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<ol class="chapter"><li class="chapter-item expanded "><a href="overview.html"><strong aria-hidden="true">1.</strong> Overview</a></li><li class="chapter-item expanded "><a href="quick_start.html"><strong aria-hidden="true">2.</strong> Quick start</a></li><li class="chapter-item expanded "><a href="trusted_setup.html"><strong aria-hidden="true">3.</strong> Trusted Setup</a></li><li class="chapter-item expanded "><a href="cryptographic_specification.html"><strong aria-hidden="true">4.</strong> Cryptographic Specification</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="system_invariants.html"><strong aria-hidden="true">4.1.</strong> System invariants</a></li><li class="chapter-item expanded "><a href="insertion.html"><strong aria-hidden="true">4.2.</strong> Insertion</a></li><li class="chapter-item expanded "><a href="circuit_and_gates.html"><strong aria-hidden="true">4.3.</strong> The Circuit and Gates</a></li><li class="chapter-item expanded "><a href="precomputation_and_updates.html"><strong aria-hidden="true">4.4.</strong> Precomputation and updates</a></li><li class="chapter-item expanded "><a href="fiat_shamir_transcript.html"><strong aria-hidden="true">4.5.</strong> The Fiat-Shamir Transcript</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.6.</strong> Proof generation</a></li><li class="chapter-item expanded "><a href="verification.html" class="active"><strong aria-hidden="true">4.7.</strong> Proof verification</a></li></ol></li><li class="chapter-item expanded "><a href="lagrange_basis_polynomial_commitment_tree.html"><strong aria-hidden="true">5.</strong> The Lagrange Basis Polynomial Commitment Tree</a></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">6.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">7.</strong> Ethereum contracts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">7.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">8.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">9.</strong> Credits</a></li></ol>
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<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
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<h1 id="proof-verification"><a class="header" href="#proof-verification">Proof verification</a></h1>
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<p>At a high level, the verifier does three main steps to verify a proof
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(<a href="proof_generation.html#466-the-proof">4.6.6</a>):</p>
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<h2 id="471-check-the-gate-openings"><a class="header" href="#471-check-the-gate-openings">4.7.1. Check the gate openings</a></h2>
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<p>Given the opening values which the prover claims to be evaluations of the
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column polynomials which correspond to the circuit, the verifier computes the
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linear combination of the evaluation of the openings based on each gate
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equations separated by the \(v\) challenge values. This linear combination
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must equal the product of the quotient opening and the evaluation of the
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vanishing polynomial, or the verifier will return false.</p>
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<h2 id="472-compute-the-multiopen-arguments-final_poly-and-final_poly_eval-values"><a class="header" href="#472-compute-the-multiopen-arguments-final_poly-and-final_poly_eval-values">4.7.2. Compute the multiopen argument's <code>final_poly</code> and <code>final_poly_eval</code> values</a></h2>
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<p>Leveraging the homomorphic properties of KZG commitments, the verifier
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reconstructs a commitment to the \(\mathsf{final}_\pi\) polynomial generated
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in the
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<a href="./proof_generation.html#466-multiopen-argument-round">prover's multiopen argument round</a>.</p>
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<p>The verifier also
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reconstructs \(\mathsf{final\_poly\_eval}\) which is the evaluation of
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\(\mathsf{final}\) at the \(x_3\) challenge.</p>
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<h2 id="473-perform-pairing-checks"><a class="header" href="#473-perform-pairing-checks">4.7.3. Perform pairing checks</a></h2>
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<p>A three-part pairing product is performed and the verifier returns true only if
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the result is 1.</p>
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<p>\(A * B * C == 1\)</p>
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<p>where:</p>
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<p>\(A = e(\mathsf{a}_1 + \mathsf{a}_2 + \mathsf{a}_3, [1]_2)\)</p>
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<p>\(\mathsf{a}_1 = C - [\mathsf{ci}]_1\)</p>
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<p>\(\mathsf{a}_2 = \chi_2(\mathsf{srs\_g1}[t] - [1]_1)\)</p>
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<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathsf">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;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">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathsf mtight">final</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;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">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0444em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord mathsf">final_poly_eval</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathsf">final</span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;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">1</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="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">s</span></span></span></span></p>
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<p>\(B = e(-[\mathsf{zi}]_1, [\mathsf{w}]_2)\)</p>
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<p>\(C = e(-[\mathsf{final}_\pi]_1 \cdot s, [1]_2 \cdot \tau)\)</p>
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<p>and \(s\) is a separator challenge extracted from the transcript.</p>
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