}[2]{\hairspace\oldfootnote{\label{#1}{#2}}}
+
+\newcommand{\crossref}[1]{\raisebox{0ex}{\autoref{#1}}\hspace{0.2em}\emph{`\nameref*{#1}\kern -0.05em'} on p.\,\pageref*{#1}}
+\newcommand{\theoremref}[1]{\raisebox{0ex}{\autoref{#1}} on p.\,\pageref*{#1}}
+\newcommand{\footnoteref}[1]{\hairspace\raisebox{0ex}{\cref{#1}}}
+
+\newcommand{\autorefprefix}{\linkstrut\S\!}
+\renewcommand{\sectionautorefname}{\autorefprefix}
+\renewcommand{\subsectionautorefname}{\autorefprefix}
+\renewcommand{\subsubsectionautorefname}{\autorefprefix}
+\renewcommand{\paragraphautorefname}{\autorefprefix}
+\renewcommand{\subparagraphautorefname}{\autorefprefix}
+
+%\let\oldhref\href
+%\renewcommand{\href}[2]{\raisebox{0ex}{\oldhref{#1}{\linkstrut\smash{#2}}}}
+%\let\oldurl\url
+%\renewcommand{\url}[1]{\href{#1}{\nolinkurl{#1}}}
 
 % <https://tex.stackexchange.com/a/60212/78411>
 \newcommand{\subsubsubsection}[1]{\paragraph{#1}\mbox{}\\}
@@ -164,6 +194,7 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 
 \newcommand{\introlist}{\needspace{15ex}}
 \newcommand{\introsection}{\needspace{35ex}}
+\newcommand{\intropart}{\needspace{55ex}}
 
 \mathchardef\mhyphen="2D
 
@@ -212,6 +243,8 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 \newcommand{\mhspace}[1]{\mbox{\hspace{#1}}}
 \newcommand{\tab}{\hspace{1.5em}}
 
+\newcommand{\raisedstrut}{\raisebox{0.3ex}{\strut}}
+
 \newcommand{\plus}{\hairspace +\hairspace}
 \newcommand{\vv}{\hspace{0.071em}\varv\hspace{0.064em}}
 \newcommand{\varvv}{\varv\kern 0.02em\varv}
@@ -271,6 +304,7 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 \newcommand{\Sprout}{\termbf{Sprout}}
 \newcommand{\SproutOrZcash}{\notsprout{\Sprout}\sprout{\Zcash}}
 \newcommand{\SproutOrNothing}{\notsprout{\Sprout}}
+\newcommand{\pSproutOrNothing}{\notsprout{(\Sprout)}}
 \newcommand{\Sapling}{\termbf{Sapling}}
 \newcommand{\NUZero}{\termbf{Overwinter}}
 \newcommand{\Bitcoin}{\termbf{Bitcoin}}
@@ -287,6 +321,9 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 \newcommand{\MAY}{\conformance{MAY}}
 \newcommand{\ALLCAPS}{\conformance{ALL CAPS}}
 
+\newcommand{\collisionResistant}{collision\hyp resistant }
+\newcommand{\collisionResistance}{collision\hyp resistance }
+
 \newcommand{\note}{\term{note}}
 \newcommand{\notes}{\term{notes}}
 \newcommand{\Note}{\titleterm{Note}}
@@ -426,6 +463,7 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 \newcommand{\auxiliaryInputs}{\term{auxiliary inputs}}
 \newcommand{\fullValidator}{\term{full validator}}
 \newcommand{\fullValidators}{\term{full validators}}
+\newcommand{\consensusRuleChange}{\term{consensus rule change}}
 \newcommand{\anchor}{\term{anchor}}
 \newcommand{\anchors}{\term{anchors}}
 \newcommand{\block}{\term{block}}
@@ -531,7 +569,7 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 \newcommand{\merkleInternalNode}{\term{internal node}}
 \newcommand{\merkleInternalNodes}{\term{internal nodes}}
 \newcommand{\MerkleInternalNodes}{\term{Internal nodes}}
-\newcommand{\merklePath}{\term{path}}
+\newcommand{\merklePath}{\term{Merkle tree path}}
 \newcommand{\merkleLayer}{\term{layer}}
 \newcommand{\merkleLayers}{\term{layers}}
 \newcommand{\merkleIndex}{\term{index}}
@@ -637,7 +675,7 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 \newcommand{\ascii}[1]{\textbf{``\texttt{#1}''}}
 \newcommand{\Justthebox}[2][-1.8ex]{\raisebox{#1}{\;\usebox{#2}\;}}
 \newcommand{\setof}[1]{\{{#1}\}}
-\newcommand{\powerset}[1]{\raisebox{-0.25ex}{\scalebox{1.25}{$\mathscr{P}$}}\kern -0.35em\left(\strut{#1}\right)}
+\newcommand{\powerset}[1]{\raisebox{-0.28ex}{\scalebox{1.25}{$\mathscr{P}$}}\kern -0.35em\left(\strut{#1}\right)}
 \newcommand{\barerange}[2]{{{#1}\,..\,{#2}}}
 \newcommand{\range}[2]{\setof{\barerange{#1}{#2}}}
 \newcommand{\rangenozero}[2]{\range{#1}{#2} \difference \setof{0}}
@@ -648,18 +686,23 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 \newcommand{\trunc}[1]{\mathsf{trunc}\!\left({#1}\right)}
 \newcommand{\ceiling}[1]{\mathsf{ceiling}\!\left({#1}\right)}
 \newcommand{\sceiling}[1]{\mathsf{ceiling}\left({#1}\right)}
-\newcommand{\vsum}[2]{\smashoperator[r]{\sum_{#1}^{#2}}}
-\newcommand{\vproduct}[2]{\smashoperator[r]{\prod_{#1}^{#2}}}
-\newcommand{\vxor}[2]{\smashoperator[r]{\bigoplus_{#1}^{#2}}}
+\newcommand{\vop}[3]{\,\raisebox{0.29ex}{\scalebox{0.89}{$\smashoperator[r]{#3_{#1}^{#2}}$\,}}}
+\newcommand{\sop}[3]{\!\scalebox{0.89}{$\scalebox{1.404}{$\strut#3$}_{#1}^{#2}$}}
+\newcommand{\vsum}[2]{\vop{#1}{#2}{\sum}}
+\newcommand{\ssum}[2]{\sop{#1}{#2}{\sum}}
+\newcommand{\vproduct}[2]{\vop{#1}{#2}{\prod}}
+\newcommand{\sproduct}[2]{\sop{#1}{#2}{\prod}}
+\newcommand{\vxor}[2]{\vop{#1}{#2}{\bigoplus}}
+\newcommand{\sxor}[2]{\sop{#1}{#2}{\bigoplus}}
 \newcommand{\xor}{\oplus}
 \newcommand{\band}{\binampersand}
 \newcommand{\suband}{\raisebox{-0.6ex}{\kern-0.06em\scalebox{0.65}{$\binampersand$}}}
-\newcommand{\bchoose}{\,?\,}
+\newcommand{\bchoose}{\;\scalebox{1.2}[1]{\textsf{?}}\;}
 \newcommand{\mult}{\cdot}
 \newcommand{\smult}{\!\cdot\!}
 \newcommand{\scalarmult}[2]{\boldsymbol{[}{#1}\boldsymbol{]}\,{#2}}
-\newcommand{\rightarrowR}{\buildrel{\scriptstyle\mathrm{R}}\over\rightarrow}
-\newcommand{\leftarrowR}{\buildrel{\scriptstyle\mathrm{R}}\over\leftarrow}
+\newcommand{\rightarrowR}{\phantom{(}\smash{\buildrel{\scriptstyle\mathrm{R}}\over\rightarrow}\phantom{)}}
+\newcommand{\leftarrowR}{\phantom{(}\smash{\buildrel{\scriptstyle\mathrm{R}}\over\leftarrow}\phantom{)}}
 \newcommand{\union}{\cup}
 \newcommand{\intersection}{\cap}
 \newcommand{\difference}{\setminus}
@@ -686,7 +729,7 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 \newcommand{\BlakeTwosGeneric}{\mathsf{BLAKE2s}}
 \newcommand{\BlakeTwos}[1]{\mathsf{BLAKE2s\kern 0.05em\mhyphen{#1}}}
 \newcommand{\BlakeTwosOf}[2]{\BlakeTwos{#1}\!\left({#2}\right)}
-\newcommand{\CRHivk}{\mathsf{CRH}^{\InViewingKey}}
+\newcommand{\CRHivk}{\mathsf{CRH^{\InViewingKey}}}
 \newcommand{\CRHivkText}{\texorpdfstring{$\CRHivk$}{CRHivk}}
 \newcommand{\CRHivkOutput}{\CRHivk\mathsf{.Output}}
 \newcommand{\CRHivkOutputLength}{\ell_{\InViewingKey}}
@@ -1054,7 +1097,7 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 \newcommand{\encCiphertexts}{\mathtt{encCiphertexts}}
 \newcommand{\randomSeed}{\mathtt{randomSeed}}
 \newcommand{\spendAuthSig}{\mathtt{spendAuthSig}}
-\newcommand{\Varies}{\textit{Varies}}
+\newcommand{\Varies}{\textit{\!Varies}}
 \newcommand{\heading}[1]{\multicolumn{1}{c|}{#1}}
 \newcommand{\type}[1]{\texttt{#1}}
 \newcommand{\compactSize}{\type{compactSize uint}}
@@ -1228,8 +1271,8 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 \newcommand{\Base}{\mathsf{Base}}
 \newcommand{\Addend}{\mathsf{Addend}}
 \newcommand{\Sum}{\mathsf{Sum}}
-\newcommand{\ainv}{a_{\mathsf{inv}}}
 \newcommand{\Ainv}{A_{\mathsf{inv}}}
+\newcommand{\Inv}[1]{{#1}_{\mathsf{inv}}}
 
 \newcommand{\repr}{\mathsf{repr}}
 \newcommand{\abst}{\mathsf{abst}}
@@ -1283,7 +1326,7 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 % Consensus rules
 
 \newcommand{\consensusrule}[1]{\needspace{3ex}\subparagraph{Consensus rule:}{#1}}
-\newenvironment{consensusrules}{\introlist\subparagraph{Consensus rules:}\begin{itemize}}{\end{itemize}}
+\newenvironment{consensusrules}{\vspace{-4ex}\introlist\subparagraph{Consensus rules:}\begin{itemize}}{\end{itemize}}
 \newcommand{\sproutspecificitem}[1]{\item \sproutspecific{#1}}
 \newcommand{\sproutonlyitem}[1]{\item \sproutonly{#1}}
 \newcommand{\saplingonwarditem}[1]{\sapling{\item {[\Sapling onward]}\, {#1}}}
@@ -1308,6 +1351,7 @@ electronic commerce and payment, financial privacy, proof of work, zero knowledg
 \newcommand{\nuzeroonwardpnote}[1]{\nuzero{[\NUZero onward]\,\,} \textbf{Note:\,} {#1}}
 \newcommand{\fact}[1]{\subparagraph{Fact:}{#1}}
 \newcommand{\facts}[1]{\subparagraph{Facts:}{#1}}
+\newcommand{\snarkcondition}[1]{\vspace{-3.5ex}\introlist\subparagraph{#1}}
 
 \newcommand{\affiliation}{\hairspace$^\dagger$\;}
 
@@ -1438,8 +1482,8 @@ used notwithstanding.}
 \introsection
 Value in \Zcash is either \transparent or \shielded. Transfers of \transparent
 value work essentially as in \Bitcoin and have the same privacy properties.
-\xShielded value is carried by \notes\hairspace\footnote{\label{notesandnullifiers}
-In \Zerocash \cite{BCG+2014}, \notes were called \quotedterm{coins}, and \nullifiers
+\xShielded value is carried by \notes\footnotewithlabel{notesandnullifiers}{In
+\Zerocash \cite{BCG+2014}, \notes were called \quotedterm{coins}, and \nullifiers
 were called \quotedterm{serial numbers}.},
 which specify an amount and \sprout{a \payingKey. The \payingKey is part of}
 \notsprout{(indirectly)}
@@ -1448,7 +1492,7 @@ As in \Bitcoin, this is associated with a private key that can be used to
 spend \notes sent to the address; in \Zcash this is called a \spendingKey.
 
