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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 (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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 (TODO)</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 (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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<nav id="sidebar" class="sidebar" aria-label="Table of contents">
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<div class="sidebar-scrollbox">
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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" class="active"><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 (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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" class="active"><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 (TODO)</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 (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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@@ -171,7 +171,7 @@ logic.</p>
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<div class="table-wrapper"><table><thead><tr><th>Row</th><th>\(\mathsf{w}_0\)</th><th>\(\mathsf{w}_1\)</th><th>\(\mathsf{w}_2\)</th><th>\(\mathsf{key}\)</th><th>\(\mathsf{c}\)</th><th>\(\mathsf{q\_mimc}\)</th></tr></thead><tbody>
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<tr><td>0</td><td>\(\mathsf{id\_nul}\)</td><td>\(\mathsf{id\_trap}\)</td><td>\(\mathsf{ext\_nul}\)</td><td>\(\mathsf{w}_0[n] + \mathsf{w}_0[0] \)</td><td>\(\mathsf{cts}[0]\)</td><td>1</td></tr>
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<tr><td>1</td><td>\((\mathsf{w}_0[0] + \mathsf{c}[0]) ^ 7\)</td><td>\((\mathsf{w}_1[0] + \mathsf{key}[0] + \mathsf{c}[0]) ^ 7\)</td><td>\((\mathsf{w}_2[0] + \mathsf{key}[0] + \mathsf{c}[0]) ^ 7\)</td><td>\(\mathsf{w}_0[n] + \mathsf{w}_0[0] \)</td><td>\(\mathsf{cts}[1]\)</td><td>1</td></tr>
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<tr><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr>
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<tr><td>\ldots</td><td>\ldots</td><td>\ldots</td><td>\ldots</td><td>\ldots</td><td>\ldots</td><td>\ldots</td></tr>
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<tr><td>\(n\)</td><td>\((\mathsf{w}_0[n - 1] + \mathsf{c}[n - 1]) ^ 7\)</td><td>\((\mathsf{w}_1[n - 1] + \mathsf{key}[n - 1] + \mathsf{c}[n - 1]) ^ 7\)</td><td>\((\mathsf{w}_2[n - 1] + \mathsf{key}[n - 1] + \mathsf{c}[n - 1]) ^ 7\)</td><td>\(\mathsf{w}_0[n] + \mathsf{w}_0[0] \)</td><td>\(\mathsf{dummy}\)</td><td>0</td></tr>
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<tr><td>128</td><td>\(b\)</td><td>\(b\)</td><td>\(b\)</td><td>\(b\)</td><td>\(b\)</td><td>\(b\)</td></tr>
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</tbody></table>
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@@ -193,12 +193,12 @@ make it computationally infeasible to brute-force their polynomial commitments.<
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<p>To understand how the logic of the circuit is encoded, consider each row of the
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table as inputs to the linear combination of the following gates, which must
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evaluate to 0 for a valid proof to be generated. In effect:</p>
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<p>\(\mathsf{gate}_0(r) + ... + \mathsf{gate}_n(r) = 0\) must be true.</p>
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<p>\(\mathsf{gate}_0(r) + \ldots + \mathsf{gate}_n(r) = 0\) must be true.</p>
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<p>Each and every gate must evaluate to 0. It is not possible for the prover to
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cheat by having some gates evaluate to some value such that the total evaluates
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to 0, since the prover will be forced to separate each gate with a challenge
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that they cannot control. Internally, the equation is actually:</p>
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<p>\(\mathsf{gate}_0(r) \cdot v_0 + ... + \mathsf{gate}_n(r) \cdot v_n = 0\) must be true.</p>
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<p>\(\mathsf{gate}_0(r) \cdot v_0 + \ldots + \mathsf{gate}_n(r) \cdot v_n = 0\) must be true.</p>
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<p>where the \(v\) values are successive powers of the hash of the public
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inputs. The prover would have to break a strong hash function to choose the
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public inputs and \(v\) values in order to cheat.</p>
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<nav id="sidebar" class="sidebar" aria-label="Table of contents">
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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 (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html" class="active"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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 (TODO)</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 (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html" class="active"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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" class="active"><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 (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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" class="active"><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 (TODO)</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 (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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@@ -207,8 +207,8 @@ generated using a <a href="https://eprint.iacr.org/2017/1050.pdf">securely run t
