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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">Orchard</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/preliminaries.html"><strong aria-hidden="true">1.1.</strong> Preliminaries</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/keys.html"><strong aria-hidden="true">2.1.</strong> Creating keys and addresses</a></li><li class="chapter-item expanded "><a href="../user/creating-notes.html"><strong aria-hidden="true">2.2.</strong> Creating notes</a></li><li class="chapter-item expanded "><a href="../user/spending-notes.html"><strong aria-hidden="true">2.3.</strong> Spending notes</a></li><li class="chapter-item expanded "><a href="../user/integration.html"><strong aria-hidden="true">2.4.</strong> Integration into an existing chain</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/keys.html"><strong aria-hidden="true">3.1.</strong> Keys and addresses</a></li><li class="chapter-item expanded "><a href="../design/actions.html"><strong aria-hidden="true">3.2.</strong> Actions</a></li><li class="chapter-item expanded "><a href="../design/commitments.html"><strong aria-hidden="true">3.3.</strong> Commitments</a></li><li class="chapter-item expanded "><a href="../design/commitment-tree.html" class="active"><strong aria-hidden="true">3.4.</strong> Commitment tree</a></li><li class="chapter-item expanded "><a href="../design/nullifiers.html"><strong aria-hidden="true">3.5.</strong> Nullifiers</a></li><li class="chapter-item expanded "><a href="../design/signatures.html"><strong aria-hidden="true">3.6.</strong> Signatures</a></li><li class="chapter-item expanded "><a href="../design/circuit.html"><strong aria-hidden="true">3.7.</strong> Circuit</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/circuit/gadgets.html"><strong aria-hidden="true">3.7.1.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../design/circuit/commit-ivk.html"><strong aria-hidden="true">3.7.2.</strong> CommitIvk</a></li><li class="chapter-item expanded "><a href="../design/circuit/note-commit.html"><strong aria-hidden="true">3.7.3.</strong> NoteCommit</a></li></ol></li></ol></li></ol>
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<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
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<h1 id="commitment-tree"><a class="header" href="#commitment-tree">Commitment tree</a></h1>
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<p>The commitment tree structure for Orchard is identical to Sapling:</p>
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<ul>
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<li>A single global commitment tree of fixed depth 32.</li>
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<li>Note commitments are appended to the tree in-order from the block.</li>
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<li>Valid Orchard anchors correspond to the global tree state at block boundaries (after all
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commitments from a block have been appended, and before any commitments from the next
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block have been appended).</li>
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</ul>
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<p>The only difference is that we instantiate <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9334em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">MerkleCRH</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9334em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathsf mtight">Orchard</span></span></span></span></span></span></span></span></span></span></span></span> with
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Sinsemilla (whereas <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9334em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">MerkleCRH</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9334em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathsf mtight" style="margin-right:0.01389em;">Sapling</span></span></span></span></span></span></span></span></span></span></span></span> used a Bowe--Hopwood Pedersen
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hash).</p>
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<h2 id="uncommitted-leaves"><a class="header" href="#uncommitted-leaves">Uncommitted leaves</a></h2>
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<p>The fixed-depth incremental Merkle trees that we use (in Sprout and Sapling, and again in
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Orchard) require specifying an "empty" or "uncommitted" leaf - a value that will never be
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appended to the tree as a regular leaf.</p>
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<ul>
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<li>For Sprout (and trees composed of the outputs of bit-twiddling hash functions), we use
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the all-zeroes array; the probability of a real note having a colliding note commitment
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is cryptographically negligible.</li>
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<li>For Sapling, where leaves are <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>-coordinates of Jubjub points, we use the value <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>
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which is not the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>-coordinate of any Jubjub point.</li>
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</ul>
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<p>Orchard note commitments are the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>-coordinates of Pallas points; thus we take the same
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approach as Sapling, using a value that is not the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>-coordinate of any Pallas point as the
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uncommitted leaf value. We use the value <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> for both Pallas and Vesta, because <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span></span></span></span> is
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not a square in either <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">q</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>:</p>
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<pre><code class="language-python">sage: p = 0x40000000000000000000000000000000224698fc094cf91b992d30ed00000001
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sage: q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001
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sage: EllipticCurve(GF(p), [0, 5]).count_points() == q
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True
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sage: EllipticCurve(GF(q), [0, 5]).count_points() == p
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True
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sage: Mod(13, p).is_square()
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False
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sage: Mod(13, q).is_square()
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False
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</code></pre>
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<blockquote>
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<p>Note: There are also no Pallas points with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>-coordinate <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>, but we map the identity to
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<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span> within the circuit. Although <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">SinsemillaCommit</span></span></span></span></span> cannot return the identity
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(the incomplete addition would return <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mrel">⊥</span></span></span></span> instead), it would arguably be confusing to
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rely on that.</p>
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</blockquote>
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<h2 id="considered-alternatives"><a class="header" href="#considered-alternatives">Considered alternatives</a></h2>
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<p>We considered splitting the commitment tree into several sub-trees:</p>
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<ul>
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<li>Bundle tree, that accumulates the commitments within a single bundle (and thus a single
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transaction).</li>
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<li>Block tree, that accumulates the bundle tree roots within a single block.</li>
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<li>Global tree, that accumulates the block tree roots.</li>
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</ul>
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<p>Each of these trees would have had a fixed depth (necessary for being able to create
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proofs). Chains that integrated Orchard could have decoupled the limits on
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commitments-per-subtree from higher-layer constraints like block size, by enabling their
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blocks and transactions to be structured internally as a series of Orchard blocks or txs
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(e.g. a Zcash block would have contained a <code>Vec<BlockTreeRoot></code>, that each were appended
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in-order).</p>
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<p>The motivation for considering this change was to improve the lives of light client wallet
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developers. When a new note is received, the wallet derives its incremental witness from
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the state of the global tree at the point when the note's commitment is appended; this
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incremental state then needs to be updated with every subsequent commitment in the block
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in-order. Wallets can't get help from the server to create these for new notes without
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leaking the specific note that was received.</p>
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<p>We decided that this was too large a change from Sapling, and that it should be possible
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to improve the Incremental Merkle Tree implementation to work around the efficiency issues
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without domain-separating the tree.</p>
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