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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"><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="../../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system.html"><strong aria-hidden="true">3.1.</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">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/comparison.html" class="active"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../../design/protocol.html"><strong aria-hidden="true">3.2.</strong> Protocol Description</a></li><li class="chapter-item expanded "><a href="../../design/implementation.html"><strong aria-hidden="true">3.3.</strong> Implementation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/implementation/proofs.html"><strong aria-hidden="true">3.3.1.</strong> Proofs</a></li><li class="chapter-item expanded "><a href="../../design/implementation/fields.html"><strong aria-hidden="true">3.3.2.</strong> Fields</a></li><li class="chapter-item expanded "><a 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chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../../background/pc-ipa.html"><strong aria-hidden="true">4.5.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../../background/recursion.html"><strong aria-hidden="true">4.6.</strong> Recursion</a></li></ol></li></ol>
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<h1 class="menu-title">The halo2 Book</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 id="comparison-to-other-work"><a class="header" href="#comparison-to-other-work">Comparison to other work</a></h1>
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<h2 id="bcms20-appendix-a2"><a class="header" href="#bcms20-appendix-a2">BCMS20 Appendix A.2</a></h2>
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<p>Appendix A.2 of <a href="https://eprint.iacr.org/2020/499">BCMS20</a> describes a polynomial commitment scheme that is similar to the
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one described in <a href="https://eprint.iacr.org/2019/1021">BGH19</a> (BCMS20 being a generalization of the original Halo paper). Halo
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2 builds on both of these works, and thus itself uses a polynomial commitment scheme that
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is very similar to the one in BCMS20.</p>
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<p>The following table provides a mapping between the variable names in BCMS20, and the
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equivalent objects in Halo 2 (which builds on the nomenclature from the Halo paper):</p>
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<div class="table-wrapper"><table><thead><tr><th style="text-align: center">BCMS20</th><th style="text-align: center">Halo 2</th></tr></thead><tbody>
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<tr><td style="text-align: center"><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;">S</span></span></span></span></td><td style="text-align: center"><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></td></tr>
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<tr><td style="text-align: center"><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></td><td style="text-align: center"><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:0.10903em;">U</span></span></span></span></td></tr>
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<tr><td style="text-align: center"><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></span></span></td><td style="text-align: center"><code>msm</code> or <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;">P</span></span></span></span></td></tr>
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<tr><td style="text-align: center"><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:0.0037em;">α</span></span></span></span></td><td style="text-align: center"><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">ι</span></span></span></span></td></tr>
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<tr><td style="text-align: center"><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 class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</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></td><td style="text-align: center"><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:0.04398em;">z</span></span></span></span></td></tr>
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<tr><td style="text-align: center"><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 class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;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></span></span></td><td style="text-align: center"><code>challenge_i</code></td></tr>
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||
<tr><td style="text-align: center"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align: center"><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:0.04398em;">z</span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span></td></tr>
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||
<tr><td style="text-align: center"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7622199999999999em;vertical-align:-0.19444em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">p</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;"><span class="mord">ˉ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align: center"><code>s_poly</code></td></tr>
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||
<tr><td style="text-align: center"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.56778em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span></span></span></span></td><td style="text-align: center"><code>s_poly_blind</code></td></tr>
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||
<tr><td style="text-align: center"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8201099999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201099999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;"><span class="mord">ˉ</span></span></span></span></span></span></span></span></span></span></td><td style="text-align: center"><code>s_poly_commitment</code></td></tr>
|
||
<tr><td style="text-align: center"><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">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span></td><td style="text-align: center"><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" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span></td></tr>
|
||
<tr><td style="text-align: center"><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:0.10903em;">U</span></span></span></span></td><td style="text-align: center"><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></span></span></td></tr>
|
||
<tr><td style="text-align: center"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align: center"><code>blind</code> / <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:0.04601em;">ξ</span></span></span></span></td></tr>
|
||
<tr><td style="text-align: center"><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 mathbf">c</span></span></span></span></td><td style="text-align: center"><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 mathbf">a</span></span></span></span></td></tr>
|
||
<tr><td style="text-align: center"><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">c</span></span></span></span></td><td style="text-align: center"><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">a</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:0.59444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><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">0</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></td></tr>
|
||
<tr><td style="text-align: center"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align: center"><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">ab</span></span></span></span></td></tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p>Halo 2's polynomial commitment scheme differs from Appendix A.2 of BCMS20 in two ways:</p>
|
||
<ol>
|
||
<li>
|
||
<p>Step 8 of the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">Open</span></span></span></span></span> algorithm computes a "non-hiding" commitment <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> prior to
|
||
the inner product argument, which opens to the same value as <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></span></span> but is a commitment to
|
||
a randomly-drawn polynomial. The remainder of the protocol involves no blinding. By
|
||
contrast, in Halo 2 we blind every single commitment that we make (even for instance
|
||
and fixed polynomials, though using a blinding factor of 1 for the fixed polynomials);
|
||
this makes the protocol simpler to reason about. As a consequence of this, the verifier
|
||
needs to handle the cumulative blinding factor at the end of the protocol, and so there
|
||
is no need to derive an equivalent to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> at the start of the protocol.</p>
|
||
<ul>
|
||
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> is also an input to the random oracle for <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 class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</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>; in Halo 2 we utilize a
|
||
transcript that has already committed to the equivalent components of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> prior to
|
||
sampling <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:0.04398em;">z</span></span></span></span>.</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>The <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">PC</span></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-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">DL</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span><span class="mord text"><span class="mord">SuccinctCheck</span></span></span></span></span> subroutine (Figure 2 of BCMS20) computes
|
||
the initial group element <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"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</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> by adding <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">]</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></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:0.03588em;">v</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</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="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>, which requires two
|
||
scalar multiplications. Instead, we subtract <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">]</span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><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">0</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> from the original commitment <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;">P</span></span></span></span>,
|
||
so that we're effectively opening the polynomial at the point to the value zero. The
|
||
computation <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">]</span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><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">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is more efficient in the context of recursion because <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"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><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">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is a
|
||
fixed base (so we can use lookup tables).</p>
|
||
</li>
|
||
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|
||
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