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506 lines
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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/actions.html"><strong aria-hidden="true">3.1.</strong> Actions</a></li><li class="chapter-item expanded "><a href="design/commitments.html"><strong aria-hidden="true">3.2.</strong> Commitments</a></li><li class="chapter-item expanded "><a href="design/commitment-tree.html"><strong aria-hidden="true">3.3.</strong> Commitment tree</a></li><li class="chapter-item expanded "><a href="design/nullifiers.html"><strong aria-hidden="true">3.4.</strong> Nullifiers</a></li><li class="chapter-item expanded "><a href="design/signatures.html"><strong aria-hidden="true">3.5.</strong> Signatures</a></li><li class="chapter-item expanded "><a href="design/circuit.html"><strong aria-hidden="true">3.6.</strong> Circuit</a></li></ol></li></ol>
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<h1 class="menu-title">The Orchard Book</h1>
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<h1><a class="header" href="#orchard-a-hrefhttpscratesiocratesorchardimg-srchttpsimgshieldsiocratesvorchardsvg-altcratesio-a" id="orchard-a-hrefhttpscratesiocratesorchardimg-srchttpsimgshieldsiocratesvorchardsvg-altcratesio-a">orchard <a href="https://crates.io/crates/orchard"><img src="https://img.shields.io/crates/v/orchard.svg" alt="Crates.io" /></a></a></h1>
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<p><strong>IMPORTANT</strong>: This library is being actively developed and should not be used in production software.</p>
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<h2><a class="header" href="#documentation" id="documentation">Documentation</a></h2>
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<ul>
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<li><a href="https://zcash.github.io/orchard/">The Orchard Book</a></li>
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<li><a href="https://docs.rs/orchard">Crate documentation</a></li>
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</ul>
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<h2><a class="header" href="#license" id="license">License</a></h2>
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<p>Copyright 2020 The Electric Coin Company.</p>
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<p>You may use this package under the Transitive Grace Period Public Licence,
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version 1.0, or at your option, any later version. See the file
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<a href="LICENSE-TGPPL"><code>LICENSE-TGPPL</code></a> for the terms of the Transitive Grace Period
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Public Licence, version 1.0.</p>
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<p>The purpose of the TGPPL is to allow commercial improvements to the package
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while ensuring that all improvements are eventually open source. See
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<a href="https://tahoe-lafs.org/%7Ezooko/tgppl.pdf">here</a> for why the TGPPL exists,
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graphically illustrated on three slides.</p>
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<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
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<h1><a class="header" href="#concepts" id="concepts">Concepts</a></h1>
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<h1><a class="header" href="#preliminaries" id="preliminaries">Preliminaries</a></h1>
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<h1><a class="header" href="#user-documentation" id="user-documentation">User Documentation</a></h1>
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<h1><a class="header" href="#creating-keys-and-addresses" id="creating-keys-and-addresses">Creating keys and addresses</a></h1>
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<h1><a class="header" href="#creating-notes" id="creating-notes">Creating notes</a></h1>
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<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
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<h1><a class="header" href="#spending-notes" id="spending-notes">Spending notes</a></h1>
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<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
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<h1><a class="header" href="#integration-into-an-existing-chain" id="integration-into-an-existing-chain">Integration into an existing chain</a></h1>
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<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
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<h1><a class="header" href="#design" id="design">Design</a></h1>
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<h2><a class="header" href="#general-design-notes" id="general-design-notes">General design notes</a></h2>
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<h3><a class="header" href="#requirements" id="requirements">Requirements</a></h3>
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<ul>
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<li>Keep the design close to Sapling, while eliminating aspects we don't like.</li>
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</ul>
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<h3><a class="header" href="#non-requirements" id="non-requirements">Non-requirements</a></h3>
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<ul>
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<li>Delegated proving with privacy from the prover.
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<ul>
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<li>We know how to do this, but it would require a discrete log equality proof, and the
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most efficient way to do this would be to do RedDSA and this at the same time, which
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means more work for e.g. hardware wallets.</li>
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</ul>
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</li>
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</ul>
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<h3><a class="header" href="#open-issues" id="open-issues">Open issues</a></h3>
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<ul>
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<li>Should we have one memo per output, or one memo per transaction, or 0..n memos?