 To each \note there is cryptographically associated a \noteCommitment, and
-a \nullifier\hairspace\cref{notesandnullifiers} (so that there is a 1:1:1 relation
+a \nullifier\footnoteref{notesandnullifiers} (so that there is a 1:1:1 relation
 between \notes, \noteCommitments, and \nullifiers). Computing the \nullifier
 requires the associated private \spendingKey\sapling{ (or the \fullViewingKey for \Sapling \notes)}.
 It is infeasible to correlate the \noteCommitment with the corresponding
@@ -1463,7 +1507,7 @@ A \transaction can contain \transparent inputs, outputs, and scripts, which all
 work as in \Bitcoin \cite{Bitc-Protocol}.
 \sprout{
 It also contains a sequence of zero or more \joinSplitDescriptions.
-Each of these describes a \joinSplitTransfer\hairspace\footnote{
+Each of these describes a \joinSplitTransfer\footnote{
 \joinSplitTransfers in \Zcash generalize \quotedterm{Mint} and \quotedterm{Pour}
 \transactions in \Zerocash; see \crossref{trstructure} for differences.}
 which takes in a \transparent value and up to two input \notes, and produces a
@@ -1564,6 +1608,7 @@ one valid \nullifier, and so attempting to spend a \note twice would reveal the
 \nullifier twice, which would cause the second \transaction to be rejected.
 
 
+\introsection
 \section{Notation}
 
 $\bit$ means the type of bit values, i.e.\ $\setof{0, 1}$.
@@ -1591,7 +1636,7 @@ $f(x, y)$ can also be written $f_x(y)$.
 
 \notsprout{
 $\fun{x \typecolon T}{e_x \typecolon U}$ means the function of type $T \rightarrow U$
-mapping formal parameter $x$ to $e_x$ (an expression depending on $x$).
+mapping formal parameter $x$ to $e_x$ (an expression depending on~$x$).
 The types $T$ and $U$ are always explicit.
 
 $\powerset{T}$ means the powerset of $T$.
@@ -1635,11 +1680,11 @@ $\zeros{\ell}$ means the sequence of $\ell$ zero bits.
 
 $a..b$, used as a subscript, means the sequence of values
 with indices $a$ through $b$ inclusive. For example,
-$\AuthPublicNew{\allNew}$ means the sequence $[\AuthPublicNew{\mathrm{1}},
-\AuthPublicNew{\mathrm{2}}, ...\,\AuthPublicNew{\NNew}]$.
+$\AuthPublicNew{\allNew}$ means the sequence
+$\vphantom{\mathsf{a_a}}\smash{[\AuthPublicNew{\mathrm{1}}, \AuthPublicNew{\mathrm{2}}, ...\,\AuthPublicNew{\NNew}]}$.
 (For consistency with the notation in \cite{BCG+2014} and in \cite{BK2016},
 this specification uses 1-based indexing and inclusive ranges,
-notwithstanding the compelling arguments to the contrary made in
+\raisedstrut notwithstanding the compelling arguments to the contrary made in
 \cite{EWD-831}.)
 
 $\range{a}{b}$ means the set or type of integers from $a$ through
@@ -1680,7 +1725,7 @@ $a \xor b$ means the bitwise-exclusive-or of $a$ and $b$,
 and $a \band b$ means the bitwise-and of $a$ and $b$. These are
 defined on integers or bit sequences according to context.
 
-$\vsum{i=1}{\mathrm{N}} a_i$ means the sum of $a_{\allN{}}$.\;
+$\!\vsum{i=1}{\mathrm{N}} a_i$ means the sum of $a_{\allN{}}$.\;
 \notsprout{$\vproduct{i=1}{\mathrm{N}} a_i$ means the product
 of $a_{\allN{}}$.\;}
 $\vxor{i=1}{\mathrm{N}} a_i$ means the bitwise exclusive-or of $a_{\allN{}}$.
@@ -1720,7 +1765,7 @@ and rational constants $\FoundersFraction$, $\PoWMaxAdjustDown$, and
 $\PoWMaxAdjustUp$ will also be defined in that section.
 
 
-\introsection
+\intropart
 \section{Concepts}
 
 \subsection{Payment Addresses and Keys} \label{addressesandkeys}
@@ -1737,8 +1782,8 @@ Arrows point from a component to any other component(s) that can be derived
 from it.
 
 \begin{center}
-\sprout{\includegraphics[scale=.7]{key_components}}
-\sapling{\includegraphics[scale=.5]{key_components_sapling}}
+\sprout{\includegraphics[scale=.7,interpolate]{key_components}}
+\sapling{\includegraphics[scale=.5,interpolate]{key_components_sapling}}
 \end{center}
 
 \sproutspecific{
@@ -1810,6 +1855,7 @@ spendable by the recipient who holds the \spendingKey corresponding
 to a given \paymentAddress.
 } %notsprout
 
+\introlist
 A \SproutOrNothing \note is a tuple $\changed{(\AuthPublic,
 \Value, \NoteAddressRand, \NoteCommitRand)}$, where:
 \begin{itemize}
@@ -1825,6 +1871,7 @@ A \SproutOrNothing \note is a tuple $\changed{(\AuthPublic,
         is a random \commitmentTrapdoor as defined in \crossref{abstractcommit}.
 \end{itemize}
 
+\introlist
 Let $\NoteTypeSprout$ be the type of a \SproutOrNothing \note, i.e.
 \begin{formulae}
   \item $\NoteTypeSprout := \changed{\PRFOutput \times \range{0}{\MAXMONEY} \times \PRFOutput
@@ -1833,6 +1880,7 @@ Let $\NoteTypeSprout$ be the type of a \SproutOrNothing \note, i.e.
 
 \sapling{
 \vspace{2ex}
+\introlist
 A \Sapling \note is a tuple $(\Diversifier, \DiversifiedTransmitPublic,
 \Value, \NoteCommitRand)$, where:
 \begin{itemize}
@@ -1846,6 +1894,7 @@ A \Sapling \note is a tuple $(\Diversifier, \DiversifiedTransmitPublic,
         is a random \commitmentTrapdoor as defined in \crossref{abstractcommit}.
 \end{itemize}
 
+\introlist
 Let $\NoteTypeSapling$ be the type of a \Sapling \note, i.e.
 
 \begin{formulae}
@@ -1862,8 +1911,10 @@ used by the \zeroKnowledgeProof when the \note is spent, to check that it exists
 on the \blockchain.
 
 \vspace{2ex}
-A \notsprout{\Sprout} \noteCommitment on a \note
+\introlist
+A \SproutOrNothing{} \noteCommitment on a \note
 $\NoteTuple{} = \changed{(\AuthPublic, \Value, \NoteAddressRand, \NoteCommitRand)}$ is computed as
+
 \begin{formulae}
   \item $\NoteCommitmentSprout(\NoteTuple{}) =
           \NoteCommitSprout{\NoteCommitRand}(\AuthPublic, \Value, \NoteAddressRand)$,
@@ -1871,11 +1922,13 @@ $\NoteTuple{} = \changed{(\AuthPublic, \Value, \NoteAddressRand, \NoteCommitRand
 \vspace{-1.5ex}
 where $\NoteCommitSprout{}$ is instantiated in \crossref{concretesproutcommit}.
 
+
 \sapling{
 \vspace{2ex}
+\introlist
 Let $\GroupJHash{}$ and $U$ be as defined in \crossref{concretegrouphashjubjub}.
 
-A \Sapling \noteCommitment on a \note
+A \Sapling{} \noteCommitment on a \note
 $\NoteTuple{} = (\Diversifier, \DiversifiedTransmitPublic, \Value, \NoteCommitRand)$ is computed as
 
 \begin{formulae}
@@ -1952,8 +2005,8 @@ of one or valid \blocks consistent with consensus rules, is called a
 \validBlockchain.
 
 Each \block in a \blockchain has a \blockHeight. The \blockHeight of the
-\genesisBlock is 0, and the \blockHeight of each subsequent \block in the
-\blockchain increments by 1.
+\genesisBlock is $0$, and the \blockHeight of each subsequent \block in the
+\blockchain increments by $1$.
 
 In order to choose the \bestValidBlockchain in its view of the
 overall \block tree, a node sums the work, as defined in \crossref{workdef}, of
@@ -2127,7 +2180,7 @@ for the whole \transaction to balance.
 \subsection{\NoteCommitmentTrees} \label{merkletree}
 
 \begin{center}
-\includegraphics[scale=.4]{incremental_merkle}
+\includegraphics[scale=.4,interpolate]{incremental_merkle}
 \end{center}
 
 \sapling{\todo{The commitment indices in the above diagram should be zero-based to reflect the \notePosition{}.}}
@@ -2191,6 +2244,7 @@ included in this \block. The \coinbaseTransaction must also pay the \foundersRew
 as described in \crossref{foundersreward}.
 
 
+\intropart
 \section{Abstract Protocol}
 
 \subsection{Abstract Cryptographic Schemes}
@@ -2199,7 +2253,7 @@ as described in \crossref{foundersreward}.
 
 \sprout{
 $\MerkleCRH \typecolon \MerkleHashSprout \times \MerkleHashSprout \rightarrow \MerkleHashSprout$
-is a collision-resistant \hashFunction used in \crossref{merklepath}.
+is a \collisionResistant \hashFunction used in \crossref{merklepath}.
 It is instantiated in \crossref{merklecrh}.
 } %sprout
 \notsprout{
@@ -2210,14 +2264,14 @@ $\MerkleCRHSapling \typecolon \MerkleLayerSapling \times \MerkleHashSapling \tim
 \rightarrow \MerkleHashSapling$
 }
 are \hashFunctions used in \crossref{merklepath}.
-\sapling{$\MerkleCRHSapling$ is collision-resistant on all its arguments, and}
-$\MerkleCRHSprout$ is collision-resistant except on its first argument.
+\sapling{$\MerkleCRHSapling$ is \collisionResistant on all its arguments, and}
+$\MerkleCRHSprout$ is \collisionResistant except on its first argument.
 Both of these functions are instantiated in \crossref{merklecrh}.
 } %notsprout
 
 \changed{
 $\hSigCRH{} \typecolon \bitseq{\RandomSeedLength} \times \typeexp{\PRFOutput}{\NOld} \times \JoinSplitSigPublic \rightarrow \hSigType$
-is a collision-resistant \hashFunction used in \crossref{joinsplitdesc}.
+is a \collisionResistant \hashFunction used in \crossref{joinsplitdesc}.
 It is instantiated in \crossref{hsigcrh}.
 
