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<title>Polynomial commitment using inner product argument - The halo2 Book</title>
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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></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"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/implementation/proofs.html"><strong aria-hidden="true">3.2.1.</strong> Proofs</a></li><li class="chapter-item expanded "><a href="../design/implementation/fields.html"><strong aria-hidden="true">3.2.2.</strong> Fields</a></li></ol></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="
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<h1 id="polynomial-commitment-using-inner-product-argument"><a class="header" href="#polynomial-commitment-using-inner-product-argument">Polynomial commitment using inner product argument</a></h1>
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<p>We want to commit to some polynomial <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">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">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:1.036108em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathbb">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><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">p</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 class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">]</span></span></span></span>, and be able to provably
deploy: 6d7e9afdb73595993161440c4747510fa2aa7cb9
2021-01-11 12:55:22 -08:00
evaluate the committed polynomial at arbitrary points. The naive solution would be for the
prover to simply send the polynomial's coefficients to the verifier: however, this
requires <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:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> communication. Our polynomial commitment scheme gets the job done using
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> communication.</p>
deploy: 53ac47fceceba6ebc2c3d54d3a1f62d4b874b8db
2021-06-05 03:42:23 -07:00
<h3 id="setup"><a class="header" href="#setup"><code>Setup</code></a></h3>
deploy: 6d7e9afdb73595993161440c4747510fa2aa7cb9
2021-01-11 12:55:22 -08:00
<p>Given a parameter <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">d</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:1.043548em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><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 mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span> we generate the common reference string
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<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;">σ</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:1.036108em;vertical-align:-0.286108em;"></span><span class="mopen">(</span><span class="mord mathbb">G</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathbf">G</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 mathbb">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><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">p</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 class="mclose">)</span></span></span></span> defining certain constants for this
deploy: 6d7e9afdb73595993161440c4747510fa2aa7cb9
2021-01-11 12:55:22 -08:00
scheme:</p>
<ul>
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68889em;vertical-align:0em;"></span><span class="mord mathbb">G</span></span></span></span> is a group of prime order <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">p</span><span class="mpunct">;</span></span></span></span></li>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72521em;vertical-align:-0.0391em;"></span><span class="mord mathbf">G</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:0.849108em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbb">G</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 mathnormal mtight">d</span></span></span></span></span></span></span></span></span></span></span> is a vector 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 mathnormal">d</span></span></span></span> random group elements;</li>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72243em;vertical-align:-0.0391em;"></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:0.68889em;vertical-align:0em;"></span><span class="mord mathbb">G</span></span></span></span> is a random group element; and</li>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.974998em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathbb">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><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">p</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> is the finite field of order <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">p</span><span class="mord">.</span></span></span></span></li>
deploy: 6d7e9afdb73595993161440c4747510fa2aa7cb9
2021-01-11 12:55:22 -08:00
</ul>
deploy: 53ac47fceceba6ebc2c3d54d3a1f62d4b874b8db
2021-06-05 03:42:23 -07:00
<h3 id="commit"><a class="header" href="#commit"><code>Commit</code></a></h3>
deploy: 6d7e9afdb73595993161440c4747510fa2aa7cb9
2021-01-11 12:55:22 -08:00
<p>The Pedersen vector 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 text"><span class="mord">Commit</span></span></span></span></span> is defined as</p>
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<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="mord text"><span class="mord">Commit</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</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:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathbf">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathbf">G</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><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span></span></span></span></span></p>
<p>for some polynomial <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">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">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:1.036108em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathbb">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><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">p</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 class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">]</span></span></span></span> and some blinding factor
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</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:0.974998em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathbb">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><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">p</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 class="mord">.</span></span></span></span> Here, each element of the vector <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;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:0.31166399999999994em;"><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">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:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.974998em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathbb">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><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">p</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> is
deploy: 6d7e9afdb73595993161440c4747510fa2aa7cb9
2021-01-11 12:55:22 -08:00
the coefficient for the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathnormal">i</span></span></span></span>th degree term of <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">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mpunct">,</span></span></span></span> and <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">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span> is of maximal degree
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">d</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:0.64444em;vertical-align:0em;"></span><span class="mord">1.</span></span></span></span></p>
deploy: 53ac47fceceba6ebc2c3d54d3a1f62d4b874b8db
2021-06-05 03:42:23 -07:00
<h3 id="open-prover-and-openverify-verifier"><a class="header" href="#open-prover-and-openverify-verifier"><code>Open</code> (prover) and <code>OpenVerify</code> (verifier)</a></h3>
deploy: 6d7e9afdb73595993161440c4747510fa2aa7cb9