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<ul>
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<li>Variable, or (1 or n), is a potential privacy leak.</li>
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<li>Need to consider the privacy issue related to light clients requesting individual
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memos vs being able to fetch all memos.</li>
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</ul>
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</li>
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</ul>
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<h3><a class="header" href="#key-structure" id="key-structure">Key structure</a></h3>
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<p>Provisional proposal: exactly the same key structure as Sapling.</p>
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<p>Group hashing uses the isogeny.</p>
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<ul>
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<li>nsk goes away; <code>nk</code> is now a field element</li>
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<li>TODO: ak / nk split enables splitting the security argument, but could consider merging.
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Merging would help with ivk derivation perf (though as a commitment now it's pretty cheap)</li>
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<li>TODO: nullifier computation</li>
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</ul>
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<p>ZIP 32 integration</p>
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<ul>
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<li>Use same Sapling design?</li>
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<li>Simpler "hardened-only" derivation structure?</li>
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<li>Improve diversifier integration / documentation</li>
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</ul>
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<h3><a class="header" href="#note-structure" id="note-structure">Note structure</a></h3>
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<ul>
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<li>TODO: UDAs: arbitrary vs whitelisted</li>
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</ul>
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<h3><a class="header" href="#typed-variables-vs-byte-encodings" id="typed-variables-vs-byte-encodings">Typed variables vs byte encodings</a></h3>
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<p>For Sapling, we have encountered multiple places where the specification uses typed
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variables to define the consensus rules, but the C++ implementation in zcashd relied on
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byte encodings to implement them. This resulted in subtly-different consensus rules being
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deployed than were intended, for example where a particular type was not round-trip
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encodable.</p>
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<p>In Orchard, we avoid this by defining the consensus rules in terms of the byte encodings
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of all variables, and being explicit about any types that are not round-trip encodable.
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This makes consensus compatibility between strongly-typed implementations (such as this
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crate) and byte-oriented implementations easier to achieve.</p>
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||
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
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<h1><a class="header" href="#actions" id="actions">Actions</a></h1>
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<p>In Sprout, we had a single proof that represented two spent notes and two new notes. This
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was necessary in order to faciliate spending multiple notes in a single transaction (to
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balance value, an output of one JoinSplit could be spent in the next one), but also
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provided a minimal level of arity-hiding: single-JoinSplit transactions all looked like
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2-in 2-out transactions, and in multi-JoinSplit transactions each JoinSplit looked like a
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1-in 1-out.</p>
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<p>In Sapling, we switched to using value commitments to balance the transaction, removing
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the min-2 arity requirement. We opted for one proof per spent note and one (much simpler)
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proof per output note, which greatly improved the performance of generating outputs, but
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removed any arity-hiding from the proofs (instead having the transaction builder pad
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transactions to 1-in, 2-out).</p>
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<p>For Orchard, we take a combined approach: we define an Orchard transaction as containing a
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bundle of actions, where each action is both a spend and an output. This provides the same
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inherent arity-hiding as multi-JoinSplit Sprout, but using Sapling value commitments to
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balance the transaction without doubling its size.</p>
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<p>TODO: Depending on the circuit cost, we <em>may</em> switch to having an action internally
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represent either a spend or an output. Externally spends and outputs would still be
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indistinguishable, but the transaction would be larger.</p>
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<h2><a class="header" href="#memo-fields" id="memo-fields">Memo fields</a></h2>
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<p>TODO: One memo per tx vs one memo per output</p>
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||
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
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<h1><a class="header" href="#commitments" id="commitments">Commitments</a></h1>
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<p>As in Sapling, we require two kinds of commitment schemes in Pollard:</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:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">p</span><span class="mord mathnormal">h</span><span class="mord mathnormal">i</span><span class="mord mathnormal">c</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord mathnormal">t</span></span></span></span> is a linearly homomorphic commitment scheme with perfect hiding, and
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strong binding reducible to DL.</li>
|
||
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord mathnormal">t</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord mathnormal">h</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">t</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord mathnormal">t</span></span></span></span> are commitment schemes with perfect hiding, and strong
|
||
binding reducible to DL.</li>
|
||
</ul>
|
||
<p>By "strong binding" we mean that the scheme is collision resistant on the input and
|
||
randomness.</p>
|
||
<p>We instantiate <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">p</span><span class="mord mathnormal">h</span><span class="mord mathnormal">i</span><span class="mord mathnormal">c</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord mathnormal">t</span></span></span></span> with a Pedersen commitment, and use it for value