 $\EquihashGen{} \typecolon (n \typecolon \PosInt) \times \PosInt \times \byteseqs \times \PosInt \rightarrow \bitseq{n}$
@@ -2259,7 +2313,9 @@ $\PRFexpand{} $&$\typecolon\; \bitseq{\SpendingKeyLength} $&$\times\; \range{0}{
 $\PRFnr{}     $&$\typecolon\; \bitseq{\ellJ}              $&$\times\; \bitseq{\ellJ} $& &$\rightarrow \GF{\ParamJ{r}} $
 \end{tabular}
 
-$\PRFexpand{}$ is used in \crossref{saplingkeycomponents}; $\PRFnr{}$ is used in \crossref{spendstatement}.
+$\PRFexpand{}$ is used in \crossref{saplingkeycomponents}.
+
+$\PRFnr{}$ is used in \crossref{spendstatement}.
 } %sapling
 
 \sprout{They}\notsprout{All of these \pseudoRandomFunctions} are instantiated in \crossref{concreteprfs}.
@@ -2268,7 +2324,7 @@ $\PRFexpand{}$ is used in \crossref{saplingkeycomponents}; $\PRFnr{}$ is used in
   \item Security definitions for \pseudoRandomFunctions are given in \cite[section 4]{BDJR2000}.
   \item In addition to being \pseudoRandomFunctions, it is required that
         $\PRFnf{x}$,\changed{ $\PRFaddr{x}$,\sprout{ and} $\PRFrho{x}$}\sapling{, and $\PRFnr{x}$}
-        be collision-resistant across all $x$ --- i.e.\ finding $(x, y) \neq (x', y')$
+        be \collisionResistant across all $x$ --- i.e.\ finding $(x, y) \neq (x', y')$
         such that $\PRFnf{x}(y) = \PRFnf{x'}(y')$ should not be feasible\changed{, and
         similarly for $\PRFaddr{}$ and $\PRFrho{}$\sapling{ and $\PRFnr{}$}}.
 \end{securityrequirements}
@@ -2383,8 +2439,7 @@ If the adversary's advantage is insignificant, then the asymmetric encryption sc
 constructed from $\KA$, $\KDF$ and $\Sym$ in \crossref{inband} will be key-private
 as defined in \cite{BBDP2001}.
 
-\pnote{
-The given definition only requires ciphertexts to be indistinguishable
+\pnote{The given definition only requires ciphertexts to be indistinguishable
 between \transmissionKeys that are outputs of $\KADerivePublic$ (which
 includes all keys generated as in \crossref{sproutkeycomponents}). If a
 \transmissionKey not in that range is used, it may be distinguishable.
@@ -2411,6 +2466,7 @@ such that for any key pair $(\sk, \vk) \leftarrowR \SigGen()$, and
 any $m \typecolon \SigMessage$ and $s \typecolon \SigSignature \leftarrowR \SigSign{\sk}(m)$,
 $\SigVerify{\vk}(m, s) = 1$.
 
+\introlist
 \Zcash uses \sprout{two}\sapling{three} signature schemes:
 
 \begin{itemize}
@@ -2484,14 +2540,14 @@ valid $\Sig$ key pair, then:
           \item $\left(\SigRandomizePublic(\pk, \SigRandomness), \SigRandomizePrivate(\sk, \SigRandomness)\right) :
                 \SigRandomness \leftarrowR \SigRandom$
         \end{formulae}
-        \vspace{-0.ex} is identical to the distribution of $\SigGen()$.
+        \vspace{-0.5ex} is identical to the distribution of $\SigGen()$.
   \item $\left(\SigRandomizePublic(\pk, \SigRandomnessId), \SigRandomizePrivate(\sk, \SigRandomnessId)\right) = (\pk, \sk)$.
 \end{itemize}
 
 The following security requirement for such signature schemes is based on that
 given in \cite[section 3]{FKMSSS2016}. Note that we require Strong Unforgeability
 with Re-randomized Keys, not Existential Unforgeability with Re-randomized Keys
-(the latter is just called ``Unforgeability under Re-randomized Keys'' in
+(the latter is called ``Unforgeability under Re-randomized Keys'' in
 \cite[Definition 8]{FKMSSS2016}). Unlike the case for $\JoinSplitSig$, we require
 security under adaptive chosen message attack with multiple messages signed using
 a given key. (Although each \note uses a different re-randomized key pair, the same
@@ -2589,12 +2645,10 @@ A \representedGroup $\GroupG{}$ consists of:
         and additive identity $\ZeroG{}$;
   \item a generator $\GenG{}$ of the subgroup of $\GroupG{}$ of order $\ParamG{r}$;
   \item a bit-length parameter $\ellG{} \typecolon \Nat$;
-  \item a representation function $\reprG{} \typecolon
-        \GroupG{} \rightarrow \bitseq{\ellG{}}$;
-  \item an abstraction function $\abstG{} \typecolon
-        \bitseq{\ellG{}} \rightarrow \GroupG{} \union \setof{\bot}$;
+  \item a representation function \smash{$\reprG{} \typecolon \GroupG{} \rightarrow \bitseq{\ellG{}}$};
+  \item an abstraction function \smash{$\abstG{} \typecolon \bitseq{\ellG{}} \rightarrow \GroupG{} \union \setof{\bot}$};
 \end{itemize}
-\vspace{-2ex}
+\vspace{-1.5ex}
 such that $\abstG{}$ is the left inverse of $\reprG{}$, i.e.
 for all $P \in \GroupG{}$, $\abstG{}(\reprG{}(P)) = P$, and
 for all $S$ not in the image of $\reprG{}$, $\abstG{}(S) = \bot$.
@@ -2644,7 +2698,7 @@ with the following security requirement.
 For a randomly selected member $\GroupGHash{\CRS}$ of the family, it is infeasible to find
 a sequence of distinct inputs $m_{\alln} \typecolon \typeexp{\bitseq{\ell}}{n}$
 and a sequence of nonzero scalars $x_{\alln} \typecolon \typeexp{\GFstar{\ParamG{r}}}{n}$
-such that $\vsum{i = 1}{n}\left(\scalarmult{x_i}{\GroupGHash{\CRS}(m_i)}\right) = \ZeroG{}$.
+such that $\ssum{i = 1}{n}\!\left(\scalarmult{x_i}{\GroupGHash{\CRS}(m_i)}\right) = \ZeroG{}$.
 }
 
 \begin{pnotes}
@@ -2674,7 +2728,7 @@ A \representedPairing $\GroupP{}$ consists of:
 \begin{itemize}
   \item a group order parameter $\ParamP{r} \typecolon \PosInt$ which must be prime;
   \item two \representedGroups $\GroupP{1..2}$, both of order $\ParamP{r}$;
-  \item a group $\GroupP{T}$ of order $\ParamP{r}$, written multiplicatively with operation
+  \item a group $\GroupP{T}$ of order $\ParamP{r}$, written multiplicatively with operation\,
         $\mult \typecolon \GroupP{T} \times \GroupP{T} \rightarrow \GroupP{T}$
         and multiplicative identity $\ParamP{\mathbf{1}}$;
   \item a pairing function
@@ -2683,11 +2737,11 @@ A \representedPairing $\GroupP{}$ consists of:
 
         \begin{itemize}
           \item (Bilinearity)\; for all $a, b \typecolon \GFstar{r}$,
-                $P \typecolon \GroupP{1}$, and $Q \typecolon \GroupP{2}$,
-                $\PairingP(\scalarmult{a}{P}, \scalarmult{b}{Q}) = \PairingP(P, Q)^{a \mult b}$, and
+                $P \typecolon \GroupP{1}$, and $Q \typecolon \GroupP{2}$,\;
+                $\PairingP(\scalarmult{a}{P}, \scalarmult{b}{Q}) = \PairingP(P, Q)^{a \mult b}$;\, and
           \item (Nondegeneracy)\; there does not exist $P \typecolon \GroupP{1} \setminus \ZeroP{1}$
-                such that for all $Q \typecolon \GroupP{2},
-                \PairingP(P, Q) = \ParamP{\mathbf{1}}$;
+                such that for all $Q \typecolon \GroupP{2},\;
+                \PairingP(P, Q) = \ParamP{\mathbf{1}}$.
         \end{itemize}
 \end{itemize}
 
@@ -3003,8 +3057,7 @@ where
   \item $\rt \typecolon \MerkleHashSapling$ is an \anchor, as defined in
         \crossref{blockchain}, for the output \treestate of a previous \block.
   \item $\nf \typecolon \bitseq{\ellJ}$ is the \nullifier for the input \note;
-  \item $\ProofSpend \typecolon \SpendProof$ is
-        the \zeroKnowledgeProof for the \spendStatement;
+  \item $\ProofSpend \typecolon \SpendProof$ is the \zeroKnowledgeProof for the \spendStatement;
   \item $\spendAuthSig \typecolon \SpendAuthSigSignature$ is a signature authorizing this spend.
 \end{itemize}
 
@@ -3101,6 +3154,7 @@ the private \joinSplitSigningKey:
 Then the encoded \transaction including $\joinSplitSig$ is submitted to the network.
 
 
+\introlist
 \subsubsection{\DummyNotes\notsprout{ (\Sprout)}} \label{dummynotes}
 
 The fields in a \joinSplitDescription allow for $\NOld$ input \notes, and
@@ -3129,6 +3183,7 @@ A \dummy output \note is constructed as normal but with zero value, and
 sent to a random \paymentAddress.
 
 \sapling{
+\introlist
 \subsubsection{Sending \Notes{} (\Sapling)} \label{saplingsend}
 
 In order to send \shielded value, the sender constructs a \transaction
@@ -3203,6 +3258,7 @@ The encoded \transaction is submitted to the network.
 } %sapling
 
 
+\introsection
 \subsection{Merkle path validity} \label{merklepath}
 
 \sprout{
@@ -3220,9 +3276,10 @@ The following discussion applies independently to the \Sprout and \Sapling \note
 } %notsprout
 
 Each \merkleNode in the \incrementalMerkleTree is associated with a \merkleHash,
-which is a bit sequence. The \merkleLayer numbered $h$, counting from
-\merkleLayer $0$ at the \merkleRoot, has $2^h$ \merkleNodes with \merkleIndices
-$0$ to $2^h-1$ inclusive.
+which is a bit sequence.
+
+The \merkleLayer numbered $h$, counting from \merkleLayer $0$ at the \merkleRoot,
+has $2^h$ \merkleNodes with \merkleIndices $0$ to $2^h-1$ inclusive.
 
 Let $\MerkleNode{h}{i}$ be the \merkleHash associated with the \merkleNode at
 \merkleIndex $i$ in \merkleLayer $h$.
@@ -3278,6 +3335,7 @@ $\SIGHASHALL$.
 }
 }
 
+\vspace{3ex}
 Let $\dataToBeSigned$ be the hash of the \transaction{} \changed{using the $\SIGHASHALL$
 \sighashType}. \changed{This \emph{excludes} all of the $\scriptSig$ fields in
 the non-\Zcash-specific parts of the \transaction.}
@@ -3298,8 +3356,10 @@ $\JoinSplitSigVerify{\text{\small\joinSplitPubKey}}(\dataToBeSigned, \joinSplitS
 % FIXME: distinguish pubkey and signature from their encodings.
 }
 
-Let $\hSig$ be computed as specified in \crossref{joinsplitdesc}, and let
-$\PRFpk{}$ be as defined in \crossref{abstractprfs}.
+\introsection
+Let $\hSig$ be computed as specified in \crossref{joinsplitdesc}.
+
+Let $\PRFpk{}$ be as defined in \crossref{abstractprfs}.
 
 For each $i \in \setofOld$, the creator of a \joinSplitDescription calculates
 $\h{i} = \PRFpk{\AuthPrivateOld{i}}(i, \hSig)$.
@@ -3319,6 +3379,7 @@ to $\joinSplitPubKey$ to sign this \transaction.
 
