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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"><strong aria-hidden="true">3.4.</strong> Commitment tree</a></li><li class="chapter-item expanded "><a href="../design/nullifiers.html" class="active"><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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<h1 id="nullifiers"><a class="header" href="#nullifiers">Nullifiers</a></h1>
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<p>The nullifier design we use for Orchard is</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" style="margin-right:0.06944em;">nf</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:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">Extract</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3322em;"><span style="top:-2.55em;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 mathbb mtight">P</span></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="mord"><span class="delimsizing size1">(</span></span><span class="mopen">[(</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:0.3361em;"><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 mtight"><span class="mord mtight"><span class="mord mathsf mtight">nk</span></span></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="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mclose">)</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:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.0556em;"></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.0556em;"></span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span></span><span class="mclose">]</span><span class="mord mathcal" style="margin-right:0.0593em;">G</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:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">cm</span></span><span class="mord"><span class="delimsizing size1">)</span></span><span class="mpunct">,</span></span></span></span></span></p>
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<p>where:</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:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> is a keyed circuit-efficient PRF (such as Rescue or Poseidon).</li>
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<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span> is unique to this output. As with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathsf">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;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;">Sig</span></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> in Sprout, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span> includes
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the nullifiers of any Orchard notes being spent in the same action. Given that an action
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consists of a single spend and a single output, we set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span> to be the nullifier of the
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spent note.</li>
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<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span></span></span></span> is sender-controlled randomness. It is not required to be unique, and in practice
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is derived from both <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span> and a sender-selected random value <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">rseed</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:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</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:1.1723em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathit">KDF</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9223em;"><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 mathnormal mtight" style="margin-right:0.03588em;">ψ</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathsf">rseed</span></span><span class="mclose">)</span><span class="mord">.</span></span></span></span></span></li>
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<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7805em;vertical-align:-0.0972em;"></span><span class="mord mathcal" style="margin-right:0.0593em;">G</span></span></span></span> is a fixed independent base.</li>
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<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">Extract</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3322em;"><span style="top:-2.55em;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 mathbb mtight">P</span></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></span></span> extracts 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 a Pallas curve point.</li>
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</ul>
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<p>This gives a note structure of</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:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">dd</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathsf">rcm</span></span><span class="mclose">)</span><span class="mord">.</span></span></span></span></span></p>
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<p>The note plaintext includes <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">rseed</span></span></span></span></span> in place of <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 mathnormal" style="margin-right:0.03588em;">ψ</span></span></span></span> and <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">rcm</span></span></span></span></span>, and
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omits <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span> (which is a public part of the action).</p>
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<h2 id="security-properties"><a class="header" href="#security-properties">Security properties</a></h2>
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<p>We care about several security properties for our nullifiers:</p>
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<ul>
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<li>
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<p><strong>Balance:</strong> can I forge money?</p>
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</li>
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<li>
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<p><strong>Note Privacy:</strong> can I gain information about notes only from the public block chain?</p>
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<ul>
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<li>This describes notes sent in-band.</li>
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</ul>
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</li>
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<li>
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<p><strong>Note Privacy (OOB):</strong> can I gain information about notes sent out-of-band, only from
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the public block chain?</p>
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<ul>
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<li>In this case, we assume privacy of the channel over which the note is sent, and that
|
||
the adversary does not have access to any notes sent to the same address which are
|
||
then spent (so that the nullifier is on the block chain somewhere).</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p><strong>Spend Unlinkability:</strong> given the incoming viewing key for an address, and not the full
|
||
viewing key, can I (possibly the sender) detect spends of any notes sent to that address?</p>
|
||
<ul>
|
||
<li>We're giving <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> to the attacker and allowing it to be the sender in order
|
||
to make this property as strong as possible: they will have <em>all</em> the notes sent to that
|
||
address.</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p><strong>Faerie Resistance:</strong> can I perform a Faerie Gold attack (i.e. cause notes to be
|
||
accepted that are unspendable)?</p>
|
||
<ul>
|
||
<li>We're giving the full viewing key to the attacker and allowing it to be the sender in
|
||
order to make this property as strong as possible: they will have <em>all</em> the notes sent
|
||
to that address, and be able to derive <em>every</em> nullifier.</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
<p>We assume (and instantiate elsewhere) the following primitives:</p>
|
||
<ul>
|
||
<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">GH</span></span></span></span></span> is a cryptographic hash into the group (such as BLAKE2s with simplified SWU),
|
||
used to derive all fixed independent bases.</li>
|
||
<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 mathnormal" style="margin-right:0.05764em;">E</span></span></span></span> is an elliptic curve (such as Pallas).</li>
|
||
<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">KDF</span></span></span></span></span> is the note encryption key derivation function.</li>
|
||
</ul>
|
||
<p>For our chosen design, our desired security properties rely on the following assumptions:</p>
|
||