|
||
commitments:</p>
|
||
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf">c</span><span class="mord mathsf" style="margin-right:0.01389em;">v</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">p</span><span class="mord mathnormal">h</span><span class="mord mathnormal">i</span><span class="mord mathnormal">c</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.724108em;"><span style="top:-2.4530000000000003em;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 mtight"><span class="mord mathsf mtight" style="margin-right:0.01389em;">r</span><span class="mord mathsf mtight">c</span><span class="mord mathsf mtight" style="margin-right:0.01389em;">v</span></span></span></span></span><span style="top:-3.113em;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">c</span><span class="mord mathsf mtight" style="margin-right:0.01389em;">v</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span></span></p>
|
||
<p>We instantiate <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord mathnormal">t</span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord mathnormal">h</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">t</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord mathnormal">t</span></span></span></span> with Sinsemilla, and use them for all other
|
||
commitments:</p>
|
||
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf">i</span><span class="mord mathsf" style="margin-right:0.01389em;">v</span><span class="mord mathsf">k</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1491079999999998em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord mathnormal">h</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">t</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em;"><span style="top:-2.4530000000000003em;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 mtight"><span class="mord mathsf mtight" style="margin-right:0.01389em;">r</span><span class="mord mathsf mtight">i</span><span class="mord mathsf mtight" style="margin-right:0.01389em;">v</span><span class="mord mathsf mtight">k</span></span></span></span></span><span style="top:-3.113em;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">i</span><span class="mord mathsf mtight" style="margin-right:0.01389em;">v</span><span class="mord mathsf mtight">k</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">n</span><span class="mord mathsf">k</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf">a</span><span class="mord mathsf">k</span></span><span class="mclose">)</span></span></span></span></span>
|
||
<span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.724108em;"><span style="top:-2.4530000000000003em;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 mtight"><span class="mord mathsf mtight" style="margin-right:0.01389em;">r</span><span class="mord mathsf mtight">c</span><span class="mord mathsf mtight">m</span></span></span></span></span><span style="top:-3.113em;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">c</span><span class="mord mathsf mtight">m</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord text"><span class="mord">rest of note</span></span><span class="mclose">)</span></span></span></span></span></p>
|
||
<p>This is the same split (and rationale) as in Sapling, but using the more PLONK-efficient
|
||
Sinsemilla instead of Bowe-Hopwood Pedersen hashes.</p>
|
||
<p>Note that we also deviate from Sapling by using <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord mathnormal">h</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">t</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord mathnormal">t</span></span></span></span> to deriving <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf">i</span><span class="mord mathsf" style="margin-right:0.01389em;">v</span><span class="mord mathsf">k</span></span></span></span></span>
|
||
instead of a full PRF. This removes an unnecessary (large) PRF primitive from the circuit,
|
||
at the cost of requiring <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">i</span><span class="mord mathsf" style="margin-right:0.01389em;">v</span><span class="mord mathsf">k</span></span></span></span></span> to be part of the full viewing key.</p>
|
||
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
|
||
<h1><a class="header" href="#commitment-tree" id="commitment-tree">Commitment tree</a></h1>
|
||
<p>One of the things we learned from Sapling is that having a single global commitment tree
|
||
makes life hard for light client wallets. When a new note is received, the wallet derives
|
||
its incremental witness from the state of the global tree at the point when the note's
|
||
commitment is appended; this incremental state then needs to be updated with every
|
||
subsequent commitment in the block in-order. It isn't efficient for a server to
|
||
pre-compute and send over the necessary incremental updates for every new note in a block,
|
||
and if a wallet requested a specific update from the server it would leak the specific
|
||
note that was received.</p>
|
||
<p>Orchard addresses this by splitting the commitment tree into several sub-trees:</p>
|
||
<ul>
|
||
<li>Bundle tree, that accumulates the commitments within a single bundle (and thus a single
|
||
transaction).</li>
|
||
<li>Block tree, that accumulates the bundle tree roots within a single block.</li>
|
||
<li>Global tree, that accumulates the block tree roots.</li>
|
||
</ul>
|
||
<p>Each of these trees has a fixed depth (necessary for being able to create proofs).</p>
|
||
<p>Chains that integrate Orchard can decouple the limits on commitments-per-subtree from
|
||
higher-layer constraints like block size, by enabling their blocks and transactions to be
|
||
structured internally as a series of Orchard blocks or txs (e.g. a Zcash block would
|
||
contain a <code>Vec<BlockTreeRoot></code>, that each get appended in-order).</p>
|
||
<p>Zcash level: we also bind these roots into the FlyClient history leaves, so that light
|
||
clients can assert they are valid independently of the full block.</p>
|
||
<p>TODO: Sean is pretty sure we can just improve the Incremental Merkle Tree implementation
|
||
to work around this, without domain-separating the tree. If we can do that instead, it may
|
||
be simpler.</p>
|
||
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
|
||
<h1><a class="header" href="#nullifiers" id="nullifiers">Nullifiers</a></h1>
|
||
<p>The nullifier design we use for Orchard is</p>
|
||
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><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:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;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 mtight"><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight">k</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" style="margin-right:0.03588em;">ψ</span><span class="mclose">)</span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></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" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mclose">]</span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span><span class="mpunct">,</span></span></span></span></span></p>
|
||
<p>where:</p>
|
||
<ul>
|
||
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span></span></span></span> is a keyed circuit-efficient hash (such as Rescue).</li>
|
||
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">G</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is a cryptographic hash into the group (such as BLAKE2s with simplified SWU).</li>
|
||
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span></span></span></span> is a fixed base, independent of any others returned by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">G</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</li>
|
||
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is a base unique to this output.
|
||