 A \joinSplitTransfer can be seen, from the perspective of the \transaction, as
 an input \changed{and an output simultaneously}.
+
 \changed{$\vpubOld$ takes value from the \transparentValuePool and}
 $\vpubNew$ adds value to the \transparentValuePool. As a result, \changed{$\vpubOld$ is
 treated like an \emph{output} value, whereas} $\vpubNew$ is treated like an
@@ -3327,8 +3388,9 @@ treated like an \emph{output} value, whereas} $\vpubNew$ is treated like an
 \changed{
 Unlike original \Zerocash \cite{BCG+2014}, \Zcash does not have
 a distinction between Mint and Pour operations. The addition of $\vpubOld$ to a
-\joinSplitDescription subsumes the functionality of both Mint and Pour. Also,
-a difference in the number of real input \notes does not by itself cause two
+\joinSplitDescription subsumes the functionality of both Mint and Pour.
+
+Also, a difference in the number of real input \notes does not by itself cause two
 \joinSplitDescriptions to be distinguishable.
 
 As stated in \crossref{joinsplitdesc}, either $\vpubOld$ or $\vpubNew$ \MUST be zero.
@@ -3347,7 +3409,9 @@ according to client implementation.
 A \transaction that contains one or more
 \joinSplitDescriptions\sapling{ or \spendDescriptions}, when entered
 into the \blockchain, appends to the \noteCommitmentTree with all constituent
-\noteCommitments. All of the constituent \nullifiers are also entered into the
+\noteCommitments.
+
+All of the constituent \nullifiers are also entered into the
 \nullifierSet of the associated \treestate. A \transaction is not valid if it
 attempts to add a \nullifier to the \nullifierSet that already exists in the set.
 
@@ -3367,22 +3431,22 @@ $\MixingPedersenHash$ is defined in \crossref{concretemixinghash}.
 
 Let $\PRFnf{}{}$\sapling{ and $\PRFnr{}{}$} be as instantiated in \crossref{concreteprfs}.
 
-\sprout{The \nullifier of a \note}\notsprout{For a \Sprout \note, the \nullifier}
+\sprout{The \nullifier of a \note}\notsprout{For a \Sprout{} \note, the \nullifier}
 is derived as $\PRFnf{\AuthPrivate}(\NoteAddressRand)$.
 
 % TODO: \scalarmults should only be in the concrete section
 \sapling{
-For a \Sapling \note, the \nullifier is derived as
+For a \Sapling{} \note, the \nullifier is derived as
 $\scalarmult{\PRFnr{\AuthProvePublic}(\NoteAddressRand)}{\AuthSignPublic}$.
 } %sapling
 
-\introsection
 
+\intropart
 \subsection{\ZkSNARKStatements} \label{snarkstatements}
 
-\subsubsection{\JoinSplitStatement{} \notsprout{(\Sprout)}} \label{joinsplitstatement}
+\subsubsection{\JoinSplitStatement{} \pSproutOrNothing} \label{joinsplitstatement}
 
-A valid instance of $\ProofJoinSplit$ assures that given a \term{primary input}:
+A valid instance of $\ProofJoinSplit$ assures that given a \primaryInput:
 
 \begin{formulae}
   \item $(\rt \typecolon \MerkleHashSprout,\\
@@ -3391,12 +3455,11 @@ A valid instance of $\ProofJoinSplit$ assures that given a \term{primary input}:
    \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})$,
+   \hparen\h{\allOld} \typecolon \smash{\typeexp{\PRFOutput}{\NOld})}$,
 \end{formulae}
-
+\vspace{-1ex}
 \introlist
-the prover knows an \term{auxiliary input}:
-
+the prover knows an \auxiliaryInput:
 \begin{formulae}
   \item $(\treepath{\allOld} \typecolon \typeexp{\typeexp{\MerkleHashSprout}{\MerkleDepthSprout}
                                                  \times \NotePositionTypeSprout}{\NOld},\\
@@ -3406,62 +3469,62 @@ the prover knows an \term{auxiliary input}:
    \hparen\changed{\NoteAddressPreRand \typecolon \bitseq{\NoteAddressPreRandLength},}\\
    \hparen\changed{\EnforceMerklePath{\allOld} \typecolon \bitseq{\NOld}})$,
 \end{formulae}
-
+\vspace{-1ex}
 \introlist
 where:
-
 \begin{formulae}
   \item for each $i \in \setofOld$: $\nOld{i} = (\AuthPublicOld{i},
 \vOld{i}, \NoteAddressRandOld{i}, \NoteCommitRandOld{i})$;
   \item for each $i \in \setofNew$: $\nNew{i} = (\AuthPublicNew{i},
 \vNew{i}, \NoteAddressRandNew{i}, \NoteCommitRandNew{i})$
 \end{formulae}
-
 \introlist
 such that the following conditions hold:
+\vspace{2ex}
 
-\subparagraph{Merkle path validity} \label{sproutmerklepathvalidity}
+\snarkcondition{Merkle path validity} \label{sproutmerklepathvalidity}
 
 for each $i \in \setofOld$ \changed{$\mid$ $\EnforceMerklePath{i} = 1$}:
-$\treepath{i}$ must be a valid \merklePath of depth given by $\MerkleDepthSprout$, as defined in
-\crossref{merklepath}, from $\NoteCommitmentSprout(\nOld{i})$ to \noteCommitmentTree root $\rt$.
+$\treepath{i}$ is a valid \merklePath (see \crossref{merklepath}) of depth
+$\MerkleDepthSprout$ from $\NoteCommitmentSprout(\nOld{i})$ to the \anchor $\rt$.
 
 \textbf{Note:} Merkle path validity covers both conditions 1. (a) and 1. (d) of the NP statement
-given in \cite[section 4.2]{BCG+2014}.
+in \cite[section 4.2]{BCG+2014}.
+\vspace{2ex}
 
 \changed{
-\subparagraph{Merkle path enforcement} \label{sproutmerklepathenforcement}
+\snarkcondition{Merkle path enforcement} \label{sproutmerklepathenforcement}
 
 for each $i \in \setofOld$, if $\vOld{i} \neq 0$ then $\EnforceMerklePath{i} = 1$.
 }
 
-\subparagraph{Balance} \label{sproutbalance}
+\snarkcondition{Balance} \label{sproutbalance}
 
-$\changed{\vpubOld\; +} \vsum{i=1}{\NOld} \vOld{i} = \vpubNew + \vsum{i=1}{\NNew} \vNew{i} \in \range{0}{2^{64}-1}$.
+$\changed{\vpubOld\; +} \ssum{i=1}{\NOld} \vOld{i} = \vpubNew + \ssum{i=1}{\NNew} \vNew{i} \in \range{0}{2^{64}-1}$.
 
-\subparagraph{\Nullifier{} integrity} \label{sproutnullifierintegrity}
+\snarkcondition{\Nullifier{} integrity} \label{sproutnullifierintegrity}
 
 for each $i \in \setofOld$:
 $\nfOld{i} = \PRFnf{\AuthPrivateOld{i}}(\NoteAddressRandOld{i})$.
 
-\subparagraph{Spend authority} \label{sproutspendauthority}
+\snarkcondition{Spend authority} \label{sproutspendauthority}
 
 for each $i \in \setofOld$:
 $\AuthPublicOld{i} = \changed{\PRFaddr{\AuthPrivateOld{i}}(0)}$.
 
-\subparagraph{Non-malleability} \label{sproutnonmalleablejs}
+\snarkcondition{Non-malleability} \label{sproutnonmalleablejs}
 
 for each $i \in \setofOld$:
 $\h{i} = \PRFpk{\AuthPrivateOld{i}}(i, \hSig)$.
 
 \changed{
-\subparagraph{Uniqueness of $\NoteAddressRandNew{i}$} \label{sproutuniquerho}
+\snarkcondition{Uniqueness of $\NoteAddressRandNew{i}$} \label{sproutuniquerho}
 
 for each $i \in \setofNew$:
 $\NoteAddressRandNew{i} = \PRFrho{\NoteAddressPreRand}(i, \hSig)$.
 }
 
-\subparagraph{Note commitment integrity} \label{sproutcommitmentintegrity}
+\snarkcondition{Note commitment integrity} \label{sproutcommitmentintegrity}
 
 for each $i \in \setofNew$: $\cmNew{i}$ = $\NoteCommitSprout(\nNew{i})$.
 
@@ -3470,6 +3533,7 @@ For details of the form and encoding of proofs, see \crossref{phgr}.
 
 
 \sapling{
+\vspace{50ex}
 \introsection
 \subsubsection{\SpendStatement{} (\Sapling)} \label{spendstatement}
 
@@ -3482,7 +3546,7 @@ A valid instance of $\ProofSpend$ assures that given a \term{primary input}:
 \end{formulae}
 
 \introlist
-the prover knows an \term{auxiliary input}:
+the prover knows an \auxiliaryInput:
 
 \begin{formulae}
   \item $(\treepath{} \typecolon \typeexp{\MerkleHash}{\MerkleDepthSapling} \times \NotePositionTypeSapling,\\
@@ -3501,20 +3565,20 @@ where $\nOld{} = (\Diversifier, \DiversifiedTransmitPublic, \vOld{}, \NoteCommit
 \introlist
 such that the following conditions hold:
 
-\subparagraph{Note commitment integrity} \label{saplingnotecommitmentintegrity}
+\snarkcondition{Note commitment integrity} \label{saplingnotecommitmentintegrity}
 
 $\pack(\cmOld{}) = \NoteCommitmentSapling(\nOld{})$.
 
-\subparagraph{Merkle path validity} \label{saplingmerklepathvalidity}
+\snarkcondition{Merkle path validity} \label{saplingmerklepathvalidity}
 
 $\treepath{}$ must be a valid \merklePath of depth $\MerkleDepthSapling$, as defined in
 \crossref{merklepath}, from $\cmOld{}$ to \noteCommitmentTree root $\rt$.
 
-\subparagraph{Value commitment integrity} \label{saplingvaluecommitmentintegrity}
+\snarkcondition{Value commitment integrity} \label{saplingvaluecommitmentintegrity}
 
 $\cvOld{} = \ValueCommit{\ValueCommitRandOld{}}(\vOld{})$.
 
-\subparagraph{Point validity checks} \label{saplingpointvalidity}
+\snarkcondition{Point validity checks} \label{saplingpointvalidity}
 
 $\AuthSignPublic, \DiversifiedTransmitBase \in \GroupJ$.
 
@@ -3522,9 +3586,7 @@ $\scalarmult{8}{\AuthSignPublic} \neq \ZeroJ$.
 
 $\scalarmult{8}{\DiversifiedTransmitBase} \neq \ZeroJ$.
 
-\subparagraph{\Nullifier{} integrity} \label{saplingnullifierintegrity}
-
-
+\snarkcondition{\Nullifier{} integrity} \label{saplingnullifierintegrity}
 
 $\nfOld{} = \scalarmult{\PRFnr{\AuthProvePublic}(\NoteAddressRand)}{\AuthSignPublic}$.
 
@@ -3535,7 +3597,7 @@ where
   \item $\NoteAddressRand = \MixingPedersenHash(\cmOld{}, \NotePosition)$
 \end{formulae}
 
-\subparagraph{Spend authority} \label{saplingspendauthority}
+\snarkcondition{Spend authority} \label{saplingspendauthority}
 
 \todo{}
 
@@ -3570,30 +3632,34 @@ is used to encrypt them. The recipient's possession of the associated
 
 All of the resulting ciphertexts are combined to form a \notesCiphertext.
 