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.2527em;vertical-align:-2.8563em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.3963em;"><span style="top:-5.3563em;"><span class="pstrut" style="height:5.3563em;"></span><span class="mtable"><span class="vertical-separator" style="height:6.2127em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-2.8563em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.3563em;"><span style="top:-5.5163em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Balance</span></span></span></span><span style="top:-4.2307em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Note Privacy</span></span></span></span><span style="top:-3.0307em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Note Privacy (OOB)</span></span></span></span><span style="top:-1.7037em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Spend Unlinkability</span></span></span></span><span style="top:-0.5037em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Faerie Resistance</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.8563em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:6.2127em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-2.8563em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.3563em;"><span style="top:-5.5163em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></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-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:0.05764em;">E</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></span><span style="top:-4.2307em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">HashDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9257em;"><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 mathnormal mtight" style="margin-right:0.05764em;">E</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 mathit mtight">KDF</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></span><span style="top:-3.0307em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Near perfect</span></span><span class="mord">‡</span></span></span><span style="top:-1.7037em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.967em;"><span style="top:-2.4065em;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:0.05764em;">E</span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">†</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2935em;"><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 class="mord"><span class="mord"><span class="mord mathit">PRF</span></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-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;">F</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></span><span style="top:-0.5037em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></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-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:0.05764em;">E</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></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.8563em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:6.2127em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-2.8563em;"></span></span></span><span style="top:-2.5em;"><span class="pstrut" style="height:5.3563em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-8.7127em;"><span class="pstrut" style="height:5.3563em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.8563em;"><span></span></span></span></span></span></span></span></span></span></p>
|
||
<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1727em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord"><span class="mord mathit">HashDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9257em;"><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 mathnormal mtight" style="margin-right:0.05764em;">E</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 mathit mtight">KDF</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></span></span> is computational Diffie-Hellman using <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">KDF</span></span></span></span></span> for
|
||
the key derivation, with one-time ephemeral keys. This assumption is heuristically weaker
|
||
than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></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-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:0.05764em;">E</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></span></span> but stronger than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DL</span></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-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:0.05764em;">E</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></span></span>.</p>
|
||
<p>We omit <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">O</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:-0.0278em;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 mathit mtight">GH</span></span></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></span></span> as a security assumption because we only rely on the random oracle
|
||
applied to fixed inputs defined by the protocol, i.e. to generate the fixed base
|
||
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7805em;vertical-align:-0.0972em;"></span><span class="mord mathcal" style="margin-right:0.0593em;">G</span></span></span></span>, not to attacker-specified inputs.</p>
|
||
<blockquote>
|
||
<p><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></span></span></span> We additionally assume that for any input <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>,
|
||
<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"><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:0.3361em;"><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 mtight"><span class="mord mtight"><span class="mord mathsf mtight">nk</span></span></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="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</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:0.7335em;vertical-align:-0.0391em;"></span><span class="mord"><span class="mord mathsf">nk</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 mathnormal" style="margin-right:0.05764em;">E</span><span class="mclose">}</span></span></span></span> gives a scalar in an adequate range for
|
||
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></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-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:0.05764em;">E</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></span></span>. (Otherwise, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> could be trivial, e.g. independent of <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">nk</span></span></span></span></span>.)</p>
|
||
<p><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></span></span></span> Statistical distance <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></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"><span class="mord mtight">−</span><span class="mord mtight">167.8</span></span></span></span></span></span></span></span></span></span></span></span> from perfect.</p>
|
||
</blockquote>
|
||
<h2 id="considered-alternatives"><a class="header" href="#considered-alternatives">Considered alternatives</a></h2>
|
||
<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord" style="color:red;"><span class="mord text" style="color:red;"><span class="mord" style="color:red;">⚠</span><span class="mord textsf" style="color:red;"> Caution</span></span></span></span></span></span>: be skeptical of the claims in this table about what
|
||
problem(s) each security property depends on. They may not be accurate and are definitely
|
||
not fully rigorous.</p>
|
||
<p>The entries in this table omit the application of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">Extract</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3322em;"><span style="top:-2.55em;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 mathbb mtight">P</span></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></span></span>,
|
||
which is an optimization to halve the nullifier length. That optimization requires its
|
||
own security analysis, but because it is a deterministic mapping, only Faerie Resistance
|
||
could be affected by it.</p>
|
||
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:19.1828em;vertical-align:-9.4564em;"></span><span class="mord sizing reset-size6 size1"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:19.4528em;"><span style="top:-21.4128em;"><span class="pstrut" style="height:21.4128em;"></span><span class="mtable"><span class="vertical-separator" style="height:38.3255em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-18.9128em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:19.4128em;"><span style="top:-21.8064em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"></span></span><span style="top:-21.2529em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">nf</span></span></span></span><span style="top:-19.7943em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"></span></span><span style="top:-19.6287em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"></span></span><span style="top:-18.9141em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathsf">nk</span></span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span><span style="top:-17.3262em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"></span></span><span style="top:-16.6116em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathsf">nk</span></span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">rnf</span></span><span class="mclose">]</span><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span></span><span style="top:-15.0237em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"></span></span><span style="top:-14.309em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord mathit">Hash</span></span><span class="mopen">([</span><span class="mord"><span class="mord mathsf">nk</span></span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span><span style="top:-12.7211em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"></span></span><span style="top:-12.0065em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord mathit">Hash</span></span><span class="mopen">([</span><span class="mord"><span class="mord mathsf">nk</span></span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">rnf</span></span><span class="mclose">]</span><span class="mord mathcal" style="margin-right:0.07382em;">I</span><span class="mclose">)</span></span></span><span style="top:-10.4186em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"></span></span><span style="top:-9.5576em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">[</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:0.3448em;"><span style="top:-2.3448em;margin-left:-0.1389em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">nk</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3496em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mclose">)]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span><span style="top:-7.9697em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"></span></span><span style="top:-7.1088em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">[</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:0.3448em;"><span