\documentclass{article} \RequirePackage{amsmath} \RequirePackage{bytefield} \RequirePackage{graphicx} \RequirePackage{newtxmath} \RequirePackage{mathtools} \RequirePackage{xspace} \RequirePackage{url} \RequirePackage{changepage} \RequirePackage[unicode,bookmarksnumbered,bookmarksopen,pdfview=Fit]{hyperref} \RequirePackage{cleveref} \RequirePackage{nameref} \RequirePackage{enumitem} \RequirePackage{tabularx} \RequirePackage{hhline} \RequirePackage[usestackEOL]{stackengine} \RequirePackage{comment} \RequirePackage[style=alphabetic,maxbibnames=99,dateabbrev=false,urldate=iso8601,backref=true,backrefstyle=none,backend=biber]{biblatex} \addbibresource{zcash.bib} % Fonts \RequirePackage{lmodern} \RequirePackage{bold-extra} \RequirePackage{quattrocento} \RequirePackage{dsfont} % 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. \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. \let\oldtexttt\texttt \let\oldmathtt\mathtt \renewcommand{\texttt}[1]{\scalebox{1.02}[1.07]{\oldtexttt{#1}}} \renewcommand{\mathtt}[1]{\scalebox{1.02}[1.07]{$\oldmathtt{#1}$}} % bold but not extended \newcommand{\textbnx}[1]{{\fontseries{b}\selectfont #1}} \crefformat{footnote}{#2\footnotemark[#1]#3} \DeclareLabelalphaTemplate{ \labelelement{\field{citekey}} } \DefineBibliographyStrings{english}{ page = {page}, pages = {pages}, backrefpage = {\mbox{$\uparrow$ p\!}}, backrefpages = {\mbox{$\uparrow$ p\!}} } \setlength{\oddsidemargin}{-0.25in} \setlength{\textwidth}{7in} \setlength{\topmargin}{-0.75in} \setlength{\textheight}{9.2in} \setlength{\parskip}{1.5ex} \setlength{\parindent}{0ex} \renewcommand{\arraystretch}{1.4} \overfullrule=2cm \setlist[itemize]{itemsep=0.5ex,topsep=0.2ex,after=\vspace{1.5ex}} \newcommand{\docversion}{Version unavailable (check protocol.ver)} \InputIfFileExists{protocol.ver}{}{} \newcommand{\doctitle}{Zcash Protocol Specification} \newcommand{\leadauthor}{Daira Hopwood} \newcommand{\coauthors}{Sean Bowe | Taylor Hornby | Nathan Wilcox} \hypersetup{ pdfborderstyle={/S/U/W 0.7}, pdfinfo={ Title={\doctitle, \docversion}, Author={\leadauthor\ | \coauthors} } } \renewcommand{\sectionautorefname}{\S\!} \renewcommand{\subsectionautorefname}{\S\!} \renewcommand{\subsubsectionautorefname}{\S\!} \newcommand{\crossref}[1]{\autoref{#1}\, \emph{`\nameref*{#1}\kern -0.05em'} on p.\,\pageref*{#1}} \newcommand{\nstrut}{\rule[-.2\baselineskip]{0pt}{\baselineskip}} \newcommand{\nsection}[1]{\section{\texorpdfstring{#1\nstrut}{#1}}} \newcommand{\nsubsection}[1]{\subsection{\texorpdfstring{#1\nstrut}{#1}}} \newcommand{\nsubsubsection}[1]{\subsubsection{\texorpdfstring{#1\nstrut}{#1}}} \mathchardef\mhyphen="2D % http://tex.stackexchange.com/a/309445/78411 \DeclareFontFamily{U}{FdSymbolA}{} \DeclareFontShape{U}{FdSymbolA}{m}{n}{ <-> s*[.4] FdSymbolA-Regular }{} \DeclareSymbolFont{fdsymbol}{U}{FdSymbolA}{m}{n} \DeclareMathSymbol{\smallcirc}{\mathord}{fdsymbol}{"60} \makeatletter \newcommand{\hollowcolon}{\mathpalette\hollow@colon\relax} \newcommand{\hollow@colon}[2]{ \mspace{0.7mu} \vbox{\hbox{$\m@th#1\smallcirc$}\nointerlineskip\kern.45ex \hbox{$\m@th#1\smallcirc$}\kern-.06ex} \mspace{1mu} } \makeatother \newcommand{\typecolon}{\;\hollowcolon\;} \newcommand{\hairspace}{~\!} \newcommand{\hfrac}[2]{\scalebox{0.8}{$\genfrac{}{}{0.5pt}{0}{#1}{#2}$}} \RequirePackage[usenames,dvipsnames]{xcolor} % https://en.wikibooks.org/wiki/LaTeX/Colors#The_68_standard_colors_known_to_dvips \newcommand{\todo}[1]{{\color{Sepia}\sf{TODO: #1}}} \newcommand{\changedcolor}{magenta} \newcommand{\setchanged}{\color{\changedcolor}} \newcommand{\changed}[1]{\texorpdfstring{{\setchanged{#1}}}{#1}} % terminology \newcommand{\term}[1]{\textsl{#1}\kern 0.05em\xspace} \newcommand{\titleterm}[1]{#1} \newcommand{\termbf}[1]{\textbf{#1}\xspace} \newcommand{\conformance}[1]{\textbnx{#1}\xspace} \newcommand{\Zcash}{\termbf{Zcash}} \newcommand{\Zerocash}{\termbf{Zerocash}} \newcommand{\Bitcoin}{\termbf{Bitcoin}} \newcommand{\ZEC}{\termbf{ZEC}} \newcommand{\zatoshi}{\term{zatoshi}} \newcommand{\MUST}{\conformance{MUST}} \newcommand{\MUSTNOT}{\conformance{MUST NOT}} \newcommand{\SHOULD}{\conformance{SHOULD}} \newcommand{\SHOULDNOT}{\conformance{SHOULD NOT}} \newcommand{\ALLCAPS}{\conformance{ALL CAPS}} \newcommand{\note}{\term{note}} \newcommand{\notes}{\term{notes}} \newcommand{\Note}{\titleterm{Note}} \newcommand{\Notes}{\titleterm{Notes}} \newcommand{\dummy}{\term{dummy}} \newcommand{\dummyNotes}{\term{dummy notes}} \newcommand{\DummyNotes}{\titleterm{Dummy Notes}} \newcommand{\commitmentScheme}{\term{commitment scheme}} \newcommand{\commitmentTrapdoor}{\term{commitment trapdoor}} \newcommand{\commitmentTrapdoors}{\term{commitment trapdoors}} \newcommand{\trapdoor}{\term{trapdoor}} \newcommand{\noteCommitment}{\term{note commitment}} \newcommand{\noteCommitments}{\term{note commitments}} \newcommand{\NoteCommitment}{\titleterm{Note Commitment}} \newcommand{\NoteCommitments}{\titleterm{Note Commitments}} \newcommand{\noteCommitmentTree}{\term{note commitment tree}} \newcommand{\NoteCommitmentTree}{\titleterm{Note Commitment Tree}} \newcommand{\noteTraceabilitySet}{\term{note traceability set}} \newcommand{\noteTraceabilitySets}{\term{note traceability sets}} \newcommand{\joinSplitDescription}{\term{JoinSplit description}} \newcommand{\joinSplitDescriptions}{\term{JoinSplit descriptions}} \newcommand{\JoinSplitDescriptions}{\titleterm{JoinSplit Descriptions}} \newcommand{\sequenceOfJoinSplitDescriptions}{\changed{sequence of} \joinSplitDescription\changed{\term{s}}\xspace} \newcommand{\joinSplitTransfer}{\term{JoinSplit transfer}} \newcommand{\joinSplitTransfers}{\term{JoinSplit transfers}} \newcommand{\JoinSplitTransfer}{\titleterm{JoinSplit Transfer}} \newcommand{\JoinSplitTransfers}{\titleterm{JoinSplit Transfers}} \newcommand{\joinSplitSignature}{\term{JoinSplit signature}} \newcommand{\joinSplitSignatures}{\term{JoinSplit signatures}} \newcommand{\joinSplitSigningKey}{\term{JoinSplit signing key}} \newcommand{\joinSplitVerifyingKey}{\term{JoinSplit verifying key}} \newcommand{\joinSplitStatement}{\term{JoinSplit statement}} \newcommand{\joinSplitStatements}{\term{JoinSplit statements}} \newcommand{\JoinSplitStatement}{\titleterm{JoinSplit Statement}} \newcommand{\joinSplitProof}{\term{JoinSplit proof}} \newcommand{\statement}{\term{statement}} \newcommand{\zeroKnowledgeProof}{\term{zero-knowledge proof}} \newcommand{\ZeroKnowledgeProofs}{\titleterm{Zero-Knowledge Proofs}} \newcommand{\provingSystem}{\term{proving system}} \newcommand{\zeroKnowledgeProvingSystem}{\term{zero-knowledge proving system}} \newcommand{\ZeroKnowledgeProvingSystem}{\titleterm{Zero-Knowledge Proving System}} \newcommand{\ppzkSNARK}{\term{preprocessing zk-SNARK}} \newcommand{\provingKey}{\term{proving key}} \newcommand{\zkProvingKeys}{\term{zero-knowledge proving keys}} \newcommand{\verifyingKey}{\term{verifying key}} \newcommand{\zkVerifyingKeys}{\term{zero-knowledge verifying keys}} \newcommand{\joinSplitParameters}{\term{JoinSplit parameters}} \newcommand{\JoinSplitParameters}{\titleterm{JoinSplit Parameters}} \newcommand{\arithmeticCircuit}{\term{arithmetic circuit}} \newcommand{\rankOneConstraintSystem}{\term{Rank 1 Constraint System}} \newcommand{\primary}{\term{primary}} \newcommand{\primaryInput}{\term{primary input}} \newcommand{\primaryInputs}{\term{primary inputs}} \newcommand{\auxiliaryInput}{\term{auxiliary input}} \newcommand{\auxiliaryInputs}{\term{auxiliary inputs}} \newcommand{\fullnode}{\term{full node}} \newcommand{\fullnodes}{\term{full nodes}} \newcommand{\anchor}{\term{anchor}} \newcommand{\anchors}{\term{anchors}} \newcommand{\block}{\term{block}} \newcommand{\blocks}{\term{blocks}} \newcommand{\blockHeader}{\term{block header}} \newcommand{\blockHeaders}{\term{block headers}} \newcommand{\BlockHeaders}{\titleterm{Block Headers}} \newcommand{\blockVersionNumber}{\term{block version number}} \newcommand{\blockTime}{\term{block time}} \newcommand{\blockHeight}{\term{block height}} \newcommand{\genesisBlock}{\term{genesis block}} \newcommand{\transaction}{\term{transaction}} \newcommand{\transactions}{\term{transactions}} \newcommand{\Transactions}{\titleterm{Transactions}} \newcommand{\transactionFee}{\term{transaction fee}} \newcommand{\transactionFees}{\term{transaction fees}} \newcommand{\transactionVersionNumber}{\term{transaction version number}} \newcommand{\coinbaseTransaction}{\term{coinbase transaction}} \newcommand{\coinbaseTransactions}{\term{coinbase transactions}} \newcommand{\CoinbaseTransactions}{\titleterm{Coinbase Transactions}} \newcommand{\transparent}{\term{transparent}} \newcommand{\xTransparent}{\term{Transparent}} \newcommand{\Transparent}{\titleterm{Transparent}} \newcommand{\transparentValuePool}{\term{transparent value pool}} \newcommand{\xprotected}{\term{protected}} \newcommand{\protectedNote}{\term{protected note}} \newcommand{\protectedNotes}{\term{protected notes}} \newcommand{\xProtected}{\term{Protected}} \newcommand{\Protected}{\titleterm{Protected}} \newcommand{\blockchainview}{\term{block chain view}} \newcommand{\blockchain}{\term{block chain}} \newcommand{\mempool}{\term{mempool}} \newcommand{\treestate}{\term{treestate}} \newcommand{\treestates}{\term{treestates}} \newcommand{\nullifier}{\term{nullifier}} \newcommand{\nullifiers}{\term{nullifiers}} \newcommand{\Nullifier}{\titleterm{Nullifier}} \newcommand{\Nullifiers}{\titleterm{Nullifiers}} \newcommand{\nullifierSet}{\term{nullifier set}} \newcommand{\NullifierSet}{\titleterm{Nullifier Set}} % Daira: This doesn't adequately distinguish between zk stuff and transparent stuff \newcommand{\paymentAddress}{\term{payment address}} \newcommand{\paymentAddresses}{\term{payment addresses}} \newcommand{\viewingKey}{\term{viewing key}} \newcommand{\viewingKeys}{\term{viewing keys}} \newcommand{\spendingKey}{\term{spending key}} \newcommand{\spendingKeys}{\term{spending keys}} \newcommand{\payingKey}{\term{paying key}} \newcommand{\transmissionKey}{\term{transmission key}} \newcommand{\transmissionKeys}{\term{transmission keys}} \newcommand{\keyTuple}{\term{key tuple}} \newcommand{\notePlaintext}{\term{note plaintext}} \newcommand{\notePlaintexts}{\term{note plaintexts}} \newcommand{\NotePlaintexts}{\titleterm{Note Plaintexts}} \newcommand{\notesCiphertext}{\term{transmitted notes ciphertext}} \newcommand{\incrementalMerkleTree}{\term{incremental Merkle tree}} \newcommand{\merkleRoot}{\term{root}} \newcommand{\merkleNode}{\term{node}} \newcommand{\merkleNodes}{\term{nodes}} \newcommand{\merkleHash}{\term{hash value}} \newcommand{\merkleHashes}{\term{hash values}} \newcommand{\merkleLeafNode}{\term{leaf node}} \newcommand{\merkleLeafNodes}{\term{leaf nodes}} \newcommand{\merkleInternalNode}{\term{internal node}} \newcommand{\merkleInternalNodes}{\term{internal nodes}} \newcommand{\MerkleInternalNodes}{\term{Internal nodes}} \newcommand{\merklePath}{\term{path}} \newcommand{\merkleLayer}{\term{layer}} \newcommand{\merkleLayers}{\term{layers}} \newcommand{\merkleIndex}{\term{index}} \newcommand{\merkleIndices}{\term{indices}} \newcommand{\zkSNARK}{\term{zk-SNARK}} \newcommand{\zkSNARKs}{\term{zk-SNARKs}} \newcommand{\libsnark}{\term{libsnark}} \newcommand{\memo}{\term{memo field}} \newcommand{\memos}{\term{memo fields}} \newcommand{\Memos}{\titleterm{Memo Fields}} \newcommand{\keyAgreementScheme}{\term{key agreement scheme}} \newcommand{\KeyAgreement}{\titleterm{Key Agreement}} \newcommand{\keyDerivationFunction}{\term{Key Derivation Function}} \newcommand{\KeyDerivation}{\titleterm{Key Derivation}} \newcommand{\encryptionScheme}{\term{encryption scheme}} \newcommand{\symmetricEncryptionScheme}{\term{authenticated one-time symmetric encryption scheme}} \newcommand{\SymmetricEncryption}{\titleterm{Authenticated One-Time Symmetric Encryption}} \newcommand{\signatureScheme}{\term{signature scheme}} \newcommand{\pseudoRandomFunction}{\term{Pseudo Random Function}} \newcommand{\pseudoRandomFunctions}{\term{Pseudo Random Functions}} \newcommand{\PseudoRandomFunctions}{\titleterm{Pseudo Random Functions}} % conventions \newcommand{\bytes}[1]{\underline{\raisebox{-0.22ex}{}\smash{#1}}} \newcommand{\zeros}[1]{[0]^{#1}} \newcommand{\bit}{\mathds{B}} \newcommand{\Nat}{\mathbb{N}} \newcommand{\PosInt}{\mathbb{N}^+} \newcommand{\Rat}{\mathbb{Q}} \newcommand{\typeexp}[2]{{#1}\vphantom{)}^{[{#2}]}} \newcommand{\bitseq}[1]{\typeexp{\bit}{#1}} \newcommand{\byteseqs}{\typeexp{\bit}{8\mult\Nat}} \newcommand{\concatbits}{\mathsf{concat}_\bit} \newcommand{\hexint}[1]{\mathbf{0x{#1}}} \newcommand{\dontcare}{\kern -0.06em\raisebox{0.1ex}{\footnotesize{$\times$}}} \newcommand{\ascii}[1]{\textbf{``\texttt{#1}"}} \newcommand{\Justthebox}[2][-1.3ex]{\;\raisebox{#1}{\usebox{#2}}\;} \newcommand{\hSigCRH}{\mathsf{hSigCRH}} \newcommand{\hSigLength}{\mathsf{\ell_{hSig}}} \newcommand{\hSigType}{\bitseq{\hSigLength}} \newcommand{\EquihashGen}[1]{\mathsf{EquihashGen}_{#1}} \newcommand{\CRH}{\mathsf{CRH}} \newcommand{\CRHbox}[1]{\SHA\left(\Justthebox{#1}\right)} \newcommand{\SHA}{\mathtt{SHA256Compress}} \newcommand{\SHAName}{\term{SHA-256 compression}} \newcommand{\FullHash}{\mathtt{SHA256}} \newcommand{\FullHashName}{\mathsf{SHA\mhyphen256}} \newcommand{\Blake}[1]{\mathsf{BLAKE2b\kern 0.05em\mhyphen{#1}}} \newcommand{\BlakeGeneric}{\mathsf{BLAKE2b}} \newcommand{\FullHashbox}[1]{\FullHash\left(\Justthebox{#1}\right)} \newcommand{\setof}[1]{\{{#1}\}} \newcommand{\range}[2]{\{{#1}\,..\,{#2}\}} \newcommand{\minimum}{\mathsf{min}} \newcommand{\floor}[1]{\mathsf{floor}\!\left({#1}\right)} \newcommand{\ceiling}[1]{\mathsf{ceiling}\!\left({#1}\right)} \newcommand{\vsum}[2]{\smashoperator[r]{\sum_{#1}^{#2}}} \newcommand{\vxor}[2]{\smashoperator[r]{\bigoplus_{#1}^{#2}}} \newcommand{\xor}{\oplus} \newcommand{\mult}{\cdot} \newcommand{\rightarrowR}{\buildrel{\scriptstyle\mathrm{R}}\over\rightarrow} \newcommand{\leftarrowR}{\buildrel{\scriptstyle\mathrm{R}}\over\leftarrow} % key pairs: \newcommand{\PaymentAddress}{\mathsf{addr_{pk}}} \newcommand{\PaymentAddressLeadByte}{\hexint{16}} \newcommand{\PaymentAddressSecondByte}{\hexint{9A}} \newcommand{\SpendingKeyLeadByte}{\hexint{AB}} \newcommand{\SpendingKeySecondByte}{\hexint{36}} \newcommand{\PtoSHAddressLeadByte}{\hexint{1B}} \newcommand{\PtoSHAddressSecondByte}{\hexint{9C}} \newcommand{\PtoPKHAddressLeadByte}{\hexint{1B}} \newcommand{\PtoPKHAddressSecondByte}{\hexint{97}} \newcommand{\PaymentAddressTestnetLeadByte}{\hexint{14}} \newcommand{\PaymentAddressTestnetSecondByte}{\hexint{51}} \newcommand{\SpendingKeyTestnetLeadByte}{\hexint{B1}} \newcommand{\SpendingKeyTestnetSecondByte}{\hexint{EB}} \newcommand{\PtoSHAddressTestnetLeadByte}{\hexint{1B}} \newcommand{\PtoSHAddressTestnetSecondByte}{\hexint{9A}} \newcommand{\PtoPKHAddressTestnetLeadByte}{\hexint{1C}} \newcommand{\PtoPKHAddressTestnetSecondByte}{\hexint{05}} \newcommand{\NotePlaintextLeadByte}{\hexint{00}} \newcommand{\AuthPublic}{\mathsf{a_{pk}}} \newcommand{\AuthPrivate}{\mathsf{a_{sk}}} \newcommand{\AuthPublicX}[1]{\mathsf{a^\mathrm{#1}_{pk}}} \newcommand{\AuthPrivateX}[1]{\mathsf{a^\mathrm{#1}_{sk}}} \newcommand{\AuthPrivateLength}{\mathsf{\ell_{\AuthPrivate}}} \newcommand{\AuthPublicOld}[1]{\mathsf{a^{old}_{pk,\mathnormal{#1}}}} \newcommand{\AuthPrivateOld}[1]{\mathsf{a^{old}_{sk,\mathnormal{#1}}}} \newcommand{\AuthPublicOldX}[1]{\mathsf{a^{old}_{pk,\mathrm{#1}}}} \newcommand{\AuthPrivateOldX}[1]{\mathsf{a^{old}_{sk,\mathrm{#1}}}} \newcommand{\AuthPublicNew}[1]{\mathsf{a^{new}_{pk,\mathnormal{#1}}}} \newcommand{\AuthPrivateNew}[1]{\mathsf{a^{new}_{sk,\mathnormal{#1}}}} \newcommand{\AddressPublicNew}[1]{\mathsf{addr^{new}_{pk,\mathnormal{#1}}}} \newcommand{\enc}{\mathsf{enc}} \newcommand{\DHSecret}[1]{\mathsf{sharedSecret}_{#1}} \newcommand{\EphemeralPublic}{\mathsf{epk}} \newcommand{\EphemeralPrivate}{\mathsf{esk}} \newcommand{\TransmitPublic}{\mathsf{pk_{enc}}} \newcommand{\TransmitPublicSup}[1]{\mathsf{pk}^{#1}_\mathsf{enc}} \newcommand{\TransmitPublicNew}[1]{\mathsf{pk^{new}_{\enc,\mathnormal{#1}}}} \newcommand{\TransmitPrivate}{\mathsf{sk_{enc}}} \newcommand{\TransmitPrivateSup}[1]{\mathsf{sk}^{#1}_\mathsf{enc}} % PRFs \newcommand{\PRF}[2]{\mathsf{{PRF}^{#2}_\mathnormal{#1}}} \newcommand{\PRFaddr}[1]{\PRF{#1}{addr}} \newcommand{\PRFnf}[1]{\PRF{#1}{\nf}} \newcommand{\PRFsn}[1]{\PRF{#1}{sn}} \newcommand{\PRFpk}[1]{\PRF{#1}{pk}} \newcommand{\PRFrho}[1]{\PRF{#1}{\NoteAddressRand}} \newcommand{\PRFOutputLength}{\mathsf{\ell_{PRF}}} \newcommand{\PRFOutput}{\bitseq{\PRFOutputLength}} % Commitments \newcommand{\Commit}[1]{\mathsf{COMM}_{#1}} \newcommand{\CommitTrapdoor}{\mathsf{COMM.Trapdoor}} \newcommand{\CommitInput}{\mathsf{COMM.Input}} \newcommand{\CommitOutput}{\mathsf{COMM.Output}} \newcommand{\NoteCommit}{\mathtt{NoteCommitment}} \newcommand{\Uncommitted}{\mathsf{Uncommitted}} % Symmetric encryption \newcommand{\Sym}{\mathsf{Sym}} \newcommand{\SymEncrypt}[1]{\mathsf{Sym.}\mathtt{Encrypt}_\mathsf{#1}} \newcommand{\SymDecrypt}[1]{\mathsf{Sym.}\mathtt{Decrypt}_\mathsf{#1}} \newcommand{\SymSpecific}{\mathsf{AEAD\_CHACHA20\_POLY1305}} \newcommand{\SymCipher}{\mathsf{ChaCha20}} \newcommand{\SymAuth}{\mathsf{Poly1305}} \newcommand{\Ptext}{\mathsf{P}} \newcommand{\Plaintext}{\mathsf{Sym.}\mathbf{P}} \newcommand{\Ctext}{\mathsf{C}} \newcommand{\Ciphertext}{\mathsf{Sym.}\mathbf{C}} \newcommand{\Key}{\mathsf{K}} \newcommand{\Keyspace}{\mathsf{Sym.}\mathbf{K}} \newcommand{\TransmitPlaintext}[1]{\Ptext^\enc_{#1}} \newcommand{\TransmitCiphertext}[1]{\Ctext^\enc_{#1}} \newcommand{\TransmitKey}[1]{\Key^\enc_{#1}} \newcommand{\Adversary}{\mathcal{A}} \newcommand{\CryptoBoxSeal}{\mathsf{crypto\_box\_seal}} % Key agreement \newcommand{\KA}{\mathsf{KA}} \newcommand{\KAPublic}{\mathsf{KA.Public}} \newcommand{\KAPrivate}{\mathsf{KA.Private}} \newcommand{\KASharedSecret}{\mathsf{KA.SharedSecret}} \newcommand{\KAFormatPrivate}{\mathsf{KA.}\mathtt{FormatPrivate}} \newcommand{\KADerivePublic}{\mathsf{KA.}\mathtt{DerivePublic}} \newcommand{\KAAgree}{\mathsf{KA.