-Let $\Sym$ be the \encryptionScheme instantiated in \crossref{concretesym}.
-
 \introlist
+\sprout{
 For both encryption and decryption,
 
-\sprout{
 \begin{itemize}
-  \item Let $\KDFSprout$ be the \keyDerivationFunction instantiated in \crossref{concretesproutkdf}.
-  \item Let $\KASprout$ be the \keyAgreementScheme instantiated in \crossref{concretesproutkeyagreement}.
-  \item Let $\hSig$ be the value computed for this \joinSplitDescription in \crossref{joinsplitdesc}.
+  \item let $\Sym$ be the \encryptionScheme instantiated in \crossref{concretesym};
+  \item let $\KDFSprout$ be the \keyDerivationFunction instantiated in \crossref{concretesproutkdf};
+  \item let $\KASprout$ be the \keyAgreementScheme instantiated in \crossref{concretesproutkeyagreement};
+  \item let $\hSig$ be the value computed for this \joinSplitDescription in \crossref{joinsplitdesc}.
 \end{itemize}
-}
+} %sprout
 \notsprout{
-\begin{itemize}
-  \item Let $\KDFSprout$\sapling{ and $\KDFSapling$} be the \keyDerivationFunctions instantiated in
-        \crossref{concretesproutkdf}.
-  \item Let $\KASprout$\sapling{ and $\KASapling$} be the \keyAgreementSchemes instantiated in
-        \crossref{concretekaandkdf}.
-  \item \sproutspecific{Let $\hSig$ be the value computed for this \joinSplitDescription in
-        \crossref{joinsplitdesc}.}
-\end{itemize}
-}
+For both encryption and decryption:
 
-\subsubsection{Encryption}
+Let $\Sym$ be the \encryptionScheme instantiated in \crossref{concretesym}.
+
+Let $\KDFSprout$\sapling{ and $\KDFSapling$} be the \keyDerivationFunctions instantiated in
+\crossref{concretesproutkdf}.
+
+Let $\KASprout$\sapling{ and $\KASapling$} be the \keyAgreementSchemes instantiated in
+\crossref{concretekaandkdf}.
+
+\sproutspecific{Let $\hSig$ be the value computed for this \joinSplitDescription in
+\crossref{joinsplitdesc}.}
+} %notsprout
+
+
+\subsubsection{Encryption \pSproutOrNothing}
 
 Let $\TransmitPublicNew{\allNew}$ be the \transmissionKeys
 for the intended recipient addresses of each new \note.
@@ -3633,7 +3699,7 @@ further security considerations, for example of how to validate a \note received
 out-of-band, which are not addressed in this document.
 }
 
-\subsubsection{Decryption by a Recipient}
+\subsubsection{Decryption by a Recipient \pSproutOrNothing}
 
 Let $\InViewingKey = (\AuthPublic, \TransmitPrivate)$ be the recipient's \incomingViewingKey,
 and let $\TransmitPublic$ be the corresponding \transmissionKey derived from
@@ -3687,6 +3753,7 @@ See \crossref{inbandrationale} for further discussion of the security and
 engineering rationale behind this encryption scheme.
 
 
+\intropart
 \section{Concrete Protocol}
 
 \subsection{Caution}
@@ -3696,7 +3763,7 @@ abstract and concrete protocol. E.g. \crossref{internalh}}
 
 \subsection{Integers, Bit Sequences, and Endianness} \label{boxnotation} \label{endian}
 
-All integers in \emph{\Zcash-specific} encodings are unsigned, have a fixed
+All integers in \Zcash-specific encodings are unsigned, have a fixed
 bit length, and are encoded in little-endian byte order \emph{unless otherwise
 specified}.
 
@@ -3734,12 +3801,12 @@ as a byte with the bits ordered from \emph{most significant} to
 is toward the left of a diagram. Where bit fields are used, the text will
 clarify their position in each case.
 
-\introlist
+\introsection
 \subsection{Constants} \label{constants}
 
 Define:
 
-\begin{formulae}
+\begin{formulae}[itemsep=\sprout{1ex}\notsprout{0.2ex}]
   \item $\MerkleDepthSprout \typecolon \Nat := \changed{29}$
 \sapling{
   \item $\MerkleDepthSapling \typecolon \Nat := 29$
@@ -3755,11 +3822,11 @@ Define:
   \item $\NoteCommitRandLength \typecolon \Nat := \changed{256}$
   \item $\changed{\RandomSeedLength \typecolon \Nat := 256}$
   \item $\AuthPrivateLength \typecolon \Nat := \changed{252}$
+  \item $\changed{\NoteAddressPreRandLength \typecolon \Nat := 252}$
 \sapling{
   \item $\SpendingKeyLength \typecolon \Nat := 256$
   \item $\DiversifierLength \typecolon \Nat := 88$
 } %sapling
-  \item $\changed{\NoteAddressPreRandLength \typecolon \Nat := 252}$
   \item $\UncommittedSprout \typecolon \bitseq{\MerkleHashLengthSprout} := \zeros{\MerkleHashLengthSprout}$
 \sapling{
   \item $\UncommittedSapling \typecolon \bitseq{\MerkleHashLengthSapling} := \ItoLEBSP{\MerkleHashLengthSapling}(1)$
@@ -3783,7 +3850,7 @@ Define:
 \end{formulae}
 
 
-\introlist
+\introsection
 \subsection{Concrete Cryptographic Schemes}
 
 \subsubsection{\HashFunctions}
@@ -3912,7 +3979,7 @@ $\MerkleCRHSprout \typecolon \MerkleLayerSprout \times \MerkleHashSprout \times
 
 \vspace{-2ex}
 \securityrequirement{
-$\SHACompress$ must be collision-resistant, and it must be infeasible to find a preimage $x$
+$\SHACompress$ must be \collisionResistant, and it must be infeasible to find a preimage $x$
 such that $\SHACompress(x) = \zeros{256}$.
 }
 
@@ -3932,7 +3999,7 @@ $\MerkleCRHSapling \typecolon \MerkleLayerSapling \times \MerkleHashSapling \tim
 
 \vspace{-2ex}
 \securityrequirement{
-$\PedersenHash$ must be collision-resistant.
+$\PedersenHash$ must be \collisionResistant\!.
 }
 
 \pnote{
@@ -3943,7 +4010,7 @@ as noted in \crossref{concretewindowedcommit}.
 } %sapling
 
 
-\introlist
+\introsection
 \subsubsubsection{\hSigText{} \HashFunction} \label{hsigcrh}
 
 \newsavebox{\hsigbox}
@@ -3973,7 +4040,7 @@ where
 $\BlakeTwobOf{256}{p, x}$ is defined in \crossref{concreteblake2}.
 
 \securityrequirement{
-$\BlakeTwobOf{256}{\ascii{ZcashComputehSig}, x}$ must be collision-resistant on $x$.
+$\BlakeTwobOf{256}{\ascii{ZcashComputehSig}, x}$ must be \collisionResistant on $x$.
 }
 
 
@@ -4015,7 +4082,7 @@ $32$ bytes, $8$-byte personalization string $p$, and input $x$.
 
 \securityrequirement{
 $\LEOStoIPOf{256}{\BlakeTwosOf{256}{\ascii{Zcashivk}, x}} \bmod 2^{251}$
-must be collision-resistant on a $512$-bit input $x$. Note that this
+must be \collisionResistant on a $64$-byte input $x$. Note that this
 does not follow from collision-resistance of $\BlakeTwos{256}$
 (and the best possible concrete security is that of a $251$-bit hash
 rather than a $256$-bit hash), but it is a reasonable assumption
@@ -4112,12 +4179,13 @@ See \crossref{cctpedersenhash} for rationale and efficient circuit implementatio
 of these functions.
 
 \securityrequirement{
-$\PedersenHash$ and $\PedersenHashToPoint$ are required to be collision-resistant
+$\PedersenHash$ and $\PedersenHashToPoint$ are required to be \collisionResistant
 between inputs of fixed length, for a given personalization input $D$.
 No other security properties commonly associated with \hashFunctions are needed.
 }
 
 \vspace{2ex}
+\introsection
 \begin{theorem} \label{thmpedersenencodeinjective}
 The encoding function $\PedersenEncode{\paramdot}$ is injective.
 \end{theorem}
@@ -4130,6 +4198,7 @@ The range of this expression is a subset of
 $\rangenozero{-\PedersenRangeOffset}{\PedersenRangeOffset}$ where
 $\PedersenRangeOffset = 4 \mult \vsum{i=0}{c-1} 2^{4 \mult i} = 4 \mult \hfrac{2^{4 \mult c}}{15}$.
 
+\introlist
 When $c = 63$, we have
 
 \begin{tabular}{@{\hskip 2em}r@{\;}l}
@@ -4139,7 +4208,7 @@ When $c = 63$, we have
 \end{tabular}
 
 so the required condition is met. This implies that there is no ``wrap around''
-and so $\vsum{j=1}{k_i} \enc(m_j) \mult 2^{4 \mult (j-1)}$ may be treated as an
+and so $\ssum{j=1}{k_i} \enc(m_j) \mult 2^{4 \mult (j-1)}$ may be treated as an
 integer expression.
 
 $\enc$ is injective. In order to prove that $\PedersenEncode{\paramdot}$ is injective,
@@ -4147,7 +4216,7 @@ consider $\PedersenEncodeNonneg{\paramdot} \typecolon \bitseq{3 \mult \range{1}{
 \range{0}{2 \smult \PedersenRangeOffset}$ such that
 $\PedersenEncodeNonneg{M_i} = \PedersenEncode{M_i} + \PedersenRangeOffset$.
 With $k_i$ and $m_j$ defined as above, we have
-$\PedersenEncodeNonneg{M_i} = \vsum{j=1}{k_i} \enc'(m_j) \mult 2^{4 \mult (j-1)}$
+$\PedersenEncodeNonneg{M_i} = \ssum{j=1}{k_i} \enc'(m_j) \mult 2^{4 \mult (j-1)}$
 where $\enc'(m_j) = \enc(m_j) + 4$ is in $\range{0}{8}$ and $\enc'$ is injective.
 Express this sum in hexadecimal; then each $m_j$ affects only one hex digit, and
 it is easy to see that $\PedersenEncodeNonneg{\paramdot}$ is injective.
@@ -4157,9 +4226,9 @@ Therefore so is $\PedersenEncode{\paramdot}$.
 Since the security proof from \cite[Appendix A]{BGG1995}
 depends only on the encoding being injective and its range not including
 zero, the proof can be adapted straightforwardly to show that $\PedersenHashToPoint$
-is collision-resistant under the same assumptions and security bounds.
+is \collisionResistant under the same assumptions and security bounds.
 Because $\ItoLEBSP{\MerkleHashLengthSapling}$ and $\ExtractJ$ are injective,
-it follows that $\PedersenHash$ is equally collision-resistant.
+it follows that $\PedersenHash$ is equally \collisionResistant\!.
 
