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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</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/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html" class="active"><strong aria-hidden="true">1.2.</strong> PLONKish Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.3.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.4.</strong> Gadgets</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/dev-tools.html"><strong aria-hidden="true">2.1.</strong> Developer tools</a></li><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.2.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.3.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.4.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.5.</strong> Tips and tricks</a></li><li class="chapter-item expanded "><a href="../user/wasm-port.html"><strong aria-hidden="true">2.6.</strong> WASM Guide</a></li></ol></li><li class="chapter-item expanded "><a href="../dev.html"><strong aria-hidden="true">3.</strong> Developer Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../dev/features.html"><strong aria-hidden="true">3.1.</strong> Feature development</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">4.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">4.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">4.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">4.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">4.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">4.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">4.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">4.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">4.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/protocol.html"><strong aria-hidden="true">4.2.</strong> Protocol Description</a></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">4.3.</strong> Implementation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/implementation/proofs.html"><strong aria-hidden="true">4.3.1.</strong> Proofs</a></li><li class="chapter-item expanded "><a href="../design/implementation/fields.html"><strong aria-hidden="true">4.3.2.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../design/implementation/selector-combining.html"><strong aria-hidden="true">4.3.3.</strong> Selector combining</a></li></ol></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">4.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/ecc.html"><strong aria-hidden="true">4.4.1.</strong> Elliptic curve cryptography</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/ecc/witnessing-points.html"><strong aria-hidden="true">4.4.1.1.</strong> Witnessing points</a></li><li class="chapter-item expanded "><a href="../design/gadgets/ecc/addition.html"><strong aria-hidden="true">4.4.1.2.</strong> Incomplete and complete addition</a></li><li class="chapter-item expanded "><a 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argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">5.6.</strong> Recursion</a></li></ol></li></ol>
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<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
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<h1 id="plonkish-arithmetization"><a class="header" href="#plonkish-arithmetization">PLONKish Arithmetization</a></h1>
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<p>The arithmetization used by Halo 2 comes from <a href="https://eprint.iacr.org/2019/953">PLONK</a>, or
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more precisely its extension UltraPLONK that supports custom gates and lookup arguments. We'll
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call it <a href="https://twitter.com/feministPLT/status/1413815927704014850"><em><strong>PLONKish</strong></em></a>.</p>
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<p><em><strong>PLONKish circuits</strong></em> are defined in terms of a rectangular matrix of values. We refer to
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<em><strong>rows</strong></em>, <em><strong>columns</strong></em>, and <em><strong>cells</strong></em> of this matrix with the conventional meanings.</p>
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<p>A PLONKish circuit depends on a <em><strong>configuration</strong></em>:</p>
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<ul>
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<li>
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<p>A finite field <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">F</span></span></span></span>, where cell values (for a given statement and witness) will be
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elements of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">F</span></span></span></span>.</p>
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</li>
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<li>
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<p>The number of columns in the matrix, and a specification of each column as being
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<em><strong>fixed</strong></em>, <em><strong>advice</strong></em>, or <em><strong>instance</strong></em>. Fixed columns are fixed by the circuit;
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advice columns correspond to witness values; and instance columns are normally used for
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public inputs (technically, they can be used for any elements shared between the prover
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and verifier).</p>
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</li>
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<li>
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<p>A subset of the columns that can participate in equality constraints.</p>
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</li>
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<li>
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<p>A <em><strong>maximum constraint degree</strong></em>.</p>
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</li>
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<li>
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<p>A sequence of <em><strong>polynomial constraints</strong></em>. These are multivariate polynomials over
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<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">F</span></span></span></span> that must evaluate to zero <em>for each row</em>. The variables in a polynomial
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constraint may refer to a cell in a given column of the current row, or a given column of
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another row relative to this one (with wrap-around, i.e. taken modulo <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>). The maximum
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degree of each polynomial is given by the maximum constraint degree.</p>
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</li>
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<li>
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<p>A sequence of <em><strong>lookup arguments</strong></em> defined over tuples of <em><strong>input expressions</strong></em>
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(which are multivariate polynomials as above) and <em><strong>table columns</strong></em>.</p>
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</li>
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</ul>
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<p>A PLONKish circuit also defines:</p>
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<ul>
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<li>
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<p>The number of rows <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> in the matrix. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> must correspond to the size of a multiplicative
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subgroup of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7713em;"></span><span class="mord"><span class="mord mathbb">F</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><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="mbin mtight">×</span></span></span></span></span></span></span></span></span></span></span>; typically a power of two.</p>
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</li>
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<li>
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<p>A sequence of <em><strong>equality constraints</strong></em>, which specify that two given cells must have equal
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values.</p>
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</li>
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<li>
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<p>The values of the fixed columns at each row.</p>
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</li>
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</ul>
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<p>From a circuit description we can generate a <em><strong>proving key</strong></em> and a <em><strong>verification key</strong></em>,
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which are needed for the operations of proving and verification for that circuit.</p>
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<blockquote>
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<p>Note that we specify the ordering of columns, polynomial constraints, lookup arguments, and
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equality constraints, even though these do not affect the meaning of the circuit. This makes
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it easier to define the generation of proving and verification keys as a deterministic
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process.</p>
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</blockquote>
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<p>Typically, a configuration will define polynomial constraints that are switched off and on by
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<em><strong>selectors</strong></em> defined in fixed columns. For example, a constraint <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</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:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord">...</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> can
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||
be switched off for a particular row <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> by setting <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</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:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>. In this case we sometimes refer
|
||
to a set of constraints controlled by a set of selector columns that are designed to be used
|
||
together, as a <em><strong>gate</strong></em>. Typically there will be a <em><strong>standard gate</strong></em> that supports generic
|
||
operations like field multiplication and division, and possibly also <em><strong>custom gates</strong></em> that
|
||
support more specialized operations.</p>
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