<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:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">G</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</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:0.980548em;vertical-align:-0.286108em;"></span><span class="mord"><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:0.3361079999999999em;"><span style="top:-2.5500000000000003em;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">S</span><span class="mord mathsf mtight">i</span><span class="mord mathsf mtight" style="margin-right:0.01389em;">g</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></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:-0.19444em;"></span><span class="mord mathnormal">ρ</span></span></span></span> includes the nullifiers of any Orchard notes being spent in the same action.
|
||
Given that an action 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:-0.19444em;"></span><span class="mord mathnormal">ρ</span></span></span></span> to
|
||
be the nullifier of the spent note.</li>
|
||
<li>For zero-valued notes, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></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:0.68333em;vertical-align:0em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span></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:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">G</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</li>
|
||
</ul>
|
||
</li>
|
||
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></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
|
||
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:-0.19444em;"></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:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">s</span><span class="mord mathsf">e</span><span class="mord mathsf">e</span><span class="mord mathsf">d</span></span></span></span></span>:
|
||
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0991079999999998em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><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 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:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">s</span><span class="mord mathsf">e</span><span class="mord mathsf">e</span><span class="mord mathsf">d</span></span><span class="mclose">)</span></span></span></span>.</li>
|
||
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span></span></span></span> is a blinding scalar, similarly generated as
|
||
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><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 mathsf mtight" style="margin-right:0.01389em;">r</span><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight" style="margin-right:0.06944em;">f</span></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:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">s</span><span class="mord mathsf">e</span><span class="mord mathsf">e</span><span class="mord mathsf">d</span></span><span class="mclose">)</span></span></span></span>.</li>
|
||
</ul>
|
||
<p>This gives a note structure of</p>
|
||
<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">d</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span><span class="mclose">)</span><span class="mord">.</span></span></span></span></span></p>
|
||
<p>The note plaintext includes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">s</span><span class="mord mathsf">e</span><span class="mord mathsf">e</span><span class="mord mathsf">d</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:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span></span></span></span>, and
|
||
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span></span></span></span>. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is omitted entirely from the action:</p>
|
||
<ul>
|
||
<li>Consensus nodes directly derive <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 mathnormal">G</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mclose">)</span></span></span></span> and provide it as a public input to the
|
||
circuit (which ignores it for zero-valued notes, as with the commitment tree anchor).</li>
|
||
<li>The recipient can recompute the correct <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> given their additional knowledge of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span>.</li>
|
||
</ul>
|
||
<h2><a class="header" href="#security-properties" id="security-properties">Security properties</a></h2>
|
||
<p>We care about several security properties for our nullifiers:</p>
|
||
<ul>
|
||
<li>
|
||
<p><strong>Balance:</strong> can I forge money?</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>Note Privacy:</strong> can I gain information about notes only from the public block chain?</p>
|
||
<ul>
|
||
<li>This describes notes sent in-band.</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p><strong>Note Privacy (OOB):</strong> can I gain information about notes sent out-of-band, only from
|
||
the public block chain?</p>
|
||
<ul>
|
||
<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:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">i</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</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:0.68333em;vertical-align:0em;"></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:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</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:6.128347em;vertical-align:-2.8141735em;"></span><span class="mord"><span class="mtable"><span class="vertical-separator" style="height:6.128347em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-2.8141735em;"></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:3.3141735000000003em;"><span style="top:-5.4741735em;"><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:-4.272842499999999em;"><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:-3.0728424999999997em;"><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:-1.7458265000000004em;"><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:-0.5458265000000002em;"><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:2.8141735em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:6.128347em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-2.8141735em;"></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:3.3141735000000003em;"><span style="top:-5.4741735em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:-4.272842499999999em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.424669em;margin-left:-0.08125em;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:-3.063em;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 mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.0728424999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:-1.7458265000000004em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9670159999999999em;"><span style="top:-2.4064690000000004em;margin-left:-0.08125em;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:-3.1809080000000005em;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:0.293531em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 style="top:-0.5458265000000002em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</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:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;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 mathnormal mtight">G</span><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;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 mathnormal mtight">G</span><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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="mclose">)</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:2.8141735em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:6.128347em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-2.8141735em;"></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:1.1166619999999998em;vertical-align:-0.275331em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.424669em;margin-left:-0.08125em;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:-3.063em;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 mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em;"><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:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</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:0.83333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;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:0.83333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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>