}\mathtt{Agree}} \newcommand{\CurveMultiply}{\mathsf{Curve25519}} \newcommand{\CurveBase}{\bytes{9}} \newcommand{\Clamp}{\mathsf{clamp_{Curve25519}}} % KDF \newcommand{\KDF}{\mathsf{KDF}} \newcommand{\kdftag}{\mathsf{kdftag}} \newcommand{\kdfinput}{\mathsf{kdfinput}} % Notes \newcommand{\Value}{\mathsf{v}} \newcommand{\ValueNew}[1]{\mathsf{v^{new}_\mathnormal{#1}}} \newcommand{\NoteTuple}[1]{\mathbf{n}_{#1}} \newcommand{\NoteType}{\mathsf{Note}} \newcommand{\NotePlaintext}[1]{\mathbf{np}_{#1}} \newcommand{\NoteCommitRand}{\mathsf{r}} \newcommand{\NoteCommitRandLength}{\mathsf{\ell_{\NoteCommitRand}}} \newcommand{\NoteCommitRandOld}[1]{\mathsf{r^{old}_\mathnormal{#1}}} \newcommand{\NoteCommitRandNew}[1]{\mathsf{r^{new}_\mathnormal{#1}}} \newcommand{\NoteAddressRand}{\mathsf{\uprho}} \newcommand{\NoteAddressRandOld}[1]{\mathsf{\uprho^{old}_\mathnormal{#1}}} \newcommand{\NoteAddressRandOldX}[1]{\mathsf{\uprho^{old}_\mathrm{#1}}} \newcommand{\NoteAddressRandNew}[1]{\mathsf{\uprho^{new}_\mathnormal{#1}}} \newcommand{\NoteAddressPreRand}{\mathsf{\upvarphi}} \newcommand{\NoteAddressPreRandLength}{\mathsf{\ell_{\NoteAddressPreRand}}} \newcommand{\NoteCommitS}{\mathsf{s}} \newcommand{\cm}{\mathsf{cm}} \newcommand{\cmOldX}[1]{\mathsf{{cm}^{old}_\mathrm{#1}}} \newcommand{\cmNew}[1]{\mathsf{{cm}^{new}_\mathnormal{#1}}} \newcommand{\snOldX}[1]{\mathsf{{sn}^{old}_\mathrm{#1}}} \newcommand{\nf}{\mathsf{nf}} \newcommand{\nfOld}[1]{\nf^\mathsf{old}_\mathnormal{#1}} \newcommand{\Memo}{\mathsf{memo}} \newcommand{\DecryptNote}{\mathtt{DecryptNote}} \newcommand{\ReplacementCharacter}{\textsf{U+FFFD}} % Money supply \newcommand{\MAXMONEY}{\mathsf{MAX\_MONEY}} \newcommand{\BlockSubsidy}{\mathsf{BlockSubsidy}} \newcommand{\MinerSubsidy}{\mathsf{MinerSubsidy}} \newcommand{\FoundersReward}{\mathsf{FoundersReward}} \newcommand{\SlowStartInterval}{\mathsf{SlowStartInterval}} \newcommand{\SlowStartShift}{\mathsf{SlowStartShift}} \newcommand{\SlowStartRate}{\mathsf{SlowStartRate}} \newcommand{\HalvingInterval}{\mathsf{HalvingInterval}} \newcommand{\MaxBlockSubsidy}{\mathsf{MaxBlockSubsidy}} \newcommand{\NumFounderAddresses}{\mathsf{NumFounderAddresses}} \newcommand{\FounderAddressChangeInterval}{\mathsf{FounderAddressChangeInterval}} \newcommand{\FoundersFraction}{\mathsf{FoundersFraction}} \newcommand{\BlockHeight}{\mathsf{height}} \newcommand{\Halving}{\mathsf{Halving}} \newcommand{\FounderAddress}{\mathsf{FounderAddress}} \newcommand{\FounderAddressList}{\mathsf{FounderAddressList}} \newcommand{\FounderAddressIndex}{\mathsf{FounderAddressIndex}} \newcommand{\RedeemScriptHash}{\mathsf{RedeemScriptHash}} \newcommand{\blockSubsidy}{\term{block subsidy}} \newcommand{\minerSubsidy}{\term{miner subsidy}} \newcommand{\foundersReward}{\term{Founders' Reward}} \newcommand{\slowStartPeriod}{\term{slow-start period}} \newcommand{\halvingInterval}{\term{halving interval}} % Signatures \newcommand{\Sig}{\mathsf{Sig}} \newcommand{\SigPublic}{\mathsf{Sig.Public}} \newcommand{\SigPrivate}{\mathsf{Sig.Private}} \newcommand{\SigMessage}{\mathsf{Sig.Message}} \newcommand{\SigSignature}{\mathsf{Sig.Signature}} \newcommand{\SigGen}{\mathsf{Sig.Gen}} \newcommand{\SigSign}[1]{\mathsf{Sig.Sign}_{#1}} \newcommand{\SigVerify}[1]{\mathsf{Sig.Verify}_{#1}} \newcommand{\JoinSplitSig}{\mathsf{JoinSplitSig}} \newcommand{\JoinSplitSigPublic}{\mathsf{JoinSplitSig.Public}} \newcommand{\JoinSplitSigPrivate}{\mathsf{JoinSplitSig.Private}} \newcommand{\JoinSplitSigMessage}{\mathsf{JoinSplitSig.Message}} \newcommand{\JoinSplitSigSignature}{\mathsf{JoinSplitSig.Signature}} \newcommand{\JoinSplitSigGen}{\mathsf{JoinSplitSig.Gen}} \newcommand{\JoinSplitSigSign}[1]{\mathsf{JoinSplitSig.Sign}_{#1}} \newcommand{\JoinSplitSigVerify}[1]{\mathsf{JoinSplitSig.Verify}_{#1}} \newcommand{\JoinSplitSigSpecific}{\mathsf{Ed25519}} \newcommand{\JoinSplitSigHashName}{\mathsf{SHA\mhyphen512}} \newcommand{\EdDSAr}{R} \newcommand{\EdDSAs}{S} \newcommand{\EdDSAR}{\bytes{R}} \newcommand{\EdDSAS}{\bytes{S}} \newcommand{\RandomSeedLength}{\mathsf{\ell_{Seed}}} \newcommand{\RandomSeedType}{\bitseq{\mathsf{\ell_{Seed}}}} \newcommand{\pksig}{\mathsf{pk_{sig}}} \newcommand{\sk}{\mathsf{sk}} \newcommand{\hSigInput}{\mathsf{hSigInput}} \newcommand{\dataToBeSigned}{\mathsf{dataToBeSigned}} % Merkle tree \newcommand{\MerkleDepth}{\mathsf{d_{Merkle}}} \newcommand{\MerkleNode}[2]{\mathsf{M}^{#1}_{#2}} \newcommand{\MerkleSibling}{\mathsf{sibling}} \newcommand{\MerkleCRH}{\mathsf{MerkleCRH}} \newcommand{\MerkleHashLength}{\mathsf{\ell_{Merkle}}} \newcommand{\MerkleHash}{\bitseq{\MerkleHashLength}} % Transactions \newcommand{\versionField}{\mathtt{version}} \newcommand{\txInCount}{\mathtt{tx\_in\_count}} \newcommand{\txIn}{\mathtt{tx\_in}} \newcommand{\txOutCount}{\mathtt{tx\_out\_count}} \newcommand{\txOut}{\mathtt{tx\_out}} \newcommand{\lockTime}{\mathtt{lock\_time}} \newcommand{\nJoinSplit}{\mathtt{nJoinSplit}} \newcommand{\vJoinSplit}{\mathtt{vJoinSplit}} \newcommand{\vpubOldField}{\mathtt{vpub\_old}} \newcommand{\vpubNewField}{\mathtt{vpub\_new}} \newcommand{\anchorField}{\mathtt{anchor}} \newcommand{\joinSplitSig}{\mathtt{joinSplitSig}} \newcommand{\joinSplitPrivKey}{\mathtt{joinSplitPrivKey}} \newcommand{\joinSplitPubKey}{\mathtt{joinSplitPubKey}} \newcommand{\nullifiersField}{\mathtt{nullifiers}} \newcommand{\commitments}{\mathtt{commitments}} \newcommand{\ephemeralKey}{\mathtt{ephemeralKey}} \newcommand{\encCiphertexts}{\mathtt{encCiphertexts}} \newcommand{\randomSeed}{\mathtt{randomSeed}} \newcommand{\Varies}{\textit{Varies}} \newcommand{\heading}[1]{\multicolumn{1}{c|}{#1}} \newcommand{\type}[1]{\texttt{#1}} \newcommand{\compactSize}{\type{compactSize uint}} \newcommand{\sighashType}{\term{SIGHASH type}} \newcommand{\sighashTypes}{\term{SIGHASH types}} \newcommand{\SIGHASHALL}{\mathsf{SIGHASH\_ALL}} \newcommand{\scriptSig}{\mathtt{scriptSig}} \newcommand{\scriptPubKey}{\mathtt{scriptPubKey}} \newcommand{\ScriptOP}[1]{\texttt{OP\_{#1}}} % Equihash and block headers \newcommand{\validEquihashSolution}{\term{valid Equihash solution}} \newcommand{\powtag}{\mathsf{powtag}} \newcommand{\powheader}{\mathsf{powheader}} \newcommand{\powcount}{\mathsf{powcount}} \newcommand{\nVersion}{\mathtt{nVersion}} \newcommand{\hashPrevBlock}{\mathtt{hashPrevBlock}} \newcommand{\hashMerkleRoot}{\mathtt{hashMerkleRoot}} \newcommand{\hashReserved}{\mathtt{hashReserved}} \newcommand{\nTime}{\mathtt{nTime}} \newcommand{\nBits}{\mathtt{nBits}} \newcommand{\nNonce}{\mathtt{nNonce}} \newcommand{\solutionSize}{\mathtt{solutionSize}} \newcommand{\solution}{\mathtt{solution}} \newcommand{\SHAd}{\term{SHA-256d}} % Proving system \newcommand{\ZK}{\mathsf{ZK}} \newcommand{\ZKProvingKey}{\mathsf{ZK.ProvingKey}} \newcommand{\ZKVerifyingKey}{\mathsf{ZK.VerifyingKey}} \newcommand{\pk}{\mathsf{pk}} \newcommand{\vk}{\mathsf{vk}} \newcommand{\ZKGen}{\mathsf{ZK.Gen}} \newcommand{\ZKProof}{\mathsf{ZK.Proof}} \newcommand{\ZKPrimary}{\mathsf{ZK.PrimaryInput}} \newcommand{\ZKAuxiliary}{\mathsf{ZK.AuxiliaryInput}} \newcommand{\ZKSatisfying}{\mathsf{ZK.SatisfyingInputs}} \newcommand{\ZKProve}[1]{\mathsf{ZK.}\mathtt{Prove}_{#1}} \newcommand{\ZKVerify}[1]{\mathsf{ZK.}\mathtt{Verify}_{#1}} \newcommand{\JoinSplit}{\text{\footnotesize\texttt{JoinSplit}}} \newcommand{\ZKJoinSplit}{\mathsf{ZK}_{\JoinSplit}} \newcommand{\ZKJoinSplitVerify}{\ZKJoinSplit\mathsf{.Verify}} \newcommand{\ZKJoinSplitProve}{\ZKJoinSplit\mathsf{.Prove}} \newcommand{\ZKJoinSplitProof}{\ZKJoinSplit\mathsf{.Proof}} \newcommand{\Proof}{\pi} \newcommand{\JoinSplitProof}{\Proof_{\JoinSplit}} \newcommand{\zkproof}{\mathtt{zkproof}} \newcommand{\POUR}{\texttt{POUR}} % JoinSplit \newcommand{\hSig}{\mathsf{h_{Sig}}} \newcommand{\hSigText}{\texorpdfstring{$\hSig$}{hSig}} \newcommand{\h}[1]{\mathsf{h_{\mathnormal{#1}}}} \newcommand{\NOld}{\mathrm{N}^\mathsf{old}} \newcommand{\NNew}{\mathrm{N}^\mathsf{new}} \newcommand{\allN}[1]{\mathrm{1}..\mathrm{N}^\mathsf{#1}} \newcommand{\allOld}{\allN{old}} \newcommand{\allNew}{\allN{new}} \newcommand{\setofOld}{\setof{\allOld}} \newcommand{\setofNew}{\setof{\allNew}} \newcommand{\vmacs}{\mathtt{vmacs}} \newcommand{\GroupG}[1]{\mathbb{G}_{#1}} \newcommand{\PointP}[1]{\mathcal{P}_{#1}} \newcommand{\GF}[1]{\mathbb{F}_{#1}} \newcommand{\GFstar}[1]{\mathbb{F}^\ast_{#1}} \newcommand{\ECtoOSP}{\mathsf{EC2OSP}} \newcommand{\ECtoOSPXL}{\mathsf{EC2OSP\mhyphen{}XL}} \newcommand{\ECtoOSPXS}{\mathsf{EC2OSP\mhyphen{}XS}} \newcommand{\ItoOSP}[1]{\mathsf{I2OSP}_{#1}} \newcommand{\ItoBSP}[1]{\mathsf{I2BSP}_{#1}} \newcommand{\FEtoIP}{\mathsf{FE2IP}} \newcommand{\BNImpl}{\mathtt{ALT\_BN128}} \newcommand{\vpubOld}{\mathsf{v_{pub}^{old}}} \newcommand{\vpubNew}{\mathsf{v_{pub}^{new}}} \newcommand{\nOld}[1]{\NoteTuple{#1}^\mathsf{old}} \newcommand{\nNew}[1]{\NoteTuple{#1}^\mathsf{new}} \newcommand{\vOld}[1]{\mathsf{v}_{#1}^\mathsf{old}} \newcommand{\vNew}[1]{\mathsf{v}_{#1}^\mathsf{new}} \newcommand{\RandomSeed}{\mathsf{randomSeed}} \newcommand{\rt}{\mathsf{rt}} \newcommand{\treepath}[1]{\mathsf{path}_{#1}} \newcommand{\Receive}{\mathsf{Receive}} \newcommand{\EnforceCommit}[1]{\mathsf{enforce}_{#1}} \newcommand{\consensusrule}[1]{\subparagraph{Consensus rule:}{#1}} \newenvironment{consensusrules}{\subparagraph{Consensus rules:}\begin{itemize}}{\end{itemize}} \newcommand{\securityrequirement}[1]{\subparagraph{Security requirement:}{#1}} \newenvironment{securityrequirements}{\subparagraph{Security requirements:}\begin{itemize}}{\end{itemize}} \newcommand{\pnote}[1]{\subparagraph{Note:}{#1}} \newenvironment{pnotes}{\subparagraph{Notes:}\begin{itemize}}{\end{itemize}} \begin{document} \title{\doctitle \\ \Large \docversion \\ \vspace{1ex} \large as intended for the \Zcash release of autumn 2016} \author{\Large \leadauthor \\ \Large \coauthors} \date{\today} \maketitle \tableofcontents \newpage \nsection{Introduction} \Zcash is an implementation of the \term{Decentralized Anonymous Payment} scheme \Zerocash \cite{BCG+2014}, with some security fixes and adjustments to terminology, functionality and performance. It bridges the existing \emph{transparent} payment scheme used by \Bitcoin \cite{Naka2008} with a \emph{protected} payment scheme secured by zero-knowledge succinct non-interactive arguments of knowledge (\zkSNARKs). Changes from the original \Zerocash are explained in \crossref{differences}, and highlighted in \changed{\changedcolor} throughout the document. Technical terms for concepts that play an important role in \Zcash are written in \term{slanted text}. \emph{Italics} are used for emphasis and for references between sections of the document. The key words \MUST, \MUSTNOT, \SHOULD, and \SHOULDNOT in this document are to be interpreted as described in \cite{RFC-2119} when they appear in \ALLCAPS. These words may also appear in this document in lower case as plain English words, absent their normative meanings. This specification is structured as follows: \begin{itemize} \item Notation | definitions of notation used throughout the document; \item Concepts | the principal abstractions needed to understand the protocol; \item Abstract Protocol | a high-level description of the protocol in terms of ideal cryptographic components; \item Concrete Protocol | how the functions and encodings of the abstract protocol are instantiated; \item Consensus Changes from \Bitcoin | how \Zcash differs from \Bitcoin at the consensus layer, including the Proof of Work; \item Differences from the \Zerocash protocol | a summary of changes from the protocol in \cite{BCG+2014}. \end{itemize} \nsubsection{Caution} \Zcash security depends on consensus. Should a program interacting with the \Zcash network diverge from consensus, its security will be weakened or destroyed. The cause of the divergence doesn't matter: it could be a bug in your program, it could be an error in this documentation which you implemented as described, or it could be that you do everything right but other software on the network behaves unexpectedly. The specific cause will not matter to the users of your software whose wealth is lost. Having said that, a specification of \emph{intended} behaviour is essential for security analysis, understanding of the protocol, and maintenance of \Zcash and related software. If you find any mistake in this specification, please contact \texttt{}. While the production \Zcash network has yet to be launched, please feel free to do so in public even if you believe the mistake may indicate a security weakness. \nsubsection{High-level Overview} The following overview is intended to give a concise summary of the ideas behind the protocol, for an audience already familiar with \blockchain-based cryptocurrencies such as \Bitcoin. It is imprecise in some aspects and is not part of the normative protocol specification. Value in \Zcash is either \transparent or \xprotected. Transfers of \transparent value work essentially as in \Bitcoin and have the same privacy properties. \xProtected value is carried by \notes\hairspace\footnote{\label{notesandnullifiers} In \Zerocash \cite{BCG+2014}, \notes were called ``coins'', and \nullifiers were called ``serial numbers''.}, which specify an amount and a \payingKey. The \payingKey is part of a \paymentAddress, which is a destination to which \notes can be sent. 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 between \notes, \noteCommitments, and \nullifiers). Computing the \nullifier requires the associated private \spendingKey. It is infeasible to correlate the \noteCommitment with the corresponding \nullifier without knowledge of at least this \spendingKey. An unspent valid \note, at a given point on the \blockchain, is one for which the \noteCommitment has been publically revealed on the \blockchain prior to that point, but the \nullifier has not. A \transaction can contain \transparent inputs, outputs, and scripts, which all work as in \Bitcoin. It also contains a sequence of zero or more \joinSplitDescriptions. Each of these describes a \joinSplitTransfer\hairspace\footnote{ \joinSplitTransfers in \Zcash generalize ``Mint'' and ``Pour'' \transactions in \Zerocash; see \crossref{trstructure} for the differences.} which takes in a \transparent value and up to two input \notes, and produces a \transparent value and up to two output \notes. The \nullifiers of the input \notes are revealed (preventing them from being spent again) and the commitments of the output \notes are revealed (allowing them to be spent in future). Each \joinSplitDescription also includes a computationally sound \zkSNARK proof, which proves that all of the following hold except with negligable probability: \begin{itemize} \item The input and output values balance (individually for each \joinSplitTransfer). \item For each input \note of non-zero value, some revealed \noteCommitment exists for that \note. \item The prover knew the private \spendingKeys of the input \notes. \item The \nullifiers and \noteCommitments are computed correctly. \item The private \spendingKeys of the input \notes are cryptographically linked to a signature over the whole \transaction, in such a way that the \transaction cannot be modified by a party who did not know these private keys. \item Each output \note is generated in such a way that it is infeasible to cause its \nullifier to collide with the \nullifier of any other \note. \end{itemize} Outside the \zkSNARK, it is also checked that the \nullifiers for the input \notes had not already been revealed (i.e.\ they had not already been spent). A \paymentAddress includes two public keys: a \payingKey matching that of \notes sent to the address, and a \transmissionKey for a key-private asymmetric encryption scheme. ``Key-private'' means that ciphertexts do not reveal information about which key they were encrypted to, except to a holder of the corresponding private key, which in this context is called the \viewingKey. This facility is used to communicate encrypted output \notes on the \blockchain to their intended recipient, who can use the \viewingKey to scan the \blockchain for \notes addressed to them and then decrypt those \notes. The basis of the privacy properties of \Zcash is that when a \note is spent, the spender only proves that some commitment for it had been revealed, without revealing which one. This implies that a spent \note cannot be linked to the \transaction in which it was created. That is, from an adversary's point of view the set of possibilities for a given \note input to a \transaction ---its \noteTraceabilitySet--- includes \emph{all} previous notes that the adversary does not control or know to have been spent. This contrasts with other proposals for private payment systems, such as CoinJoin \cite{Bitcoin-CoinJoin} or CryptoNote \cite{vanS2014}, that are based on mixing of a limited number of transactions and that therefore have smaller \noteTraceabilitySets. The \nullifiers are necessary to prevent double-spending: each note only has 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. \nsection{Notation} The notation $\bit$ means the type of bit values, i.e. $\setof{0, 1}$. The notation $\Nat$ means the set of nonnegative integers. $\PosInt$ means the set of positive integers. $\Rat$ means the set of rationals. The notation $x \typecolon T$ is used to specify that $x$ has type $T$. A cartesian product type is denoted by $S \times T$, and a function type by $S \rightarrow T$. An argument to a function can determine other argument or result types. The type of a randomized algorithm is denoted by $S \rightarrowR T$. The domain of a randomized algorithm may be $()$, indicating that it requires no arguments. Given $f \typecolon S \rightarrowR T$ and $s \typecolon S$, sampling a variable $x \typecolon T$ from the output of $f$ applied to $s$ is denoted by $x \leftarrowR f(s)$. Initial arguments to a function or randomized algorithm may be written as subscripts, e.g.\ if $x \typecolon X$, $y \typecolon Y$, and $f \typecolon X \times Y \rightarrow Z$, then an invocation of $f(x, y)$ can also be written $f_x(y)$. The notation $\typeexp{T}{\ell}$, where $T$ is a type and $\ell$ is an integer, means the type of sequences of length $\ell$ with elements in $T$. For example, $\bitseq{\ell}$ means the set of sequences of $\ell$ bits. The notation $T \subseteq U$ indicates that $T$ is an inclusive subset or subtype of $U$. $\byteseqs$ means the set of bit sequences constrained to be of length a multiple of 8 bits. The notation $\hexint{}$ followed by a string of \textbf{boldface} hexadecimal digits means the corresponding integer converted from hexadecimal. The notation $\ascii{...}$ means the given string represented as a sequence of bytes in US-ASCII. For example, $\ascii{abc}$ represents the byte sequence $[\hexint{61}, \hexint{62}, \hexint{63}]$. The notation $a..b$, used as a subscript, means the sequence of values with indices $a$ through $b$ inclusive. For example, $\AuthPublicNew{\allNew}$ means the sequence $[\AuthPublicNew{\mathrm{1}}, \AuthPublicNew{\mathrm{2}}, ...\,\AuthPublicNew{\NNew}]$. (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 \cite{EWD-831}.) The notation $\range{a}{b}$ means the set or type of integers from $a$ through $b$ inclusive. The notation $[f(x)$ for $x$ from $a$ up to $b\,]$ means the sequence formed by evaluating $f$ on each integer from $a$ to $b$ inclusive, in ascending order. Similarly, $[f(x)$ for $x$ from $a$ down to $b\,]$ means the sequence formed by evaluating $f$ on each integer from $a$ to $b$ inclusive, in descending order. The notation $a\,||\,b$ means the concatenation of sequences $a$ then $b$. The notation $\concatbits(S)$ means the sequence of bits obtained by concatenating the elements of $S$ viewed as bit sequences. If the elements of $S$ are byte sequences, they are converted to bit sequences with the \emph{most significant} bit of each byte first. The notation $\GF{n}$ means the finite field with $n$ elements, and $\GFstar{n}$ means its group under multiplication. $\GF{n}[z]$ means the ring of polynomials over $z$ with coefficients in $\GF{n}$. The notation $a \mult b$ means the result of multiplying $a$ and $b$. This may refer to multiplication of integers, rationals, or finite field elements according to context. The notation $a^b$, for $a$ an integer or finite field element and $b$ an integer, means the result of raising $a$ to the exponent $b$. The notation $a \bmod q$, for $a \typecolon \Nat$ and $q \typecolon \PosInt$, means the remainder on dividing $a$ by $q$. The notation $a \xor b$ means the bitwise exclusive-or of $a$ and $b$, defined either on integers or bit sequences according to context. The notation $\vsum{i=1}{\mathrm{N}} a_i$ means the sum of $a_{\allN{}}$.