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setup</a>.</p>
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<p>These points are defined as such:</p>
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<ul>
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<li>\(\mathsf{srs\_g1}\): \([g_1, g_1^{\tau}, ..., g_1^{\tau^{n + 1}}]\)</li>
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<li>\(\mathsf{srs\_g2}\): \([g_2, g_2^{\tau}, ..., g_2^{\tau^{n + 1}}]\)</li>
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<li>\(\mathsf{srs\_g1}\): \([g_1, g_1 \cdot {\tau}, \ldots, g_1 \cdot {\tau \cdot {n + 1}}]\)</li>
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<li>\(\mathsf{srs\_g2}\): \([g_2, g_2 \cdot {\tau}, \ldots, g_2 \cdot {\tau \cdot {n + 1}}]\)</li>
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</ul>
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<p>Where \(g_1\) is defined in 1.3 and \(g_2\) is defined in 1.4.</p>
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<h3 id="4-the-mimc7-hash-function"><a class="header" href="#4-the-mimc7-hash-function">4. The MiMC7 hash function</a></h3>
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@@ -248,7 +248,7 @@ modulo the field order of \(\mathbb{F}_r\).</li>
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<li>Return \(\mathsf{rd}[n - 1] + k\).</li>
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</ol>
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<h4 id="43-the-mimc7-multi_hash-algorithm"><a class="header" href="#43-the-mimc7-multi_hash-algorithm">4.3. The MiMC7 <code>multi_hash</code> algorithm</a></h4>
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<p>To hash multiple field elements \(x_0, ..., x_n\), we use the <code>multi_hash()</code>
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<p>To hash multiple field elements \(x_0, \ldots, x_n\), we use the <code>multi_hash()</code>
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algorithm. The inputs to <code>multi_hash()</code> are the array of said field elements
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and a key \(k\).</p>
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<ol>
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@@ -282,12 +282,15 @@ is crucial to understanding how the circuit construction works.</p>
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<p>Semcaulk uses the KZG commitment scheme described in
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<a href="https://www.iacr.org/archive/asiacrypt2010/6477178/6477178.pdf">KZG10</a>.</p>
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<p>Given a polynomial \(\phi\) with \(l\) coefficients
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\([\phi_0, ..., \phi_{l - 1}]\), one
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\([\phi_0, \ldots, \phi_{l - 1}]\), one
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can use \(\mathsf{srs\_g1}\) to
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produce a commitment in the form of a \(\mathbb{G}_1\) point, or
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\(\mathsf{srs\_g2}\) to produce a commitment in the form of a
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\(\mathbb{G}_2\) point.</p>
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<p>\(\mathsf{commit}(\phi, \mathsf{srs}) = \sum_{i=1}^{l} \mathsf{srs}[i] \cdot \phi_i \)</p>
|
||||
<p>An alternative notation for \(\mathsf{commit}\) is:</p>
|
||||
<p>\([\phi]_1\) where the commitment is a \(\mathbb{G}_1\) point or
|
||||
\([\phi]_2\) where the commitment is a \(\mathbb{G}_2\) point.</p>
|
||||
<h3 id="6-lagrange-basis-polynomials"><a class="header" href="#6-lagrange-basis-polynomials">6. Lagrange basis polynomials</a></h3>
|
||||
<p>Lagrange basis polynomials are an important concept and are used in several
|
||||
parts of the protocol. To understand them, we must first define roots of unity
|
||||
@@ -302,11 +305,11 @@ order \(p\) are field elements where for each element \(x\), \(x^n = 1\).</p>
|
||||
0x0000000000000000B3C4D79D41A91758CB49C3517C4604A520CFF123608FC9CB
|
||||
</code></pre>
|
||||
<p>Another name for the \(n\) roots of unity is the evaluation domain of size \(n\)
|
||||
for a given finite field. They are commonly denoted as \(\{1, \omega, ...,
|
||||
for a given finite field. They are commonly denoted as \(\{1, \omega, \ldots,
|
||||
\omega^{n-1}\}\).</p>
|
||||
<h4 id="62-lagrange-basis-polynomials"><a class="header" href="#62-lagrange-basis-polynomials">6.2. Lagrange basis polynomials</a></h4>
|
||||
<p>Given an evaluation domain of size \(n\), the Lagrange basis polynomials of
|
||||
this domain are the \(n\) polynomials \([L_0, ..., L_n]\) such that
|
||||
this domain are the \(n\) polynomials \([L_0, \ldots, L_n]\) such that
|
||||
\(L_i(\omega^{i - 1} = 1)\) and \(L_i(\omega^{j} = 0)\) for all
|
||||
\(j \neq i - 1\). For example:</p>
|
||||
<ul>
|
||||
@@ -332,7 +335,7 @@ capacity of the instance of Semacaulk in question. These elements are ordered
|
||||
with the users' identity commitments followed by nothing-up-my-sleeve values.</p>
|
||||
<p>An <em>empty accumulator</em> is simply a commitment to \(t\) nothing-up-my-sleeve
|
||||
values.</p>
|
||||
<p>Given the vector of values \([v_0, ..., v_t]\), the accumulator \(C\) is computed
|
||||
<p>Given the vector of values \([v_0, \ldots, v_t]\), the accumulator \(C\) is computed
|
||||
as such:</p>
|
||||
<p>\(\sum_{i=1}^{t} \mathsf{commit}(L_i, \mathsf{srs\_g1}) \cdot v_i\)</p>
|
||||