2021-01-11 12:55:22 -08:00
<p>The modified inner product argument is an argument of knowledge for the relation</p>
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.6799999999999997em;vertical-align:-0.5899999999999999em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0899999999999999em;"><span style="top:-3.6799999999999997em;"><span class="pstrut" style="height:3.6799999999999997em;"></span><span class="boxpad"><span class="mord"><span class="mord"><span class="mopen">{((</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">x</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="mclose">)</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord mathbf">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</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 class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mopen">⟨</span><span class="mord mathbf">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathbf">G</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="mopen">[</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mclose">]</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;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mopen">⟨</span><span class="mord mathbf">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathbf">b</span><span class="mclose">⟩}</span></span></span></span></span><span style="top:-3.09em;"><span class="pstrut" style="height:3.6799999999999997em;"></span><span class="stretchy fbox" style="height:1.6799999999999997em;border-style:solid;border-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5899999999999999em;"><span></span></span></span></span></span><span class="mpunct">,</span></span></span></span></span></p>
<p>where <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 mathbf">b</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="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><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">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">x</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 mtight"><span class="mord mathnormal mtight">d</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> is composed of increasing powers of the
deploy: 6d7e9afdb73595993161440c4747510fa2aa7cb9
2021-01-11 12:55:22 -08:00
evaluation point <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 class="mord">.</span></span></span></span> This allows a prover to demonstrate to a verifier that the
polynomial contained “inside” the 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> evaluates to <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> at <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">x</span><span class="mpunct">,</span></span></span></span> and moreover,
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
that the committed polynomial has maximum degree <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">d</span><span class="mord">−1.</span></span></span></span></p>
deploy: 6d7e9afdb73595993161440c4747510fa2aa7cb9
2021-01-11 12:55:22 -08:00
<p>The inner product argument proceeds in <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:0.03148em;">k</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:0.93858em;vertical-align:-0.24414em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.20696799999999996em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">d</span></span></span></span> rounds. For our purposes, it is
sufficient to know about its final outputs, while merely providing intuition about the
intermediate rounds. (Refer to Section 3 in the <a href="https://eprint.iacr.org/2019/1021.pdf">Halo</a> paper for a full explanation.)</p>
deploy: 20c347b269af58c878e2e4207a3b4ab0d9193c6e
2021-07-13 09:48:02 -07:00
<p>Before beginning the argument, the verifier selects a random group element <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> and sends it
to the prover. We initialize the argument at round <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:0.03148em;">k</span><span class="mpunct">,</span></span></span></span> with the vectors
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><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="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose 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:0.63888em;vertical-align:-0.19444em;"></span><span class="mord mathbf">a</span><span class="mpunct">,</span></span></span></span> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">G</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><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="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose 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:0.68611em;vertical-align:0em;"></span><span class="mord mathbf">G</span></span></span></span> and
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><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="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose 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:0.69444em;vertical-align:0em;"></span><span class="mord mathbf">b</span><span class="mord">.</span></span></span></span> In each round <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</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:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</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:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span></span></span></span>:</p>
deploy: 6d7e9afdb73595993161440c4747510fa2aa7cb9
2021-01-11 12:55:22 -08:00
<ul>
<li>the prover computes two values <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></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:0.311664em;"><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:0.05724em;">j</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> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;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:0.05724em;">j</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> by taking some inner product of
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><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="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">G</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><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="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><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="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span>. Note that are in some
sense &quot;cross-terms&quot;: the lower half 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 mathbf">a</span></span></span></span> is used with the higher half of
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord mathbf">G</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 mathbf">b</span></span></span></span>, and vice versa:</li>
deploy: 6d7e9afdb73595993161440c4747510fa2aa7cb9
2021-01-11 12:55:22 -08:00
</ul>
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.4096em;vertical-align:-1.4548em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.9548em;"><span style="top:-3.9548em;"><span class="pstrut" style="height:3.0448em;"></span><span class="mord"><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:0.311664em;"><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:0.05724em;">j</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 style="top:-2.25em;"><span class="pstrut" style="height:3.0448em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;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:0.05724em;">j</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 class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4548em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.9548em;"><span style="top:-3.9548em;"><span class="pstrut" style="height:3.0448em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mopen">⟨</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:1.0448em;"><span style="top:-2.3986920000000005em;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 mathbf mtight">lo</span></span></span></span><span style="top:-3.2198em;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="mopen mtight">(</span><span class="mord mathbf mtight">j</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013079999999999em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathbf">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.3986920000000005em;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 mathbf mtight">hi</span></span></span></span><span style="top:-3.2198em;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="mopen mtight">(</span><sp