\; $\vxor{i=1}{\mathrm{N}} a_i$ means the bitwise exclusive-or of $a_{\allN{}}$. The binary relations $<$, $\leq$, $=$, $\geq$, and $>$ have their conventional meanings on integers and rationals, and are defined lexicographically on sequences of integers. The notation $\floor{x}$ means the largest integer $\leq x$. $\ceiling{x}$ means the smallest integer $\geq x$. The symbol $\bot$ is used to indicate unavailable information or a failed decryption. The following integer constants will be instantiated in \crossref{constants}: $\MerkleDepth$, $\NOld$, $\NNew$, $\MerkleHashLength$, $\hSigLength$, $\PRFOutputLength$, $\NoteCommitRandLength$, $\RandomSeedLength$, $\AuthPrivateLength$, $\NoteAddressPreRandLength$, $\MAXMONEY$, $\SlowStartInterval$, $\HalvingInterval$, $\MaxBlockSubsidy$, $\NumFounderAddresses$. The bit sequence constant $\Uncommitted \typecolon \bitseq{\MerkleHashLength}$ and the rational constant $\FoundersFraction \typecolon \Rat$ will also be defined in that section. \nsection{Concepts} \nsubsection{Payment Addresses and Keys} A \keyTuple $(\AuthPrivate, \TransmitPrivate, \PaymentAddress)$ is generated by users who wish to receive payments under this scheme. The \viewingKey $\TransmitPrivate$ and the \paymentAddress $\PaymentAddress = (\AuthPublic, \TransmitPublic)$ are derived from the \spendingKey $\AuthPrivate$. The following diagram depicts the relations between key components. Arrows point from a component to any other component(s) that can be derived from it. \begin{center} \includegraphics[scale=.7]{key_components} \end{center} The composition of \paymentAddresses\changed{, \viewingKeys,} and \spendingKeys is a cryptographic protocol detail that should not normally be exposed to users. However, user-visible operations should be provided to obtain a \paymentAddress or \viewingKey from a \spendingKey. Users can accept payment from multiple parties with a single \paymentAddress $\PaymentAddress$ and the fact that these payments are destined to the same payee is not revealed on the \blockchain, even to the paying parties. \emph{However} if two parties collude to compare a \paymentAddress they can trivially determine they are the same. In the case that a payee wishes to prevent this they should create a distinct \paymentAddress for each payer. \pnote{ It is conventional in cryptography to refer to the key used to encrypt a message in an asymmetric encryption scheme as the ``public key". However, the public key used as the \transmissionKey component of an address ($\TransmitPublic$) need not be publically distributed; it has the same distribution as the \paymentAddress itself. As mentioned above, limiting the distribution of the \paymentAddress is important for some use cases. This also helps to reduce reliance of the overall protocol on the security of the cryptosystem used for \note encryption (see \crossref{inband}), since an adversary would have to know $\TransmitPublic$ in order to exploit a hypothetical weakness in that cryptosystem. } \nsubsection{\Notes} A \note (denoted $\NoteTuple{}$) is a tuple $\changed{(\AuthPublic, \Value, \NoteAddressRand, \NoteCommitRand)}$. It represents that a value $\Value$ is spendable by the recipient who holds the \spendingKey $\AuthPrivate$ corresponding to $\AuthPublic$, as described in the previous section. \begin{itemize} \item $\AuthPublic \typecolon \PRFOutput$ is the \payingKey of the recipient; \item $\Value \typecolon \range{0}{\MAXMONEY}$ is an integer representing the value of the \note in \zatoshi ($1$ \ZEC = $10^8$ \zatoshi); \item $\NoteAddressRand \typecolon \PRFOutput$ is used as input to $\PRFnf{\AuthPrivate}$ to derive the \nullifier of the \note; \item $\NoteCommitRand \typecolon \bitseq{\NoteCommitRandLength}$ is a random bit sequence used as a \commitmentTrapdoor as defined in \crossref{abstractcomm}. \end{itemize} Let $\NoteType$ be the type of a \note, i.e. \changed{ $\PRFOutput \times \range{0}{\MAXMONEY} \times \PRFOutput \times \bitseq{\NoteCommitRandLength}$}. Creation of new \notes is described in \crossref{send}. When \notes are sent, only a commitment (see \crossref{abstractcomm}) to the above values is disclosed publically. This allows the value and recipient to be kept private, while the commitment is used by the \zeroKnowledgeProof when the \note is spent, to check that it exists on the \blockchain. The \noteCommitment is computed as $\NoteCommit(\NoteTuple{}) = \Commit{\NoteCommitRand}(\AuthPublic, \Value, \NoteAddressRand)$, where $\Commit{}$ is instantiated in \crossref{concretecomm}. A \nullifier (denoted $\nf$) is derived from the $\NoteAddressRand$ component of a \note and the recipient's \spendingKey, using a \pseudoRandomFunction (see \crossref{abstractprfs}). Specifically it is derived as $\PRFnf{\AuthPrivate}(\NoteAddressRand)$ where $\PRFnf{}{}$ is instantiated in \crossref{concreteprfs}. A \note is spent by proving knowledge of $\NoteAddressRand$ and $\AuthPrivate$ in zero knowledge while publically disclosing its \nullifier $\nf$, allowing $\nf$ to be used to prevent double-spending. \nsubsubsection{\NotePlaintexts{} and \Memos} Transmitted \notes are stored on the \blockchain in encrypted form, together with a \noteCommitment $\cm$. The \notePlaintexts in a \joinSplitDescription are encrypted to the respective \transmissionKeys $\TransmitPublicNew{\allNew}$, and the result forms part of a \notesCiphertext (see \crossref{inband} for further details). Each \notePlaintext (denoted $\NotePlaintext{}$) consists of $(\Value, \NoteAddressRand, \NoteCommitRand\changed{, \Memo})$. The first three of these fields are as defined earlier. \changed{ $\Memo$ represents a \memo associated with this \note. The usage of the \memo is by agreement between the sender and recipient of the \note. } \nsubsection{Transactions, Blocks, and the Block Chain} \label{blockchain} At a given point in time, the \blockchainview of each \fullnode consists of a sequence of one or more valid \blocks. Each \block consists of a sequence of one or more \transactions. To each \transaction there is associated an initial \treestate, which consists of a \noteCommitmentTree (\crossref{merkletree}), \nullifierSet (\crossref{nullifierset}), and data structures associated with \Bitcoin such as the UTXO (Unspent Transaction Output) set. Inputs to a \transaction insert value into a \transparentValuePool, and outputs remove value from this pool. As in \Bitcoin, the remaining value in the pool is available to miners as a fee. An \anchor is a Merkle tree root of a \noteCommitmentTree. It uniquely identifies a \noteCommitmentTree state given the assumed security properties of the Merkle tree's hash function. Since the \nullifierSet is always updated together with the \noteCommitmentTree, this also identifies a particular state of the \nullifierSet. In a given node's \blockchainview, \treestates are chained as follows: \begin{itemize} \item The input \treestate of the first \block is the empty \treestate. \item The input \treestate of the first \transaction of a \block is the final \treestate of the immediately preceding \block. \item The input \treestate of each subsequent \transaction in a \block is the output \treestate of the immediately preceding \transaction. \item The final \treestate of a \block is the output \treestate of its last \transaction. \end{itemize} \todo{\joinSplitDescriptions also have input and output \treestates.} We rely on Bitcoin-style consensus for \fullnodes to eventually converge on their views of valid \blocks, and therefore of the sequence of \treestates in those \blocks. \nsubsection{\JoinSplitTransfers{} and Descriptions} \label{joinsplit} A \joinSplitDescription is data included in a \transaction that describes a \joinSplitTransfer, i.e.\ a \xprotected value transfer. This kind of value transfer is the primary \Zcash-specific operation performed by \transactions; it uses, but should not be confused with, the \joinSplitStatement used for the \zkSNARK proof and verification. A \joinSplitTransfer spends $\NOld$ \notes $\nOld{\allOld}$ and \transparent input $\vpubOld$, and creates $\NNew$ \notes $\nNew{\allNew}$ and \transparent output $\vpubNew$. Each \transaction is associated with a \sequenceOfJoinSplitDescriptions. The input and output values of each \joinSplitTransfer{} \MUST balance exactly. The \changed{total $\vpubNew$ value adds to, and the total} $\vpubOld$ value subtracts from the \transparentValuePool of the containing \transaction. \todo{Describe the interaction of \transparent value flows with the \joinSplitDescription's \changed{$\vpubOld$ and} $\vpubNew$.} \changed{ The \anchor of each \joinSplitDescription in a \transaction must refer to either some earlier \block's final \treestate, or to the output \treestate of any prior \joinSplitDescription in the same \transaction. } These conditions act as constraints on the blocks that a \fullnode will accept into its \blockchainview. \nsubsection{\NoteCommitmentTree} \label{merkletree} \begin{center} \includegraphics[scale=.4]{incremental_merkle} \end{center} The \noteCommitmentTree is an \incrementalMerkleTree of fixed depth used to store \noteCommitments that \joinSplitTransfers produce. Just as the \term{unspent transaction output set} (UTXO set) used in \Bitcoin, it is used to express the existence of value and the capability to spend it. However, unlike the UTXO set, it is \emph{not} the job of this tree to protect against double-spending, as it is append-only. Blocks in the \blockchain are associated (by all nodes) with the \merkleRoot of this tree after all of its constituent \joinSplitDescriptions' \noteCommitments have been entered into the \noteCommitmentTree associated with the previous \block. \todo{Make this more precise.} Each \merkleNode in the \incrementalMerkleTree is associated with a \merkleHash of size $\MerkleHashLength$ bytes. The \merkleLayer numbered $h$, counting from \merkleLayer $0$ at the \merkleRoot, has $2^h$ \merkleNodes with \merkleIndices $0$ to $2^h-1$ inclusive. The \merkleHash associated with the \merkleNode at \merkleIndex $i$ in \merkleLayer $h$ is denoted $\MerkleNode{h}{i}$. \nsubsection{\NullifierSet} \label{nullifierset} Each \fullnode maintains a \nullifierSet alongside the \noteCommitmentTree and UTXO set. As valid \transactions containing \joinSplitTransfers are processed, the \nullifiers revealed in \joinSplitDescriptions are inserted into this \nullifierSet. If a \joinSplitDescription reveals a \nullifier that already exists in the \fullnode's \blockchainview, the containing transaction will be rejected, since it would otherwise result in a double-spend. \nsubsection{Block Subsidy and Founders' Reward} \label{subsidyconcepts} Like \Bitcoin, \Zcash creates currency when \blocks are mined. The value created on mining a \block is called the \blockSubsidy. It is composed of a \minerSubsidy and a \foundersReward. As in \Bitcoin, the miner of a \block also receives \transactionFees. The amount of the \blockSubsidy and \minerSubsidy depends on the \blockHeight. The \blockHeight of the \genesisBlock is 0, and the \blockHeight of each subsequent \block in the \blockchain increments by 1. The calculations of the \blockSubsidy, \minerSubsidy, and \foundersReward for a given \blockHeight are given in \crossref{subsidies}. \nsubsection{\CoinbaseTransactions} The first \transaction in a block must be a \coinbaseTransaction, which should collect and spend any \minerSubsidy and \transactionFees paid by \transactions included in this \block. The \coinbaseTransaction must also pay the \foundersReward as described in \crossref{coinbases}. \nsection{Abstract Protocol} \nsubsection{Abstract Cryptographic Functions} \nsubsubsection{Hash Functions} \label{abstracthashes} $\MerkleCRH \typecolon \MerkleHash \times \MerkleHash \rightarrow \MerkleHash$ is a collision-resistant hash function used in \crossref{merklepath}. It is instantiated in \crossref{merklecrh}. \changed{ $\hSigCRH{} \typecolon \bitseq{\RandomSeedLength} \times \typeexp{\PRFOutput}{\NOld} \times \JoinSplitSigPublic \rightarrow \hSigType$ is a collision-resistant hash function used in \crossref{joinsplitdesc}. It is instantiated in \crossref{hsigcrh}. $\EquihashGen{} \typecolon (n \typecolon \PosInt) \times \PosInt \times \byteseqs \times \PosInt \rightarrow \bitseq{n}$ is another hash function, used in \crossref{equihash} to generate input to the Equihash solver. The first two arguments, representing the Equihash parameters $n$ and $k$, are written subscripted. It is instantiated in \crossref{equihashgen}. } \nsubsubsection{\PseudoRandomFunctions} \label{abstractprfs} $\PRF{x}{}$ is a \pseudoRandomFunction keyed by $x$. \changed{Four} \emph{independent} $\PRF{x}{}$ are needed in our protocol: \begin{tabular}{@{\hskip 2em}l@{\;}l@{\;}l@{\;}l@{\;}l} $\PRFaddr{} $&$\typecolon\; \bitseq{\AuthPrivateLength} $&$\times\; \range{0}{255} $& &$\rightarrow \PRFOutput $\\ $\PRFnf{} $&$\typecolon\; \bitseq{\AuthPrivateLength} $&$\times\; \PRFOutput $& &$\rightarrow \PRFOutput $\\ $\PRFpk{} $&$\typecolon\; \bitseq{\AuthPrivateLength} $&$\times\; \setofOld $&$\times\; \hSigType $&$\rightarrow \PRFOutput $\\ $\PRFrho{} $&$\typecolon\; \bitseq{\NoteAddressPreRandLength} $&$\times\; \setofNew $&$\times\; \hSigType $&$\rightarrow \PRFOutput $ \end{tabular} These are used in \crossref{jsstatement}; $\PRFaddr{}$ is also used to derive a \paymentAddress from a \spendingKey in \crossref{keycomponents}. They are instantiated in \crossref{concreteprfs}. \securityrequirement{ In addition to being \pseudoRandomFunctions, it is required that $\PRFnf{x}$ \changed{, $\PRFaddr{x}$, and $\PRFrho{x}$} be collision-resistant across all $x$ --- i.e.\ it should not be feasible to find $(x, y) \neq (x', y')$ such that $\PRFnf{x}(y) = \PRFnf{x'}(y')$\changed{, and similarly for $\PRFaddr{}$ and $\PRFrho{}$}. } \pnote{$\PRFnf{}$ was called $\PRFsn{}$ in \Zerocash \cite{BCG+2014}.} \nsubsubsection{\SymmetricEncryption} \label{abstractsym} Let $\Sym$ be an \symmetricEncryptionScheme with keyspace $\Keyspace$, encrypting plaintexts in $\Plaintext$ to produce ciphertexts in $\Ciphertext$. $\SymEncrypt{} \typecolon \Keyspace \times \Plaintext \rightarrow \Ciphertext$ is the encryption algorithm. $\SymDecrypt{} \typecolon \Keyspace \times \Ciphertext \rightarrow \Plaintext \cup \setof{\bot}$ is the corresponding decryption algorithm, such that for any $\Key \in \Keyspace$ and $\Ptext \in \Plaintext$, $\SymDecrypt{\Key}(\SymEncrypt{\Key}(\Ptext)) = \Ptext$. $\bot$ is used to represent the decryption of an invalid ciphertext. \securityrequirement{ $\Sym$ must be one-time (INT-CTXT $\wedge$ IND-CPA)-secure. ``One-time'' here means that an honest protocol participant will almost surely encrypt only one message with a given key; however, the attacker may make many adaptive chosen ciphertext queries for a given key. The security notions INT-CTXT and IND-CPA are as defined in \cite{BN2007}. } \nsubsubsection{\KeyAgreement} \label{abstractkeyagreement} A \keyAgreementScheme is a cryptographic protocol in which two parties agree a shared secret, each using their private key and the other party's public key. A \keyAgreementScheme $\KA$ defines a type of public keys $\KAPublic$, a type of private keys $\KAPrivate$, and a type of shared secrets $\KASharedSecret$. Let $\KAFormatPrivate \typecolon \PRFOutput \rightarrow \KAPrivate$ be a function that converts a bit string of length $\PRFOutputLength$ to a $\KA$ private key. Let $\KADerivePublic \typecolon \KAPrivate \rightarrow \KAPublic$ be a function that derives the $\KA$ public key corresponding to a given $\KA$ private key. Let $\KAAgree \typecolon \KAPrivate \times \KAPublic \rightarrow \KASharedSecret$ be the agreement function. \pnote{ The range of $\KADerivePublic$ may be a strict subset of $\KAPublic$. } \begin{securityrequirements} \item $\KAFormatPrivate$ must preserve sufficient entropy from its input to be used as a secure $\KA$ private key. \item The key agreement and the KDF defined in the next section must together satisfy a suitable adaptive security assumption along the lines of \cite[section 3]{Bern2006} or \cite[Definition 3]{ABR1999}. \end{securityrequirements} More precise formalization of these requirements is beyond the scope of this specification. \nsubsubsection{\KeyDerivation} \label{abstractkdf} A \keyDerivationFunction is defined for a particular \keyAgreementScheme and \symmetricEncryptionScheme; it takes the shared secret produced by the key agreement and additional arguments, and derives a key suitable for the encryption scheme. Let $\KDF \typecolon \setofNew \times \hSigType \times \KASharedSecret \times \KAPublic \times \KAPublic \rightarrow \Keyspace$ be a \keyDerivationFunction suitable for use with $\KA$, deriving keys for $\SymEncrypt{}$. \securityrequirement{ In addition to adaptive security of the key agreement and KDF, the following security property is required: Let $\TransmitPrivateSup{1}$ and $\TransmitPrivateSup{2}$ each be chosen uniformly and independently at random from $\KAPrivate$. Let $\TransmitPublicSup{j} := \KADerivePublic(\TransmitPrivateSup{j})$. An adversary can adaptively query a function $Q \typecolon \range{1}{2} \times \hSigType \rightarrow \KAPublic \times \Keyspace_{\allNew}$ where $Q_j(\hSig)$ is defined as follows: \begin{enumerate} \item Choose $\EphemeralPrivate$ uniformly at random from $\KAPrivate$. \item Let $\EphemeralPublic := \KADerivePublic(\EphemeralPrivate)$. \item For $i \in \setofNew$, let $\Key_i := \KDF(i, \hSig, \KAAgree(\EphemeralPrivate, \TransmitPublicSup{j}), \EphemeralPublic, \TransmitPublicSup{j}))$. \item Return $(\EphemeralPublic, \Key_{\allNew})$. \end{enumerate} Then the adversary must make another query to $Q_j$ with random unknown $j \in \range{1}{2}$, and guess $j$ with probability greater than chance. } If the adversary's advantage is negligible, then the asymmetric encryption scheme 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 between \transmissionKeys that are outputs of $\KADerivePublic$ (which includes all keys generated as in \crossref{keycomponents}). If a \transmissionKey not in that range is used, it may be distinguishable. This is not considered to be a significant security weakness. } \nsubsubsection{Signatures} \label{abstractsig} A signature scheme $\Sig$ defines: \begin{itemize} \item a type of signing keys $\SigPrivate$; \item a type of verifying keys $\SigPublic$; \item a type of messages $\SigMessage$; \item a type of signatures $\SigSignature$; \item a randomized key pair generation algorithm $\SigGen \typecolon () \rightarrowR \SigPrivate \times \SigPublic$; \item a randomized signing algorithm $\SigSign{} \typecolon \SigPrivate \times \SigMessage \rightarrowR \SigSignature$; \item a verifying algorithm $\SigVerify{} \typecolon \SigPublic \times \SigMessage \times \SigSignature \rightarrow \bit$; \end{itemize} such that for any key pair $(\sk, \vk) \leftarrowR \SigGen()$, and any $m \typecolon \SigMessage$ and $s \typecolon \SigSignature \leftarrowR \SigSign{\sk}(m)$, $\SigVerify{\vk}(m, s) = 1$. \Zcash uses two signature schemes, one used for signatures that can be verified by script operations such as \ScriptOP{CHECKSIG} and \ScriptOP{CHECKMULTISIG} as in \Bitcoin, and one called $\JoinSplitSig$ which is used to sign \transactions that contain at least one \joinSplitDescription. The latter is instantiated in \crossref{concretesig}. The following defines only the security properties needed for $\JoinSplitSig$. \securityrequirement{ $\JoinSplitSig$ must be Strongly Unforgeable under (non-adaptive) Chosen Message Attack (SU-CMA), as defined for example in \cite[Definition 6]{BDEHR2011}. This allows an adversary to obtain signatures on chosen messages, and then requires it to be infeasible for the adversary to forge a previously unseen valid \mbox{(message, signature)} pair without access to the signing key. } \begin{pnotes} \item Since a fresh key pair is generated for every \transaction containing a \joinSplitDescription and is only used for one signature (see \crossref{nonmalleability}), a one-time signature scheme would suffice for $\JoinSplitSig$. This is also the reason why only security against \emph{non-adaptive} chosen message attack is needed. In fact the instantiation of $\JoinSplitSig$ uses a scheme designed for security under adaptive attack even when multiple signatures are signed under the same key. \item SU-CMA security requires it to be infeasible for the adversary, not knowing the private key, to forge a distinct signature on a previously seen message. That is, \joinSplitSignatures are intended to be nonmalleable in the sense of \cite{BIP-62}. \end{pnotes} \nsubsubsection{Commitment} \label{abstractcomm} A \commitmentScheme is a function that, given a random \commitmentTrapdoor and an input, can be used to commit to the input in such a way that: \begin{itemize} \item no information is revealed about it without the \trapdoor (``hiding''), \item given the \trapdoor and input, the commitment can be verified to ``open'' to that input and no other (``binding''). \end{itemize} \vspace{-3ex} A \commitmentScheme $\Commit{}$ defines a type of inputs $\CommitInput$, a type of commitments $\CommitOutput$, and a type of \commitmentTrapdoors $\CommitTrapdoor$. Let $\Commit{} \typecolon \CommitTrapdoor \times \CommitInput \rightarrow \CommitOutput$ be a function satisfying the security requirements of computational hiding and computational binding, as defined in \todo{need reference}. \nsubsubsection{\ZeroKnowledgeProvingSystem} \label{abstractzk} A \zeroKnowledgeProvingSystem is a cryptographic protocol that allows proving a particular \statement, dependent on \primary and \auxiliaryInputs, in zero knowledge --- that is, without revealing information about the \auxiliaryInputs other than that implied by the \statement. The type of \zeroKnowledgeProvingSystem needed by \Zcash is a \ppzkSNARK. A \ppzkSNARK instance $\ZK$ defines: \begin{itemize} \item a type of \zkProvingKeys, $\ZKProvingKey$; \item a type of \zkVerifyingKeys, $\ZKVerifyingKey$; \item a type of \primaryInputs $\ZKPrimary$; \item a type of \auxiliaryInputs $\ZKAuxiliary$; \item a type of proofs $\ZKProof$; \item a type $\ZKSatisfying \subseteq \ZKPrimary \times \ZKAuxiliary$ of inputs satisfying the \statement; \item a randomized key pair generation algorithm $\ZKGen \typecolon () \rightarrowR \ZKProvingKey \times \ZKVerifyingKey$; \item a proving algorithm $\ZKProve{} \typecolon \ZKProvingKey \times \ZKSatisfying \rightarrow \ZKProof$; \item a verifying algorithm $\ZKVerify{} \typecolon \ZKVerifyingKey \times \ZKPrimary \times \ZKProof \rightarrow \bit$; \end{itemize} The security requirements below are supposed to hold with overwhelming probability for $(\pk, \vk) \leftarrowR \ZKGen()$. \begin{securityrequirements} \item \textbf{Completeness:} An honestly generated proof will convince a verifier: for any $(x, w) \in \ZKSatisfying$, if $\ZKProve{\pk}(x, w)$ outputs $\Proof$, then $\ZKVerify{\vk}(x, \Proof) = 1$. \item \textbf{Proof of Knowledge:} For any adversary $\Adversary$ able to find an $x \typecolon \ZKPrimary$ and proof $\Proof \typecolon \ZKProof$ such that $\ZKVerify{\vk}(x, \Proof) = 1$, there is an efficient extractor $E_{\Adversary}$ such that if $E_{\Adversary}(\vk, \pk)$ returns $w$, then the probability that $(x, w) \not\in \ZKSatisfying$ is negligable. \item \textbf{Statistical Zero Knowledge:} An honestly generated proof is statistical zero knowledge. \todo{Full definition.