|
||
<blockquote>
|
||
<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></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:0.43056em;vertical-align:0em;"></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 mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;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 mtight"><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight">k</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:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord"><span class="mord mathsf">n</span><span class="mord mathsf">k</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></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:0.83333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;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:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</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:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf">n</span><span class="mord mathsf">k</span></span></span></span></span>.)</p>
|
||
</blockquote>
|
||
<h2><a class="header" href="#considered-alternatives" id="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:0.69444em;vertical-align:0em;"></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><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:17.98846799999999em;vertical-align:-8.724234em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.264233999999991em;"><span style="top:-11.224234em;"><span class="pstrut" style="height:11.224234em;"></span><span class="mtable"><span class="vertical-separator" style="height:17.948467999999995em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-8.724233999999997em;"></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:9.224234em;"><span style="top:-11.384233999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span></span></span><span style="top:-10.182902999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathsf">n</span><span class="mord mathsf">k</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:-8.981571999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathsf">n</span><span class="mord mathsf">k</span></span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mclose">]</span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span></span></span><span style="top:-7.780240999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mopen">[</span><span class="mord"><span class="mord mathsf">n</span><span class="mord mathsf">k</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:-6.578909999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mopen">[</span><span class="mord"><span class="mord mathsf">n</span><span class="mord mathsf">k</span></span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mclose">]</span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span><span class="mclose">)</span></span></span><span style="top:-5.251893999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;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 mtight"><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight">k</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" style="margin-right:0.03588em;">ψ</span><span class="mclose">)</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:-3.9248779999999965em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;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 mtight"><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight">k</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" style="margin-right:0.03588em;">ψ</span><span class="mclose">)</span><span class="mclose">]</span><span class="mord"><span class="mord mathcal" style="margin-right:0.0593em;">G</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></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:-2.597861999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;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 mtight"><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight">k</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" style="margin-right:0.03588em;">ψ</span><span class="mclose">)</span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span></span></span><span style="top:-1.2708459999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;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 mtight"><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight">k</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="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mclose">)</span><span class="mclose">]</span><span class="mord"><span class="mord mathcal" style="margin-right:0.0593em;">G</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span></span></span><span style="top:0.056169999999998776em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;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 mtight"><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight">k</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="mclose">]</span><span class="mord"><span class="mord mathcal" style="margin-right:0.0593em;">G</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span></span></span><span style="top:1.383185999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;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 mtight"><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight">k</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="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mclose">)</span><span class="mclose">]</span><span class="mord"><span class="mord mathcal" style="margin-right:0.0593em;">G</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-2.4168920000000003em;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 mtight"><span class="mord mathsf mtight" style="margin-right:0.01389em;">r</span><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight" style="margin-right:0.06944em;">f</span></span></span></span></span><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 class="mord mathsf mtight">n</span><span class="mord mathsf mtight" style="margin-right:0.06944em;">f</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em;"><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:0.16666666666666666em;"></span><span class="mord mathnormal">ρ</span><span class="mclose">)</span></span></span><span style="top:2.710201999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;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 mtight"><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight">k</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="mclose">]</span><span class="mord"><span class="mord mathcal" style="margin-right:0.0593em;">G</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-2.4168920000000003em;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 mtight"><span class="mord mathsf mtight" style="margin-right:0.01389em;">r</span><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight" style="margin-right:0.06944em;">f</span></span></span></span></span><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 class="mord mathsf mtight">n</span><span class="mord mathsf mtight" style="margin-right:0.06944em;">f</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em;"><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:0.16666666666666666em;"></span><span class="mord mathnormal">ρ</span><span class="mclose">)</span></span></span><span style="top:4.037217999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;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 mtight"><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight">k</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="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mclose">)</span><span class="mclose">]</span><span class="mord"><span class="mord mathcal" style="margin-right:0.0593em;">G</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mclose">]</span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span></span></span><span style="top:5.364233999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;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 mtight"><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight">k</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="mclose">]</span><span