<h4 id="71-updating-the-accumulator"><a class="header" href="#71-updating-the-accumulator">7.1 Updating the accumulator</a></h4>
|
||||
@@ -361,8 +364,14 @@ function.</li>
|
||||
</ul>
|
||||
<h3 id="9-evaluation-domains"><a class="header" href="#9-evaluation-domains">9. Evaluation domains</a></h3>
|
||||
<h4 id="91-the-subgroup-domain"><a class="header" href="#91-the-subgroup-domain">9.1. The subgroup domain</a></h4>
|
||||
<p>An evaluation domain with size 128. We denote this constant as
|
||||
\(\mathsf{SUBGROUP\_SIZE}\).</p>
|
||||
<h4 id="92-the-extended-coset-domain"><a class="header" href="#92-the-extended-coset-domain">9.2. The extended coset domain</a></h4>
|
||||
<p>An evaluation domain with size 8 * 128. We denote this as
|
||||
\(\mathsf{EXTENDED\_DOMAIN\_FACTOR} * \mathsf{SUBGROUP\_SIZE} = 1024\).</p>
|
||||
<h4 id="93-the-table-domain"><a class="header" href="#93-the-table-domain">9.3. The table domain</a></h4>
|
||||
<p>An evaluation domain with a size of at least 1024. This is the upper limit on
|
||||
the number of elements that the accumulator can hold.</p>
|
||||
|
||||
</main>
|
||||
|
||||
|
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@@ -82,7 +82,7 @@
|
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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 (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html" class="active"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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222
fiat_shamir_transcript.html
Normal file
222
fiat_shamir_transcript.html
Normal file
@@ -0,0 +1,222 @@
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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 (TODO)</a></li><li class="chapter-item expanded "><a href="fiat_shamir_transcript.html" class="active"><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 (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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<h1 id="the-fiat-shamir-transcript"><a class="header" href="#the-fiat-shamir-transcript">The Fiat-Shamir Transcript</a></h1>
|
||||
<p>A transcript is an abstraction over the <a href="https://en.wikipedia.org/wiki/Fiat%E2%80%93Shamir_heuristic">Fiat-Shamir
|
||||
heuristic</a>. Both
|
||||
the prover and verifier use the transcript to deterministically generate
|
||||
<em>challenge variables</em> based on the public inputs and proof data.</p>
|
||||
<p>Another way to think about the transcript is as a state machine where the state
|
||||
is a single data buffer. Every time a challenge is requested, it hashes the
|
||||
buffer replaces the contents of the buffer with the hash, and returns a value
|
||||
derived from the hash. The transcript can also be updated with abitrary data by
|
||||
appending the update to the buffer.</p>
|
||||
<p>Our transcript implements this concept with the following functions:</p>
|
||||
<ul>
|
||||
<li><code>new_transcript</code>: returns a new transcript whose buffer is 32 bytes of <code>0</code>
|
||||
values.</li>
|
||||
<li><code>update_with_f</code>: accepts a single \(\mathbb{F}_r\) value, converts it to a
|
||||
big-endian byte array, and appends it to the buffer.</li>
|
||||
<li><code>update_with_g1</code>: accepts a single \(\mathbb{G}_1\) point, converts its
|
||||
\(x\) and \(y\) points to big-endian byte arrays, and appends them to the
|
||||
buffer in the aforementioned order.</li>
|
||||
<li><code>update_with_g2</code>: accepts a single \(\mathbb{G}_2\) point, converts its
|
||||
\(x_0\), \(x_1\), \(y_0\), and \(y_1\) points into big-endian byte
|
||||
arrays, and appends them to the buffer in the aforementioned order.</li>
|
||||
<li><code>get_challenge</code>: hashes the buffer with Keccak256, replaces the buffer with
|
||||
the hash, converts the hash into a \(\mathbb{F}_r\) element (treating it
|
||||
as a big-endian buffer), and returns the \(\mathbb{F}_r\) element.</li>
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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 (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html" class="active"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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 (TODO)</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 (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html" class="active"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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<ol class="chapter"><li class="chapter-item expanded "><a href="overview.html" class="active"><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 (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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<ol class="chapter"><li class="chapter-item expanded "><a href="overview.html" class="active"><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 (TODO)</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 (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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" class="active"><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 (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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" class="active"><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 (TODO)</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 (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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 (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html" class="active"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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 (TODO)</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 (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html" class="active"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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<ol