} \end{securityrequirements} These definitions are derived from those in \cite[Appendix C]{BCTV2014}, adapted to state concrete rather than asymptotic security. ($\ZKProve{}$ corresponds to $P$, $\ZKVerify{}$ corresponds to $V$, and $\ZKSatisfying$ corresponds to $\mathcal{R}_C$ in the notation of that appendix.) The Proof of Knowledge definition is a way to formalize the property that it is infeasible to find a new proof $\Proof$ where $\ZKVerify{\vk}(x, \Proof) = 1$ without \emph{knowing} an \auxiliaryInput $w$ such that $(x, w) \in \ZKSatisfying$. (It is possible to replay proofs, but informally, a proof for a given $(x, w)$ gives no information that helps to find a proof for other $(x, w)$.) The \provingSystem is instantiated in \crossref{proofs}. $\ZKJoinSplit$ refers to this \provingSystem specialized to the \joinSplitStatement given in \crossref{jsstatement}. In this case we omit the key subscripts on $\ZKJoinSplitVerify$ and $\ZKJoinSplitProve$, taking them to be the particular \provingKey and \verifyingKey defined by the \joinSplitParameters in \crossref{jsparameters}. \nsubsection{Key Components} \label{keycomponents} Let $\KA$ be a \keyAgreementScheme, instantiated in \crossref{concretekeyagreement}. A new \spendingKey $\AuthPrivate$ is generated by choosing a bit string uniformly at random from $\bitseq{\AuthPrivateLength}$. \changed{ $\AuthPublic$, $\TransmitPrivate$ and $\TransmitPublic$ are derived from $\AuthPrivate$ as follows:} {\hfuzz=50pt \begin{equation*} \begin{aligned} \AuthPublic &:= \changed{\PRFaddr{\AuthPrivate}(0)} \\ \TransmitPrivate &:= \changed{\KAFormatPrivate(\PRFaddr{\AuthPrivate}(1))} \\ \TransmitPublic &:= \changed{\KADerivePublic(\TransmitPrivate)} \end{aligned} \end{equation*} } \nsubsection{\JoinSplitDescriptions} \label{joinsplitdesc} A \joinSplitTransfer, as specified in \crossref{joinsplit}, is encoded in \transactions as a \joinSplitDescription. Each \transaction includes a sequence of zero or more \joinSplitDescriptions. When this sequence is non-empty, the \transaction also includes encodings of a $\JoinSplitSig$ public verification key and signature. Each \joinSplitDescription consists of $(\vpubOld, \vpubNew, \rt, \nfOld{\allOld}, \cmNew{\allNew}, \EphemeralPublic, \RandomSeed, \h{\allOld}, \JoinSplitProof, \TransmitCiphertext{\allNew})$ where \begin{itemize} \item \changed{$\vpubOld \typecolon \range{0}{\MAXMONEY}$ is the value that the \joinSplitTransfer removes from the \transparentValuePool}; \item $\vpubNew \typecolon \range{0}{\MAXMONEY}$ is the value that the \joinSplitTransfer inserts into the \transparentValuePool; \item $\rt \typecolon \MerkleHash$ is an \anchor, as defined in \crossref{blockchain}, for the output \treestate of either a previous \block, or a previous \joinSplitTransfer in this \transaction. \item $\nfOld{\allOld} \typecolon \typeexp{\PRFOutput}{\NOld}$ is the sequence of \nullifiers for the input \notes; \item $\cmNew{\allNew} \typecolon \typeexp{\CommitOutput}{\NNew}$ is the sequence of \noteCommitments for the output \notes; \item \changed{$\EphemeralPublic \typecolon \KAPublic$ is a key agreement public key, used to derive the key for encryption of the \notesCiphertext (\crossref{inband})}; \item \changed{$\RandomSeed \typecolon \RandomSeedType$ is a seed that must be chosen independently at random for each \joinSplitDescription}; \item $\h{\allOld} \typecolon \typeexp{\PRFOutput}{\NOld}$ is a sequence of tags that bind $\hSig$ to each $\AuthPrivate$ of the input \notes; \item $\JoinSplitProof \typecolon \ZKJoinSplitProof$ is the \zeroKnowledgeProof for the \joinSplitStatement; \item $\TransmitCiphertext{\allNew} \typecolon \typeexp{\Ciphertext}{\NNew}$ is a sequence of ciphertext components for the encrypted output \notes. \end{itemize} The $\ephemeralKey$ and $\encCiphertexts$ fields together form the \notesCiphertext. The value $\hSig$ is also computed from \changed{$\RandomSeed$, $\nfOld{\allOld}$, and} the $\joinSplitPubKey$ of the containing \transaction: \begin{itemize} \item[] $\hSig := \hSigCRH(\changed{\RandomSeed, \nfOld{\allOld},\,} \joinSplitPubKey)$. \end{itemize} $\hSigCRH$ is instantiated in \crossref{hsigcrh}. \begin{consensusrules} \item Elements of a \joinSplitDescription{} \MUST have the types given above (for example: $0 \leq \vpubOld \leq \MAXMONEY$ and $0 \leq \vpubNew \leq \MAXMONEY$). \item Either $\vpubOld$ or $\vpubNew$ \MUST be zero. \item The proof $\Proof_{\JoinSplit}$ \MUST be valid given a \primaryInput formed from the other fields and $\hSig$. I.e. it must be the case that $\ZKJoinSplitVerify((\rt, \nfOld{\allOld}, \cmNew{\allNew}, \vpubOld, \vpubNew, \hSig, \h{\allOld}), \Proof_{\JoinSplit}) = 1$. \end{consensusrules} \nsubsection{Sending \Notes} \label{send} In order to send \xprotected value, the sender constructs a \transaction containing one or more \joinSplitDescriptions. This involves first generating a new $\JoinSplitSig$ key pair: \hskip 1.5em $(\joinSplitPrivKey, \joinSplitPubKey) \leftarrowR \JoinSplitSigGen()$. For each \joinSplitDescription, the sender chooses $\RandomSeed$ uniformly at random on $\bitseq{\RandomSeedLength}$, and selects the input \notes. At this point there is sufficient information to compute $\hSig$, as described in the previous section. \changed{The sender also chooses $\NoteAddressPreRand$ uniformly at random on $\bitseq{\NoteAddressPreRandLength}$.} Then it creates each output \note with index $i \typecolon \setofNew$ as follows: \begin{itemize} \item Choose $\NoteCommitRandNew{i}$ uniformly at random on $\bitseq{\NoteCommitRandLength}$. \changed{ \item Compute $\NoteAddressRandNew{i} := \PRFrho{\NoteAddressPreRand}(i, \hSig)$. } \item Encrypt the \note to the recipient \transmissionKey $\TransmitPublicNew{i}$, as described in \crossref{inband}, giving the ciphertext component $\TransmitCiphertext{i}$. \end{itemize} In order to minimize information leakage, the sender \SHOULD randomize the order of the input \notes and of the output \notes. Other considerations relating to information leakage from the structure of \transactions are beyond the scope of this specification. After generating all of the \joinSplitDescriptions, the sender obtains the $\dataToBeSigned$ (\crossref{nonmalleability}), and signs it with the private \joinSplitSigningKey: \hskip 1.5em $\joinSplitSig \leftarrowR \JoinSplitSigSign{\text{\small\joinSplitPrivKey}}(\dataToBeSigned)$ Then the encoded \transaction including $\joinSplitSig$ is submitted to the network. \nsubsubsection{\DummyNotes} \label{dummynotes} The fields in a \joinSplitDescription allow for $\NOld$ input \notes, and $\NNew$ output \notes. In practice, we may wish to encode a \joinSplitTransfer with fewer input or output \notes. This is achieved using \dummyNotes. \changed{ A \dummy input \note, with index $i$ in the \joinSplitDescription, is constructed as follows: \begin{itemize} \item Generate a new random \spendingKey $\AuthPrivateOld{i}$ and derive its \payingKey $\AuthPublicOld{i}$. \item Set $\vOld{i} := 0$. \item Choose $\NoteAddressRandOld{i}$ uniformly at random on $\PRFOutput$. \item Choose $\NoteCommitRandOld{i}$ uniformly at random on $\bitseq{\NoteCommitRandLength}$. \item Compute $\nfOld{i} := \PRFnf{\AuthPrivateOld{i}}(\NoteAddressRandOld{i})$. \item Construct a \dummy \merklePath $\treepath{i}$ for use in the \auxiliaryInput to the \joinSplitStatement (this will not be checked). \item When generating the \joinSplitProof\!\!, set $\EnforceCommit{i}$ to 0. \end{itemize} } A \dummy output \note is constructed as normal but with zero value, and sent to a random \paymentAddress. \nsubsection{Merkle path validity} \label{merklepath} The depth of the \noteCommitmentTree is $\MerkleDepth$ (defined in \crossref{constants}). Each \merkleNode in the \incrementalMerkleTree is associated with a \merkleHash, which is a byte 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 $h$. The \merkleNodes at \merkleLayer $\MerkleDepth$ are called \merkleLeafNodes. When a \noteCommitment is added to the tree, it occupies the \merkleLeafNode \merkleHash $\MerkleNode{\MerkleDepth}{i}$ for the next available $i$. As-yet unused \merkleLeafNodes are associated with a distinguished \merkleHash $\Uncommitted$. It is assumed to be infeasible to find a preimage \note $\NoteTuple{}$ such that $\NoteCommit(\NoteTuple{}) = \Uncommitted$. The \merkleNodes at \merkleLayers $0$ to $\MerkleDepth-1$ inclusive are called \merkleInternalNodes, and are associated with $\MerkleCRH$ outputs. \MerkleInternalNodes are computed from their children in the next \merkleLayer as follows: for $0 \leq h < \MerkleDepth$ and $0 \leq i < 2^h$, \hskip 2em $\MerkleNode{h}{i} := \MerkleCRH(\MerkleNode{h+1}{2i}, \MerkleNode{h+1}{2i+1})$. A \merklePath from \merkleLeafNode $\MerkleNode{\MerkleDepth}{i}$ in the \incrementalMerkleTree is the sequence \hskip 2em $[\hairspace\MerkleNode{h}{\MerkleSibling(h, i)} \text{ for } h \text{ from } \MerkleDepth \text{ down to } 1\hairspace]$, where \hskip 2em $\MerkleSibling(h, i) = \floor{\frac{i}{2^{\MerkleDepth-h}}} \xor 1$ Given such a \merklePath, it is possible to verify that \merkleLeafNode $\MerkleNode{\MerkleDepth}{i}$ is in a tree with a given \merkleRoot $\rt = \MerkleNode{0}{0}$. \nsubsection{Non-malleability} \label{nonmalleability} \Bitcoin defines several \sighashTypes that cover various parts of a transaction. \changed{In \Zcash, all of these \sighashTypes are extended to cover the \Zcash-specific fields $\nJoinSplit$, $\vJoinSplit$, and (if present) $\joinSplitPubKey$, described in \crossref{txnencoding}. They \emph{do not} cover the field $\joinSplitSig$. \consensusrule{ If $\nJoinSplit > 0$, the \transaction{} \MUSTNOT use \sighashTypes other than $\SIGHASHALL$. } } 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.} In order to ensure that a \joinSplitDescription is cryptographically bound to the \transparent inputs and outputs corresponding to $\vpubNew$ and $\vpubOld$, and to the other \joinSplitDescriptions in the same \transaction, an ephemeral $\JoinSplitSig$ key pair is generated for each \transaction, and the $\dataToBeSigned$ is signed with the private signing key of this key pair. The corresponding public verification key is included in the \transaction encoding as $\joinSplitPubKey$. $\JoinSplitSig$ is instantiated in \crossref{concretesig}. \changed{ If $\nJoinSplit$ is zero, the $\joinSplitPubKey$ and $\joinSplitSig$ fields are omitted. Otherwise, a \transaction has a correct \joinSplitSignature if and only if $\JoinSplitSigVerify{\text{\small\joinSplitPubKey}}(\dataToBeSigned, \joinSplitSig) = 1$. % FIXME: distinguish pubkey and signature from their encodings. } The condition enforced by the \joinSplitStatement specified in \crossref{nonmalleablepour} ensures that a holder of all of $\AuthPrivateOld{\allOld}$ for each \joinSplitDescription has authorized the use of the private signing key corresponding to $\joinSplitPubKey$ to sign this \transaction. \nsubsection{Balance} 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 \emph{input} value. \changed{ \pnote{ 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, \joinSplitDescriptions are indistinguishable regardless of the number of real input \notes. } As stated in \crossref{joinsplitdesc}, either $\vpubOld$ or $\vpubNew$ \MUST be zero. No generality is lost because, if a \transaction in which both $\vpubOld$ and $\vpubNew$ were nonzero were allowed, it could be replaced by an equivalent one in which $\minimum(\vpubOld, \vpubNew)$ is subtracted from both of these values. This restriction helps to avoid unnecessary distinctions between \transactions according to client implementation. } \nsubsection{\NoteCommitments{} and \Nullifiers} A \transaction that contains one or more \joinSplitDescriptions, when entered into the blockchain, appends to the \noteCommitmentTree with all constituent \noteCommitments. All of the constituent \nullifiers are also entered into the \nullifierSet of the \blockchainview \emph{and} \mempool. A \transaction is not valid if it attempts to add a \nullifier to the \nullifierSet that already exists in the set. \nsubsection{\JoinSplitStatement} \label{jsstatement} A valid instance of $\JoinSplitProof$ assures that given a \term{primary input}: \begin{itemize} \item[] $(\rt \typecolon \MerkleHash, \nfOld{\allOld} \typecolon \typeexp{\PRFOutput}{\NOld}, \cmNew{\allNew} \typecolon \typeexp{\CommitOutput}{\NNew}, \changed{\vpubOld \typecolon \range{0}{2^{64}-1},}\, \vpubNew \typecolon \range{0}{2^{64}-1},\\ \hphantom{(} \hSig \typecolon \hSigType, \h{\allOld} \typecolon \typeexp{\PRFOutput}{\NOld})$, \end{itemize} the prover knows an \term{auxiliary input}: \begin{itemize} \item[] $(\treepath{\allOld} \typecolon \typeexp{\typeexp{\MerkleHash}{\MerkleDepth}}{\NOld}, \nOld{\allOld} \typecolon \typeexp{\NoteType}{\NOld}, \AuthPrivateOld{\allOld} \typecolon \typeexp{\bitseq{\AuthPrivateLength}}{\NOld}, \nNew{\allNew} \typecolon \typeexp{\NoteType}{\NOld}\changed{,}\\ \hphantom{(} \changed{\NoteAddressPreRand \typecolon \bitseq{\NoteAddressPreRandLength}, \EnforceCommit{\allOld} \typecolon \bitseq{\NOld}})$, \end{itemize} where: \begin{itemize} \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{itemize} such that the following conditions hold: \subparagraph{Merkle path validity} \label{merklepathvalidity} for each $i \in \setofOld$ \changed{$\mid$ $\EnforceCommit{i} = 1$}: $\treepath{i}$ must be a valid \merklePath of depth $\MerkleDepth$, as defined in \crossref{merklepath}, from $\NoteCommit(\nOld{i})$ to \noteCommitmentTree root $\rt$. \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}. \changed{ \subparagraph{Commitment Enforcement} for each $i \in \setofOld$, if $\vOld{i} \neq 0$ then $\EnforceCommit{i} = 1$. } \subparagraph{Balance} $\changed{\vpubOld\; +} \vsum{i=1}{\NOld} \vOld{i} = \vpubNew + \vsum{i=1}{\NNew} \vNew{i} \in \range{0}{2^{64}-1}$. \subparagraph{\Nullifier{} integrity} for each $i \in \setofNew$: $\nfOld{i} = \PRFnf{\AuthPrivateOld{i}}(\NoteAddressRandOld{i})$. \subparagraph{Spend authority} \label{spendauthority} for each $i \in \setofOld$: $\AuthPublicOld{i} = \changed{\PRFaddr{\AuthPrivateOld{i}}(0)}$. \subparagraph{Non-malleability} \label{nonmalleablepour} for each $i \in \setofOld$: $\h{i} = \PRFpk{\AuthPrivateOld{i}}(i, \hSig)$. \changed{ \subparagraph{Uniqueness of $\NoteAddressRandNew{i}$} \label{uniquerho} for each $i \in \setofNew$: $\NoteAddressRandNew{i} = \PRFrho{\NoteAddressPreRand}(i, \hSig)$. } \subparagraph{Commitment integrity} for each $i \in \setofNew$: $\cmNew{i}$ = $\NoteCommit(\nNew{i})$. \vspace{2.5ex} For details of the form and encoding of proofs, see \crossref{proofs}. \nsubsection{In-band secret distribution} \label{inband} In order to transmit the secret $\Value$, $\NoteAddressRand$, and $\NoteCommitRand$ (necessary for the recipient to later spend) \changed{and also a \memo} to the recipient \emph{without} requiring an out-of-band communication channel, the \transmissionKey $\TransmitPublic$ is used to encrypt these secrets. The recipient's possession of the associated \keyTuple $(\AuthPrivate, \TransmitPrivate, \PaymentAddress)$ is used to reconstruct the original \note \changed{ and \memo}. All of the resulting ciphertexts are combined to form a \notesCiphertext. For both encryption and decryption, \begin{itemize} \item Let $\Sym$ be the \encryptionScheme instantiated in \crossref{concretesym}. \item Let $\KDF$ be the \keyDerivationFunction instantiated in \crossref{concretekdf}. \item Let $\KA$ be the \keyAgreementScheme instantiated in \crossref{concretekeyagreement}. \item Let $\hSig$ be the value computed for this \joinSplitDescription in \crossref{joinsplitdesc}. \end{itemize} \nsubsubsection{Encryption} Let $\TransmitPublicNew{\allNew}$ be the \transmissionKeys for the intended recipient addresses of each new \note. Let $\NotePlaintext{\allNew}$ be the \notePlaintexts as defined in \crossref{notept}. Then to encrypt: \begin{itemize} \changed{ \item Generate a new $\KA$ (public, private) key pair $(\EphemeralPublic, \EphemeralPrivate)$. \item For $i \in \setofNew$, \begin{itemize} \item Let $\TransmitPlaintext{i}$ be the raw encoding of $\NotePlaintext{i}$. \item Let $\DHSecret{i} := \KAAgree(\EphemeralPrivate, \TransmitPublicNew{i})$. \item Let $\TransmitKey{i} := \KDF(i, \hSig, \DHSecret{i}, \EphemeralPublic, \TransmitPublicNew{i})$. \item Let $\TransmitCiphertext{i} := \SymEncrypt{\TransmitKey{i}}(\TransmitPlaintext{i})$. \end{itemize} } \end{itemize} The resulting \notesCiphertext is $\changed{(\EphemeralPublic, \TransmitCiphertext{\allNew})}$. \pnote{ It is technically possible to replace $\TransmitCiphertext{i}$ for a given \note with a random (and undecryptable) dummy ciphertext, relying instead on out-of-band transmission of the \note to the recipient. In this case the ephemeral key \MUST still be generated as a random public key (rather than a random bit string) to ensure indistinguishability from other \joinSplitDescriptions. This mode of operation raises further security considerations, for example of how to validate a \note received out-of-band, which are not addressed in this document. } \nsubsubsection{Decryption by a Recipient} Let $\PaymentAddress = (\AuthPublic, \TransmitPublic)$ be the recipient's \paymentAddress, and let $\TransmitPrivate$ be the recipient's \viewingKey. Let $\cmNew{\allNew}$ be the \noteCommitments of each output coin. Then for each $i \in \setofNew$, the recipient will attempt to decrypt that ciphertext component as follows: \changed{ \begin{itemize} \item Let $\DHSecret{i} := \KAAgree(\TransmitPrivate, \EphemeralPublic)$. \item Let $\TransmitKey{i} := \KDF(i, \hSig, \DHSecret{i}, \EphemeralPublic, \TransmitPublicNew{i})$. \item Return $\DecryptNote(\TransmitKey{i}, \TransmitCiphertext{i}, \cmNew{i}, \AuthPublic).