style="top:-2.3448em;margin-left:-0.1389em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">nk</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3496em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mclose">)]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">rnf</span></span><span class="mclose">]</span><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span></span><span style="top:-5.5209em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"></span></span><span style="top:-4.6599em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">[</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:0.3448em;"><span style="top:-2.3448em;margin-left:-0.1389em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">nk</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3496em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mclose">)]</span><span class="mord mathcal" style="margin-right:0.0593em;">G</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span><span style="top:-3.072em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"></span></span><span style="top:-2.211em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">[</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:0.3448em;"><span style="top:-2.3448em;margin-left:-0.1389em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">nk</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3496em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mclose">)]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord mathsf">cm</span></span></span></span><span style="top:-0.6231em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"></span></span><span style="top:0.2379em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">[</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:0.3448em;"><span style="top:-2.3448em;margin-left:-0.1389em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">nk</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3496em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mclose">)]</span><span class="mord mathcal" style="margin-right:0.0593em;">G</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord mathsf">cm</span></span></span></span><span style="top:1.8258em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"></span></span><span style="top:2.6868em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">[</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:0.3448em;"><span style="top:-2.3448em;margin-left:-0.1389em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">nk</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3496em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mclose">)]</span><span class="mord mathcal" style="margin-right:0.0593em;">G</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord mathsf">cm</span></span></span></span><span style="top:4.2747em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"></span></span><span style="top:5.1357em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">[</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:0.3448em;"><span style="top:-2.3448em;margin-left:-0.1389em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">nk</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3496em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mclose">)]</span><span class="mord"><span class="mord mathcal" style="margin-right:0.0593em;">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2306em;"><span style="top:-2.3em;margin-left:-0.0593em;margin-right:0.1em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">rnf</span></span><span class="mclose">]</span><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span></span><span style="top:6.7236em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"></span></span><span style="top:7.5845em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">[</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:0.3448em;"><span style="top:-2.3448em;margin-left:-0.1389em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">nk</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3496em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mclose">)]</span><span class="mord"><span class="mord mathcal" style="margin-right:0.0593em;">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2306em;"><span style="top:-2.3em;margin-left:-0.0593em;margin-right:0.1em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">rnf</span></span><span class="mclose">]</span><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span></span><span style="top:9.1724em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"></span></span><span style="top:10.0334em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">[(</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:0.3448em;"><span style="top:-2.3448em;margin-left:-0.1389em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">nk</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3496em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.0818em;"></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.0818em;"></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord mathnormal">p</span></span><span class="mclose">]</span><span class="mord"><span class="mord mathcal" style="margin-right:0.0593em;">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2306em;"><span style="top:-2.3em;margin-left:-0.0593em;margin-right:0.1em;"><span class="pstrut" style="height:2.5em;"></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span></span></span></span><span style="top:11.6213em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"></span></span><span style="top:12.4823em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.2453em;"></span><span class="mopen">[</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:0.3448em;"><span style="top:-2.3448em;margin-left:-0.1389em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">nk</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3496em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mclose">)]</span><span class="mord mathcal" style="margin-right:0.0593em;">G</span><span class="mspace" style="margin-right:-0.2453em;"></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:-0.2453em;"></span><span class="mspace" style="margin-right:0.3271em;"></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:1.0984em;"><span style="top:-2.208em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight" style="margin-right:0.06944em;">rnf</span></span></span></span><span style="top:-3.0984em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight" style="margin-right:0.06944em;">nf</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4864em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal">ρ</span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.2453em;"></span></span></span><span style="top:14.0702em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"></span></span><span style="top:14.9312em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">[</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:0.3448em;"><span style="top:-2.3448em;margin-left:-0.1389em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight">nk</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3496em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mclose">)]</span><span class="mord mathcal" style="margin-right:0.0593em;">G</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.3271em;"></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:1.0984em;"><span style="top:-2.208em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight" style="margin-right:0.06944em;">rnf</span></span></span></span><span style="top:-3.0984em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathsf mtight" style="margin-right:0.06944em;">nf</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4864em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal">ρ</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:18.9128em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:38.3255em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-18.9128em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:18.8593em;"><span style="top:-21.2529em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Note</span></span></span></span><span style="top:-18.9141em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">dd</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf">rcm</span></span><span class="mclose">)</span></span></span><span style="top:-16.6116em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">dd</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">rnf</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf">rcm</span></span><span class="mclose">)</span></span></span><span style="top:-14.309em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">dd</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf">rcm</span></span><span class="mclose">)</span></span></span><span style="top:-12.0065em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">dd</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">rnf</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf">rcm</span></span><span class="mclose">)</span></span></span><span style="top:-9.5576em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">dd</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf">rcm</span></span><span class="mclose">)</span></span></span><span style="top:-7.1088em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">dd</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">rnf</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf">rcm</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:-0.2453em;"></span></span></span><span style="top:-4.6599em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">dd</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf">rcm</span></span><span class="mclose">)</span></span></span><span style="top:-2.211em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">dd</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf">rcm</span></span><span class="mclose">)</span></span></span><span style="top:0.2379em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">dd</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf">rcm</span></span><span class="mclose">)</span></span></span><span