class="chapter"><li class="chapter-item expanded "><a href="overview.html" class="active"><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 (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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<ol class="chapter"><li class="chapter-item expanded "><a href="overview.html" class="active"><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 (TODO)</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 (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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 (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html" class="active"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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 (TODO)</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 (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html" class="active"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
|
||||
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|
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||||
@@ -82,7 +82,7 @@
|
||||
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||||
<nav id="sidebar" class="sidebar" aria-label="Table of contents">
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<div class="sidebar-scrollbox">
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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" class="active"><strong aria-hidden="true">4.4.</strong> Precomputation and updates (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
|
||||
<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" class="active"><strong aria-hidden="true">4.4.</strong> Precomputation and updates (TODO)</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 (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
|
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@@ -163,15 +163,19 @@ precomputation.</p>
|
||||
<h2 id="precomputed-data"><a class="header" href="#precomputed-data">Precomputed data</a></h2>
|
||||
<p>Precomputed data consists of the following:</p>
|
||||
<ol>
|
||||
<li>\(\mathsf{mimc\_cts}\):</li>
|
||||
<li>\(\mathsf{mimc\_cts}\)</li>
|
||||
<li>\(\mathsf{mimc\_cts\_coset\_evals}\)</li>
|
||||
<li>\(\mathsf{zh\_inverse\_coset\_evals}\)</li>
|
||||
<li>\(\mathsf{q\_mimc}\) </li>
|
||||
<li>\(\mathsf{q\_mimc\_coset\_evals}\) </li>
|
||||
<li>\(\mathsf{l0\_coset\_evals}\) </li>
|
||||
<li>\(\mathsf{w_1\_mapping}\) </li>
|
||||
<li>\(\mathsf{w_2\_mapping}\) </li>
|
||||
<li>\({\mathsf{W}_1}^{i}\) </li>
|
||||
<li>\({\mathsf{W}_2}^{i}\) </li>
|
||||
</ol>
|
||||
<p>\({\mathsf{W}_1}^{i}\) and \({\mathsf{W}_2}^{i}\) are precomputed for the
|
||||
index \(i\), which denotes the secret position of the prover's identity
|
||||
commitment in the accumulator.</p>
|
||||
<h3 id="mathsfmimc_cts"><a class="header" href="#mathsfmimc_cts">\(\mathsf{mimc\_cts}\)</a></h3>
|
||||
<p>A polynomial over the multiplicative subgroup which evaluates to the MiMC7
|
||||
round constants at each root of unity. The subgroup size is the number of MiMC7
|
||||
@@ -181,15 +185,22 @@ rounds defined in
|
||||
<p>A vector of \(\mathbb{F}_r\) elements that are the evaluations of the MiMC7
|
||||
round constants over the extended coset (TODO: define what a coset is)</p>
|
||||
<h3 id="mathsfzh_inverse_coset_evals"><a class="header" href="#mathsfzh_inverse_coset_evals">\(\mathsf{zh\_inverse\_coset\_evals}\)</a></h3>
|
||||
<p>A polynomial over the extended coset domain which ... (TODO)</p>
|
||||
<p>A vector of \(\mathbb{F}_r\) elements that are the field inversions of the
|
||||
(coefficients of the vanishing polynomial over the coset??? TODO)</p>
|
||||
<h3 id="mathsfq_mimc"><a class="header" href="#mathsfq_mimc">\(\mathsf{q\_mimc}\)</a></h3>
|
||||
<p>A polynomial whose evaluations at the roots of unity over the subgroup domain
|
||||
of size 128 are \(n = 91\) <code>1</code> values, followed by zeroes. It represents the
|
||||
\(\mathsf{q\_mimc}\) <a href="./circuit_and_gates.html">selector column</a>.</p>
|
||||
<h3 id="mathsfq_mimc_coset_evals"><a class="header" href="#mathsfq_mimc_coset_evals">\(\mathsf{q\_mimc\_coset\_evals}\)</a></h3>
|
||||
<p>A vector of \(\mathbb{F}_r\) elements that are the evaluations of the
|
||||
\(\mathsf{q\_mimc}\) polynomial coefficients over the coset?? (TODO) </p>
|
||||
<h3 id="mathsfl0_coset_evals"><a class="header" href="#mathsfl0_coset_evals">\(\mathsf{l0\_coset\_evals}\)</a></h3>
|
||||
<h3 id="mathsfw_1_mapping"><a class="header" href="#mathsfw_1_mapping">\(\mathsf{w_1\_mapping}\)</a></h3>
|
||||
<h3 id="mathsfw_2_mapping"><a class="header" href="#mathsfw_2_mapping">\(\mathsf{w_2\_mapping}\)</a></h3>
|
||||
<p>Where \(L_0\) is the 0th Lagrange basis polynomial over the subgroup
|
||||
evaluation domain, this is a vector of its evaluations over the coset (?? TODO)</p>
|
||||
<h3 id="mathsfw_1i"><a class="header" href="#mathsfw_1i">\({\mathsf{W}_1}^{i}\)</a></h3>
|
||||
<p>As defined in the <a href="https://eprint.iacr.org/2022/957.pdf">Caulk+ paper, section 3</a>.</p>
|
||||
<h3 id="mathsfw_2i"><a class="header" href="#mathsfw_2i">\({\mathsf{W}_2}^{i}\)</a></h3>
|
||||
<p>As defined in the <a href="https://eprint.iacr.org/2022/957.pdf">Caulk+ paper, section 3</a>.</p>
|
||||
|
||||
</main>
|
||||
|
||||
@@ -199,7 +210,7 @@ of size 128 are \(n = 91\) <code>1</code> values, followed by zeroes. It represe
|
||||
<i class="fa fa-angle-left"></i>
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</a>
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<i class="fa fa-angle-right"></i>
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</a>
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|
||||