$ \end{itemize} $\DecryptNote(\TransmitKey{i}, \TransmitCiphertext{i}, \cmNew{i}, \AuthPublic)$ is defined as follows: \begin{itemize} \item Let $\TransmitPlaintext{i} := \SymDecrypt{\TransmitKey{i}}(\TransmitCiphertext{i})$. \item If $\TransmitPlaintext{i} = \bot$, return $\bot$. \item Extract $\NotePlaintext{i} = (\ValueNew{i}, \NoteAddressRandNew{i}, \NoteCommitRandNew{i}, \Memo_i)$ from $\TransmitPlaintext{i}$. \item If $\NoteCommit((\AuthPublic, \ValueNew{i}, \NoteAddressRandNew{i}, \NoteCommitRandNew{i})) \neq \cmNew{i}$, return $\bot$, else return $\NotePlaintext{i}$. \end{itemize} } To test whether a \note is unspent in a particular \blockchainview also requires the \spendingKey $\AuthPrivate$; the coin is unspent if and only if $\nf = \PRFnf{\AuthPrivate}(\NoteAddressRand)$ is not in the \nullifierSet for that \blockchainview. \begin{pnotes} \item The decryption algorithm corresponds to step 3 (b) i. and ii. (first bullet point) of the $\Receive$ algorithm shown in \cite[Figure 2]{BCG+2014}. \item A \note can change from being unspent to spent on a given \blockchainview, as \transactions are added to that view. Also, blockchain reorganisations can cause the \transaction in which a \note was output to no longer be on the consensus blockchain. \end{pnotes} See \crossref{inbandrationale} for further discussion of the security and engineering rationale behind this encryption scheme. \nsection{Concrete Protocol} \nsubsection{Caution} \todo{Explain the kind of things that can go wrong with linkage between abstract and concrete protocol. E.g. \crossref{internalh}} \nsubsection{Integers, Bit Sequences, and Endianness} \label{boxnotation} All integers in \emph{\Zcash-specific} encodings are unsigned, have a fixed bit length, and are encoded in little-endian byte order \emph{unless otherwise specified}. In bit layout diagrams, each box of the diagram represents a sequence of bits. Diagrams are read from left-to-right, with lines read from top-to-bottom; the breaking of boxes across lines has no significance. The bit length is given explicitly in each box, except for the case of a single bit, or for the notation $\zeros{n}$ which represents the sequence of $n$ zero bits. The entire diagram represents the sequence of \emph{bytes} formed by first concatenating these bit sequences, and then treating each subsequence of 8 bits as a byte with the bits ordered from \emph{most significant} to \emph{least significant}. Thus the \emph{most significant} bit in each byte is toward the left of a diagram. Where bit fields are used, the text will clarify their position in each case. \begin{comment} \todo{Update example for big-bit-endian order.} \newsavebox{\exampleabox} \begin{lrbox}{\exampleabox} \setchanged \begin{bytefield}[bitwidth=1.3em]{32} \bitbox{1}{0} & \bitbox{1}{1} & \bitbox{1}{0} & \bitbox{1}{0} & \bitbox{16}{16 bit $\hexint{ABCD}$} & \bitbox{12}{12 bit $\hexint{123}$} & \end{bytefield} \end{lrbox} \newsavebox{\examplebbox} \begin{lrbox}{\examplebbox} \setchanged \begin{bytefield}[bitwidth=1.3em]{32} \bitbox{4}{4 bit $\hexint{2}$} & \bitbox{4}{4 bit $\hexint{D}$} & \bitbox{4}{4 bit $\hexint{C}$} & \bitbox{4}{4 bit $\hexint{B}$} & \bitbox{4}{4 bit $\hexint{A}$} & \bitbox{4}{4 bit $\hexint{3}$} & \bitbox{4}{4 bit $\hexint{2}$} & \bitbox{4}{4 bit $\hexint{1}$} & \end{bytefield} \end{lrbox} \newsavebox{\examplecbox} \begin{lrbox}{\examplecbox} \setchanged \begin{bytefield}[bitwidth=1.3em]{32} \bitbox{8}{8 bit $\hexint{D2}$} & \bitbox{8}{8 bit $\hexint{BC}$} & \bitbox{8}{8 bit $\hexint{3A}$} & \bitbox{8}{8 bit $\hexint{12}$} & \end{bytefield} \end{lrbox} For example, the following diagrams are all equivalent: \begin{itemize} \item[] $\Justthebox{\exampleabox}$ \item[] $\Justthebox{\examplebbox}$ \item[] $\Justthebox{\examplecbox}$ \end{itemize} and represent the byte sequence $[\hexint{D2}, \hexint{BC}, \hexint{3A}, \hexint{12}]$. \end{comment} \nsubsection{Constants} \label{constants} Define: \begin{itemize} \item[] $\MerkleDepth \typecolon \Nat := \changed{29}$ \item[] $\NOld \typecolon \Nat := 2$ \item[] $\NNew \typecolon \Nat := 2$ \item[] $\MerkleHashLength \typecolon \Nat := 256$ \item[] $\hSigLength \typecolon \Nat := 256$ \item[] $\PRFOutputLength \typecolon \Nat := 256$ \item[] $\NoteCommitRandLength \typecolon \Nat := \changed{256}$ \item[] $\changed{\RandomSeedLength \typecolon \Nat := 256}$ \item[] $\AuthPrivateLength \typecolon \Nat := \changed{252}$ \item[] $\changed{\NoteAddressPreRandLength \typecolon \Nat := 252}$ \item[] $\Uncommitted \typecolon \bitseq{\MerkleHashLength} := \zeros{\MerkleHashLength}$ \item[] $\MAXMONEY \typecolon \Nat := \changed{2.1 \mult 10^{15}}$ (\zatoshi) \item[] $\SlowStartInterval \typecolon \Nat := 20000$ \item[] $\HalvingInterval \typecolon \Nat := 840000$ \item[] $\MaxBlockSubsidy \typecolon \Nat := 1.25 \mult 10^9$ (\zatoshi) \item[] $\NumFounderAddresses \typecolon \Nat := 48$ \item[] $\FoundersFraction \typecolon \Rat := \frac{1}{5}$. \end{itemize} \nsubsection{Concrete Cryptographic Functions} \nsubsubsection{Merkle Tree Hash Function} \label{merklecrh} $\MerkleCRH$ is used to hash \incrementalMerkleTree \merkleHashes. It is instantiated by the $\SHAName$ function, which takes a 512-bit block and produces a 256-bit hash. \cite{NIST2015} \newsavebox{\merklebox} \begin{lrbox}{\merklebox} \begin{bytefield}[bitwidth=0.04em]{512} \bitbox{256}{$256$-bit $\mathsf{left}$} & \bitbox{256}{$256$-bit $\mathsf{right}$} \end{bytefield} \end{lrbox} \hskip 2em $\MerkleCRH(\mathsf{left}, \mathsf{right}) := \CRHbox{\merklebox}$. \pnote{ $\SHA$ is not the same as the $\FullHashName$ function, which hashes arbitrary-length sequences. } \securityrequirement{ $\SHA$ must be collision-resistant, and it must be infeasible to find a preimage $x$ such that $\SHA(x) = \zeros{256}$. } \nsubsubsection{\hSigText{} Hash Function} \label{hsigcrh} \newsavebox{\hsigbox} \begin{lrbox}{\hsigbox} \setchanged \begin{bytefield}[bitwidth=0.04em]{1024} \bitbox{256}{$256$-bit $\RandomSeed$} \bitbox{256}{\hfill $256$-bit $\nfOld{\mathrm{1}}$\hfill...\;} & \bitbox{256}{$256$-bit $\nfOld{\NOld}$} & \bitbox{300}{$256$-bit $\joinSplitPubKey$} \end{bytefield} \end{lrbox} $\hSigCRH$ is used to compute the value $\hSig$ in \crossref{joinsplitdesc}. \changed{ \hskip 1.5em $\hSigCRH(\RandomSeed, \nfOld{\allOld}, \joinSplitPubKey) := \Blake{256}(\ascii{ZcashComputehSig},\; \hSigInput)$ where \hskip 1.5em $\hSigInput := \Justthebox{\hsigbox}$. } $\Blake{256}(p, x)$ refers to unkeyed $\Blake{256}$ \cite{ANWW2013} in sequential mode, with an output digest length of $32$ bytes, 16-byte personalization string $p$, and input $x$. This is not the same as $\Blake{512}$ truncated to $256$ bits, because the digest length is encoded in the parameter block. \securityrequirement{ $\Blake{256}(\ascii{ZcashComputehSig}, x)$ must be collision-resistant. } \nsubsubsection{Equihash Generator} \label{equihashgen} $\EquihashGen{n, k}$ is a specialized hash function that maps an input and an index to an output of length $n$ bits. It is used in \crossref{equihash}. \newsavebox{\powtagbox} \begin{lrbox}{\powtagbox} \begin{bytefield}[bitwidth=0.16em]{128} \bitbox{64}{64-bit $\ascii{ZcashPoW}$} \bitbox{32}{32-bit $n$} \bitbox{32}{32-bit $k$} \end{bytefield} \end{lrbox} \newsavebox{\powcountbox} \begin{lrbox}{\powcountbox} \begin{bytefield}[bitwidth=0.16em]{32} \bitbox{32}{32-bit $g$} \end{bytefield} \end{lrbox} Let $\powtag := \Justthebox{\powtagbox}$. Let $\powcount(g) := \Justthebox{\powcountbox}$. \vspace{2ex} % Blech. Dijkstra was right \cite{EWD831}. Let $\EquihashGen{n, k}(S, i) := T_{h+1\hairspace..\hairspace h+n}$, where \begin{itemize} \item $m := \floor{\frac{512}{n}}$; \item $h := (i-1 \bmod m) \mult n$; \item $T := \Blake{(\mathnormal{n \mult m})}(\powtag,\, S \,||\, \powcount(\floor{\frac{i-1}{m}}))$. \end{itemize} Indices of bits in $T$ are 1-based. $\Blake{\ell}(p, x)$ refers to unkeyed $\Blake{\ell}$ \cite{ANWW2013} in sequential mode, with an output digest length of $\ell/8$ bytes, 16-byte personalization string $p$, and input $x$. This is not the same as $\Blake{512}$ truncated to $\ell$ bits, because the digest length is encoded in the parameter block. \securityrequirement{ $\Blake{\ell}(\powtag, x)$ must generate output that is sufficiently unpredictable to avoid short-cuts to the Equihash solution process. It would suffice to model it as a random oracle. } \pnote{ When $\EquihashGen{}$ is evaluated for sequential indices (as in \crossref{equihash}), the number of calls to $\BlakeGeneric$ can be reduced by a factor of $\floor{\frac{512}{n}}$ in the best case (which is a factor of 2 for $n = 200$). } \nsubsubsection{\PseudoRandomFunctions} \label{concreteprfs} The \changed{four} independent PRFs described in \crossref{abstractprfs} are all instantiated using the $\SHAName$ function: \newcommand{\iminusone}{\hspace{0.3pt}\scriptsize{$i$\hspace{0.6pt}-1}} \newsavebox{\addrbox} \begin{lrbox}{\addrbox} \setchanged \begin{bytefield}[bitwidth=0.06em]{512} \bitbox{18}{$1$} & \bitbox{18}{$1$} & \bitbox{18}{$0$} & \bitbox{18}{$0$} & \bitbox{224}{$252$-bit $x$} & \bitbox{56}{$8$-bit $t$} & \bitbox{200}{$\zeros{248}$} \end{bytefield} \end{lrbox} \newsavebox{\nfbox} \begin{lrbox}{\nfbox} \setchanged \begin{bytefield}[bitwidth=0.06em]{512} \bitbox{18}{$1$} & \bitbox{18}{$1$} & \bitbox{18}{$1$} & \bitbox{18}{$0$} & \bitbox{224}{$252$-bit $\AuthPrivate$} & \bitbox{256}{$256$-bit $\NoteAddressRand$} \end{bytefield} \end{lrbox} \newsavebox{\pkbox} \begin{lrbox}{\pkbox} \setchanged \begin{bytefield}[bitwidth=0.06em]{512} \bitbox{18}{$0$} & \bitbox{18}{\iminusone} & \bitbox{18}{$0$} & \bitbox{18}{$0$} & \bitbox{224}{$252$-bit $\AuthPrivate$} & \bitbox{256}{$256$-bit $\hSig$} \end{bytefield} \end{lrbox} \newsavebox{\rhobox} \begin{lrbox}{\rhobox} \setchanged \begin{bytefield}[bitwidth=0.06em]{512} \bitbox{18}{$0$} & \bitbox{18}{\iminusone} & \bitbox{18}{$1$} & \bitbox{18}{$0$} & \bitbox{224}{$252$-bit $\NoteAddressPreRand$} & \bitbox{256}{$256$-bit $\hSig$} \end{bytefield} \end{lrbox} \vspace{-2ex} \begin{equation*} \begin{aligned} \setchanged \PRFaddr{x}(t) &\setchanged := \CRHbox{\addrbox} \\ \PRFnf{\AuthPrivate}(\NoteAddressRand) &:= \CRHbox{\nfbox} \\ \PRFpk{\AuthPrivate}(i, \hSig) &:= \CRHbox{\pkbox} \\ \setchanged \PRFrho{\NoteAddressPreRand}(i, \hSig) &\setchanged := \CRHbox{\rhobox} \end{aligned} \end{equation*} \begin{securityrequirements} \item The $\SHAName$ function must be collision-resistant. \item The $\SHAName$ function 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. \end{securityrequirements} \changed{ \pnote{ The first four bits --i.e.\ the most significant four bits of the first byte-- are used to distinguish different uses of $\SHA$, ensuring that the functions are independent. In addition to the inputs shown here, the bits $\mathtt{1011}$ in this position are used to distinguish uses of the full $\FullHashName$ hash function --- see \crossref{concretecomm}. (The specific bit patterns chosen here are motivated by the possibility of future extensions that either increase $\NOld$ and/or $\NNew$ to 3, or that add an additional bit to $\AuthPrivate$ to encode a new key type, or that require an additional PRF.) } } \nsubsubsection{\SymmetricEncryption} \label{concretesym} \changed{ Let $\Keyspace := \bitseq{256}$, $\Plaintext := \byteseqs$, and $\Ciphertext := \byteseqs$. Let $\SymEncrypt{\Key}(\Ptext)$ be authenticated encryption using $\SymSpecific$ \cite{RFC-7539} encryption of plaintext $\Ptext \in \Plaintext$, with empty ``associated data", all-zero nonce $\zeros{96}$, and 256-bit key $\Key \in \Keyspace$. Similarly, let $\SymDecrypt{\Key}(\Ctext)$ be $\SymSpecific$ decryption of ciphertext $\Ctext \in \Ciphertext$, with empty ``associated data", all-zero nonce $\zeros{96}$, and 256-bit key $\Key \in \Keyspace$. The result is either the plaintext byte sequence, or $\bot$ indicating failure to decrypt. } \pnote{ The ``IETF" definition of $\SymSpecific$ from \cite{RFC-7539} is used; this uses a 32-bit block count and a 96-bit nonce, rather than a 64-bit block count and 64-bit nonce as in the original definition of $\SymCipher$. } \nsubsubsection{\KeyAgreement} \label{concretekeyagreement} \changed{ The \keyAgreementScheme specified in \crossref{abstractkeyagreement} is instantiated using Curve25519 \cite{Bern2006} as follows. Let $\KAPublic$ and $\KASharedSecret$ be the type of Curve25519 public keys (i.e.\ a sequence of 32 bytes), and let $\KAPrivate$ be the type of Curve25519 secret keys. Let $\CurveMultiply(\bytes{n}, \bytes{q})$ be the result of point multiplication of the Curve25519 public key represented by the byte sequence $\bytes{q}$ by the Curve25519 secret key represented by the byte sequence $\bytes{n}$, as defined in \cite[section 2]{Bern2006}. Let $\CurveBase$ be the public byte sequence representing the Curve25519 base point. Let $\Clamp(\bytes{x})$ take a 32-byte sequence $\bytes{x}$ as input and return a byte sequence representing a Curve25519 private key, with bits ``clamped'' as described in \cite[section 3]{Bern2006}: ``clear bits $0, 1, 2$ of the first byte, clear bit $7$ of the last byte, and set bit $6$ of the last byte.'' Here the bits of a byte are numbered such that bit $b$ has numeric weight $2^b$. Define $\KAFormatPrivate(x) := \Clamp(x)$. Define $\KAAgree(n, q) := \CurveMultiply(n, q)$. } \nsubsubsection{\KeyDerivation} \label{concretekdf} \newsavebox{\kdftagbox} \begin{lrbox}{\kdftagbox} \setchanged \begin{bytefield}[bitwidth=0.16em]{128} \bitbox{64}{$64$-bit $\ascii{ZcashKDF}$} & \bitbox{32}{$8$-bit $i\!-\!1$} \bitbox{56}{$\zeros{56}$} \end{bytefield} \end{lrbox} \newsavebox{\kdfinputbox} \begin{lrbox}{\kdfinputbox} \setchanged \begin{bytefield}[bitwidth=0.04em]{1024} \bitbox{256}{$256$-bit $\hSig$} \bitbox{256}{$256$-bit $\DHSecret{i}$} & \bitbox{256}{$256$-bit $\EphemeralPublic$} & \bitbox{256}{$256$-bit $\TransmitPublicNew{i}$} & \end{bytefield} \end{lrbox} \changed{ The \keyDerivationFunction specified in \crossref{abstractkdf} is instantiated using $\Blake{256}$ as follows: \hskip 1.5em $\KDF(i, \hSig, \DHSecret{i}, \EphemeralPublic, \TransmitPublicNew{i}) := \Blake{256}(\kdftag, \kdfinput)$ where: \hskip 1.5em $\kdftag := \Justthebox{\kdftagbox}$ \hskip 1.5em $\kdfinput := \Justthebox{\kdfinputbox}$. } $\Blake{256}(p, x)$ refers to unkeyed $\Blake{256}$ \cite{ANWW2013} in sequential mode, with an output digest length of $32$ bytes, 16-byte personalization string $p$, and input $x$. This is not the same as $\Blake{512}$ truncated to $256$ bits, because the digest length is encoded in the parameter block. \nsubsubsection{Signatures} \label{concretesig} $\JoinSplitSig$ is specified in \crossref{abstractsig}. \changed{It is instantiated as $\JoinSplitSigSpecific$ \cite{BDL+2012}, with the additional requirement that $\EdDSAs$ (the integer represented by $\EdDSAS$) must be less than the prime $\ell = 2^{252} + 27742317777372353535851937790883648493$, otherwise the signature is considered invalid. $\JoinSplitSigSpecific$ is defined as using $\JoinSplitSigHashName$ internally. } \newsavebox{\sigbox} \begin{lrbox}{\sigbox} \setchanged \begin{bytefield}[bitwidth=0.075em]{512} \bitbox{256}{$256$-bit $\EdDSAR$} \bitbox{256}{$256$-bit $\EdDSAS$} \end{bytefield} \end{lrbox} \changed{ The encoding of a signature is: } \begin{itemize} \item[] $\Justthebox{\sigbox}$ \end{itemize} \changed{ where $\EdDSAR$ and $\EdDSAS$ are as defined in \cite{BDL+2012}. The encoding of a public key is as defined in \cite{BDL+2012}. } \nsubsubsection{Commitment} \label{concretecomm} \newsavebox{\cmbox} \begin{lrbox}{\cmbox} \setchanged \begin{bytefield}[bitwidth=0.032em]{840} \bitbox{24}{$1$} & \bitbox{24}{$0$} & \bitbox{24}{$1$} & \bitbox{24}{$1$} & \bitbox{24}{$0$} & \bitbox{24}{$0$} & \bitbox{24}{$0$} & \bitbox{24}{$0$} & \bitbox{256}{$256$-bit $\AuthPublic$} & \bitbox{128}{$64$-bit $\Value$} & \bitbox{256}{$256$-bit $\NoteAddressRand$} \bitbox{256}{$256$-bit $\NoteCommitRand$} & \end{bytefield} \end{lrbox} The commitment scheme $\Commit{}$ specified in \crossref{abstractcomm} is instantiated using $\FullHashName$ as follows: \hskip 1em $\Commit{\NoteCommitRand}(\Value, \AuthPublic, \NoteAddressRand) := \FullHashbox{\cmbox}$. \pnote{ The leading byte of the $\FullHash$ input is $\hexint{B0}$. } \todo{Security requirements on $\FullHashName$.} \nsubsection{\NotePlaintexts{} and \Memos} \label{notept} Transmitted \notes are stored on the blockchain in encrypted form, together with a \noteCommitment $\cm$. The \notePlaintexts associated with a \joinSplitDescription are encrypted to the respective \transmissionKeys $\TransmitPublicNew{\allNew}$, and the result forms part of a \notesCiphertext (see \crossref{inband} for further details). Each \notePlaintext (denoted $\NotePlaintext{}$) consists of $(\Value, \NoteAddressRand, \NoteCommitRand\changed{, \Memo})$. The first three of these fields are as defined earlier. \changed{$\Memo$ is a 512-byte \memo associated with this \note. The usage of the \memo is by agreement between the sender and recipient of the \note. The \memo{} \SHOULD be encoded either as: \begin{itemize} \item a UTF-8 human-readable string \cite{Unicode}, padded by appending zero bytes; or \item an arbitrary sequence of 512 bytes starting with a byte value of $\hexint{F5}$ or greater, which is therefore not a valid UTF-8 string. \end{itemize} In the former case, wallet software is expected to strip any trailing zero bytes and then display the resulting \mbox{UTF-8} string to the recipient user, where applicable. Incorrect UTF-8-encoded byte sequences should be displayed as replacement characters (\ReplacementCharacter). In the latter case, the contents of the \memo{} \SHOULDNOT be displayed. A start byte of $\hexint{F5}$ is reserved for use by automated software by private agreement. A start byte of $\hexint{F6}$ or greater is reserved for use in future \Zcash protocol extensions. } The encoding of a \notePlaintext consists of, in order: \begin{equation*} \begin{bytefield}[bitwidth=0.029em]{1608} \changed{ \bitbox{192}{$8$-bit $\NotePlaintextLeadByte$} &}\bitbox{192}{$64$-bit $\Value$} & \bitbox{256}{$256$-bit $\NoteAddressRand$} & \bitbox{256}{\changed{$256$}-bit $\NoteCommitRand$} & \changed{\bitbox{800}{$\Memo$ ($512$ bytes)}} \end{bytefield} \end{equation*} \begin{itemize} \changed{ \item A byte, $\NotePlaintextLeadByte$, indicating this version of the encoding of a \notePlaintext. } \item 8 bytes specifying $\Value$. \item 32 bytes specifying $\NoteAddressRand$. \item \changed{32} bytes specifying $\NoteCommitRand$. \changed{ \item 512 bytes specifying $\Memo$. } \end{itemize} \nsubsection{Encodings of Addresses and Keys} This section describes how \Zcash encodes \paymentAddresses\changed{, \viewingKeys,} and \spendingKeys. Addresses and keys can be encoded as a byte sequence; this is called the \term{raw encoding}. This byte sequence can then be further encoded using Base58Check. The Base58Check layer is the same as for upstream \Bitcoin addresses \cite{Bitcoin-Base58}. $\SHAName$ outputs are always represented as sequences of 32 bytes. The language consisting of the following encoding possibilities is prefix-free. \nsubsubsection{\Transparent{} Payment Addresses} \label{transparentaddrencoding} \xTransparent payment addresses are either P2SH (Pay to Script Hash) \cite{BIP-13} or P2PKH (Pay