deploy: 6d7e9afdb73595993161440c4747510fa2aa7cb9
2021-01-11 12:55:22 -08:00
<ul>
<li>the verifier issues a random challenge <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><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:0.05724em;">j</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>;</li>
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<li>the prover uses <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><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:0.05724em;">j</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> to compress the lower and higher halves of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><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="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span>,
deploy: 6d7e9afdb73595993161440c4747510fa2aa7cb9
2021-01-11 12:55:22 -08:00
thus producing a new vector of half the original length
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.938em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><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="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mclose 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:1.3461079999999999em;vertical-align:-0.3013079999999999em;"></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:1.0448em;"><span style="top:-2.3986920000000005em;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 mathbf mtight">hi</span></span></span></span><span style="top:-3.2198em;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="mopen mtight">(</span><span class="mord mathbf mtight">j</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013079999999999em;"><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><span class="base"><span class="strut" style="height:1.267211em;vertical-align:-0.403103em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.864108em;"><span style="top:-2.4330050000000005em;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:0.05724em;">j</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><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.403103em;"><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><span class="base"><span class="strut" style="height:1.3461079999999999em;vertical-align:-0.3013079999999999em;"></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:1.0448em;"><span style="top:-2.3986920000000005em;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 mathbf mtight">lo</span></span></span></span><span style="top:-3.2198em;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="mopen mtight">(</span><span class="mord
The vectors <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">G</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><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="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><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="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> are similarly compressed to give
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">G</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><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="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><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="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span>.</li>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><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="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">G</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><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="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><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="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> are input to the
next round <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</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:0.64444em;vertical-align:0em;"></span><span class="mord">1.</span></span></span></span></li>
deploy: 6d7e9afdb73595993161440c4747510fa2aa7cb9
2021-01-11 12:55:22 -08:00
</ul>
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<p>Note that at the end of the last round <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</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:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mpunct">,</span></span></span></span> we are left with <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:0.8879999999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><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="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span>,
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">G</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:0.8879999999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">G</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><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="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">b</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:1.0824399999999998em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathbf">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><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="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span> each of length 1. The intuition is that
deploy: 6d7e9afdb73595993161440c4747510fa2aa7cb9
2021-01-11 12:55:22 -08:00
these final scalars, together with the challenges <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><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:0.05724em;">j</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 class="mclose">}</span></span></span></span> and &quot;cross-terms&quot;
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mopen">{</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:0.311664em;"><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:0.05724em;">j</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 class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;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:0.05724em;">j</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 class="mclose">}</span></span></span></span> from each round, encode the compression in each round. Since the prover did
not know the challenges <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><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:0.05724em;">j</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 class="mclose">}</span></span></span></span> in advance, they would have been unable to manipulate
the round compressions. Thus, checking a constraint on these final terms should enforce
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
that the compression had been performed correctly, and that the original <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>
deploy: 6d7e9afdb73595993161440c4747510fa2aa7cb9
2021-01-11 12:55:22 -08:00
satisfied the relation before undergoing compression.</p>
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<p>Note that <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">G</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">b</span></span></span></span> are simply rearrangements of the publicly known <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 mathbf">G</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathbf">b</span><span class="mpunct">,</span></span></span></span>
deploy: 6d7e9afdb73595993161440c4747510fa2aa7cb9
2021-01-11 12:55:22 -08:00
with the round challenges <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><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:0.05724em;">j</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 class="mclose">}</span></span></span></span> mixed in: this means the verifier can compute <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">G</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">b</span></span></span></span>
independently and verify that the prover had provided those same values.</p>
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