style="top:2.6868em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">dd</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf">rcm</span></span><span class="mclose">)</span></span></span><span style="top:5.1357em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">dd</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">rnf</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf">rcm</span></span><span class="mclose">)</span></span></span><span style="top:7.5845em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">dd</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">rnf</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf">rcm</span></span><span class="mclose">)</span></span></span><span style="top:10.0334em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">dd</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf">rcm</span></span><span class="mclose">)</span></span></span><span style="top:12.4823em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">dd</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">rnf</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf">rcm</span></span><span class="mclose">)</span></span></span><span style="top:14.9312em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">dd</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.06944em;">rnf</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2453em;"></span><span class="mord"><span class="mord mathsf">rcm</span></span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:18.9128em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:38.3255em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-18.9128em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:18.8593em;"><span style="top:-21.2529em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.2453em;"></span><span class="mord text"><span class="mord">Balance</span></span><span class="mspace" style="margin-right:-0.2453em;"></span></span></span><span style="top:-18.9141em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:-16.6116em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:-14.309em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:-12.0065em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:-9.5576em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:-7.1088em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6599em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.211em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:0.2379em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:2.6868em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:5.1357em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:7.5845em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:10.0334em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:12.4823em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:14.9312em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:18.9128em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:38.3255em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-18.9128em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:18.8593em;"><span style="top:-21.2529em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Note Privacy</span></span></span></span><span style="top:-18.9141em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">HashDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0873em;"><span style="top:-2.208em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.0873em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">KDF</span></span><span class="mspace mtight" style="margin-right:-0.2453em;"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4753em;"><span></span></span></span></span></span></span></span></span><span style="top:-16.6116em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">HashDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0873em;"><span style="top:-2.208em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.0873em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">KDF</span></span><span class="mspace mtight" style="margin-right:-0.2453em;"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4753em;"><span></span></span></span></span></span></span></span></span><span style="top:-14.309em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">HashDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0873em;"><span style="top:-2.208em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.0873em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">KDF</span></span><span class="mspace mtight" style="margin-right:-0.2453em;"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4753em;"><span></span></span></span></span></span></span></span></span><span style="top:-12.0065em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">HashDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0873em;"><span style="top:-2.208em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.0873em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">KDF</span></span><span class="mspace mtight" style="margin-right:-0.2453em;"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4753em;"><span></span></span></span></span></span></span></span></span><span style="top:-9.5576em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">HashDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0873em;"><span style="top:-2.208em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.0873em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">KDF</span></span><span class="mspace mtight" style="margin-right:-0.2453em;"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4753em;"><span></span></span></span></span></span></span></span></span><span style="top:-7.1088em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">HashDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0873em;"><span style="top:-2.208em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.0873em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">KDF</span></span><span class="mspace mtight" style="margin-right:-0.2453em;"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4753em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6599em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">HashDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0873em;"><span style="top:-2.208em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.0873em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">KDF</span></span><span class="mspace mtight" style="margin-right:-0.2453em;"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4753em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.211em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">HashDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0873em;"><span style="top:-2.208em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.0873em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">KDF</span></span><span class="mspace mtight" style="margin-right:-0.2453em;"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4753em;"><span></span></span></span></span></span></span></span></span><span style="top:0.2379em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">HashDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0873em;"><span style="top:-2.208em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.0873em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">KDF</span></span><span class="mspace mtight" style="margin-right:-0.2453em;"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4753em;"><span></span></span></span></span></span></span></span></span><span style="top:2.6868em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">HashDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0873em;"><span style="top:-2.208em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.0873em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">KDF</span></span><span class="mspace mtight" style="margin-right:-0.2453em;"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4753em;"><span></span></span></span></span></span></span></span></span><span style="top:5.1357em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">HashDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0873em;"><span style="top:-2.208em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.0873em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">KDF</span></span><span class="mspace mtight" style="margin-right:-0.2453em;"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4753em;"><span></span></span></span></span></span></span></span></span><span style="top:7.5845em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">HashDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0873em;"><span style="top:-2.208em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.0873em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">KDF</span></span><span class="mspace mtight" style="margin-right:-0.2453em;"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4753em;"><span></span></span></span></span></span></span></span></span><span style="top:10.0334em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">HashDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0873em;"><span style="top:-2.208em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.0873em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">KDF</span></span><span class="mspace mtight" style="margin-right:-0.2453em;"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4753em;"><span></span></span></span></span></span></span></span></span><span style="top:12.4823em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">HashDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0873em;"><span style="top:-2.208em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.0873em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">KDF</span></span><span class="mspace mtight" style="margin-right:-0.2453em;"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4753em;"><span></span></span></span></span></span></span></span></span><span style="top:14.9312em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">HashDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0873em;"><span style="top:-2.208em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.0873em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">KDF</span></span><span