 \vspace{2ex}
 \begin{theorem} \label{thmnohashtouncommittedsapling}
@@ -4167,10 +4236,10 @@ $\UncommittedSapling = \ItoLEBSP{\MerkleHashLengthSapling}(1)$ is not in the ran
 \end{theorem}
 
 \begin{proof}
-By the definition of $\PedersenHash$, $\ItoLEBSP{\MerkleHashLengthSapling}(1)$ can be in the
-range of $\PedersenHash$ only if there exist $D \typecolon \byteseq{8}$ and $M \typecolon \bitseq{\PosInt}$
-such that $\ExtractJ(\PedersenHashToPoint(D, M)) = 1$.
-The latter can only be the affine-Edwards $u$-coordinate of a point in $\GroupJ$.
+By the definition of $\PedersenHash$, $\ItoLEBSP{\smash{\MerkleHashLengthSapling}}(1)$
+can be in the range of $\PedersenHash$ only if there exist $D \typecolon \byteseq{8}$ and
+$M \typecolon \bitseq{\PosInt}$ such that $\ExtractJ(\PedersenHashToPoint(D, M)) = 1$.
+The latter can only be the affine-Edwards $u$-coordinate of a point in $\strut\GroupJ$.
 We show that there are no points in $\GroupJ$ with affine-Edwards $u$-coordinate $1$.
 Suppose for a contradiction that $(u, \varv) \in \GroupJ$ for $u = 1$ and some
 $\varv \typecolon \GF{\ParamJ{r}}$. By writing the curve equation as
@@ -4200,9 +4269,10 @@ We define $\MixingPedersenHash \typecolon \GroupJ \times \range{0}{\ParamJ{r}-1}
 The function
 \begin{formulae}
   \item $\fun{(r, M, x) \typecolon \range{0}{\ParamJ{r}-1} \times \bitseq{\PosInt} \times
-           \range{0}{\ParamJ{r}-1}}{\MixingPedersenHash(\WindowedPedersenCommit{r}(M), x) \typecolon \GroupJ}$.
+           \range{0}{\ParamJ{r}-1}}{\MixingPedersenHash(\WindowedPedersenCommit{r}(M), x) \typecolon \GroupJ}$
 \end{formulae}
-must be collision-resistant on $(r, M, x)$.
+\vspace{-1ex}
+must be \collisionResistant on $(r, M, x)$.
 }
 
 See \crossref{cctmixinghash} for efficient circuit implementation of this function.
@@ -4337,7 +4407,7 @@ defined in \crossref{concretesha256}:
 \end{equation*}
 
 \begin{securityrequirements}
-  \item The \shaCompressFunction must be collision-resistant.
+  \item The \shaCompressFunction must be \collisionResistant\!.
   \item The \shaCompressFunction must be a PRF when keyed by the bits
         corresponding to $x$, $\AuthPrivate$ or $\NoteAddressPreRand$
         in the above diagrams, with input in the remaining bits.
@@ -4346,10 +4416,10 @@ defined in \crossref{concretesha256}:
 \changed{
 \pnote{
 The first four bits --i.e.\ the most significant four bits of the first byte--
-are used to distinguish different uses of $\SHACompress$, ensuring that the functions
-are independent. In addition to the inputs shown here, the bits $\mathtt{1011}$
+are used to separate distinct uses of $\SHACompress$, ensuring that the functions
+are independent. As well as the inputs shown here, bits $\mathtt{1011}$
 in this position are used to distinguish uses of the full $\SHAFull$ hash
-function --- see \crossref{concretesproutcommit}.
+function; see \crossref{concretesproutcommit}.
 
 (The specific bit patterns chosen here were motivated by the possibility of future
 extensions that might have increased $\NOld$ and/or $\NNew$ to 3, or added an
@@ -4401,7 +4471,7 @@ It is instantiated using the $\BlakeTwobGeneric$ \hashFunction defined in
 
 \begin{formulae}
   \item $\PRFexpand{\SpendingKey}() :=
-           \LEOStoIPOf{512}{\BlakeTwobOf{512}{\ascii{Zcash\_ExpandSeed}, \Justthebox{\expandbox}}} \bmod \ParamJ{r}$
+           \LEOStoIPOf{512}{\BlakeTwobOf{512}{\ascii{Zcash\_ExpandSeed}, \Justthebox{\expandbox}}} \pmod{\ParamJ{r}}$
 \end{formulae}
 
 \securityrequirement{
@@ -4583,8 +4653,9 @@ the type of $\JubjubCurve$ secret keys. \todo{expand this}
 \sapling{
 \subsubsubsection{\Sapling \KeyDerivation} \label{concretesaplingkdf}
 
-The $\KDFSapling$ \keyDerivationFunction specified in \crossref{abstractkdf}
-is instantiated using $\BlakeTwob{256}$ as follows:
+The $\KDFSapling$ \keyDerivationFunction is specified in \crossref{abstractkdf}.
+
+It is instantiated using $\BlakeTwob{256}$ as follows:
 
 \begin{formulae}
   \item $\KDFSapling(\OutputIndex, \DHSecret{}, \EphemeralPublic) :=
@@ -4668,17 +4739,17 @@ It is instantiated as EdJubjub, which is defined as $\EdDSA$ \cite{BJLSY2015} ov
 \newsavebox{\cmbox}
 \begin{lrbox}{\cmbox}
 \setchanged
-\begin{bytefield}[bitwidth=0.031em]{840}
-    \sbitbox{24}{$1$} &
-    \sbitbox{24}{$0$} &
-    \sbitbox{24}{$1$} &
-    \sbitbox{24}{$1$} &
-    \sbitbox{24}{$0$} &
-    \sbitbox{24}{$0$} &
-    \sbitbox{24}{$0$} &
-    \sbitbox{24}{$0$} &
+\begin{bytefield}[bitwidth=0.027em]{840}
+    \sbitbox{28}{$1$} &
+    \sbitbox{28}{$0$} &
+    \sbitbox{28}{$1$} &
+    \sbitbox{28}{$1$} &
+    \sbitbox{28}{$0$} &
+    \sbitbox{28}{$0$} &
+    \sbitbox{28}{$0$} &
+    \sbitbox{28}{$0$} &
     \sbitbox{256}{$256$-bit $\AuthPublic$} &
-    \sbitbox{128}{$64$-bit $\Value$} &
+    \sbitbox{140}{$64$-bit $\Value$} &
     \sbitbox{256}{$256$-bit $\NoteAddressRand$} &
     \sbitbox{256}{$256$-bit $\NoteCommitRand$}
 \end{bytefield}
@@ -4688,7 +4759,7 @@ The commitment scheme $\NoteCommitSprout{}$ specified in \crossref{abstractcommi
 instantiated using $\SHAFull$ as follows:
 
 \begin{formulae}[leftmargin=1em]
-  \item $\NoteCommitSprout{\NoteCommitRand}(\AuthPublic, \Value, \NoteAddressRand) := \SHAFullBox{\cmbox}$.
+  \item $\NoteCommitSprout{\NoteCommitRand}(\AuthPublic, \Value, \NoteAddressRand) := \SHAFullBox{\cmbox}$
 \end{formulae}
 
 \pnote{
@@ -4696,7 +4767,7 @@ The leading byte of the $\SHAFull$ input is $\hexint{B0}$.
 }
 
 \begin{securityrequirements}
-  \item The \shaCompressFunction must be collision-resistant.
+  \item The \shaCompressFunction must be \collisionResistant\!.
   \item The \shaCompressFunction must be a PRF when keyed by the bits corresponding
         to the position of $\NoteCommitRand$ in the second block of $\SHAFull$
         input, with input to the PRF in the remaining bits of the block and
@@ -4752,15 +4823,15 @@ $\MerkleDepthSapling < 64$.
 
 The windowed Pedersen commitments defined in the preceding section are
 highly efficient, but they do not support the homomorphic property we
-need when instantiating $\ValueCommit{}$ (see \crossref{spendsandoutputs}
-and \crossref{saplingbalance}).
+need when instantiating $\ValueCommit{}$ (see \crossref{saplingbalance} and
+\crossref{spendsandoutputs}).
 
 In order to support this property, we also define \quotedterm{homomorphic}
 \xPedersenCommitments as follows:
 
 \begin{formulae}
   \item $\HomomorphicPedersenCommit{\ValueCommitRand}(D, \Value) :=
-          \scalarmult{\Value}{\FindGroupJHashOf{D}, \ascii{v}} + \scalarmult{\ValueCommitRand}{\FindGroupJHashOf{D, \ascii{r}}}$
+           \scalarmult{\Value}{\FindGroupJHashOf{D, \ascii{v}}} + \scalarmult{\ValueCommitRand}{\FindGroupJHashOf{D, \ascii{r}}}$
 \end{formulae}
 
 
@@ -4828,10 +4899,10 @@ Let $\GenG{1} \typecolon \GroupGstar{1} := (1, 2)$.
 \begin{tabular}{@{}l@{}r@{}l@{}}
 Let $\GenG{2} \typecolon \GroupGstar{2} :=\;$
 % are these the right way round?
-&$(11559732032986387107991004021392285783925812861821192530917403151452391805634$ & $\mult\, t\;+$ \\
-&$ 10857046999023057135944570762232829481370756359578518086990519993285655852781$ & $,           $ \\
-&$  4082367875863433681332203403145435568316851327593401208105741076214120093531$ & $\mult\, t\;+$ \\
-&$  8495653923123431417604973247489272438418190587263600148770280649306958101930$ & $).          $
+&$(11559732032986387107991004021392285783925812861821192530917403151452391805634$ & $\,\mult\, t\;+$ \\
+&$ 10857046999023057135944570762232829481370756359578518086990519993285655852781$ & $,             $ \\
+&$  4082367875863433681332203403145435568316851327593401208105741076214120093531$ & $\,\mult\, t\;+$ \\
+&$  8495653923123431417604973247489272438418190587263600148770280649306958101930$ & $).            $
 \end{tabular}
 
 $\GenG{1}$ and $\GenG{2}$ are generators of $\GroupG{1}$ and $\GroupG{2}$ respectively.
@@ -4955,7 +5026,7 @@ The \representedPairing $\BLSCurve$ is defined in this section. Parameters are t
 \cite{Bowe2017}.
 
 \introlist
-Let $\ParamS{q} :=\;$\scalebox{0.812}[1]{$4002409555221667393417789825735904156556882819939007885332058136124031650490837864442687629129015664037894272559787$}.
+Let $\ParamS{q} :=\;$\scalebox{0.81}[1]{$4002409555221667393417789825735904156556882819939007885332058136124031650490837864442687629129015664037894272559787$.}
 
 Let $\ParamS{r} := 52435875175126190479447740508185965837690552500527637822603658699938581184513$.
 
@@ -4992,10 +5063,10 @@ Let $\GenS{1} \typecolon \GroupSstar{1} := (1, 2)$.
 \begin{tabular}{@{}l@{}r@{}l@{}}
 Let $\GenS{2} \typecolon \GroupSstar{2} :=\;$
 % are these the right way round?
-&$(11559732032986387107991004021392285783925812861821192530917403151452391805634$ & $\mult\, t\;+$ \\
-&$ 10857046999023057135944570762232829481370756359578518086990519993285655852781$ & $,           $ \\
-&$  4082367875863433681332203403145435568316851327593401208105741076214120093531$ & $\mult\, t\;+$ \\
-&$  8495653923123431417604973247489272438418190587263600148770280649306958101930$ & $).          $
+&$(11559732032986387107991004021392285783925812861821192530917403151452391805634$ & $\,\mult\, t\;+$ \\
+&$ 10857046999023057135944570762232829481370756359578518086990519993285655852781$ & $,             $ \\
+&$  4082367875863433681332203403145435568316851327593401208105741076214120093531$ & $\,\mult\, t\;+$ \\
+&$  8495653923123431417604973247489272438418190587263600148770280649306958101930$ & $).            $
 \end{tabular}
 
 $\GenS{1}$ and $\GenS{2}$ are generators of $\GroupS{1}$ and $\GroupS{2}$ respectively.
@@ -5206,7 +5277,7 @@ Let $\FindGroupJHashOf{D, M} =
 \begin{pnotes}
   \item The $\BlakeTwos{256}$ chaining variable after processing $\CRS$ may be precomputed.
   \item For random input, $\FindGroupJHash$ returns $\bot$ with probability approximately $2^{-256}$.
-        The uses of $\FindGroupJHash$ in the \Zcash protocol never return $\bot$.
+        In the \Zcash protocol, uses of $\FindGroupJHash$ never return $\bot$.
 \end{pnotes}
 }
 
@@ -5443,7 +5514,7 @@ The encoding of a \Sapling \notePlaintext consists of:
 The encoding of $\Diversifier$ and $\DiversifiedTransmitPublic$ is identical
 to the raw encoding of a \Sapling \paymentAddress in \crossref{saplingpaymentaddrencoding}.
 }
-}
+} %sapling
 
 
 \subsection{Encodings of Addresses and Keys} \label{addressandkeyencoding}
@@ -5652,7 +5723,8 @@ The raw encoding of an \incomingViewingKey consists of, in order:
 $\TransmitPrivate$ \MUST be ``clamped'' using $\KASproutFormatPrivate$ as specified
 in \crossref{sproutkeycomponents}. That is, a decoded \incomingViewingKey{} \MUST be
 considered invalid if $\TransmitPrivate \neq \KASproutFormatPrivate(\TransmitPrivate)$.
-($\KASproutFormatPrivate$ is defined in \crossref{concretesproutkeyagreement}.)
+
+$\KASproutFormatPrivate$ is defined in \crossref{concretesproutkeyagreement}.
 