@@ -213,7 +224,7 @@ of size 128 are \(n = 91\) <code>1</code> values, followed by zeroes. It represe
|
||||
<i class="fa fa-angle-left"></i>
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</a>
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<a rel="next" href="proof_generation.html" class="nav-chapters next" title="Next chapter" aria-label="Next chapter" aria-keyshortcuts="Right">
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<a rel="next" href="fiat_shamir_transcript.html" class="nav-chapters next" title="Next chapter" aria-label="Next chapter" aria-keyshortcuts="Right">
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<i class="fa fa-angle-right"></i>
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</a>
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</nav>
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99
print.html
99
print.html
@@ -83,7 +83,7 @@
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<nav id="sidebar" class="sidebar" aria-label="Table of contents">
|
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<div class="sidebar-scrollbox">
|
||||
<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 (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
|
||||
<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 (TODO)</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 (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
|
||||
</div>
|
||||
<div id="sidebar-resize-handle" class="sidebar-resize-handle"></div>
|
||||
</nav>
|
||||
@@ -338,8 +338,8 @@ generated using a <a href="https://eprint.iacr.org/2017/1050.pdf">securely run t
|
||||
setup</a>.</p>
|
||||
<p>These points are defined as such:</p>
|
||||
<ul>
|
||||
<li>\(\mathsf{srs\_g1}\): \([g_1, g_1^{\tau}, ..., g_1^{\tau^{n + 1}}]\)</li>
|
||||
<li>\(\mathsf{srs\_g2}\): \([g_2, g_2^{\tau}, ..., g_2^{\tau^{n + 1}}]\)</li>
|
||||
<li>\(\mathsf{srs\_g1}\): \([g_1, g_1 \cdot {\tau}, \ldots, g_1 \cdot {\tau \cdot {n + 1}}]\)</li>
|
||||
<li>\(\mathsf{srs\_g2}\): \([g_2, g_2 \cdot {\tau}, \ldots, g_2 \cdot {\tau \cdot {n + 1}}]\)</li>
|
||||
</ul>
|
||||
<p>Where \(g_1\) is defined in 1.3 and \(g_2\) is defined in 1.4.</p>
|
||||
<h3 id="4-the-mimc7-hash-function"><a class="header" href="#4-the-mimc7-hash-function">4. The MiMC7 hash function</a></h3>
|
||||
@@ -379,7 +379,7 @@ modulo the field order of \(\mathbb{F}_r\).</li>
|
||||
<li>Return \(\mathsf{rd}[n - 1] + k\).</li>
|
||||
</ol>
|
||||
<h4 id="43-the-mimc7-multi_hash-algorithm"><a class="header" href="#43-the-mimc7-multi_hash-algorithm">4.3. The MiMC7 <code>multi_hash</code> algorithm</a></h4>
|
||||
<p>To hash multiple field elements \(x_0, ..., x_n\), we use the <code>multi_hash()</code>
|
||||
<p>To hash multiple field elements \(x_0, \ldots, x_n\), we use the <code>multi_hash()</code>
|
||||
algorithm. The inputs to <code>multi_hash()</code> are the array of said field elements
|
||||
and a key \(k\).</p>
|
||||
<ol>
|
||||
@@ -413,12 +413,15 @@ is crucial to understanding how the circuit construction works.</p>
|
||||
<p>Semcaulk uses the KZG commitment scheme described in
|
||||
<a href="https://www.iacr.org/archive/asiacrypt2010/6477178/6477178.pdf">KZG10</a>.</p>
|
||||
<p>Given a polynomial \(\phi\) with \(l\) coefficients
|
||||
\([\phi_0, ..., \phi_{l - 1}]\), one
|
||||
\([\phi_0, \ldots, \phi_{l - 1}]\), one
|
||||
can use \(\mathsf{srs\_g1}\) to
|
||||
produce a commitment in the form of a \(\mathbb{G}_1\) point, or
|
||||
\(\mathsf{srs\_g2}\) to produce a commitment in the form of a
|
||||
\(\mathbb{G}_2\) point.</p>
|
||||
<p>\(\mathsf{commit}(\phi, \mathsf{srs}) = \sum_{i=1}^{l} \mathsf{srs}[i] \cdot \phi_i \)</p>
|
||||
<p>An alternative notation for \(\mathsf{commit}\) is:</p>
|
||||
<p>\([\phi]_1\) where the commitment is a \(\mathbb{G}_1\) point or
|
||||
\([\phi]_2\) where the commitment is a \(\mathbb{G}_2\) point.</p>
|
||||
<h3 id="6-lagrange-basis-polynomials"><a class="header" href="#6-lagrange-basis-polynomials">6. Lagrange basis polynomials</a></h3>
|
||||
<p>Lagrange basis polynomials are an important concept and are used in several
|
||||
parts of the protocol. To understand them, we must first define roots of unity
|
||||
@@ -433,11 +436,11 @@ order \(p\) are field elements where for each element \(x\), \(x^n = 1\).</p>
|
||||
0x0000000000000000B3C4D79D41A91758CB49C3517C4604A520CFF123608FC9CB
|
||||
</code></pre>
|
||||
<p>Another name for the \(n\) roots of unity is the evaluation domain of size \(n\)
|
||||
for a given finite field. They are commonly denoted as \(\{1, \omega, ...,
|
||||
for a given finite field. They are commonly denoted as \(\{1, \omega, \ldots,
|
||||
\omega^{n-1}\}\).</p>
|
||||
<h4 id="62-lagrange-basis-polynomials"><a class="header" href="#62-lagrange-basis-polynomials">6.2. Lagrange basis polynomials</a></h4>
|
||||
<p>Given an evaluation domain of size \(n\), the Lagrange basis polynomials of
|
||||
this domain are the \(n\) polynomials \([L_0, ..., L_n]\) such that
|
||||
this domain are the \(n\) polynomials \([L_0, \ldots, L_n]\) such that
|
||||
\(L_i(\omega^{i - 1} = 1)\) and \(L_i(\omega^{j} = 0)\) for all
|
||||
\(j \neq i - 1\). For example:</p>
|
||||
<ul>
|
||||
@@ -463,7 +466,7 @@ capacity of the instance of Semacaulk in question. These elements are ordered
|
||||
with the users' identity commitments followed by nothing-up-my-sleeve values.</p>
|
||||
<p>An <em>empty accumulator</em> is simply a commitment to \(t\) nothing-up-my-sleeve
|
||||
values.</p>
|
||||
<p>Given the vector of values \([v_0, ..., v_t]\), the accumulator \(C\) is computed
|
||||
<p>Given the vector of values \([v_0, \ldots, v_t]\), the accumulator \(C\) is computed
|
||||
as such:</p>
|
||||
<p>\(\sum_{i=1}^{t} \mathsf{commit}(L_i, \mathsf{srs\_g1}) \cdot v_i\)</p>
|
||||