class="mspace mtight" style="margin-right:-0.2453em;"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4753em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:18.9128em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:38.3255em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-18.9128em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:18.8593em;"><span style="top:-21.2529em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.2453em;"></span><span class="mord text"><span class="mord">Note Priv OOB</span></span><span class="mspace" style="margin-right:-0.2453em;"></span></span></span><span style="top:-18.9141em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:-16.6116em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:-14.309em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:-12.0065em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:-9.5576em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:-7.1088em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:-4.6599em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:-2.211em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2337em;"><span style="top:-2.1599em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.2337em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mbin mtight">†</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5345em;"><span></span></span></span></span></span></span></span></span><span style="top:0.2379em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2337em;"><span style="top:-2.1599em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.2337em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mbin mtight">†</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5345em;"><span></span></span></span></span></span></span></span></span><span style="top:2.6868em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2337em;"><span style="top:-2.1599em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.2337em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mbin mtight">†</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5345em;"><span></span></span></span></span></span></span></span></span><span style="top:5.1357em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:7.5845em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:10.0334em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Near perfect</span></span><span class="mord">‡</span></span></span><span style="top:12.4823em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:14.9312em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:18.9128em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:38.3255em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-18.9128em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:18.8593em;"><span style="top:-21.2529em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.2453em;"></span><span class="mord text"><span class="mord">Spend Unlinkability</span></span><span class="mspace" style="margin-right:-0.2453em;"></span></span></span><span style="top:-18.9141em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:-16.6116em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:-14.309em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">Pre</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">Hash</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3496em;"><span></span></span></span></span></span></span></span></span><span style="top:-12.0065em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">Pre</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">Hash</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3496em;"><span></span></span></span></span></span></span></span></span><span style="top:-9.5576em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2337em;"><span style="top:-2.1599em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.2337em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mbin mtight">†</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5345em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">PRF</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:-7.1088em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2337em;"><span style="top:-2.1599em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.2337em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mbin mtight">†</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5345em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">PRF</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6599em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2337em;"><span style="top:-2.1599em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.2337em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mbin mtight">†</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5345em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">PRF</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.211em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2337em;"><span style="top:-2.1599em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.2337em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mbin mtight">†</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5345em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">PRF</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:0.2379em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2337em;"><span style="top:-2.1599em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.2337em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mbin mtight">†</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5345em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">PRF</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:2.6868em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2337em;"><span style="top:-2.1599em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.2337em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mbin mtight">†</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5345em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">PRF</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:5.1357em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2337em;"><span style="top:-2.1599em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.2337em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mbin mtight">†</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5345em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">PRF</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:7.5845em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2337em;"><span style="top:-2.1599em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.2337em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mbin mtight">†</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5345em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">PRF</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:10.0334em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2337em;"><span style="top:-2.1599em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.2337em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mbin mtight">†</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5345em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">PRF</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:12.4823em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2337em;"><span style="top:-2.1599em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.2337em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mbin mtight">†</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5345em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">PRF</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:14.9312em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2337em;"><span style="top:-2.1599em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span><span style="top:-3.2337em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mbin mtight">†</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5345em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">PRF</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:18.9128em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:38.3255em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-18.9128em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:18.8593em;"><span style="top:-21.2529em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Faerie Resistance</span></span></span></span><span style="top:-18.9141em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-left:-0.0278em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">GH</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:-16.6116em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-left:-0.0278em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">GH</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:-14.309em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.2453em;"></span><span class="mord"><span class="mord"><span class="mord mathit">Coll</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">Hash</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3496em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-left:-0.0278em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">GH</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:-0.2453em;"></span></span></span><span style="top:-12.0065em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.2453em;"></span><span class="mord"><span class="mord"><span class="mord mathit">Coll</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">Hash</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3496em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-left:-0.0278em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">GH</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:-0.2453em;"></span></span></span><span