 \pnote{
 For addresses on the production network, the lead bytes and encoded length
@@ -5727,6 +5799,7 @@ For \incomingViewingKeys on the test network, the \humanReadablePart is \ascii{z
 }
 
 
+\introsection
 \subsubsection{\SproutOrNothing \SpendingKeys} \label{sproutspendingkeyencoding}
 
 A \SproutOrNothing \spendingKey consists of $\AuthPrivate$, which is a sequence of
@@ -5851,20 +5924,20 @@ At the time of writing, two upgrades are planned: \NUZero, and
 from \Sprout to \NUZero and then \Sapling.
 
 The upgrade mechanism is described in \cite{ZIP-200}.
-The content of the \NUZero upgrade is described in \cite{ZIP-201},
+The specifications of the \NUZero upgrade are described in \cite{ZIP-201},
 \cite{ZIP-202}, \cite{ZIP-203}, and \cite{ZIP-143}.
 
 \NUZero and \Sapling will each be introduced in a
 ``bilateral hard fork''. In this kind of fork,
 
-\begin{enumerate}
+\begin{itemize}
   \item there is a \blockHeight at which the fork takes effect;
   \item \blocks and \transactions that are valid according to
         the post-fork rules are not valid before the forking \block;
   \item \blocks and \transactions that are valid according to
         the pre-fork rules are no longer valid in or after the
         forking \block.
-\end{enumerate}
+\end{itemize}
 
 Full support for each upgrade is indicated by a minimum version
 of the peer-to-peer protocol. Before the planned fork \blockHeight,
@@ -5904,7 +5977,7 @@ for use by \Sapling \transactions.
 }
 
 
-\introsection
+\intropart
 \section{Consensus Changes from \Bitcoin}
 
 \subsection{Encoding of \Transactions} \label{txnencoding}
@@ -6204,7 +6277,7 @@ $4$ & $\nBitsField$ & \type{uint32\_t} & An encoded version of the \targetThresh
 \header hash must be less than or equal to, in the same nBits format used by \Bitcoin.
 \cite{Bitc-nBits} \\ \hline
 
-$32$ & $\nNonce$ & \type{char[32]} & An arbitrary field miners change to modify the
+$32$ & $\nNonce$ & \type{char[32]} & An arbitrary field that miners can change to modify the
 \header hash in order to produce a hash less than or equal to the \targetThreshold. \\ \hline
 
 $3$ & $\solutionSize$ & \compactSize & The size of an Equihash solution in bytes (always $1344$). \\ \hline
@@ -6315,11 +6388,10 @@ The Equihash parameters for the production and test networks are $n = 200, k = 9
 
 The Generalized Birthday Problem is defined as follows: given a sequence
 $X_\barerange{1}{\mathrm{N}}$ of $n$-bit strings, find $2^k$ distinct $X_{i_j}$ such that
-$\vxor{j=1}{2^k} X_{i_j} = 0$.
+$\sxor{j=1}{2^k} X_{i_j} = 0$.
 
-\introlist
 In Equihash, $\mathrm{N} = 2^{\frac{n}{k+1}+1}$, and the sequence $X_\barerange{1}{\mathrm{N}}$ is
-derived from the \blockHeader and a nonce:
+derived from the \blockHeader and a nonce.
 
 \newsavebox{\powheaderbox}
 \begin{lrbox}{\powheaderbox}
@@ -6336,6 +6408,7 @@ derived from the \blockHeader and a nonce:
 
 Let $\powheader := \Justthebox[-11.5ex]{\powheaderbox}$
 
+\vspace{1ex}
 For $i \in \range{1}{N}$, let $X_i = \EquihashGen{n, k}(\powheader, i)$.
 
 $\EquihashGen{}$ is instantiated in \crossref{equihashgen}.
@@ -6346,21 +6419,24 @@ as in \crossref{endian}.
 A \validEquihashSolution is then a sequence $i \typecolon \range{1}{N}^{2^k}$ that
 satisfies the following conditions:
 
+\vspace{-4ex}
 \subparagraph{Generalized Birthday condition}
 
 $\vxor{j=1}{2^k} X_{i_j} = 0$.
 
+\vspace{-2ex}
 \subparagraph{Algorithm Binding conditions}
-
+\vspace{-2ex}
 \introlist
 \begin{itemize}
   \item For all $r \in \range{1}{k\!-\!1}$, for all $w \in \range{0}{2^{k-r}\!-\!1}:
-        \vxor{j=1}{2^r} X_{i_{w \mult 2^r + j}}$ has $\frac{n \mult r}{k+1}$ leading zeros; and
+        \smash{\vxor{j=1}{2^r}} X_{i_{w \mult 2^r + j}}$ has $\frac{n \mult r}{k+1}$ leading zeros; and
   \item For all $r \in \range{1}{k}$, for all $w \in \range{0}{2^{k-r}\!-\!1}:
         i_{w \mult 2^r + 1 .. w \mult 2^r + 2^{r-1}} <
         i_{w \mult 2^r + 2^{r-1} + 1 .. w \mult 2^r + 2^r}$ lexicographically.
 \end{itemize}
 
+\vspace{-2ex}
 \begin{pnotes}
   \item This does not include a difficulty condition, because here we are
         defining validity of an Equihash solution independent of difficulty.
@@ -6578,10 +6654,12 @@ $\MaxBlockSubsidy$, and $\FoundersFraction$ are instantiated in \crossref{consta
 The \foundersReward is paid by a \transparent output in the \coinbaseTransaction, to
 one of $\NumFounderAddresses$ \transparent addresses, depending on the \blockHeight.
 
-\renewcommand{\arraystretch}{0.95}
+\renewcommand{\arraystretch}{1}
 
+\vspace{1ex}
 For the production network, $\FounderAddressList_\barerange{\mathrm{1}}{\NumFounderAddresses}$ is:
 
+\scalebox{0.95}{
 \begin{tabular}{@{\hskip 2.5em}l@{\;}l}
 [& \ascii{t3Vz22vK5z2LcKEdg16Yv4FFneEL1zg9ojd}, \ascii{t3cL9AucCajm3HXDhb5jBnJK2vapVoXsop3}, \\
  & \ascii{t3fqvkzrrNaMcamkQMwAyHRjfDdM2xQvDTR}, \ascii{t3TgZ9ZT2CTSK44AnUPi6qeNaHa2eC7pUyF}, \\
@@ -6608,10 +6686,13 @@ For the production network, $\FounderAddressList_\barerange{\mathrm{1}}{\NumFoun
  & \ascii{t3WDhPfik343yNmPTqtkZAoQZeqA83K7Y3f}, \ascii{t3PSn5TbMMAEw7Eu36DYctFezRzpX1hzf3M}, \\
  & \ascii{t3R3Y5vnBLrEn8L6wFjPjBLnxSUQsKnmFpv}, \ascii{t3Pcm737EsVkGTbhsu2NekKtJeG92mvYyoN}\, ]
 \end{tabular}
+} %scalebox
 
+\vspace{1ex}
 \introlist
 For the test network, $\FounderAddressList_\barerange{\mathrm{1}}{\NumFounderAddresses}$ is:
 
+\scalebox{0.96}{
 \begin{tabular}{@{\hskip 2.5em}l@{\;}l}
 [& \ascii{t2UNzUUx8mWBCRYPRezvA363EYXyEpHokyi}, \ascii{t2N9PH9Wk9xjqYg9iin1Ua3aekJqfAtE543}, \\
  & \ascii{t2NGQjYMQhFndDHguvUw4wZdNdsssA6K7x2}, \ascii{t2ENg7hHVqqs9JwU5cgjvSbxnT2a9USNfhy}, \\
@@ -6638,8 +6719,8 @@ For the test network, $\FounderAddressList_\barerange{\mathrm{1}}{\NumFounderAdd
  & \ascii{t29pHDBWq7qN4EjwSEHg8wEqYe9pkmVrtRP}, \ascii{t2Ez9KM8VJLuArcxuEkNRAkhNvidKkzXcjJ}, \\
  & \ascii{t2D5y7J5fpXajLbGrMBQkFg2mFN8fo3n8cX}, \ascii{t2UV2wr1PTaUiybpkV3FdSdGxUJeZdZztyt}\, ]
 \end{tabular}
-
-\renewcommand{\arraystretch}{1}
+} %scalebox
+\renewcommand{\arraystretch}{\defaultarraystretch}
 
 \pnote{For the test network only, the addresses from index 4 onward have been changed from
 what was implemented at launch. This reflects a hard fork on the test network, starting
@@ -6948,7 +7029,7 @@ dependent on the position of the \noteCommitment in the \Sapling{}
 $\NoteAddressRand = \cm + \scalarmult{\NotePosition}{\NotePositionBase}$,
 where $\NotePositionBase$ is a generator independent of the generators
 used in $\NoteCommitSaplingAlg$. Therefore, $\NoteAddressRand$ commits uniquely
-to the \note and its position, and this commitment is collision-resistant
+to the \note and its position, and this commitment is \collisionResistant
 by the same argument used to prove collision resistance of \xPedersenHashes.
 Note that it is possible for two distinct \Sapling \positionedNotes (having
 different $\NoteAddressRand$ values and \nullifiers, but different
@@ -6984,12 +7065,11 @@ The motivation for the nested construction in \Zerocash
 was to allow Mint transactions to be publically verified without requiring
 a \zeroKnowledgeProof (as described under step 3 in
 \cite[section 1.3]{BCG+2014}). Since \Zcash combines ``Mint'' and ``Pour''
-transactions into generalized
-\notsprout{\joinSplitTransfers (for \Sprout), \sapling{or \spendTransfers and
-\outputTransfers (for \Sapling)}, and each transfer always uses a \zeroKnowledgeProof\!\!,
-\Zcash does not require the nesting.}
-\sprout{\joinSplitTransfers, and each transfer always uses a \zeroKnowledgeProof\!\!,
-it does not require the nesting.}
+transactions into generalized \notsprout{\joinSplitTransfers (for \Sprout),
+\sapling{or \spendTransfers and \outputTransfers (for \Sapling)}, and each
+transfer always uses a \zeroKnowledgeProof\!\!,
+\Zcash does not require the nesting.}\sprout{\joinSplitTransfers,
+and each transfer always uses a \zeroKnowledgeProof\!\!, it does not require the nesting.}
 A side benefit is that this reduces the cost of computing the
 \noteCommitments: \notsprout{for \Sprout} it reduces the number of $\SHACompress$
 evaluations needed to compute each \noteCommitment from three to two,
@@ -7184,7 +7264,7 @@ $\AuthPrivate$ is $252$ bits, and $\TransmitPrivate$ is no shorter than $\AuthPr
 \subsection{Omission in \Zerocash security proof} \label{crprf}
 
 The abstract \Zerocash protocol requires $\PRFaddr{}$ only to be a PRF;
-it is not specified to be collision-resistant. This reveals a flaw in
+it is not specified to be \collisionResistant\!. This reveals a flaw in
 the proof of the Balance property.
 
 Suppose that an adversary finds a collision on $\PRFaddr{}$ such that
@@ -7220,7 +7300,7 @@ A similar error occurs in the argument for the ``$\Adversary$ violates
 Condition II'' case.
 