to Public Key Hash) \cite{Bitcoin-P2PKH} addresses. The raw encoding of a P2SH address consists of: \begin{equation*} \begin{bytefield}[bitwidth=0.1em]{176} \bitbox{80}{$8$-bit $\PtoSHAddressLeadByte$} \bitbox{80}{$8$-bit $\PtoSHAddressSecondByte$} \bitbox{160}{$160$-bit script hash} \end{bytefield} \end{equation*} \begin{itemize} \item Two bytes $[\PtoSHAddressLeadByte, \PtoSHAddressSecondByte]$, indicating this version of the raw encoding of a P2SH address on the production network. (Addresses on the test network use $[\PtoSHAddressTestnetLeadByte, \PtoSHAddressTestnetSecondByte]$ instead.) \item 160 bits specifying a script hash \cite{Bitcoin-P2SH}. \end{itemize} The raw encoding of a P2PKH address consists of: \begin{equation*} \begin{bytefield}[bitwidth=0.1em]{176} \bitbox{80}{$8$-bit $\PtoPKHAddressLeadByte$} \bitbox{80}{$8$-bit $\PtoPKHAddressSecondByte$} \bitbox{160}{$160$-bit public key hash} \end{bytefield} \end{equation*} \begin{itemize} \item Two bytes $[\PtoPKHAddressLeadByte, \PtoPKHAddressSecondByte]$, indicating this version of the raw encoding of a P2PKH address on the production network. (Addresses on the test network use $[\PtoPKHAddressTestnetLeadByte, \PtoPKHAddressTestnetSecondByte]$ instead.) \item 160 bits specifying a public key hash, which is a RIPEMD-160 hash \cite{RIPEMD160} of a SHA-256 hash \cite{NIST2015} of an uncompressed ECDSA key encoding. \end{itemize} \begin{pnotes} \item In \Bitcoin a single byte is used for the version field identifying the address type. In \Zcash two bytes are used. For addresses on the production network, this fixes the first two characters of the Base58Check encoding to be \ascii{r3} for P2SH addresses, or \ascii{r1} for P2PKH addresses. (This does \emph{not} imply that a \transparent \Zcash address can be parsed in the same way as a \Bitcoin address just by removing the \ascii{r}.) \item \Zcash does not yet support Hierarchical Deterministic Wallet addresses \cite{BIP-32}. \end{pnotes} \nsubsubsection{\Transparent{} Private Keys} \label{transparentkeyencoding} These are encoded in the same way as in \Bitcoin \cite{Bitcoin-Base58}, for both the production and test networks. \nsubsubsection{\Protected{} Payment Addresses} \label{paymentaddrencoding} A \paymentAddress consists of $\AuthPublic$ and $\TransmitPublic$. $\AuthPublic$ is a $\SHAName$ output. $\TransmitPublic$ is a \changed{Bern2006} public key, for use with the encryption scheme defined in \crossref{inband}. The raw encoding of a \paymentAddress consists of: \begin{equation*} \begin{bytefield}[bitwidth=0.07em]{520} \changed{ \bitbox{80}{$8$-bit $\PaymentAddressLeadByte$} \bitbox{80}{$8$-bit $\PaymentAddressSecondByte$} &}\bitbox{256}{$256$-bit $\AuthPublic$} & \bitbox{256}{\changed{$256$}-bit $\TransmitPublic$} \end{bytefield} \end{equation*} \begin{itemize} \changed{ \item Two bytes $[\PaymentAddressLeadByte, \PaymentAddressSecondByte]$, indicating this version of the raw encoding of a \Zcash \paymentAddress on the production network. (Addresses on the test network use $[\PaymentAddressTestnetLeadByte, \PaymentAddressTestnetSecondByte]$ instead.) } \item 256 bits specifying $\AuthPublic$. \item \changed{256 bits} specifying $\TransmitPublic$, \changed{using the normal encoding of a Curve25519 public key \cite{Bern2006}}. \end{itemize} \nsubsubsection{Spending Keys} \label{spendingkeyencoding} A \spendingKey consists of $\AuthPrivate$, which is a sequence of \changed{252} bits. The raw encoding of a \spendingKey consists of, in order: \begin{equation*} \begin{bytefield}[bitwidth=0.07em]{264} \changed{ \bitbox{80}{$8$-bit $\SpendingKeyLeadByte$} \bitbox{80}{$8$-bit $\SpendingKeySecondByte$} \bitbox{32}{$\zeros{4}$} & &}\bitbox{252}{\changed{$252$}-bit $\AuthPrivate$} \end{bytefield} \end{equation*} \begin{itemize} \changed{ \item Two bytes $[\SpendingKeyLeadByte, \SpendingKeySecondByte]$, indicating this version of the raw encoding of a \Zcash \spendingKey on the production network. (Addresses on the test network use $[\SpendingKeyTestnetLeadByte, \SpendingKeyTestnetSecondByte]$ instead.) \item 4 zero padding bits. } \item \changed{252} bits specifying $\AuthPrivate$. \end{itemize} \changed{ The zero padding occupies the most significant 4 bits of the third byte. \pnote{ If an implementation represents $\AuthPrivate$ internally as a sequence of 32 bytes with the 4 bits of zero padding intact, it will be in the correct form for use as an input to $\PRFaddr{}$, $\PRFnf{}$, and $\PRFpk{}$ without need for bit-shifting. Future key representations may make use of these padding bits. } } \nsubsection{\ZeroKnowledgeProvingSystem} \label{proofs} \Zcash uses \zkSNARKs generated by its fork of \libsnark \cite{libsnark-fork} with the \provingSystem described in \cite{BCTV2015}, which is a refinement of the systems in \cite{PGHR2013} and \cite{BCGTV2013}. The pairing implementation is $\BNImpl$. Let $q = 21888242871839275222246405745257275088696311157297823662689037894645226208583$. Let $r = 21888242871839275222246405745257275088548364400416034343698204186575808495617$. Let $b = 3$. ($q$ and $r$ are prime.) The pairing is of type $\GroupG{1} \times \GroupG{2} \rightarrow \GroupG{T}$, where: \begin{itemize} \item $\GroupG{1}$ is a Barreto--Naehrig curve over $\GF{q}$ with equation $y^2 = x^3 + b$. This curve has embedding degree 12 with respect to $r$. \item $\GroupG{2}$ is the subgroup of order $r$ in the twisted Barreto-Naehrig curve over $\GF{q^2}$ with equation $y^2 = x^3 + \frac{b}{x \mult i}$. We represent elements of $\GF{q^2}$ as polynomials $a_1 \mult t + a_0 \typecolon \GF{q}[t]$, modulo the irreducible polynomial $t^2 + 1$. \item $\GroupG{T}$ is $\mu_r$, the subgroup of $r^\mathrm{th}$ roots of unity in $\GFstar{q^{12}}$. \end{itemize} Let $\PointP{1} \typecolon \GroupG{1} = (1, 2)$. \begin{tabular}{@{}l@{}r@{}l@{}} Let $\PointP{2} \typecolon \GroupG{2} =\;$ % are these the right way round? &$(11559732032986387107991004021392285783925812861821192530917403151452391805634$ & $\mult\, t\;+$ \\ &$ 10857046999023057135944570762232829481370756359578518086990519993285655852781$ & $, $ \\ &$ 4082367875863433681332203403145435568316851327593401208105741076214120093531$ & $\mult\, t\;+$ \\ &$ 8495653923123431417604973247489272438418190587263600148770280649306958101930$ & $). $ \end{tabular} $\PointP{1}$ and $\PointP{2}$ are generators of $\GroupG{1}$ and $\GroupG{2}$ respectively. A proof consists of a tuple $(\Proof_A \typecolon \GroupG{1},\; \Proof'_A \typecolon \GroupG{1},\; \Proof_B \typecolon \GroupG{2},\; \Proof'_B \typecolon \GroupG{1},\; \Proof_C \typecolon \GroupG{1},\; \Proof'_C \typecolon \GroupG{1},\; \Proof_K \typecolon \GroupG{1},\; \Proof_H \typecolon \GroupG{1})$. It is computed using the parameters above as described in \cite[Appendix B]{BCTV2015}. \pnote{ Many details of the \provingSystem are beyond the scope of this protocol document. For example, the \arithmeticCircuit verifying the \joinSplitStatement, or its expression as a \rankOneConstraintSystem, are not specified here. In practice it will be necessary to use the specific proving and verification keys generated for the \Zcash production \blockchain (see \crossref{jsparameters}), and a \provingSystem implementation that is interoperable with the \Zcash fork of \libsnark, to ensure compatibility. } \nsubsubsection{Encoding of Points} \label{pointencoding} \newsavebox{\gonebox} \begin{lrbox}{\gonebox} \setchanged \begin{bytefield}[bitwidth=0.05em]{264} \bitbox{20}{$0$} \bitbox{20}{$0$} \bitbox{20}{$0$} \bitbox{20}{$0$} \bitbox{20}{$0$} \bitbox{20}{$0$} \bitbox{20}{$1$} \bitbox{80}{$1$-bit $\tilde{y}$} \bitbox{256}{$256$-bit $\ItoOSP{32}(x)$} \end{bytefield} \end{lrbox} \newsavebox{\gtwobox} \begin{lrbox}{\gtwobox} \setchanged \begin{bytefield}[bitwidth=0.05em]{520} \bitbox{20}{$0$} \bitbox{20}{$0$} \bitbox{20}{$0$} \bitbox{20}{$0$} \bitbox{20}{$1$} \bitbox{20}{$0$} \bitbox{20}{$1$} \bitbox{80}{$1$-bit $\tilde{y}$} \bitbox{512}{$512$-bit $\ItoOSP{64}(x)$} \end{bytefield} \end{lrbox} Define $\ItoOSP{} \typecolon (k \typecolon \Nat) \times \range{0}{256^k\!-\!1} \rightarrow \typeexp{\range{0}{255}}{k}$ such that $\ItoOSP{\ell}(n)$ is the sequence of $\ell$ bytes representing $n$ in big-endian order. For a point $P \typecolon \GroupG{1} = (x_P, y_P)$: \begin{itemize} \item The field elements $x_P$ and $y_P \typecolon \GF{q}$ 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} For a point $P \typecolon \GroupG{2} = (x_P, y_P)$: \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 $\tilde{y} = \begin{cases} 1, &\text{if } y > y' \\0, &\text{otherwise.} \end{cases}$ \item $P$ is encoded as $\Justthebox{\gtwobox}$. \end{itemize} \subparagraph{Non-normative notes:} \begin{itemize} \item The use of big-endian byte order is different from the encoding of most other integers in this protocol. The above encodings are consistent with the definition of $\ECtoOSP{}$ for compressed curve points in \cite[section 5.5.6.2]{IEEE2004}. The LSB compressed form (i.e.\ $\ECtoOSPXL$) is used for points on $\GroupG{1}$, and the SORT compressed form (i.e.\ $\ECtoOSPXS$) for points on $\GroupG{2}$. \item Testing $y > y'$ for the compression of $\GroupG{2}$ points is equivalent to testing whether $(a_{y,1}, a_{y,0}) > (a_{-y,1}, a_{-y,0})$ in lexicographic order. \item Algorithms for decompressing points from the above encodings are given in \cite[Appendix A.12.8]{IEEE2000} for $\GroupG{1}$, and \cite[Appendix A.12.11]{IEEE2004} for $\GroupG{2}$. \end{itemize} When computing square roots in $\GF{q}$ or $\GF{q^2}$ in order to decompress a point encoding, the implementation \MUSTNOT assume that the square root exists, or that the encoding represents a point on the curve. \nsubsubsection{Encoding of \ZeroKnowledgeProofs} \label{proofencoding} \newsavebox{\proofbox} \begin{lrbox}{\proofbox} \setchanged \begin{bytefield}[bitwidth=0.021em]{2368} \bitbox{264}{264-bit $\Proof_A$} \bitbox{264}{264-bit $\Proof'_A$} \bitbox{520}{520-bit $\Proof_B$} \bitbox{264}{264-bit $\Proof'_B$} \bitbox{264}{264-bit $\Proof_C$} \bitbox{264}{264-bit $\Proof'_C$} \bitbox{264}{264-bit $\Proof_K$} \bitbox{264}{264-bit $\Proof_H$} \end{bytefield} \end{lrbox} A proof is encoded by concatenating the encodings of its elements: \vspace{1.5ex} \hskip 0.2em $\Justthebox{\proofbox}$ \vspace{1ex} The resulting proof size is 296 bytes. \vspace{0.8ex} In addition to the steps to verify a proof given in \cite[Appendix B]{BCTV2015}, the verifier \MUST check, for the encoding of each element, that: \begin{itemize} \item the lead byte is of the required form; \item the remaining bytes encode a big-endian representation of an integer in $\range{0}{q\!-\!1}$ or (in the case of $\Proof_B$) $\range{0}{q^2\!-\!1}$; \item the encoding represents a point on the relevant curve. \end{itemize} \nsubsection{\JoinSplitParameters} \label{jsparameters} For the testnet in release v0.11.2.z9 and later, the $\FullHashName$ hashes of the \provingKey and \verifyingKey for the \joinSplitStatement, encoded in \libsnark format, are: \begin{verbatim} 226913bbdc48b70834f8e044d194ddb61c8e15329f67cdc6014f4e5ac11a82ab z9-proving.key 4c151c562fce2cdee55ac0a0f8bd9454eb69e6a0db9a8443b58b770ec29b37f5 z9-verifying.key \end{verbatim} The \Zcash production \blockchain will use parameters obtained by a multi-party computation, which has yet to be performed. \nsection{Consensus Changes from \Bitcoin} \nsubsection{Encoding of \Transactions} \label{txnencoding} The \Zcash \transaction format is as follows: \begin{center} \hbadness=10000 \begin{tabularx}{0.92\textwidth}{|c|l|p{10.7em}|X|} \hline Bytes & \heading{Name} & \heading{Data Type} & \heading{Description} \\ \hhline{|=|=|=|=|} 4 & $\versionField$ & \type{uint32\_t} & Transaction version number; either 1 or 2. \\ \hline \Varies & $\txInCount$ & \compactSize & Number of \transparent inputs in this transaction. \\ \hline \Varies & $\txIn$ & $\txIn$ & \xTransparent inputs, encoded as in \Bitcoin. \\ \hline \Varies & $\txOutCount$ & \compactSize & Number of \transparent outputs in this transaction. \\ \hline \Varies & $\txOut$ & $\txOut$ & \xTransparent outputs, encoded as in \Bitcoin. \\ \hline 4 & $\lockTime$ & \type{uint32\_t} & A Unix epoch time or block number, encoded as in \Bitcoin. \\ \hline \Varies\;$\dagger$ & $\nJoinSplit$ & \compactSize & The number of \joinSplitDescriptions in $\vJoinSplit$. \\ \hline \Longunderstack{1802 $\mult$ \\ $\nJoinSplit\,\dagger$} & $\vJoinSplit$ & \type{JoinSplitDescription} \type{[$\nJoinSplit$]} & A \sequenceOfJoinSplitDescriptions, each encoded as described in \crossref{joinsplitencoding}. \\ \hline 32 $\ddagger$ & $\joinSplitPubKey$ & \type{char[32]} & An encoding of a $\JoinSplitSig$ public verification key. \\ \hline 64 $\ddagger$ & $\joinSplitSig$ & \type{char[64]} & A signature on a prefix of the \transaction encoding, to be verified using $\joinSplitPubKey$. \\ \hline \end{tabularx} \end{center} $\dagger$ The $\nJoinSplit$ and $\vJoinSplit$ fields are present if and only if $\versionField > 1$. $\ddagger$ The $\joinSplitPubKey$ and $\joinSplitSig$ fields are present if and only if $\versionField > 1$ and $\nJoinSplit > 0$. The encoding of $\joinSplitPubKey$ and the data to be signed are specified in \crossref{nonmalleability}. The changes relative to \Bitcoin version 1 transactions as described in \cite{Bitcoin-Format} are: \begin{itemize} \item The \transactionVersionNumber{} can be either 1 or 2. A version 1 \transaction is equivalent to a version 2 \transaction with $\nJoinSplit = 0$. Software that parses \blocks{} \MUSTNOT assume, when an encoded \block starts with an $\versionField$ field representing a value other than 1 or 2 (e.g.\ future versions potentially introduced by hard forks), that it will be parseable according to this format. \item The $\nJoinSplit$, $\vJoinSplit$, $\joinSplitPubKey$, and $\joinSplitSig$ fields have been added. \end{itemize} Software that creates \transactions{} \SHOULD use version 1 for \transactions with no \joinSplitDescriptions. \pnote{ A \transactionVersionNumber of 2 does not have the same meaning as in \Bitcoin, where it is associated with support for \ScriptOP{CHECKSEQUENCEVERIFY} as specified in \cite{BIP-68}. \Zcash was forked from \Bitcoin v0.11.2 and does not currently support BIP 68, or the related BIPs 9, 112 and 113. } \nsubsection{Encoding of \JoinSplitDescriptions} \label{joinsplitencoding} An abstract \joinSplitDescription, as described in \crossref{joinsplit}, is encoded in a \transaction as an instance of a \type{JoinSplitDescription} type as follows: \begin{center} \hbadness=2000 \begin{tabularx}{0.92\textwidth}{|c|l|l|X|} \hline Bytes & \heading{Name} & \heading{Data Type} & \heading{Description} \\ \hhline{|=|=|=|=|} \setchanged 8 &\setchanged $\vpubOldField$ &\setchanged \type{uint64\_t} &\mbox{}\setchanged A value $\vpubOld$ that the \joinSplitTransfer removes from the \transparentValuePool. \\ \hline 8 & $\vpubNewField$ & \type{uint64\_t} & A value $\vpubNew$ that the \joinSplitTransfer inserts into the \transparentValuePool. \\ \hline 32 & $\anchorField$ & \type{char[32]} & A merkle root $\rt$ of the \noteCommitmentTree at some block height in the past, or the merkle root produced by a previous \joinSplitTransfer in this \transaction. \\ \hline 64 & $\nullifiersField$ & \type{char[32][$\NOld$]} & A sequence of \nullifiers of the input \notes $\nfOld{\allOld}$. \\ \hline 64 & $\commitments$ & \type{char[32][$\NNew$]} & A sequence of \noteCommitments for the output \notes $\cmNew{\allNew}$. \\ \hline \setchanged 32 &\setchanged $\ephemeralKey$ &\setchanged \type{char[32]} &\mbox{}\setchanged A Curve25519 public key $\EphemeralPublic$. \\ \hline \setchanged 32 &\setchanged $\randomSeed$ &\setchanged \type{char[32]} &\mbox{}\setchanged A 256-bit seed that must be chosen independently at random for each \joinSplitDescription. \\ \hline 64 & $\vmacs$ & \type{char[32][$\NOld$]} & A sequence of message authentication tags $\h{\allOld}$ that bind $\hSig$ to each $\AuthPrivate$ of the $\joinSplitDescription$. \\ \hline 296 & $\zkproof$ & \type{char[296]} & An encoding of the \zeroKnowledgeProof $\JoinSplitProof$ (see \crossref{proofencoding}). \\ \hline 1202 & $\encCiphertexts$ & \type{char[601][$\NNew$]} & A sequence of ciphertext components for the encrypted output \notes, $\TransmitCiphertext{\allNew}$. \\ \hline \end{tabularx} \end{center} The $\ephemeralKey$ and $\encCiphertexts$ fields together form the \notesCiphertext. \nsubsection{\BlockHeaders} The \Zcash \blockHeader format is as follows: \begin{center} \hbadness=1000 \begin{tabularx}{0.92\textwidth}{|c|l|p{10.7em}|X|} \hline Bytes & \heading{Name} & \heading{Data Type} & \heading{Description} \\ \hhline{|=|=|=|=|} 4 & $\nVersion$ & \type{uint32\_t} & The \blockVersionNumber indicates which set of \block validation rules to follow. The current and only defined \blockVersionNumber for \Zcash is $4$. \\ \hline 32 & $\hashPrevBlock$ & \type{char[32]} & A $\SHAd$ hash in internal byte order of the previous \block's header. This ensures no previous \block can be changed without also changing this \block's header. \\ \hline 32 & $\hashMerkleRoot$ & \type{char[32]} & A $\SHAd$ hash in internal byte order. The merkle root is derived from the hashes of all \transactions included in this \block, ensuring that none of those \transactions can be modified without modifying the header. \\ \hline 32 & $\hashReserved$ & \type{char[32]} & A reserved field which should be ignored. \\ \hline 4 & $\nTime$ & \type{uint32\_t} & The \blockTime is a Unix epoch time when the miner started hashing the header (according to the miner). This \MUST be greater than or equal to the median time of the previous 11 blocks. A \fullnode{} \MUSTNOT accept \blocks with headers more than two hours in the future according to its clock. \\ \hline 4 & $\nBits$ & \type{uint32\_t} & An encoded version of the target threshold this \block's header hash must be less than or equal to, in the same nBits format used by \Bitcoin. \cite{Bitcoin-nBits} \\ \hline 32 & $\nNonce$ & \type{char[32]} & An arbitrary field miners change to modify the header hash in order to produce a hash below the target threshold. \\ \hline 3 & $\solutionSize$ & \compactSize & The size of an Equihash solution in bytes (always 1344). \\ \hline 1344 & $\solution$ & \type{char[1344]} & The Equihash solution, which \MUST be valid according to \crossref{equihash}. \\ \hline \end{tabularx} \end{center} The changes relative to \Bitcoin version 4 blocks as described in \cite{Bitcoin-Block} are: \begin{itemize} \item The \blockVersionNumber{} \MUST be 4. Previous versions are not supported. Software that parses blocks \MUSTNOT assume, when an encoded \block starts with an $\nVersion$ field representing a value other than 4 (e.g.\ future versions potentially introduced by hard forks), that it will be parseable according to this format. \item The $\hashReserved$, $\solutionSize$, and $\solution$ fields have been added. \item The type of the $\nNonce$ field has changed from \type{uint32\_t} to \type{char[32]}. \end{itemize} \begin{pnotes} \item There is no relation between the values of the $\versionField$ field of a \transaction, and the $\nVersion$ field of a \blockHeader. \item Like other serialized fields of type $\compactSize$, the $\solutionSize$ field \MUST be encoded with the minimum number of bytes (3 in this case), and other encodings \MUST be rejected. This is necessary to avoid a potential attack in which a miner could test several distinct encodings of each Equihash solution against the difficulty filter, rather than only the single intended encoding. \end{pnotes} \nsubsection{Proof of Work} \Zcash uses Equihash \cite{BK2016} as its Proof of Work. Motivations for changing the Proof of Work from \SHAd used by \Bitcoin are described in \cite{WG2016}. A \block satisfies the Proof of Work if and only if: \begin{itemize} \item The $\solution$ field encodes a \validEquihashSolution according to \crossref{equihash}. \item The \blockHeader satisfies the difficulty check according to \crossref{difficulty}. \end{itemize} \nsubsubsection{Equihash} \label{equihash} An instance of the Equihash algorithm is parameterized by positive integers $n$ and $k$, such that $n$ is a multiple of $k+1$. We assume $k \geq 3$. 