<h4 id="71-updating-the-accumulator"><a class="header" href="#71-updating-the-accumulator">7.1 Updating the accumulator</a></h4>
|
||||
@@ -492,8 +495,14 @@ function.</li>
|
||||
</ul>
|
||||
<h3 id="9-evaluation-domains"><a class="header" href="#9-evaluation-domains">9. Evaluation domains</a></h3>
|
||||
<h4 id="91-the-subgroup-domain"><a class="header" href="#91-the-subgroup-domain">9.1. The subgroup domain</a></h4>
|
||||
<p>An evaluation domain with size 128. We denote this constant as
|
||||
\(\mathsf{SUBGROUP\_SIZE}\).</p>
|
||||
<h4 id="92-the-extended-coset-domain"><a class="header" href="#92-the-extended-coset-domain">9.2. The extended coset domain</a></h4>
|
||||
<p>An evaluation domain with size 8 * 128. We denote this as
|
||||
\(\mathsf{EXTENDED\_DOMAIN\_FACTOR} * \mathsf{SUBGROUP\_SIZE} = 1024\).</p>
|
||||
<h4 id="93-the-table-domain"><a class="header" href="#93-the-table-domain">9.3. The table domain</a></h4>
|
||||
<p>An evaluation domain with a size of at least 1024. This is the upper limit on
|
||||
the number of elements that the accumulator can hold.</p>
|
||||
<div style="break-before: page; page-break-before: always;"></div><h1 id="system-invariants"><a class="header" href="#system-invariants">System invariants</a></h1>
|
||||
<p>An <a href="https://mathworld.wolfram.com/Invariant.html">invariant</a> is a property of a
|
||||
system which remains unmodified even after operations or transformations are
|
||||
@@ -562,7 +571,7 @@ logic.</p>
|
||||
<div class="table-wrapper"><table><thead><tr><th>Row</th><th>\(\mathsf{w}_0\)</th><th>\(\mathsf{w}_1\)</th><th>\(\mathsf{w}_2\)</th><th>\(\mathsf{key}\)</th><th>\(\mathsf{c}\)</th><th>\(\mathsf{q\_mimc}\)</th></tr></thead><tbody>
|
||||
<tr><td>0</td><td>\(\mathsf{id\_nul}\)</td><td>\(\mathsf{id\_trap}\)</td><td>\(\mathsf{ext\_nul}\)</td><td>\(\mathsf{w}_0[n] + \mathsf{w}_0[0] \)</td><td>\(\mathsf{cts}[0]\)</td><td>1</td></tr>
|
||||
<tr><td>1</td><td>\((\mathsf{w}_0[0] + \mathsf{c}[0]) ^ 7\)</td><td>\((\mathsf{w}_1[0] + \mathsf{key}[0] + \mathsf{c}[0]) ^ 7\)</td><td>\((\mathsf{w}_2[0] + \mathsf{key}[0] + \mathsf{c}[0]) ^ 7\)</td><td>\(\mathsf{w}_0[n] + \mathsf{w}_0[0] \)</td><td>\(\mathsf{cts}[1]\)</td><td>1</td></tr>
|
||||
<tr><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr>
|
||||
<tr><td>\ldots</td><td>\ldots</td><td>\ldots</td><td>\ldots</td><td>\ldots</td><td>\ldots</td><td>\ldots</td></tr>
|
||||
<tr><td>\(n\)</td><td>\((\mathsf{w}_0[n - 1] + \mathsf{c}[n - 1]) ^ 7\)</td><td>\((\mathsf{w}_1[n - 1] + \mathsf{key}[n - 1] + \mathsf{c}[n - 1]) ^ 7\)</td><td>\((\mathsf{w}_2[n - 1] + \mathsf{key}[n - 1] + \mathsf{c}[n - 1]) ^ 7\)</td><td>\(\mathsf{w}_0[n] + \mathsf{w}_0[0] \)</td><td>\(\mathsf{dummy}\)</td><td>0</td></tr>
|
||||
<tr><td>128</td><td>\(b\)</td><td>\(b\)</td><td>\(b\)</td><td>\(b\)</td><td>\(b\)</td><td>\(b\)</td></tr>
|
||||
</tbody></table>
|
||||
@@ -584,12 +593,12 @@ make it computationally infeasible to brute-force their polynomial commitments.<
|
||||
<p>To understand how the logic of the circuit is encoded, consider each row of the
|
||||
table as inputs to the linear combination of the following gates, which must
|
||||
evaluate to 0 for a valid proof to be generated. In effect:</p>
|
||||
<p>\(\mathsf{gate}_0(r) + ... + \mathsf{gate}_n(r) = 0\) must be true.</p>
|
||||
<p>\(\mathsf{gate}_0(r) + \ldots + \mathsf{gate}_n(r) = 0\) must be true.</p>
|
||||
<p>Each and every gate must evaluate to 0. It is not possible for the prover to
|
||||
cheat by having some gates evaluate to some value such that the total evaluates
|
||||
to 0, since the prover will be forced to separate each gate with a challenge
|
||||
that they cannot control. Internally, the equation is actually:</p>
|
||||
<p>\(\mathsf{gate}_0(r) \cdot v_0 + ... + \mathsf{gate}_n(r) \cdot v_n = 0\) must be true.</p>
|
||||
<p>\(\mathsf{gate}_0(r) \cdot v_0 + \ldots + \mathsf{gate}_n(r) \cdot v_n = 0\) must be true.</p>
|
||||
<p>where the \(v\) values are successive powers of the hash of the public
|
||||
inputs. The prover would have to break a strong hash function to choose the
|
||||
public inputs and \(v\) values in order to cheat.</p>
|
||||
@@ -684,15 +693,19 @@ precomputation.</p>
|
||||
<h2 id="precomputed-data"><a class="header" href="#precomputed-data">Precomputed data</a></h2>
|
||||
<p>Precomputed data consists of the following:</p>
|
||||
<ol>
|
||||
<li>\(\mathsf{mimc\_cts}\):</li>
|
||||
<li>\(\mathsf{mimc\_cts}\)</li>
|
||||
<li>\(\mathsf{mimc\_cts\_coset\_evals}\)</li>
|
||||
<li>\(\mathsf{zh\_inverse\_coset\_evals}\)</li>
|
||||
<li>\(\mathsf{q\_mimc}\) </li>
|
||||
<li>\(\mathsf{q\_mimc\_coset\_evals}\) </li>
|
||||
<li>\(\mathsf{l0\_coset\_evals}\) </li>
|
||||
<li>\(\mathsf{w_1\_mapping}\) </li>
|
||||
<li>\(\mathsf{w_2\_mapping}\) </li>
|
||||
<li>\({\mathsf{W}_1}^{i}\) </li>
|
||||
<li>\({\mathsf{W}_2}^{i}\) </li>
|
||||
</ol>
|
||||
<p>\({\mathsf{W}_1}^{i}\) and \({\mathsf{W}_2}^{i}\) are precomputed for the
|
||||
index \(i\), which denotes the secret position of the prover's identity
|
||||
commitment in the accumulator.</p>
|
||||
<h3 id="mathsfmimc_cts"><a class="header" href="#mathsfmimc_cts">\(\mathsf{mimc\_cts}\)</a></h3>
|
||||
<p>A polynomial over the multiplicative subgroup which evaluates to the MiMC7
|
||||
round constants at each root of unity. The subgroup size is the number of MiMC7
|
||||
@@ -702,16 +715,70 @@ rounds defined in
|
||||
<p>A vector of \(\mathbb{F}_r\) elements that are the evaluations of the MiMC7
|
||||
round constants over the extended coset (TODO: define what a coset is)</p>
|
||||
<h3 id="mathsfzh_inverse_coset_evals"><a class="header" href="#mathsfzh_inverse_coset_evals">\(\mathsf{zh\_inverse\_coset\_evals}\)</a></h3>
|
||||
<p>A polynomial over the extended coset domain which ... (TODO)</p>
