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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" class="active"><strong aria-hidden="true">3.3.</strong> Commitments</a></li><li class="chapter-item expanded "><a href="../design/commitment-tree.html"><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="commitments"><a class="header" href="#commitments">Commitments</a></h1>
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<p>As in Sapling, we require two kinds of commitment schemes in Orchard:</p>
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<ul>
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<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathit">HomomorphicCommit</span></span></span></span></span> is a linearly homomorphic commitment scheme with perfect hiding,
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and strong binding reducible to DL.</li>
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<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathit">Commit</span></span></span></span></span> and <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 mathit">ShortCommit</span></span></span></span></span> are commitment schemes with perfect hiding, and
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strong binding reducible to DL.</li>
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</ul>
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<p>By "strong binding" we mean that the scheme is collision resistant on the input and
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randomness.</p>
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<p>We instantiate <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathit">HomomorphicCommit</span></span></span></span></span> with a Pedersen commitment, and use it for
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value commitments:</p>
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<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">cv</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0084em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathit">HomomorphicCommit</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7584em;"><span style="top:-2.453em;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 mtight"><span class="mord mathsf mtight" style="margin-right:0.01389em;">rcv</span></span></span></span></span><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 mtight"><span class="mord mathsf mtight" style="margin-right:0.01389em;">cv</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span></span></p>
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<p>We instantiate <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathit">Commit</span></span></span></span></span> and <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 mathit">ShortCommit</span></span></span></span></span> with Sinsemilla, and use them
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for all other commitments:</p>
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<p><span class="katex-display"><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">ivk</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1834em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathit">ShortCommit</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9334em;"><span style="top:-2.453em;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 mtight"><span class="mord mathsf mtight">rivk</span></span></span></span></span><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 mtight"><span class="mord mathsf mtight">ivk</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathsf">ak</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathsf">nk</span></span><span class="mclose">)</span></span></span></span></span>
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<span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord mathsf">cm</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathit">Commit</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7473em;"><span style="top:-2.453em;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 mtight"><span class="mord mathsf mtight">rcm</span></span></span></span></span><span style="top:-3.1362em;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 mtight"><span class="mord mathsf mtight">cm</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord text"><span class="mord">rest of note</span></span><span class="mclose">)</span></span></span></span></span></p>
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<p>This is the same split (and rationale) as in Sapling, but using the more PLONK-efficient
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Sinsemilla instead of Bowe--Hopwood Pedersen hashes.</p>
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<p>Note that for <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">ivk</span></span></span></span></span>, we also deviate from Sapling in two ways:</p>
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<ul>
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<li>We use <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 mathit">ShortCommit</span></span></span></span></span> to derive <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">ivk</span></span></span></span></span> instead of a full PRF. This removes an
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unnecessary (large) PRF primitive from the circuit, at the cost of requiring <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">rivk</span></span></span></span></span> to be
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part of the full viewing key.</li>
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<li>We define <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">ivk</span></span></span></span></span> as an integer in <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">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0359em;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:0.13889em;">P</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>; that is, we exclude <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">ivk</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>. For
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Sapling, we relied on BLAKE2s to make <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">ivk</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> infeasible to produce, but it was still
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technically possible. For Orchard, we get this by construction:
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<ul>
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<li><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> is not a valid x-coordinate for any Pallas point.</li>
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<li><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">SinsemillaShortCommit</span></span></span></span></span> internally maps points to field elements by replacing the identity (which
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has no affine coordinates) with <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 <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> is defined using incomplete addition, and
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thus will never produce the identity.</li>
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