style="top:-9.5576em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-left:-0.0278em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">GH</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:-7.1088em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-left:-0.0278em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">GH</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6599em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-left:-0.0278em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">GH</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.211em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-left:-0.0278em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">GH</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:0.2379em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:2.6868em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:5.1357em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">Coll</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:7.5845em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">Coll</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.3271em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:10.0334em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord" style="color:red;"><span class="mord text" style="color:red;"><span class="mord" style="color:red;">broken</span></span></span></span></span><span style="top:12.4823em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span><span style="top:14.9312em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">DL</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:18.9128em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:38.3255em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-18.9128em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:18.8593em;"><span style="top:-21.2529em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Reason not to use</span></span></span></span><span style="top:-18.9141em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">No SU for DL-breaking</span></span><span class="mspace" style="margin-right:-0.2453em;"></span></span></span><span style="top:-16.6116em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">No SU for DL-breaking</span></span><span class="mspace" style="margin-right:-0.2453em;"></span></span></span><span style="top:-14.309em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">Coll</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">Hash</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3496em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord"> for FR</span></span></span></span><span style="top:-12.0065em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">Coll</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6944em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">Hash</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3496em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord"> for FR</span></span></span></span><span style="top:-9.5576em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Perf. (2 var-base)</span></span></span></span><span style="top:-7.1088em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.2453em;"></span><span class="mord text"><span class="mord">Perf. (1 var+1 fix-base)</span></span><span class="mspace" style="margin-right:-0.2453em;"></span></span></span><span style="top:-4.6599em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mspace" style="margin-right:-0.2453em;"></span><span class="mord text"><span class="mord">Perf. (1 var+1 fix-base)</span></span><span class="mspace" style="margin-right:-0.2453em;"></span></span></span><span style="top:-2.211em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">NP(OOB) not perfect</span></span></span></span><span style="top:0.2379em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">NP(OOB) not perfect</span></span></span></span><span style="top:2.6868em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">NP(OOB) not perfect</span></span></span></span><span style="top:5.1357em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">Coll</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord"> for FR</span></span></span></span><span style="top:7.5845em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathit">Coll</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3448em;margin-right:0.1em;"><span class="pstrut" style="height:2.6833em;"></span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3385em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord"> for FR</span></span></span></span><span style="top:10.0334em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">broken for FR</span></span></span></span><span style="top:12.4823em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Perf. (2 fix-base)</span></span></span></span><span style="top:14.9312em;"><span class="pstrut" style="height:3.2337em;"></span><span class="mord"><span class="mord text"><span class="mord">Perf. (2 fix-base)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:18.9128em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:38.3255em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-18.9128em;"></span></span></span><span style="top:-2.5em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-4.9489em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-7.3978em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-9.8466em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-12.2955em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-14.7444em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-17.1933em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-19.6422em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-22.091em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-24.5399em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-26.9888em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-29.4377em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-31.7402em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-34.0427em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-36.3453em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-38.6478em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-38.8134em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-40.8255em;"><span class="pstrut" style="height:21.4128em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:18.9128em;"><span></span></span></span></span></span></span></span></span></span></p>
|
||
<p>In the above alternatives:</p>
|
||
<ul>
|
||
<li>
|
||
<p><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">Hash</span></span></span></span></span> is a keyed circuit-efficient hash (such as Rescue).</p>
|
||
</li>
|
||
<li>
|
||
<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span></span></span> is an fixed independent base, independent of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7805em;vertical-align:-0.0972em;"></span><span class="mord mathcal" style="margin-right:0.0593em;">G</span></span></span></span> and any others
|
||
returned by <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">GH</span></span></span></span></span>.</p>
|
||
</li>
|
||
<li>
|
||
<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.0593em;">G</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:-0.0593em;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;">v</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></span></span> is a pair of fixed independent bases (independent of all others), where
|
||
the specific choice of base depends on whether the note has zero value.</p>
|
||
</li>
|
||
<li>
|
||
<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is a base unique to this output.</p>
|
||
<ul>
|
||
<li>For non-zero-valued notes, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</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 mathit">GH</span></span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mclose">)</span></span></span></span>. As with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathsf">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;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;">Sig</span></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> in Sprout,
|
||
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span> includes the nullifiers of any Orchard notes being spent in the same action.</li>
|
||
<li>For zero-valued notes, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is constrained by the circuit to a fixed base independent
|
||
of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span></span></span> and any others returned by <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">GH</span></span></span></span></span>.</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
<h2 id="rationale"><a class="header" href="#rationale">Rationale</a></h2>
|
||
<p>In order to satisfy the <strong>Balance</strong> security property, we require that the circuit must be
|
||
able to enforce that only one nullifier is accepted for a given note. As in Sprout and
|
||
Sapling, we achieve this by ensuring that the nullifier deterministically depends only on
|
||
values committed to (directly or indirectly) by the note commitment. As in Sapling,
|
||
this involves arguing that:</p>
|
||
<ul>
|
||
<li>There can be only one <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> for a given <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">addr</span></span></span></span></span>. This is true because
|
||
the circuit checks that <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 mathsf">p</span><span class="mord"><span class="mord mathsf">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathsf mtight">d</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><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="mopen">[</span><span class="mord"><span class="mord mathsf">ivk</span></span><span class="mclose">]</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0139em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathsf mtight">d</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></span></span>, and the mapping
|
||
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7054em;vertical-align:-0.011em;"></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:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathsf">ivk</span></span><span class="mclose">]</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0139em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathsf mtight">d</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></span></span> is an injection for any <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0139em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathsf mtight">d</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></span></span>.