|
||||
<p>A vector of \(\mathbb{F}_r\) elements that are the field inversions of the
|
||||
(coefficients of the vanishing polynomial over the coset??? TODO)</p>
|
||||
<h3 id="mathsfq_mimc"><a class="header" href="#mathsfq_mimc">\(\mathsf{q\_mimc}\)</a></h3>
|
||||
<p>A polynomial whose evaluations at the roots of unity over the subgroup domain
|
||||
of size 128 are \(n = 91\) <code>1</code> values, followed by zeroes. It represents the
|
||||
\(\mathsf{q\_mimc}\) <a href="./circuit_and_gates.html">selector column</a>.</p>
|
||||
<h3 id="mathsfq_mimc_coset_evals"><a class="header" href="#mathsfq_mimc_coset_evals">\(\mathsf{q\_mimc\_coset\_evals}\)</a></h3>
|
||||
<p>A vector of \(\mathbb{F}_r\) elements that are the evaluations of the
|
||||
\(\mathsf{q\_mimc}\) polynomial coefficients over the coset?? (TODO) </p>
|
||||
<h3 id="mathsfl0_coset_evals"><a class="header" href="#mathsfl0_coset_evals">\(\mathsf{l0\_coset\_evals}\)</a></h3>
|
||||
<h3 id="mathsfw_1_mapping"><a class="header" href="#mathsfw_1_mapping">\(\mathsf{w_1\_mapping}\)</a></h3>
|
||||
<h3 id="mathsfw_2_mapping"><a class="header" href="#mathsfw_2_mapping">\(\mathsf{w_2\_mapping}\)</a></h3>
|
||||
<p>Where \(L_0\) is the 0th Lagrange basis polynomial over the subgroup
|
||||
evaluation domain, this is a vector of its evaluations over the coset (?? TODO)</p>
|
||||
<h3 id="mathsfw_1i"><a class="header" href="#mathsfw_1i">\({\mathsf{W}_1}^{i}\)</a></h3>
|
||||
<p>As defined in the <a href="https://eprint.iacr.org/2022/957.pdf">Caulk+ paper, section 3</a>.</p>
|
||||
<h3 id="mathsfw_2i"><a class="header" href="#mathsfw_2i">\({\mathsf{W}_2}^{i}\)</a></h3>
|
||||
<p>As defined in the <a href="https://eprint.iacr.org/2022/957.pdf">Caulk+ paper, section 3</a>.</p>
|
||||
<div style="break-before: page; page-break-before: always;"></div><h1 id="the-fiat-shamir-transcript"><a class="header" href="#the-fiat-shamir-transcript">The Fiat-Shamir Transcript</a></h1>
|
||||
<p>A transcript is an abstraction over the <a href="https://en.wikipedia.org/wiki/Fiat%E2%80%93Shamir_heuristic">Fiat-Shamir
|
||||
heuristic</a>. Both
|
||||
the prover and verifier use the transcript to deterministically generate
|
||||
<em>challenge variables</em> based on the public inputs and proof data.</p>
|
||||
<p>Another way to think about the transcript is as a state machine where the state
|
||||
is a single data buffer. Every time a challenge is requested, it hashes the
|
||||
buffer replaces the contents of the buffer with the hash, and returns a value
|
||||
derived from the hash. The transcript can also be updated with abitrary data by
|
||||
appending the update to the buffer.</p>
|
||||
<p>Our transcript implements this concept with the following functions:</p>
|
||||
<ul>
|
||||
<li><code>new_transcript</code>: returns a new transcript whose buffer is 32 bytes of <code>0</code>
|
||||
values.</li>
|
||||
<li><code>update_with_f</code>: accepts a single \(\mathbb{F}_r\) value, converts it to a
|
||||
big-endian byte array, and appends it to the buffer.</li>
|
||||
<li><code>update_with_g1</code>: accepts a single \(\mathbb{G}_1\) point, converts its
|
||||
\(x\) and \(y\) points to big-endian byte arrays, and appends them to the
|
||||
buffer in the aforementioned order.</li>
|
||||
<li><code>update_with_g2</code>: accepts a single \(\mathbb{G}_2\) point, converts its
|
||||
\(x_0\), \(x_1\), \(y_0\), and \(y_1\) points into big-endian byte
|
||||
arrays, and appends them to the buffer in the aforementioned order.</li>
|
||||
<li><code>get_challenge</code>: hashes the buffer with Keccak256, replaces the buffer with
|
||||
the hash, converts the hash into a \(\mathbb{F}_r\) element (treating it
|
||||
as a big-endian buffer), and returns the \(\mathbb{F}_r\) element.</li>
|
||||
</ul>
|
||||
<div style="break-before: page; page-break-before: always;"></div><h1 id="proof-generation"><a class="header" href="#proof-generation">Proof generation</a></h1>
|
||||
<h2 id="1-assignment-round"><a class="header" href="#1-assignment-round">1. Assignment Round</a></h2>
|
||||
<p>Given the public and private inputs, the prover generates the assignment table
|
||||
(see <a href="./circuit_and_gates.html">4.3</a>). Each column is represented as a
|
||||
polynomial over the multiplicative subgroup generated via Lagrange
|
||||
interpolation of each row value as a coefficient (TODO: check this language).</p>
|
||||
<ul>
|
||||
<li>\(\mathsf{w}_0\)</li>
|
||||
<li>\(\mathsf{w}_1\)</li>
|
||||
<li>\(\mathsf{w}_2\)</li>
|
||||
<li>\(\mathsf{key}\)</li>
|
||||
</ul>
|
||||
<p>The prover then computes KZG commitments to each of the above polynomials:</p>
|
||||
<ul>
|
||||
<li>\([\mathsf{w}_0]_1\)</li>
|
||||
<li>\([\mathsf{w}_1]_1\)</li>
|
||||
<li>\([\mathsf{w}_2]_1\)</li>
|
||||
<li>\([\mathsf{key}]_1\)</li>
|
||||
</ul>
|
||||
<p>The prover also computes:</p>
|
||||
<p>\(A = \mathsf{w}_1 + \mathsf{w}_1(\gamma^{91}X) + 2 \cdot \mathsf{key}\)</p>
|
||||
<p>where \(\mathsf{w}_1(\gamma^{91}X)\) is .... TODO</p>
|
||||
<div style="break-before: page; page-break-before: always;"></div><h1 id="proof-verification"><a class="header" href="#proof-verification">Proof verification</a></h1>
|
||||
<div style="break-before: page; page-break-before: always;"></div><h1 id="mechanism-of-operation"><a class="header" href="#mechanism-of-operation">Mechanism of Operation</a></h1>
|
||||
<p>The goal of Semaphore is to enable users to:</p>
|
||||
|
||||
@@ -82,7 +82,7 @@
|
||||
|
||||
<nav id="sidebar" class="sidebar" aria-label="Table of contents">
|
||||
<div class="sidebar-scrollbox">
|
||||
<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 (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html" class="active"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