class="mord"><span class="mord mathcal" style="margin-right:0.0593em;">G</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mclose">]</span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:8.724233999999996em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:17.948467999999995em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-8.724233999999997em;"></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:9.224234em;"><span style="top:-11.384233999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mord mathnormal">o</span><span class="mord mathnormal">t</span><span class="mord mathnormal">e</span></span></span><span style="top:-10.182902999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">d</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span><span class="mclose">)</span></span></span><span style="top:-8.981571999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">d</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span><span class="mclose">)</span></span></span><span style="top:-7.780240999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">d</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span><span class="mclose">)</span></span></span><span style="top:-6.578909999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">d</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span><span class="mclose">)</span></span></span><span style="top:-5.251893999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">d</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span><span class="mclose">)</span></span></span><span style="top:-3.9248779999999965em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">d</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span><span class="mclose">)</span></span></span><span style="top:-2.597861999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">d</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span><span class="mclose">)</span></span></span><span style="top:-1.2708459999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">d</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span><span class="mclose">)</span></span></span><span style="top:0.056169999999998776em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">d</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span><span class="mclose">)</span></span></span><span style="top:1.383185999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">d</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span><span class="mclose">)</span></span></span><span style="top:2.710201999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">d</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span><span class="mclose">)</span></span></span><span style="top:4.037217999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">d</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span><span class="mclose">)</span></span></span><span style="top:5.364233999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">d</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:8.724233999999996em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:17.948467999999995em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-8.724233999999997em;"></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:9.224234em;"><span style="top:-11.384233999999998em;"><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:-10.182902999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:-8.981571999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:-7.780240999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:-6.578909999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:-5.251893999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:-3.9248779999999965em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:-2.597861999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:-1.2708459999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:0.056169999999998776em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:1.383185999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:2.710201999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:4.037217999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:5.364233999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:8.724233999999996em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:17.948467999999995em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-8.724233999999997em;"></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:9.224234em;"><span style="top:-11.384233999999998em;"><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:-10.182902999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.424669em;margin-left:-0.08125em;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:-3.063em;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 mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em;"><span></span></span></span></span></span></span></span></span><span style="top:-8.981571999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.424669em;margin-left:-0.08125em;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:-3.063em;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 mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em;"><span></span></span></span></span></span></span></span></span><span style="top:-7.780240999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.424669em;margin-left:-0.08125em;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:-3.063em;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 mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em;"><span></span></span></span></span></span></span></span></span><span style="top:-6.578909999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.424669em;margin-left:-0.08125em;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:-3.063em;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 mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em;"><span></span></span></span></span></span></span></span></span><span style="top:-5.251893999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.424669em;margin-left:-0.08125em;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:-3.063em;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 mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.9248779999999965em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.424669em;margin-left:-0.08125em;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:-3.063em;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 mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.597861999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.424669em;margin-left:-0.08125em;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:-3.063em;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 mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.2708459999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.424669em;margin-left:-0.08125em;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:-3.063em;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 mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em;"><span></span></span></span></span></span></span></span></span><span style="top:0.056169999999998776em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.424669em;margin-left:-0.08125em;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:-3.063em;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 mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em;"><span></span></span></span></span></span></span></span></span><span style="top:1.383185999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.424669em;margin-left:-0.08125em;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:-3.063em;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 mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em;"><span></span></span></span></span></span></span></span></span><span style="top:2.710201999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.424669em;margin-left:-0.08125em;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:-3.063em;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 mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em;"><span></span></span></span></span></span></span></span></span><span