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 $X_{1..\mathrm{N}}$ of $n$-bit strings, find $2^k$ distinct $X_{i_j}$ such that $\vxor{j=1}{2^k} X_{i_j} = 0$. In Equihash, $\mathrm{N} = 2^{\frac{n}{k+1}+1}$, and the sequence $X_{1..\mathrm{N}}$ is derived from the \blockHeader and a nonce: \newsavebox{\powheaderbox} \begin{lrbox}{\powheaderbox} \begin{bytefield}[bitwidth=0.064em]{1152} \bitbox{128}{32-bit $\nVersion$} \bitbox{256}{256-bit $\hashPrevBlock$} \bitbox{256}{256-bit $\hashMerkleRoot$} \\ \bitbox{256}{256-bit $\hashReserved$} \bitbox{128}{32-bit $\nTime$} \bitbox{128}{32-bit $\nBits$} \\ \bitbox{256}{256-bit $\nNonce$} \end{bytefield} \end{lrbox} Let $\powheader := \Justthebox[-11.5ex]{\powheaderbox}$ For $i \in \range{1}{N}$, let $X_i = \EquihashGen{n, k}(\powheader, i)$. $\EquihashGen{}$ is instantiated in \crossref{equihashgen}. Define $\ItoBSP{} \typecolon (u \typecolon \Nat) \times \range{0}{2^u\!-\!1} \rightarrow \bitseq{u}$ such that $\ItoBSP{u}(x)$ is the sequence of $u$ bits representing $x$ in big-endian order. A \validEquihashSolution is then a sequence $i \typecolon \range{1}{N}^{2^k}$ that satisfies the following conditions: \subparagraph{Generalized Birthday condition} $\vxor{j=1}{2^k} X_{i_j} = 0$. \subparagraph{Algorithm Binding conditions} For all $r \in \range{1}{k\!-\!1}$, for all $w \in \range{0}{2^{k-r}\!-\!1}$: \begin{itemize} \item $\vxor{j=1}{2^r} X_{i_{w \mult 2^r + j}}$ has $\frac{n \mult r}{k+1}$ leading zeroes; and \item $i_{w \mult 2^r + 1 .. w \mult 2^r + 2^{r-1}} < i_{w \mult 2^r + 2^{r-1} + 1 .. w \mult 2^r + 2^r}$ lexicographically. \end{itemize} \pnote{ This does not include a difficulty condition, because here we are defining validity of an Equihash solution independent of difficulty. } An Equihash solution with $n = 200$ and $k = 9$ is encoded in the $\solution$ field of a \blockHeader as follows: \newsavebox{\solutionbox} \begin{lrbox}{\solutionbox} \begin{bytefield}[bitwidth=0.45em]{105} \bitbox{21}{$\ItoBSP{21}(i_1-1)$} \bitbox{21}{$\ItoBSP{21}(i_2-1)$} \bitbox{42}{$\cdots$} \bitbox{21}{$\ItoBSP{21}(i_{512}-1)$} \end{bytefield} \end{lrbox} \newcommand{\zb}{\bitbox{1}{$0$}} \newcommand{\ob}{\bitbox{1}{$1$}} \newsavebox{\eqexamplebox} \begin{lrbox}{\eqexamplebox} \begin{bytefield}[bitwidth=0.75em]{63} \bitbox{21}{$\ItoBSP{21}(68)$} \bitbox{21}{$\ItoBSP{21}(41)$} \bitbox{21}{$\ItoBSP{21}(2^{21}-1)$} \\ \zb\zb\zb\zb\zb\zb\zb\zb\zb\zb\zb\zb\zb\zb\ob\zb\zb\zb\ob\zb\zb \zb\zb\zb\zb\zb\zb\zb\zb\zb\zb\zb\zb\zb\zb\zb\ob\zb\ob\zb\zb\ob \ob\ob\ob\ob\ob\ob\ob\ob\ob\ob\ob\ob\ob\ob\ob\ob\ob\ob\ob\ob\ob \\ \bitbox{8}{8-bit $0$} \bitbox{8}{8-bit $2$} \bitbox{8}{8-bit $32$} \bitbox{8}{8-bit $0$} \bitbox{8}{8-bit $10$} \bitbox{8}{8-bit $127$} \bitbox{8}{8-bit $255$} \bitbox{7}{$\cdots$} \end{bytefield} \end{lrbox} \hskip 1.5em $\Justthebox{\solutionbox}$ \vspace{1ex} Recall from \crossref{boxnotation} that bits in the above diagram are ordered from most to least significant in each byte. For example, if the first 3 elements of $i$ are $[69, 42, 2^{21}]$, then the corresponding bit array is: \hskip 1.5em $\Justthebox{\eqexamplebox}$ and so the first 7 bytes of $\solution$ would be $[0, 2, 32, 0, 10, 127, 255]$. \pnote{ $\ItoBSP{}$ is big-endian, while integer field encodings in $\powheader$ and in the instantiation of $\EquihashGen{}$ are little-endian. The rationale for this is that little-endian serialization of \blockHeaders is consistent with \Bitcoin, but using little-endian ordering of bits in the solution encoding would require bit-reversal (as opposed to only shifting). } \nsubsubsection{Difficulty filter} \label{difficulty} The difficulty filter is unchanged from \Bitcoin, and is calculated using \SHAd on the whole \blockHeader (including $\solutionSize$ and $\solution$). \nsubsubsection{Difficulty adjustment} \label{diffadjustment} \Zcash uses a difficulty adjustment algorithm based on DigiShield v3/v4, with simplifications and altered parameters, to adjust difficulty to target the desired 2.5-minute block time. Unlike \Bitcoin, the difficulty adjustment occurs after every block. \todo{Describe the algorithm.} \nsubsection{Calculation of Block Subsidy and Founders' Reward} \label{subsidies} \crossref{subsidyconcepts} defines the \blockSubsidy, \minerSubsidy, and \foundersReward. Their amounts in \zatoshi are calculated from the \blockHeight using the formulae below. The constants $\SlowStartInterval$, $\HalvingInterval$, $\MaxBlockSubsidy$, and $\FoundersFraction$ are instantiated in \crossref{constants}. \vspace{2ex} \hskip 1em $\SlowStartShift \typecolon \Nat := \hfrac{\SlowStartInterval}{2}$ \hskip 1em $\SlowStartRate \typecolon \Nat := \hfrac{\MaxBlockSubsidy}{\SlowStartInterval}$ \hskip 1em $\Halving(\BlockHeight) := \floor{\hfrac{\BlockHeight - \SlowStartShift}{\HalvingInterval}}$ \hskip 1em $\BlockSubsidy(\BlockHeight) := \begin{cases} \SlowStartRate \mult \BlockHeight,&\!\!\text{if } \BlockHeight < \hfrac{\SlowStartInterval}{2} \\[1.4ex] \SlowStartRate \mult (\BlockHeight + 1),&\!\!\text{if } \hfrac{\SlowStartInterval}{2} \leq \BlockHeight < \SlowStartInterval \\[1.4ex] \floor{\hfrac{\MaxBlockSubsidy}{2^{\Halving(\BlockHeight)}}},&\!\!\text{otherwise} \end{cases}$ \hskip 1em $\FoundersReward(\BlockHeight) := \begin{cases} \BlockSubsidy(\BlockHeight) \mult \FoundersFraction,&\!\!\!\text{if } \BlockHeight < \SlowStartShift + \HalvingInterval \\ 0,&\!\!\!\text{otherwise} \end{cases}$ \hskip 1em $\MinerSubsidy(\BlockHeight) := \BlockSubsidy(\BlockHeight) - \FoundersReward(\BlockHeight)$. \nsubsection{Coinbase outputs} \label{coinbases} \todo{Coinbase maturity rule.} \todo{Any tx with a coinbase input must have no \transparent outputs (vout).} The \foundersReward is paid by a \transparent output in the \coinbaseTransaction, to one of $\NumFounderAddresses$ \transparent addresses, depending on the \blockHeight. Let $\SlowStartShift$ be defined as in the previous section. \renewcommand{\arraystretch}{0.95} For mainnet, $\FounderAddressList_{\mathrm{1}..\NumFounderAddresses}$ is \todo{}. For testnet, $\FounderAddressList_{\mathrm{1}..\NumFounderAddresses}$ is: \begin{tabular}{@{\hskip 2.5em}l@{\;}l} [& \ascii{2N2e2FRfP9D1dRN1oRWkH7pbFM69eGNAuQ4}, \ascii{2NBW8WsA2jUussoJbRv82UXH1BYopkjYqcd}, \\ & \ascii{2N1MudZmwDFTcYiLCZfrcsnhHwaSTTigbcN}, \ascii{2MxfUJXWKz9D8X3mcMpVcdEJKdJ6zFukca9}, \\ & \ascii{2N8iUwMCpU16VYpKQ1HRM6xfut5FZwGwieM}, \ascii{2N9hyafTvJVrykBvZDw79j1brozwZNySwPP}, \\ & \ascii{2NFx7tRozsp3kT1M4w4tL9FfnEj8RovzbzN}, \ascii{2NAqoH96V1RtmK72LEZpJNX1uxhJ5yejRiK}, \\ & \ascii{2MyV7hoV28KS8Uam2Z8nzY3xeo7R3T3TLUr}, \ascii{2N8Tn19hMoCD4EmCwpg1V8qupVkQLVVPhav}, \\ & \ascii{2NA5UeJU9zAQkSMyy3xpDcjfp4CEyKfzXKp}, \ascii{2NBERNyXy46CfM9yewGeof4yzC3vkwYnhgS}, \\ & \ascii{2N7fnpAswHb4mnPm2ZjWX3eKkF8hABAYBtQ}, \ascii{2N9MXGsz7uYaY5ciax6tSMDG7sjZUoLhJTC}, \\ & \ascii{2N5PwzPQFFmLut2XWGQWAmpwKsF8VzUoPtr}, \ascii{2MvZdDpNP8hWyEqg6zKW9B62YTJqcUwjHr5}, \\ & \ascii{2Mx4KfKJ37EDc3A43Frzof1iEjSe91JUX7d}, \ascii{2NBMSdXjZ7YqREmwxEtgGryY59KBpqMSs1d}, \\ & \ascii{2N9RbfE4ZCJ3Nx68vPfmvH2M6Q3qicJhagb}, \ascii{2N4xwfFkFj4DR4NWNbynzP2aJmVcEFnA2DB}, \\ & \ascii{2Mx4TyAwedmsRuDkvMNYGqrcCZfQTfCvxAp}, \ascii{2Mx4HSVsxEqXjLxn8igJzmCrFdG9XhnNvtf}, \\ & \ascii{2MtLM4SP7LJbBZ5rA5ZG8kAVz9UNrNKuoFB}, \ascii{2N7SPq83Cbmwuwv5rjNBzVd9QtJKAxxKj8M}, \\ & \ascii{2MwYkbE4U4p9XBsCrupDDkdcDH9L9xvc9Bn}, \ascii{2MyaeCHpVmckokUi67YP1QK9L3Dkx3Pt86F}, \\ & \ascii{2N7URNgBPXGjqnuPHiynCa6qMMhKm6YEaHr}, \ascii{2N2eNwGVwj4WwbEdJg7YZDgrnYvDv1ZSNbB}, \\ & \ascii{2MuWAG6BqLM1mtZc67Fv1aKgGwkNQ2akDGt}, \ascii{2N7XH82MbGwpzbc7PM2aK5CU14bSJvK7Etz}, \\ & \ascii{2MuPX8Ke5TvDDQ1nkqpaPMgYWPyWbFp18Jn}, \ascii{2NFBST7oK9yw9PaXaq5QhdyYwp5HpHz9m81}, \\ & \ascii{2MuSeMBUrttbjvDZAeQjTrrDeoP197qj2kG}, \ascii{2N6JU8JNGGAUFknTCuLSuDEEhZJqMfFsH88}, \\ & \ascii{2N4P2MrwtwbiHymQm1RASoVoiH3sFrBpmXa}, \ascii{2MyhFiVXvVVxUNc8Qh9ppV7jG4NsKpnxige}, \\ & \ascii{2N5dLXUho2GtjuHMWuqixLrHLCwUMcYxd7s}, \ascii{2N9NhfSiYBt3fhETFR6mQc3uxreEy7simSg}, \\ & \ascii{2NBEEWPY3v38uuC7n1tMtviEY7ND2XzfgSG}, \ascii{2NCWWj6oREJiMmfJ2bV5sbm1xchMwQfAZ5r}, \\ & \ascii{2N4ACsVCKMvJmtEb3Pd3xkqhJ3rLT4mYx1r}, \ascii{2MtmMdabcwRJmenswaYtWA675df854KhUxD}, \\ & \ascii{2N2h27Dd87eiGcm7ajvu4hJpXjTm9GkzvLZ}, \ascii{2NGE19agRXU1EAK3PCLZWXERkpqyUexhk9r}, \\ & \ascii{2N63112wMnBsXTaBFjbCTjW9LuyTXQmvEdw}, \ascii{2NBkHxgkYZbU56zsoLNsP5WZVfMtBK6X8WK}, \\ & \ascii{2N5pK7NfKo6d9qBmsKggpwuvQeMxGf65SLH}, \ascii{2N5jHzgCg9a9uAcLaT2jij8WKTZzWbVNC5c}\, ] \end{tabular} \renewcommand{\arraystretch}{1} Define: \begin{itemize} \item[] $\FounderAddressChangeInterval := \ceiling{\hfrac{\SlowStartShift + \HalvingInterval}{\NumFounderAddresses}}$ \item[] $\FounderAddressIndex(\BlockHeight) := 1 + \floor{\hfrac{\BlockHeight}{\FounderAddressChangeInterval}}$. \end{itemize} Each address representation in $\FounderAddressList$ denotes a \transparent P2SH multisig address. Let $\RedeemScriptHash(\BlockHeight)$ be the standard redeem script hash, as defined in \cite{Bitcoin-Multisig}, for the P2SH multisig address with Base58Check representation given by $\FounderAddressList_{\,\FounderAddressIndex(\BlockHeight)}$ \consensusrule{ A \coinbaseTransaction for \blockHeight $\BlockHeight \in \range{1}{\SlowStartShift + \HalvingInterval - 1}$ \MUST include at least one output that pays exactly $\FoundersReward(\BlockHeight)$ \zatoshi with a standard P2SH script of the form \ScriptOP{HASH160} \;$\RedeemScriptHash(\BlockHeight)$\; \ScriptOP{EQUAL} as its $\scriptPubKey$. } \begin{pnotes} \item No \foundersReward is required to be paid for $\BlockHeight \geq \SlowStartShift + \HalvingInterval$ (i.e.\ after the first halving), or for $\BlockHeight = 0$ (i.e.\ the genesis block). \item The \foundersReward addresses are not treated specially in any other way, and there can be other outputs to them, in \coinbaseTransactions or otherwise. In particular, it is valid for a \coinbaseTransaction with $\BlockHeight \in \range{1}{\SlowStartShift + \HalvingInterval - 1}$ to have other outputs, possibly to the same address, that do not meet the criterion in the above consensus rule, as long as at least one output meets it. \end{pnotes} \nsubsection{Bitcoin Improvement Proposals} \label{bips} In general, Bitcoin Improvement Proposals (BIPs) do not apply to \Zcash unless otherwise specified in this section. All of the BIPs referenced below should be interpreted by replacing ``BTC'', or ``bitcoin'' used as a currency unit, with ``ZEC''; and ``satoshi'' with ``zatoshi''. The following BIPs apply, otherwise unchanged, to \Zcash: \cite{BIP-11}, \cite{BIP-14}, \cite{BIP-31}, \cite{BIP-35}, \cite{BIP-37}, \cite{BIP-61}. The following BIPs apply starting from the \genesisBlock, i.e.\ any activation rules or exceptions for particular \blocks in the \Bitcoin \blockchain are to be ignored: \cite{BIP-16}, \cite{BIP-30}, \cite{BIP-34}, \cite{BIP-65}, \cite{BIP-66}. \cite{BIP-13} applies with the changes to address version bytes described in \crossref{transparentaddrencoding}. \begin{comment} \cite{BIP-22} and \cite{BIP-23} apply with some protocol changes, which are to be specified in a Zcash Improvement Proposal. The following BIPs can be used unchanged, but do not define consensus rules: \cite{BIP-69}, \cite{BIP-126}. The following BIPs can be used by replacing the URI scheme \ascii{bitcoin:} with \ascii{zcash:}, and the MIME types starting with \ascii{bitcoin-} with corresponding types starting with \ascii{zcash-}: \cite{BIP-21}, \cite{BIP-70}, \cite{BIP-71}, \cite{BIP-72}, \cite{BIP-73}. (Note that this URI scheme and these MIME types are not formally allocated, and would require an RFC in order to do so.) \end{comment} \nsection{Differences from the Zerocash paper} \label{differences} \nsubsection{Transaction Structure} \label{trstructure} \Zerocash introduces two new operations, which are described in the paper as new transaction types, in addition to the original transaction type of the cryptocurrency on which it is based (e.g.\ \Bitcoin). In \Zcash, there is only the original \Bitcoin transaction type, which is extended to contain a sequence of zero or more \Zcash-specific operations. This allows for the possibility of chaining transfers of \xprotected value in a single \Zcash \transaction, e.g.\ to spend a \protectedNote that has just been created. (In \Zcash, we refer to value stored in UTXOs as \transparent, and value stored in \joinSplitTransfer output \notes as \xprotected.) This was not possible in the \Zerocash design without using multiple transactions. It also allows \transparent and \xprotected transfers to happen atomically --- possibly under the control of nontrivial script conditions, at some cost in distinguishability. \todo{Describe changes to signing.} \nsubsection{\Memos} \Zcash adds a \memo sent from the creator of a \joinSplitDescription to the recipient of each output \note. This feature is described in more detail in \crossref{notept}. \nsubsection{Unification of Mints and Pours} In the original \Zerocash protocol, there were two kinds of transaction relating to \protectedNotes: \begin{itemize} \item a ``Mint'' transaction takes value from \transparent UTXOs as input and produces a new \protectedNote as output. \item a ``Pour'' transaction takes up to $\NOld$ \protectedNotes as input, and produces up to $\NNew$ \protectedNotes and a \transparent UTXO as output. \end{itemize} Only ``Pour'' transactions included a \zkSNARK proof. In \Zcash, the sequence of operations added to a \transaction (described in \crossref{trstructure}) consists only of \joinSplitTransfers. A \joinSplitTransfer is a Pour operation generalized to take a \transparent UTXO as input, allowing \joinSplitTransfers to subsume the functionality of Mints. An advantage of this is that a \Zcash \transaction that takes input from an UTXO can produce up to $\NNew$ output \notes, improving the indistinguishability properties of the protocol. A related change conceals the input arity of the \joinSplitTransfer: an unused (zero-value) input is indistinguishable from an input that takes value from a \note. This unification also simplifies the fix to the Faerie Gold attack described below, since no special case is needed for Mints. \nsubsection{Faerie Gold attack and fix} \label{faeriegold} When a \protectedNote is created in \Zerocash, the creator is supposed to choose a new $\NoteAddressRand$ value at random. The \nullifier of the \note is derived from its \spendingKey ($\AuthPrivate$) and $\NoteAddressRand$. The \noteCommitment is derived from the recipient address component $\AuthPublic$, the value $\Value$, and the commitment trapdoor $\NoteCommitRand$, as well as $\NoteAddressRand$. However nothing prevents creating multiple \notes with different $\Value$ and $\NoteCommitRand$ (hence different \noteCommitments) but the same $\NoteAddressRand$. An adversary can use this to mislead a \note recipient, by sending two \notes both of which are verified as valid by $\Receive$ (as defined in \cite[Figure 2]{BCG+2014}), but only one of which can be spent. We call this a ``Faerie Gold'' attack --- referring to various Celtic legends in which faeries pay mortals in what appears to be gold, but which soon after reveals itself to be leaves, gorse blossoms, gingerbread cakes, or other less valuable things \cite{LG2004}. This attack does not violate the security definitions given in \cite{BCG+2014}. The issue could be framed as a problem either with the definition of Completeness, or the definition of Balance: \begin{itemize} \item The Completeness property asserts that a validly received \note can be spent provided that its \nullifier does not appear on the ledger. This does not take into account the possibility that distinct \notes, which are validly received, could have the same \nullifier. That is, the security definition depends on a protocol detail --\nullifiers-- that is not part of the intended abstract security property, and that could be implemented incorrectly. \item The Balance property only asserts that an adversary cannot obtain \emph{more} funds than they have minted or received via payments. It does not prevent an adversary from causing others' funds to decrease. In a Faerie Gold attack, an adversary can cause spending of a \note to reduce (to zero) the effective value of another \note for which the attacker does not know the \spendingKey, which violates an intuitive conception of global balance. \end{itemize} These problems with the security definitions need to be repaired, but doing so is outside the scope of this specification. Here we only describe how \Zcash addresses the immediate attack. It would be possible to address the attack by requiring that a recipient remember all of the $\NoteAddressRand$ values for all \notes they have ever received, and reject duplicates (as proposed in \cite{GGM2016}). However, this requirement would interfere with the intended \Zcash feature that a holder of a \spendingKey can recover access to (and be sure that they are able to spend) all of their funds, even if they have forgotten everything but the \spendingKey. Instead, \Zcash enforces that an adversary must choose distinct values for each $\NoteAddressRand$, by making use of the fact that all of the \nullifiers in \joinSplitDescriptions that appear in a valid \blockchainview must be distinct. This is true regardless of whether the \nullifiers corresponded to real or dummy notes (see \crossref{dummynotes}). The \nullifiers are used as input to $\hSigCRH$ to derive a public value $\hSig$ which uniquely identifies the transaction, as described in \crossref{joinsplitdesc}. ($\hSig$ was already used in \Zerocash in a way that requires it to be unique in order to maintain indistinguishability of \joinSplitDescriptions; adding the \nullifiers to the input of the hash used to calculate it has the effect of making this uniqueness property robust even if the \transaction creator is an adversary.) The $\NoteAddressRand$ value for each output \note is then derived from a random private seed $\NoteAddressPreRand$ and $\hSig$ using $\PRFrho{\NoteAddressPreRand}$. The correct construction of $\NoteAddressRand$ for each output \note is enforced by the \joinSplitStatement (see \crossref{uniquerho}). Now even if the creator of a \joinSplitDescription does not choose $\NoteAddressPreRand$ randomly, uniqueness of \nullifiers and collision resistance of both $\hSigCRH$ and $\PRFrho{}$ will ensure that the derived $\NoteAddressRand$ values are unique, at least for any two \joinSplitDescriptions that get into a valid \blockchainview. This is sufficient to prevent the Faerie Gold attack. \nsubsection{Internal hash collision attack and fix} \label{internalh} The \Zerocash security proof requires that the composition of $\Commit{\NoteCommitRand}$ and $\Commit{\NoteCommitS}$ is a computationally binding commitment to its inputs $\AuthPublic$, $\Value$, and $\NoteAddressRand$. However, the instantiation of $\Commit{\NoteCommitRand}$ and $\Commit{\NoteCommitS}$ in section 5.1 of the paper did not meet the definition of a binding commitment at a 128-bit security level. Specifically, the internal hash of $\AuthPublic$ and $\NoteAddressRand$ is truncated to 128 bits (motivated by providing statistical hiding security). This allows an attacker, with a work factor on the order of $2^{64}$, to find distinct values of $\NoteAddressRand$ with colliding outputs of the truncated hash, and therefore the same \noteCommitment. This would have allowed such an attacker to break the Balance property by double-spending \notes, potentially creating arbitrary amounts of currency for themself \cite{HW2016}. \Zcash uses a simpler construction with a single $\FullHashName$ evaluation for the commitment. 