 The flaw is not exploitable for the actual instantiations of $\PRFaddr{}$
-in \Zerocash and \SproutOrZcash, which \emph{are} collision-resistant assuming
+in \Zerocash and \SproutOrZcash, which \emph{are} \collisionResistant assuming
 that $\SHACompress$ is.
 
 The proof can be straightforwardly repaired. The intuition is that we can rely
@@ -7262,6 +7342,7 @@ distinct openings of the \noteCommitment when Condition I or II is violated.
 \end{itemize}
 
 
+\introsection
 \section{Acknowledgements}
 
 The inventors of \Zerocash are Eli Ben-Sasson, Alessandro Chiesa,
@@ -7294,9 +7375,18 @@ Daira Hopwood, Sean Bowe, and Jack Grigg.
 }
 
 
-\introsection
+\intropart
 \section{Change History}
 
+\subparagraph{2018.0-beta-15}
+
+\begin{itemize}
+    \item Change the \texttt{Makefile} to avoid multiple reloads in PDF readers while
+          rebuilding the PDF.
+    \item Spacing and pagination improvements.
+\end{itemize}
+
+\introlist
 \subparagraph{2018.0-beta-14}
 
 \begin{itemize}
@@ -7316,7 +7406,7 @@ Daira Hopwood, Sean Bowe, and Jack Grigg.
 \sapling{
     \item Change how $(\AuthSignPrivate, \AuthProvePrivate)$ are derived from the \spendingKey
           $\SpendingKey$ to ensure they are on the full range of $\GF{\ParamJ{r}}$.
-    \item Change $\PRFnr{}$ to produce output computationally indistinguishable from uniform on
+    \item Change $\PRF{}{\mathsf{nr}}$ to produce output computationally indistinguishable from uniform on
           $\GF{\ParamJ{r}}$.
     \item Change $\UncommittedSapling$ to be a $u$-coordinate for which there is no point on the curve.
     \item Appendix A updates:
@@ -7344,7 +7434,7 @@ Daira Hopwood, Sean Bowe, and Jack Grigg.
 }
 \sapling{
     \item Add a section on re-randomizable signatures.
-    \item Add definition of $\PRFnr{}$.
+    \item Add definition of $\PRF{}{\mathsf{nr}}$.
     \item Work-in-progress on \Sapling statements.
     \item Rename \quotedterm{raw} to \quotedterm{homomorphic} \xPedersenCommitments.
     \item Add packing modulo the field size and range checks to Appendix A.
@@ -7377,7 +7467,7 @@ Daira Hopwood, Sean Bowe, and Jack Grigg.
 \sapling{
     \item Refer to \NUZero and \Sapling just as ``upgrades'' in the abstract, not as
           the next ``minor version'' and ``major version''.
-    \item $\PRFnr{}$ must be collision-resistant.
+    \item $\PRF{}{\mathsf{nr}}$ must be \collisionResistant\!.
     \item Correct an error in the \xPedersenHash specification.
     \item Use a named variable, $c$, for chunks per segment in the \xPedersenHash
           specification, and change its value from $61$ to $63$. Add a proof
@@ -7805,7 +7895,7 @@ Daira Hopwood, Sean Bowe, and Jack Grigg.
     \item Clarify endianness, and that uses of $\BlakeTwobGeneric$ are unkeyed.
     \item Minor correction to what \sighashTypes cover.
     \item Add ``as intended for the \Zcash release of summer 2016" to title page.
-    \item Require $\PRFaddr{}$ to be collision-resistant (see \crossref{crprf}).
+    \item Require $\PRFaddr{}$ to be \collisionResistant (see \crossref{crprf}).
     \item Add specification of path computation for the \incrementalMerkleTree.
     \item Add a note in \crossref{sproutmerklepathvalidity} about how this condition
           corresponds to conditions in the \Zerocash paper.
@@ -7820,6 +7910,7 @@ Daira Hopwood, Sean Bowe, and Jack Grigg.
 \end{itemize}
 
 
+\intropart
 \section{References}
 
 \begingroup
@@ -7831,7 +7922,7 @@ Daira Hopwood, Sean Bowe, and Jack Grigg.
 
 \notsprout{
 
-\introsection
+\intropart
 \vspace{20ex}
 \appendix
 \phantomsection
@@ -7994,17 +8085,17 @@ boolean-constrained, can be implemented as:
 
 Since only nonzero elements of $\GF{\ParamS{r}}$ have a multiplicative inverse, the
 assertion $a \neq 0$ can be implemented by witnessing the inverse,
-$\ainv = a^{-1} \pmod{\ParamS{r}}$:
+$\Inv{a} = a^{-1} \pmod{\ParamS{r}}$:
 
 \begin{formulae}
-  \item $\constraint{\ainv}{a}{1}$
+  \item $\constraint{\Inv{a}}{a}{1}$
 \end{formulae}
 
 A global optimization allows to use a single inverse computation outside
 the circuit for any number of nonzero constraints. Suppose that we have
 $n$ variables (or \linearCombinations) that are supposed to be nonzero:
 $a_\barerange{0}{n-1}$. Multiply these together (using $n\!-\!1$ constraints)
-to give $a^* = \vproduct{i=0}{n-1} a_i$; then, constrain $a^*$ to be nonzero.
+to give $a^* = \sproduct{i=0}{n-1} a_i$; then, constrain $a^*$ to be nonzero.
 This works because the product $a^*$ is nonzero if and only if all of
 $a_\barerange{0}{n-1}$ are nonzero.
 
@@ -8081,7 +8172,7 @@ Let $a = \vsum{i=0}{n-1} a_i \mult 2^i$, and suppose we want to constrain
 $a \leq c$ for some \emph{constant} $c = \vsum{i=0}{n-1} c_i \mult 2^i$.
 
 Without loss of generality we can assume that $c_{n-1} = 1$, because if it
-were not then we would reduce $n$.
+were not then we would decrease $n$ accordingly.
 
 Note that since $a$ and $c$ are provided in binary representation, their
 bit length $n$ is not limited by the field element size. We \emph{do not} assume
@@ -8091,12 +8182,14 @@ Suppose $c$ has $k$ bits set to $1$, and let $j_\barerange{0}{k-1}$ be the
 indices of those bits in ascending order. Let $t$ be the minimum of $k-1$ and
 the number of trailing $1$ bits in $c$.
 
+\introlist
 Let $\Pi_{j_{k-1}} = a_{j_{k-1}}$. For $z \in \range{t}{k-2}$, constrain:
 
 \begin{formulae}
   \item $\constraint{\Pi_{j_{z+1}}}{a_{j_z}}{\Pi_{j_z}}$
 \end{formulae}
 
+\introlist
 For $i \in \range{0}{n-1}$:
 \begin{itemize}
   \item if $c_i = 0$, constrain $\constraint{1 - \Pi_{j_z} - a_i}{a_i}{0}$ where $j_z$ is the least element of $j$ greater than $i$;
@@ -8384,7 +8477,7 @@ also, the $u$-coordinate of the result is zero if-and-only-if the intermediate v
 $A$ is zero.
 
 \begin{formulae}
-  \item // Curve check.
+  \item // Curve equation check.
   \item $\constraint{u}{u}{uu}$
   \item $\constraint{\varv}{\varv}{\varvv}$
   \item $\constraint{\ParamJ{d} \smult uu}{\varvv}{\ParamJ{a} \smult uu + \varvv - 1}$
@@ -8456,8 +8549,8 @@ $750$ constraints.
 \pnote{
 It would be more efficient to use arithmetic on the Montgomery curve, as in
 \crossref{cctpedersenhash}. However since there are only three instances of
-fixed-base scalar multiplication in the \spendCircuit and two in the \outputCircuit
-\footnote{A Pedersen commitment uses fixed-base scalar multiplication as a subcomponent.},
+fixed-base scalar multiplication in the \spendCircuit and two in the
+\outputCircuit\footnote{A Pedersen commitment uses fixed-base scalar multiplication as a subcomponent.},
 the additional complexity was not considered justified for \Sapling.
 }
 
@@ -8476,7 +8569,7 @@ Given $k = \vsum{i=0}{250} k_i \smult 2^i$, we calculate $R = \scalarmult{k}{B}$
   \item let $\Base^u_0 = \SelectuOf{B}$
   \item let $\Base^{\vv}_0\hairspace = B_{\vv}$
   \item let $\Acc^u_0 = k_0 \bchoose B^u : 0$
-  \item let $\Acc^{\vv}\_0hairspace = k_0 \bchoose B^{\vv} : 1$
+  \item let $\Acc^{\vv}_0\hairspace = k_0 \bchoose B^{\vv} : 1$
         \vspace{1ex}
   \item for $i$ from $1$ up to $250$:
   \item \tab let $\Base_i = \scalarmult{2}{\Base_{i-1}}$
@@ -8674,7 +8767,7 @@ The unpacking need not be canonical in the sense discussed in \crossref{cctmodpa
 that is, it is \emph{not} necessary to ensure that the previous-layer or sibling
 bit-sequence inputs represent integers in the range $\range{0}{\ParamS{r}-1}$.
 Since the root of the Merkle tree is calculated outside the circuit using the
-canonical representations, and since the \xPedersenHashes are collision-resistant
+canonical representations, and since the \xPedersenHashes are \collisionResistant
 on arbitrary bit-sequence inputs, an attempt by an adversarial prover to use a
 non-canonical input would result in the wrong root being calculated, and the
 overall path check would fail.
@@ -8727,11 +8820,11 @@ as follows:
 
 \begin{formulae}
   \item $\HomomorphicPedersenCommit{\ValueCommitRand}(D, \Value) =
-          \scalarmult{\Value}{\FindGroupJHashOf{D, \ascii{v}}} + \scalarmult{\ValueCommitRand}{\FindGroupJHashOf{D, \ascii{r}}}$
+           \scalarmult{\Value}{\FindGroupJHashOf{D, \ascii{v}}} + \scalarmult{\ValueCommitRand}{\FindGroupJHashOf{D, \ascii{r}}}$
 \end{formulae}
 
-In the case that we need for $\ValueCommit{}$, $\Value$ has $64$ bits
-\footnote{It would be sufficient to use $51$ bits, which accomodates the range
+In the case that we need for $\ValueCommit{}$, $\Value$ has $64$
+bits\footnote{It would be sufficient to use $51$ bits, which accomodates the range
 $\range{0}{\MAXMONEY}$, but the \Sapling circuit uses $64$.}.
 This can be straightforwardly implemented in ... constraints.
 
diff --git a/protocol/zcash.bib b/protocol/zcash.bib
index 3c668aa5..09ad72c0 100644
--- a/protocol/zcash.bib
+++ b/protocol/zcash.bib
@@ -131,7 +131,7 @@ Proceedings of the 9th International Conference on Theory and Practice in Public
    title={High-speed high-security signatures},
    journal={Journal of Cryptographic Engineering},
    volume={2},
-   pages={77-89},
+   pages={77--89},
    date={2011-09-26},
    url={http://cr.yp.to/papers.html#ed25519},
    urldate={2016-08-14},
@@ -315,7 +315,7 @@ L. Hernández Encinas and C. Sánchez Ávila},
 (23--25 August 2010, Atlanta, GA, USA, 23--25 August 2010)},
    isbn={978-1-4244-7407-3},
    year={2010},
-   pages={1-4},
+   pages={1--4},
    publisher={IEEE},
    doi={10.1109/ISIAS.2010.5604194},
    url={https://digital.csic.es/bitstream/10261/32674/1/Gayoso_A%20Comparison%20of%20the%20Standardized%20Versions%20of%20ECIES.pdf},
@@ -731,7 +731,7 @@ generic composition paradigm},
    month={February},
    year={2004},
    publisher={TarcherPerigee},
-   pages={109-110},
+   pages={109--110},
    isbn={1-58542-206-1}
 }