|
||
(<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> is in the base field of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span></span></span></span>, which must be smaller than its scalar field,
|
||
as is the case for Pallas.)</li>
|
||
<li>There can be only one <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">nk</span></span></span></span></span> for a given <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>. This is true because the
|
||
circuit checks that <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>
|
||
where <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> is binding (see <a href="commitments.html">Commitments</a>).</li>
|
||
</ul>
|
||
<h3 id="use-of-ρ"><a class="header" href="#use-of-ρ">Use of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span></a></h3>
|
||
<p><strong>Faerie Resistance</strong> requires that nullifiers be unique. This is primarily achieved by
|
||
taking a unique value (checked for uniqueness by the public consensus rules) as an input
|
||
to the nullifier. However, it is also necessary to ensure that the transformations applied
|
||
to this value preserve its uniqueness. Meanwhile, to achieve <strong>Spend Unlinkability</strong>, we
|
||
require that the nullifier does not reveal any information about the unique value it is
|
||
derived from.</p>
|
||
<p>The design alternatives fall into two categories in terms of how they balance these
|
||
requirements:</p>
|
||
<ul>
|
||
<li>
|
||
<p>Publish a unique value <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span> at note creation time, and blind that value within the
|
||
nullifier computation.</p>
|
||
<ul>
|
||
<li>This is similar to the approach taken in Sprout and Sapling, which both implemented
|
||
nullifiers as PRF outputs; Sprout uses the compression function from SHA-256, while
|
||
Sapling uses BLAKE2s.</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>Derive a unique base <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> from some unique value, publish that unique base at note
|
||
creation time, and then blind the base (either additively or multiplicatively) during
|
||
nullifier computation.</p>
|
||
</li>
|
||
</ul>
|
||
<p>For <strong>Spend Unlinkability</strong>, the only value unknown to the adversary is <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">nk</span></span></span></span></span>, and
|
||
the cryptographic assumptions only involve the first term (other terms like <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></span></span>
|
||
or <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"><span class="mord mathsf" style="margin-right:0.06944em;">rnf</span></span><span class="mclose">]</span><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span></span></span> cannot be extracted directly from the observed nullifiers,
|
||
but can be subtracted from them). We therefore ensure that the first term does not commit
|
||
directly to the note (to avoid a DL-breaking adversary from immediately breaking <strong>SU</strong>).</p>
|
||
<p>We were considering using a design involving <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> with the goal of eliminating all usages
|
||
of a PRF inside the circuit, for two reasons:</p>
|
||
<ul>
|
||
<li>Instantiating <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathit">PRF</span></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-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;">F</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></span></span> with a traditional hash function is expensive in the
|
||
circuit.</li>
|
||
<li>We didn't want to solely rely on an algebraic hash function satisfying <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathit">PRF</span></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-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;">F</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></span></span>
|
||
to achieve <strong>Spend Unlinkability</strong>.</li>
|
||
</ul>
|
||
<p>However, those designs rely on both <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">O</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:-0.0278em;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 mathit mtight">GH</span></span></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></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DL</span></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-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:0.05764em;">E</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></span></span> for
|
||
<strong>Faerie Resistance</strong>, while still requiring <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></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-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:0.05764em;">E</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></span></span> for <strong>Spend Unlinkability</strong>.
|
||
(There are two designs for which this is not the case, but they rely on
|
||
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2605em;vertical-align:-0.2935em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.967em;"><span style="top:-2.4065em;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:0.05764em;">E</span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">†</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2935em;"><span></span></span></span></span></span></span></span></span></span> for <strong>Note Privacy (OOB)</strong> which was not acceptable).</p>
|
||
<p>By contrast, several designs involving <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span> (including the chosen design) have weaker
|
||
assumptions for <strong>Faerie Resistance</strong> (only relying on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DL</span></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-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:0.05764em;">E</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></span></span>), and
|
||
<strong>Spend Unlinkability</strong> does not require <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathit">PRF</span></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-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;">F</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></span></span> to hold: they can fall back
|
||
on the same <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathit">DDH</span></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-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:0.05764em;">E</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></span></span> assumption as the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> designs (along with an additional
|
||
assumption about the output of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> which is easily satisfied).</p>
|
||
<h3 id="use-of-ψ"><a class="header" href="#use-of-ψ">Use of <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 mathnormal" style="margin-right:0.03588em;">ψ</span></span></span></span></a></h3>
|
||
<p>Most of the designs include either a multiplicative blinding term <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 mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, or an
|
||
additive blinding term <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"><span class="mord mathsf" style="margin-right:0.06944em;">rnf</span></span><span class="mclose">]</span><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span></span></span>, in order to achieve perfect
|
||
<strong>Note Privacy (OOB)</strong> (to an adversary who does not know the note). The chosen design is
|
||
effectively using <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 mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mclose">]</span><span class="mord mathcal" style="margin-right:0.0593em;">G</span></span></span></span> for this purpose; a DL-breaking adversary only
|
||