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 a generalized \joinSplitTransfer which always uses a \zeroKnowledgeProof, it does not require the nesting. A side benefit is that this reduces the number of $\SHA$ evaluations needed to compute each \noteCommitment from three to two, saving a total of four $\SHA$ evaluations in the \joinSplitStatement. \pnote{ \Zcash \noteCommitments are not statistically hiding, so \Zcash does not support the ``everlasting anonymity'' property described in \cite[section 8.1]{BCG+2014}, even when used as described in that section. While it is possible to define a statistically hiding, computationally binding commitment scheme for this use at a 128-bit security level, the overhead of doing so within the \joinSplitStatement was not considered to justify the benefits. } \nsubsection{Changes to PRF inputs and truncation} \label{truncation} The format of inputs to the PRFs instantiated in \crossref{concreteprfs} has changed relative to \Zerocash. There is also a requirement for another PRF, $\PRFrho{}$, which must be domain-separated from the others. In the \Zerocash protocol, $\NoteAddressRandOld{i}$ is truncated from 256 to 254 bits in the input to $\PRFsn{}$ (which corresponds to $\PRFnf{}$ in \Zcash). Also, $\hSig$ is truncated from 256 to 253 bits in the input to $\PRFpk{}$. These truncations are not taken into account in the security proofs. Both truncations affect the validity of the proof sketch for Lemma D.2 in the proof of Ledger Indistinguishability in \cite[Appendix D]{BCG+2014}. In more detail: \begin{itemize} \item In the argument relating $\mathbf{H}$ and $\Game_2$, it is stated that in $\Game_2$, ``for each $i \in \setof{1, 2}, \mathsf{sn}_i := \PRFsn{\AuthPrivate}(\NoteAddressRand)$ for a random (and not previously used) $\NoteAddressRand$''. It is also argued that ``the calls to $\PRFsn{\AuthPrivate}$ are each by definition unique''. The latter assertion depends on the fact that $\NoteAddressRand$ is ``not previously used''. However, the argument is incorrect because the truncated input to $\PRFsn{\AuthPrivate}$, i.e. $[\NoteAddressRand]_{254}$, may repeat even if $\NoteAddressRand$ does not. \item In the same argument, it is stated that ``with overwhelming probability, $\hSig$ is unique''. In fact what is required to be unique is the truncated input to $\PRFpk{}$, i.e. $[\hSig]_{253} = [\CRH(\pksig)]_{253}$. In practice this value will be unique under a plausible assumption on $\CRH$ provided that $\pksig$ is chosen randomly, but no formal argument for this is presented. \end{itemize} Note that $\NoteAddressRand$ is truncated in the input to $\PRFsn{}$ but not in the input to $\Commit{\NoteCommitRand}$, which further complicates the analysis. As further evidence that it is essential for the proofs to explicitly take any such truncations into account, consider a slightly modified protocol in which $\NoteAddressRand$ is truncated in the input to $\Commit{\NoteCommitRand}$ but not in the input to $\PRFsn{}$. In that case, it would be possible to violate balance by creating two \notes for which $\NoteAddressRand$ differs only in the truncated bits. These \notes would have the same \noteCommitment but different \nullifiers, so it would be possible to spend the same value twice. For resistance to Faerie Gold attacks as described in \crossref{faeriegold}, \Zcash depends on collision resistance of both $\hSigCRH$ and $\PRFrho{}$ (instantiated using $\Blake{256}$ and $\SHA$ respectively). Collision resistance of a truncated hash does not follow from collision resistance of the original hash, even if the truncation is only by one bit. This motivated avoiding truncation along any path from the inputs to the computation of $\hSig$ to the uses of $\NoteAddressRand$. Since the PRFs are instantiated using $\SHA$ which has an input block size of 512 bits (of which 256 bits are used for the PRF input and 4 bits are used for domain separation), it was necessary to reduce the size of the PRF key to 252 bits. The key is set to $\AuthPrivate$ in the case of $\PRFaddr{}$, $\PRFnf{}$, and $\PRFpk{}$, and to $\NoteAddressPreRand$ (which does not exist in \Zerocash) for $\PRFrho{}$, and so those values have been reduced to 252 bits. This is preferable to requiring reasoning about truncation, and 252 bits is quite sufficient for security of these cryptovalues. \nsubsection{In-band secret distribution} \label{inbandrationale} \Zerocash specified ECIES (referencing Certicom's SEC 1 standard) as the encryption scheme used for the in-band secret distribution. This has been changed to a scheme based on Curve25519 key agreement, and the authenticated encryption algorithm $\SymSpecific$. This scheme is still loosely based on ECIES, and on the $\CryptoBoxSeal$ scheme defined in libsodium \cite{libsodium-Seal}. The motivations for this change were as follows: \begin{itemize} \item The \Zerocash paper did not specify the curve to be used. We believe that Curve25519 has significant side-channel resistance, performance, implementation complexity, and robustness advantages over most other available curve choices, as explained in \cite{Bern2006}. \item ECIES permits many options, which were not specified. There are at least --counting conservatively-- 576 possible combinations of options and algorithms over the four standards (ANSI X9.63, IEEE Std 1363a-2004, ISO/IEC 18033-2, and SEC 1) that define ECIES variants \cite{MAEA2010}. \item Although the \Zerocash paper states that ECIES satisfies key privacy (as defined in \cite{BBDP2001}), it is not clear that this holds for all curve parameters and key distributions. For example, if a group of non-prime order is used, the distribution of ciphertexts could be distinguishable depending on the order of the points representing the ephemeral and recipient public keys. Public key validity is also a concern. Curve25519 key agreement is defined in a way that avoids these concerns due to the curve structure and the ``clamping'' of private keys. \item Unlike the DHAES/DHIES proposal on which it is based \cite{ABR1999}, ECIES does not require a representation of the sender's ephemeral public key to be included in the input to the KDF, which may impair the security properties of the scheme. (The Std 1363a-2004 version of ECIES \cite{IEEE2004} has a ``DHAES mode'' that allows this, but the representation of the key input is underspecified, leading to incompatible implementations.) The scheme we use has both the ephemeral and recipient public key encodings --which are unambiguous for Curve25519-- and also $\hSig$ and a nonce as described below, as input to the KDF. Note that because $\TransmitPublic$ is included in the KDF input, being able to break the Elliptic Curve Diffie-Hellman Problem on Curve25519 (without breaking $\SymSpecific$ as an authenticated encryption scheme or $\Blake{256}$ as a KDF) would not help to decrypt the \notesCiphertext unless $\TransmitPublic$ is known or guessed. \item The KDF also takes a public seed $\hSig$ as input. This can be modeled as using a different ``randomness extractor'' for each \joinSplitTransfer, which limits degradation of security with the number of \joinSplitTransfers. This facilitates security analysis as explained in \cite{DGKM2011} --- see section 7 of that paper for a security proof that can be applied to this construction under the assumption that single-block $\Blake{256}$ is a ``weak PRF''. Note that $\hSig$ is authenticated, by the ZK proof, as having been chosen with knowledge of $\AuthPrivateOld{\allOld}$, so an adversary cannot modify it in a ciphertext from someone else's transaction for use in a chosen-ciphertext attack without detection. \item The scheme used by \Zcash includes an optimization that uses the same ephemeral key (with different nonces) for the two ciphertexts encrypted in each \joinSplitDescription. \end{itemize} The security proofs of \cite{ABR1999} can be adapted straightforwardly to the resulting scheme. Although DHAES as defined in that paper does not pass the recipient public key or a public seed to the hash function $H$, this does not impair the proof because we can consider $H$ to be the specialization of our KDF to a given recipient key and seed. It is necessary to adapt the ``HDH independence'' assumptions and the proof slightly to take into account that the ephemeral key is reused for two encryptions. Note that the 256-bit key for $\SymSpecific$ maintains a high concrete security level even under attacks using parallel hardware \cite{Bern2005} in the multi-user setting \cite{Zave2012}. This is especially necessary because the privacy of \Zcash transactions may need to be maintained far into the future, and upgrading the encryption algorithm would not prevent a future adversary from attempting to decrypt ciphertexts encrypted before the upgrade. Other cryptovalues that could be attacked to break the privacy of transactions are also sufficiently long to resist parallel brute force in the multi-user setting: $\AuthPrivate$ is 252 bits, and $\TransmitPrivate$ is no shorter than $\AuthPrivate$. \nsubsection{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 the proof of the Balance property. Suppose that an adversary finds a collision on $\PRFaddr{}$ such that $\AuthPrivateX{1}$ and $\AuthPrivateX{2}$ are distinct \spendingKeys for the same $\AuthPublic$. Because the \noteCommitment is to $\AuthPublic$, but the \nullifier is computed from $\AuthPrivate$ (and $\NoteAddressRand$), the adversary is able to double-spend the note, once with each $\AuthPrivate$. This is not detected because each spend reveals a different \nullifier. The \joinSplitStatements are still valid because they can only check that the $\AuthPrivate$ in the witness is \emph{some} preimage of the $\AuthPublic$ used in the \noteCommitment. The error is in the proof of Balance in \cite[Appendix D.3]{BCG+2014}. For the ``$\Adversary$ violates Condition I'' case, the proof says: \begin{itemize} \item[``(i)] If $\cmOldX{1} = \cmOldX{2}$, then the fact that $\snOldX{1} \neq \snOldX{2}$ implies that the witness $a$ contains two distinct openings of $\cmOldX{1}$ (the first opening contains $(\AuthPrivateOldX{1}, \NoteAddressRandOldX{1})$, while the second opening contains $(\AuthPrivateOldX{2}, \NoteAddressRandOldX{2})$). This violates the binding property of the commitment scheme $\Commit{}$." \end{itemize} In fact the openings do not contain $\AuthPrivateOld{i}$; they contain $\AuthPublicOld{i}$. 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 \Zcash, which \emph{are} collision-resistant assuming that $\SHA$ is. The proof can be straightforwardly repaired. The intuition is that we can rely on collision resistance of $\PRFaddr{}$ (on both its arguments) to argue that distinctness of $\AuthPrivateOldX{1}$ and $\AuthPrivateOldX{2}$, together with constraint 1(b) of the \joinSplitStatement (see \crossref{spendauthority}), implies distinctness of $\AuthPublicOldX{1}$ and $\AuthPublicOldX{2}$, therefore distinct openings of the \noteCommitment when Condition I or II is violated. \nsubsection{Miscellaneous} \begin{itemize} \item The paper defines a \note as $((\AuthPublic, \TransmitPublic), \Value, \NoteAddressRand, \NoteCommitRand, \NoteCommitS, \cm)$, whereas this specification defines it as $(\AuthPublic, \Value, \NoteAddressRand, \NoteCommitRand)$. The instantiation of $\Commit{\NoteCommitS}$ in section 5.1 of the paper did not actually use $\NoteCommitS$, and neither does the new instantiation of $\Commit{}$ in \Zcash. $\TransmitPublic$ is also not needed as part of a \note: it is not an input to $\Commit{}$ nor is it constrained by the \Zerocash \POUR{} \statement or the \Zcash \joinSplitStatement. $\cm$ can be computed from the other fields. \item The length of proof encodings given in the paper is 288 bytes. This differs from the 296 bytes specified in \crossref{proofencoding}, because the paper did not take into account the need to encode compressed $y$-coordinates. The fork of \libsnark used by \Zcash uses a different format to upstream \libsnark, in order to follow \cite{IEEE2004}. \item The range of monetary values differs. In \Zcash, this range is $\range{0}{\MAXMONEY}$; in \Zerocash it is $\range{0}{2^{64}-1}$. (The \joinSplitStatement still only directly enforces that the sum of amounts in a given \joinSplitTransfer is in the latter range; this enforcement is technically redundant given that the Balance property holds.) \end{itemize} \nsection{Acknowledgements} The inventors of \Zerocash are Eli Ben-Sasson, Alessandro Chiesa, Christina Garman, Matthew Green, Ian Miers, Eran Tromer, and Madars Virza. The authors would like to thank everyone with whom they have discussed the \Zerocash protocol design; in addition to the inventors, this includes Mike Perry, Isis Lovecruft, Leif Ryge, Andrew Miller, Zooko Wilcox, Samantha Hulsey, Jack Grigg, Simon Liu, Ariel Gabizon, jl777, Ben Blaxill, Alex Balducci, Jake Tarren, Solar Designer, Ling Ren, Alison Stevenson, John Tromp, Paige Peterson, Maureen Walsh, Jay Graber, Jack Gavigan, and no doubt others. \Zcash has benefited from security audits performed by NCC Group and Coinspect. The Faerie Gold attack was found by Zooko Wilcox. The internal hash collision attack was found by Taylor Hornby. The error in the \Zerocash proof of Balance relating to collision-resistance of $\PRFaddr{}$ was found by Daira Hopwood. The errors in the proof of Ledger Indistinguishability mentioned in \crossref{truncation} were also found by Daira Hopwood. \nsection{Change history} \subparagraph{2016.0-beta-1.8} \begin{itemize} \item Specify the lead bytes for \transparent P2SH and P2PKH addresses. \item Add a section on which BIPs apply to \Zcash. \item Change the representation type of $\vpubOldField$ and $\vpubNewField$ to \type{uint64\_t}. (This is not a consensus change because the type of $\vpubOld$ and $\vpubNew$ was already specified to be $\range{0}{\MAXMONEY}$; it just better reflects the implementation.) \item Correct the representation type of the \block $\nVersion$ field to \type{uint32\_t}. \end{itemize} \subparagraph{2016.0-beta-1.7} \begin{itemize} \item Clarify the consensus rule for payment of the \foundersReward, in response to an issue raised by the NCC audit. \end{itemize} \subparagraph{2016.0-beta-1.6} \begin{itemize} \item Fix an error in the definition of the sortedness condition for Equihash: it is the sequences of indices that are sorted, not the sequences of hashes. \item Correct the number of bytes in the encoding of $\solutionSize$. \item Update the section on encoding of \transparent addresses. (The precise prefixes are not decided yet.) \item Clarify why $\Blake{\ell}$ is different from truncated $\Blake{512}$. \item Clarify a note about SU-CMA security for signatures. \item Add a note about $\PRFnf{}$ corresponding to $\PRFsn{}$ in \Zerocash. \item Add a paragraph about key length in \crossref{inbandrationale}. \item Add acknowledgements for John Tromp, Paige Peterson, Maureen Walsh, Jay Graber, and Jack Gavigan. \end{itemize} \subparagraph{2016.0-beta-1.5} \begin{itemize} \item Update the \foundersReward address list. \item Add some clarifications based on Eli Ben-Sasson's review. \end{itemize} \subparagraph{2016.0-beta-1.4} \begin{itemize} \item Specify the \blockSubsidy, \minerSubsidy, and the \foundersReward. \item Specify \coinbaseTransaction outputs to \foundersReward addresses. \item Improve notation (for example ``$\mult$'' for multiplication and ``$\typeexp{T}{\ell}$'' for sequence types) to avoid ambiguity. \end{itemize} \subparagraph{2016.0-beta-1.3} \begin{itemize} \item Correct the omission of $\solutionSize$ from the \blockHeader format. \item Document that \compactSize{} encodings must be canonical. \item Add a note about conformance language in the introduction. \item Add acknowledgements for Solar Designer, Ling Ren and Alison Stevenson, and for the NCC Group and Coinspect security audits. \end{itemize} \subparagraph{2016.0-beta-1.2} \begin{itemize} \item Remove $\mathsf{GeneralCRH}$ in favour of specifying $\hSigCRH$ and $\EquihashGen{}$ directly in terms of $\BlakeGeneric$. \item Correct the security requirement for $\EquihashGen{}$. \end{itemize} \subparagraph{2016.0-beta-1.1} \begin{itemize} \item Add a specification of abstract signatures. \item Clarify what is signed in the ``Sending Notes'' section. \item Specify ZK parameter generation as a randomized algorithm, rather than as a distribution of parameters. \end{itemize} \subparagraph{2016.0-beta-1} \begin{itemize} \item Major reorganisation to separate the abstract cryptographic protocol from the algorithm instantiations. \item Add type declarations. \item Add a ``High-level Overview'' section. \item Add a section specifying the \zeroKnowledgeProvingSystem and the encoding of proofs. Change the encoding of points in proofs to follow IEEE Std 1363[a]. \item Add a section on consensus changes from \Bitcoin, and the specification of Equihash. \item Complete the ``Differences from the \Zerocash paper'' section. \item Correct the Merkle tree depth to 29. \item Change the length of \memos to 512 bytes. \item Switch the \joinSplitSignature scheme to Ed25519, with consequent changes to the computation of $\hSig$. \item Fix the lead bytes in \paymentAddress and \spendingKey encodings to match the implemented protocol. \item Add a consensus rule about the ranges of $\vpubOld$ and $\vpubNew$. \item Clarify cryptographic security requirements and added definitions relating to the in-band secret distribution. \item Add various citations: the ``Fixing Vulnerabilities in the Zcash Protocol'' and ``Why Equihash?'' blog posts, several crypto papers for security definitions, the \Bitcoin whitepaper, the CryptoNote whitepaper, and several references to \Bitcoin documentation. \item Reference the extended version of the \Zerocash paper rather than the Oakland proceedings version. \item Add \joinSplitTransfers to the Concepts section. \item Add a section on Coinbase Transactions. \item Add acknowledgements for Jack Grigg, Simon Liu, Ariel Gabizon, jl777, Ben Blaxill, Alex Balducci, and Jake Tarren. \item Fix a \texttt{Makefile} compatibility problem with the escaping behaviour of \texttt{echo}. \item Switch to \texttt{biber} for the bibliography generation, and add backreferences. \item Make the date format in references more consistent. \item Add visited dates to all URLs in references. \item Terminology changes. \end{itemize} \subparagraph{2016.0-alpha-3.1} \begin{itemize} \item Change main font to Quattrocento. \end{itemize} \subparagraph{2016.0-alpha-3} \begin{itemize} \item Change version numbering convention (no other changes). \end{itemize} \subparagraph{2.0-alpha-3} \begin{itemize} \item Allow anchoring to any previous output \treestate in the same \transaction, rather than just the immediately preceding output \treestate. \item Add change history. \end{itemize} \subparagraph{2.0-alpha-2} \begin{itemize} \item Change from truncated $\Blake{512}$ to $\Blake{256}$. \item Clarify endianness, and that uses of $\BlakeGeneric$ 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 Add specification of path computation for the \incrementalMerkleTree. \item Add a note in \crossref{merklepathvalidity} about how this condition corresponds to conditions in the \Zerocash paper. \item Changes to terminology around keys. \end{itemize} \subparagraph{2.0-alpha-1} \begin{itemize} \item First version intended for public review. \end{itemize} \nsection{References} \begingroup \hfuzz=2pt \renewcommand{\section}[2]{} \renewcommand{\emph}[1]{\textit{#1}} \printbibliography \endgroup \end{document}