|
||||
<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 (TODO)</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" class="active"><strong aria-hidden="true">4.6.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
|
||||
</div>
|
||||
<div id="sidebar-resize-handle" class="sidebar-resize-handle"></div>
|
||||
</nav>
|
||||
@@ -144,12 +144,33 @@
|
||||
<div id="content" class="content">
|
||||
<main>
|
||||
<h1 id="proof-generation"><a class="header" href="#proof-generation">Proof generation</a></h1>
|
||||
<h2 id="1-assignment-round"><a class="header" href="#1-assignment-round">1. Assignment Round</a></h2>
|
||||
<p>Given the public and private inputs, the prover generates the assignment table
|
||||
(see <a href="./circuit_and_gates.html">4.3</a>). Each column is represented as a
|
||||
polynomial over the multiplicative subgroup generated via Lagrange
|
||||
interpolation of each row value as a coefficient (TODO: check this language).</p>
|
||||
<ul>
|
||||
<li>\(\mathsf{w}_0\)</li>
|
||||
<li>\(\mathsf{w}_1\)</li>
|
||||
<li>\(\mathsf{w}_2\)</li>
|
||||
<li>\(\mathsf{key}\)</li>
|
||||
</ul>
|
||||
<p>The prover then computes KZG commitments to each of the above polynomials:</p>
|
||||
<ul>
|
||||
<li>\([\mathsf{w}_0]_1\)</li>
|
||||
<li>\([\mathsf{w}_1]_1\)</li>
|
||||
<li>\([\mathsf{w}_2]_1\)</li>
|
||||
<li>\([\mathsf{key}]_1\)</li>
|
||||
</ul>
|
||||
<p>The prover also computes:</p>
|
||||
<p>\(A = \mathsf{w}_1 + \mathsf{w}_1(\gamma^{91}X) + 2 \cdot \mathsf{key}\)</p>
|
||||
<p>where \(\mathsf{w}_1(\gamma^{91}X)\) is .... TODO</p>
|
||||
|
||||
</main>
|
||||
|
||||
<nav class="nav-wrapper" aria-label="Page navigation">
|
||||
<!-- Mobile navigation buttons -->
|
||||
<a rel="prev" href="precomputation_and_updates.html" class="mobile-nav-chapters previous" title="Previous chapter" aria-label="Previous chapter" aria-keyshortcuts="Left">
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||||
<a rel="prev" href="fiat_shamir_transcript.html" class="mobile-nav-chapters previous" title="Previous chapter" aria-label="Previous chapter" aria-keyshortcuts="Left">
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||||
<i class="fa fa-angle-left"></i>
|
||||
</a>
|
||||
|
||||
@@ -163,7 +184,7 @@
|
||||
</div>
|
||||
|
||||
<nav class="nav-wide-wrapper" aria-label="Page navigation">
|
||||
<a rel="prev" href="precomputation_and_updates.html" class="nav-chapters previous" title="Previous chapter" aria-label="Previous chapter" aria-keyshortcuts="Left">
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<a rel="prev" href="fiat_shamir_transcript.html" class="nav-chapters previous" title="Previous chapter" aria-label="Previous chapter" aria-keyshortcuts="Left">
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<i class="fa fa-angle-left"></i>
|
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</a>
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|
||||
|
||||
@@ -82,7 +82,7 @@
|
||||
|
||||
<nav id="sidebar" class="sidebar" aria-label="Table of contents">
|
||||
<div class="sidebar-scrollbox">
|
||||
<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" class="active"><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 (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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" class="active"><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 (TODO)</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 (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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" class="active"><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 (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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" class="active"><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 (TODO)</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 (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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" class="active"><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 (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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" class="active"><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 (TODO)</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 (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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 (TODO)</a></li><li class="chapter-item expanded "><a href="proof_generation.html"><strong aria-hidden="true">4.5.</strong> Proof generation (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html" class="active"><strong aria-hidden="true">4.6.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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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 (TODO)</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 (TODO)</a></li><li class="chapter-item expanded "><a href="verification.html" class="active"><strong aria-hidden="true">4.7.</strong> Proof verification (TODO)</a></li></ol></li><li class="chapter-item expanded "><a href="mechanism_of_operation.html"><strong aria-hidden="true">5.</strong> Mechanism of Operation</a></li><li class="chapter-item expanded "><a href="ethereum_contracts.html"><strong aria-hidden="true">6.</strong> Ethereum contracts (TODO)</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="gas_costs.html"><strong aria-hidden="true">6.1.</strong> Gas costs</a></li></ol></li><li class="chapter-item expanded "><a href="performance_benchmarks.html"><strong aria-hidden="true">7.</strong> Performance and Benchmarks</a></li><li class="chapter-item expanded "><a href="credits.html"><strong aria-hidden="true">8.</strong> Credits</a></li></ol>
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