style="top:4.037217999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.424669em;margin-left:-0.08125em;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:-3.063em;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 mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em;"><span></span></span></span></span></span></span></span></span><span style="top:5.364233999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.424669em;margin-left:-0.08125em;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:-3.063em;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 mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em;"><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:8.724233999999996em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:17.948467999999995em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-8.724233999999997em;"></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:9.224234em;"><span style="top:-11.384233999999998em;"><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:-10.182902999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:-8.981571999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:-7.780240999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:-6.578909999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:-5.251893999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:-3.9248779999999965em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:-2.597861999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 style="top:-1.2708459999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 style="top:0.056169999999998776em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 style="top:1.383185999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:2.710201999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:4.037217999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Perfect</span></span></span></span><span style="top:5.364233999999995em;"><span class="pstrut" style="height:3em;"></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:8.724233999999996em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:17.948467999999995em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-8.724233999999997em;"></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:9.224234em;"><span style="top:-11.384233999999998em;"><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:-10.182902999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;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:-8.981571999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;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:-7.780240999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;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 class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 style="top:-6.578909999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;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 class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 style="top:-5.251893999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9670159999999999em;"><span style="top:-2.4064690000000004em;margin-left:-0.08125em;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:-3.1809080000000005em;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:0.293531em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 style="top:-3.9248779999999965em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9670159999999999em;"><span style="top:-2.4064690000000004em;margin-left:-0.08125em;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:-3.1809080000000005em;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:0.293531em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 style="top:-2.597861999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9670159999999999em;"><span style="top:-2.4064690000000004em;margin-left:-0.08125em;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:-3.1809080000000005em;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:0.293531em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 style="top:-1.2708459999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9670159999999999em;"><span style="top:-2.4064690000000004em;margin-left:-0.08125em;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:-3.1809080000000005em;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:0.293531em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 style="top:0.056169999999998776em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9670159999999999em;"><span style="top:-2.4064690000000004em;margin-left:-0.08125em;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:-3.1809080000000005em;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:0.293531em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 style="top:1.383185999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9670159999999999em;"><span style="top:-2.4064690000000004em;margin-left:-0.08125em;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:-3.1809080000000005em;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:0.293531em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 style="top:2.710201999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9670159999999999em;"><span style="top:-2.4064690000000004em;margin-left:-0.08125em;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:-3.1809080000000005em;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:0.293531em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 style="top:4.037217999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9670159999999999em;"><span style="top:-2.4064690000000004em;margin-left:-0.08125em;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:-3.1809080000000005em;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:0.293531em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 style="top:5.364233999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9670159999999999em;"><span style="top:-2.4064690000000004em;margin-left:-0.08125em;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:-3.1809080000000005em;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:0.293531em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:8.724233999999996em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:17.948467999999995em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-8.724233999999997em;"></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:9.224234em;"><span style="top:-11.384233999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Faerie Resistance</span></span></span></span><span style="top:-10.182902999999998em;"><span class="pstrut" style="height:3em;"></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:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;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 mathnormal mtight">G</span><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:-8.981571999999998em;"><span class="pstrut" style="height:3em;"></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:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;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 mathnormal mtight">G</span><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:-7.780240999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></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:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;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 mathnormal mtight">G</span><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:-6.578909999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></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:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;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 mathnormal mtight">G</span><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:-5.251893999999997em;"><span class="pstrut" style="height:3em;"></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:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;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 mathnormal mtight">G</span><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:-3.9248779999999965em;"><span class="pstrut" style="height:3em;"></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:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;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 