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<p>halo2 uses the following lookup technique, which allows for lookups in arbitrary sets, and
is arguably simpler than Plookup.</p>
<h2><aclass="header"href="#note-on-language"id="note-on-language">Note on Language</a></h2>
<p>In addition to the <ahref="../design.html#note-on-language">general notes on language</a>:</p>
<ul>
<li>We call the <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal"style="margin-right:0.07153em;">Z</span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> polynomial (the grand product argument polynomial for the permutation
argument) the "permutation product" column.</li>
<p>We express lookups in terms of a "subset argument" over a table with <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.849108em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord">2</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.849108em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight"style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span> rows (numbered
from 0), and columns <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal">A</span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span>.</p>
<p>The goal of the subset argument is to enforce that every cell in <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal">A</span></span></span></span> is equal to <em>some</em>
cell in <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span>. This means that more than one cell in <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal">A</span></span></span></span> can be equal to the <em>same</em> cell in
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span>, and some cells in <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span> don't need to be equal to any of the cells in <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal">A</span></span></span></span>.</p>
<ul>
<li><spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span> might be fixed, but it doesn't need to be. That is, we can support looking up values
in either fixed or variable tables (where the latter includes advice columns).</li>
<li><spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal">A</span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span> can contain duplicates. If the sets represented by <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal">A</span></span></span></span> and/or <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span> are not
naturally of size <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.849108em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord">2</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.849108em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight"style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span>, we extend <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span> with duplicates and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal">A</span></span></span></span> with dummy values known
to be in <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span>.
<ul>
<li>Alternatively we could add a "lookup selector" that controls which elements of the <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal">A</span></span></span></span>
column participate in lookups. This would modify the occurrence of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal">A</span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> in the
permutation rule below to replace <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal">A</span></span></span></span> with, say, <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.83333em;vertical-align:-0.15em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.30110799999999993em;"><spanstyle="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight">0</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> if a lookup is not selected.</li>
</ul>
</li>
</ul>
<p>Let <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.84444em;vertical-align:-0.15em;"></span><spanclass="mord"><spanclass="mord">ℓ</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.31166399999999994em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">i</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> be the Lagrange basis polynomial that evaluates to <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">1</span></span></span></span> at row <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.65952em;vertical-align:0em;"></span><spanclass="mord mathnormal">i</span></span></span></span>, and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">0</span></span></span></span>
otherwise.</p>
<p>We start by allowing the prover to supply permutation columns of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal">A</span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span>. Let's call
these <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal">A</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>, respectively. We can enforce that they are permutations using a
permutation argument with product column <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.07153em;">Z</span></span></span></span> with the rules:</p>
<p>This is a version of the permutation argument which allows <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal">A</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> to be
permutations of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal">A</span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span>, respectively, but doesn't specify the exact permutations.
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><spanclass="mord mathnormal"style="margin-right:0.05278em;">β</span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical-align:-0.19444em;"></span><spanclass="mord mathnormal"style="margin-right:0.05556em;">γ</span></span></span></span> are separate challenges so that we can combine these two permutation
arguments into one without worrying that they might interfere with each other.</p>
<p>The goal of these permutations is to allow <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal">A</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> to be arranged by the prover in a
particular way:</p>
<ol>
<li>All the cells of column <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal">A</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> are arranged so that like-valued cells are vertically
adjacent to each other. This could be done by some kind of sorting algorithm, but all
that matters is that like-valued cells are on consecutive rows in column <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal">A</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>, and that
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal">A</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> is a permutation of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal">A</span></span></span></span>.</li>
<li>The first row in a sequence of like values in <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal">A</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> is the row that has the
corresponding value in <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span><spanclass="mord">.</span></span></span></span> Apart from this constraint, <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> is any arbitrary
permutation of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span>.</li>
<p>Together these constraints effectively force every element in <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal">A</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> (and thus <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal">A</span></span></span></span>) to equal
at least one element in <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> (and thus <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span>). Proof: by induction on prefixes of the rows.</p>
<li>There is the original column <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal">A</span></span></span></span> and the fixed column <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span>.</li>
<li>There is a permutation product column <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.07153em;">Z</span></span></span></span>.</li>
<li>There are the two permutations <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal">A</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>.</li>
<p>halo2's lookup argument implementation generalizes the above technique in the following
ways:</p>
<ul>
<li><spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal">A</span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span> can be extended to multiple columns, combined using a random challenge. <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal">A</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>
and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.751892em;vertical-align:0em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span><spanclass="msupsub"><spanclass="vlist-t"><spanclass="vlist-r"><spanclass="vlist"style="height:0.751892em;"><spanstyle="top:-3.063em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mtight"><spanclass="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> stay as single columns.
<ul>
<li>The commitments to the columns of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span> can be precomputed, then combined cheaply once
the challenge is known by taking advantage of the homomorphic property of Pedersen
<li>The columns of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal">A</span></span></span></span> can be given as arbitrary polynomial expressions using relative
references. These will be substituted into the product column constraint, subject to
the maximum degree bound. This potentially saves one or more advice columns.</li>
subset argument, i.e. to constrain <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathcal">R</span><spanclass="mopen">(</span><spanclass="mord mathnormal">x</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord mathnormal"style="margin-right:0.03588em;">y</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord">...</span><spanclass="mclose">)</span></span></span></span> in each row, consider
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathcal">R</span></span></span></span> as a set of tuples <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.05764em;">S</span></span></span></span> (using the method of the previous point), and check
that <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mopen">(</span><spanclass="mord mathnormal">x</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord mathnormal"style="margin-right:0.03588em;">y</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord">...</span><spanclass="mclose">)</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">∈</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathcal">R</span></span></span></span>.
<li>In the case where <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathcal">R</span></span></span></span> represents a function, this implicitly also checks