learns <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="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:0.3361em;"><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 mtight"><span class="mord mtight"><span class="mord mathsf mtight">nk</span></span></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="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mclose">)</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:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.4444em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.3333em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span>. This reduces <strong>Note Privacy (OOB)</strong> from
|
||
perfect to statistical, but given that <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 mathnormal" style="margin-right:0.03588em;">ψ</span></span></span></span> is from a distribution statistically close
|
||
to uniform on <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 mathnormal" style="margin-right:0.03588em;">q</span><span class="mclose">)</span></span></span></span>, this is statistically close to better than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></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"><span class="mord mtight">−</span><span class="mord mtight">128</span></span></span></span></span></span></span></span></span></span></span></span>. The benefit
|
||
is that it does not require an additional scalar multiplication, making it more efficient
|
||
inside the circuit.</p>
|
||
<p><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 mathnormal" style="margin-right:0.03588em;">ψ</span></span></span></span>'s derivation has two motivations:</p>
|
||
<ul>
|
||
<li>Deriving from a random value <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">rseed</span></span></span></span></span> enables multiple derived values to be
|
||
conveyed to the recipient within an action (such as the ephemeral secret <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">esk</span></span></span></span></span>,
|
||
per <a href="https://zips.z.cash/zip-0212">ZIP 212</a>), while keeping the note plaintext short.</li>
|
||
<li>Mixing <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span> into the derivation ensures that the sender can't repeat <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 mathnormal" style="margin-right:0.03588em;">ψ</span></span></span></span> across two
|
||
notes, which could have enabled spend linkability attacks in some designs.</li>
|
||
</ul>
|
||
<p>The note that is committed to, and which the circuit takes as input, only includes <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 mathnormal" style="margin-right:0.03588em;">ψ</span></span></span></span>
|
||
(i.e. the circuit does not check the derivation from <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">rseed</span></span></span></span></span>). However, an
|
||
adversarial sender is still constrained by this derivation, because the recipient
|
||
recomputes <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 mathnormal" style="margin-right:0.03588em;">ψ</span></span></span></span> during note decryption. If an action were created using an arbitrary
|
||
<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 mathnormal" style="margin-right:0.03588em;">ψ</span></span></span></span> (for which the adversary did not have a corresponding <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">rseed</span></span></span></span></span>), the
|
||
recipient would derive a note commitment that did not match the action's commitment field,
|
||
and reject it (as in Sapling).</p>
|
||
<h3 id="use-of-cm"><a class="header" href="#use-of-cm">Use of <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></span></span></a></h3>
|
||
<p>The nullifier commits to the note value via <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></span></span> for two reasons:</p>
|
||
<ul>
|
||
<li>It domain-separates nullifiers for zero-valued notes from other notes. This is necessary
|
||
because we do not require zero-valued notes to exist in the commitment tree.</li>
|
||
<li>Designs that bind the nullifier to <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="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:0.3361em;"><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 mtight"><span class="mord mtight"><span class="mord mathsf mtight">nk</span></span></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="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mclose">)</span></span></span></span> require <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathit">Coll</span></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-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;">F</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></span></span> to
|
||
achieve <strong>Faerie Resistance</strong> (and similarly where <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">Hash</span></span></span></span></span> is applied to a value
|
||
derived from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>). Adding <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></span></span> to the nullifier avoids this assumption: all of
|
||
the bases used to derive <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></span></span> are fixed and independent of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7805em;vertical-align:-0.0972em;"></span><span class="mord mathcal" style="margin-right:0.0593em;">G</span></span></span></span>, and so
|
||
the nullifier can be viewed as a Pedersen hash where the input includes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span> directly.</li>
|
||
</ul>
|
||
<p>The <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9223em;"></span><span class="mord"><span class="mord"><span class="mord mathit">Commit</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9223em;"><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" style="margin-right:0.06944em;">nf</span></span></span></span></span></span></span></span></span></span></span></span></span> variants were considered to avoid directly depending on
|
||
<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></span></span> (which in its native type is a base field element, not a group element). We
|
||
decided instead to follow Sapling by defining an intermediate representation of
|
||
<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></span></span> as a group element, that is only used in nullifier computation. The circuit
|
||
already needs to compute <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></span></span>, so this improves performance by removing an
|
||
additional commitment calculation from the circuit.</p>
|
||
<p>We also considered variants that used a choice of fixed bases <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.0593em;">G</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:-0.0593em;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;">v</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></span></span> to provide
|
||
domain separation for zero-valued notes. The most performant design (similar to the chosen
|
||
design) does not achieve <strong>Faerie Resistance</strong> for an adversary that knows the recipient's
|
||
full viewing key (<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 mathnormal" style="margin-right:0.03588em;">ψ</span></span></span></span> could be brute-forced to cancel out <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="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:0.3361em;"><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 mtight"><span class="mord mtight"><span class="mord mathsf mtight">nk</span></span></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="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mclose">)</span></span></span></span>,
|
||
causing a collision), and the other variants require assuming <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathit">Coll</span></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-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;">F</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></span></span> as
|
||
mentioned above.</p>
|
||
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