mathnormal mtight">G</span><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:-2.597861999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</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:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;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 mathnormal mtight">G</span><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;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 mathnormal mtight">G</span><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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="mclose">)</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:-1.2708459999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:0.056169999999998776em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:1.383185999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:2.710201999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:4.037217999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:5.364233999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;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 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:8.724233999999996em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:17.948467999999995em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-8.724233999999997em;"></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:9.224234em;"><span style="top:-11.384233999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Rejected because</span></span></span></span><span style="top:-10.182902999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">No SU for DL-breaking</span></span></span></span><span style="top:-8.981571999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">No SU for DL-breaking</span></span></span></span><span style="top:-7.780240999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 text"><span class="mord"> for FR</span></span></span></span><span style="top:-6.578909999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 text"><span class="mord"> for FR</span></span></span></span><span style="top:-5.251893999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord text"><span class="mord"> variable-base scalar mults</span></span></span></span><span style="top:-2.5978619999999983em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 text"><span class="mord"> for NP(OOB)</span></span></span></span><span style="top:-1.2708459999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 text"><span class="mord"> for NP(OOB)</span></span></span></span><span style="top:0.056169999999998776em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 text"><span class="mord"> for NP(OOB)</span></span></span></span><span style="top:1.383185999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 text"><span class="mord"> for FR</span></span></span></span><span style="top:2.710201999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 text"><span class="mord"> for FR</span></span></span></span><span style="top:4.037217999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 text"><span class="mord"> for FR</span></span></span></span><span style="top:5.364233999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;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 mathnormal mtight" style="margin-right:0.08125em;">H</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span><span class="mord mathnormal mtight">h</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 text"><span class="mord"> for FR</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:8.724233999999996em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span style="top:-2.5em;"><span class="pstrut" style="height:11.224234em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.8270159999999995em;"><span class="pstrut" style="height:11.224234em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-5.154032em;"><span class="pstrut" style="height:11.224234em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-6.481047999999999em;"><span class="pstrut" style="height:11.224234em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-7.808063999999997em;"><span class="pstrut" style="height:11.224234em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-9.135079999999995em;"><span class="pstrut" style="height:11.224234em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-10.462095999999995em;"><span class="pstrut" style="height:11.224234em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-11.789111999999992em;"><span class="pstrut" style="height:11.224234em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-13.116127999999993em;"><span class="pstrut" style="height:11.224234em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-14.443143999999993em;"><span class="pstrut" style="height:11.224234em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-15.644474999999993em;"><span class="pstrut" style="height:11.224234em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-16.845805999999996em;"><span class="pstrut" style="height:11.224234em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-18.047136999999992em;"><span class="pstrut" style="height:11.224234em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-19.248467999999992em;"><span class="pstrut" style="height:11.224234em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-20.44846799999999em;"><span class="pstrut" style="height:11.224234em;"></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:8.724234em;"><span></span></span></span></span></span></span></span></span></span></p>
|
||
<p>In the above alternatives:</p>
|
||
<ul>
|
||
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78055em;vertical-align:-0.09722em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.0593em;">G</span></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:0.68333em;vertical-align:0em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span></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:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">G</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</li>
|
||
</ul>
|
||
<p>The <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.849108em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><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 class="mord mathsf mtight">n</span><span class="mord mathsf mtight" style="margin-right:0.06944em;">f</span></span></span></span></span></span></span></span></span></span></span></span></span> variants enabled nullifier domain separation based on note
|
||
value, without directly depending on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf">c</span><span class="mord mathsf">m</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:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span></span></span></span> as a group element, that is only used in
|
||
nullifier computation.</p>
|
||
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
|
||
<h1><a class="header" href="#signatures" id="signatures">Signatures</a></h1>
|
||
<p>Orchard signatures are an instantiation of RedDSA with a cofactor of 1.</p>
|
||
<p>TODO:</p>
|
||
<ul>
|
||
<li>Should it be possible to sign partial transactions?
|
||
<ul>
|
||
<li>If we're going to merge down all the signatures into a single one, and also want this, we need to ensure there's a feasible MPC.</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
|
||
<h1><a class="header" href="#circuit" id="circuit">Circuit</a></h1>
|
||
|
||
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|
||
|
||
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||
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||
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||
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||
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||
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|
||
</div>
|
||
|
||
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||
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||
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||
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||
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||
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