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<ol class="chapter"><li class="chapter-item expanded "><a href="0_0_zero_network.html"><strong aria-hidden="true">1.</strong> Zero Network</a></li><li class="chapter-item expanded "><a href="1_0_overview.html"><strong aria-hidden="true">2.</strong> Overview</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="1_1_what_is_privacy.html"><strong aria-hidden="true">2.1.</strong> What is Privacy</a></li><li class="chapter-item expanded "><a href="1_2_hide_transfer_amount.html"><strong aria-hidden="true">2.2.</strong> Hide Transfer Amount</a></li><li class="chapter-item expanded "><a href="1_3_gas_limit.html"><strong aria-hidden="true">2.3.</strong> Gas Limit</a></li><li class="chapter-item expanded "><a href="1_4_zero_knowledge_scheme.html"><strong aria-hidden="true">2.4.</strong> Zero Knowledge Scheme</a></li><li class="chapter-item expanded "><a href="1_5_transaction_constraints.html"><strong aria-hidden="true">2.5.</strong> Transaction Constraints</a></li></ol></li><li class="chapter-item expanded "><a href="2_0_transaction_constraints.html"><strong aria-hidden="true">3.</strong> Transaction Constraints</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="2_1_confidential_transfer.html"><strong aria-hidden="true">3.1.</strong> Confidential Transfer</a></li><li class="chapter-item expanded "><a href="2_2_confidential_smart_contract.html"><strong aria-hidden="true">3.2.</strong> Confidential Smart Contract</a></li></ol></li><li class="chapter-item expanded "><a href="3_0_primitive.html"><strong aria-hidden="true">4.</strong> Primitive</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="3_1_crypto.html"><strong aria-hidden="true">4.1.</strong> Crypto</a></li><li class="chapter-item expanded "><a href="3_2_jubjub.html"><strong aria-hidden="true">4.2.</strong> Jubjub</a></li><li class="chapter-item expanded "><a href="3_3_bls12_381.html"><strong aria-hidden="true">4.3.</strong> Bls12 381</a></li><li class="chapter-item expanded "><a href="3_4_elgamal.html"><strong aria-hidden="true">4.4.</strong> ElGamal</a></li><li class="chapter-item expanded "><a href="3_5_pairing.html"><strong aria-hidden="true">4.5.</strong> Pairing</a></li></ol></li><li class="chapter-item expanded "><a href="4_0_pallet.html"><strong aria-hidden="true">5.</strong> Pallet</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="4_1_plonk.html"><strong aria-hidden="true">5.1.</strong> Plonk</a></li><li class="chapter-item expanded "><a href="4_2_encrypted_balance.html"><strong aria-hidden="true">5.2.</strong> Encrypted Balance</a></li><li class="chapter-item expanded "><a href="4_3_confidential_transfer.html"><strong aria-hidden="true">5.3.</strong> Confidential Transfer</a></li></ol></li><li class="chapter-item expanded "><a href="5_0_related_tools.html"><strong aria-hidden="true">6.</strong> Related Tools</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="5_1_stealth_address.html"><strong aria-hidden="true">6.1.</strong> Stealth Address</a></li><li class="chapter-item expanded "><a href="5_2_pedersen_commitment.html"><strong aria-hidden="true">6.2.</strong> Pedersen Commitment</a></li><li class="chapter-item expanded "><a href="5_3_non_interactive_zero_knowlege_proof.html"><strong aria-hidden="true">6.3.</strong> Non Interactive Zero Knowledge Proof</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="5_3_1_qap.html" class="active"><strong aria-hidden="true">6.3.1.</strong> QAP</a></li><li class="chapter-item expanded "><a href="5_3_2_polynomial_commitment.html"><strong aria-hidden="true">6.3.2.</strong> Polynomial Commitment</a></li><li class="chapter-item expanded "><a href="5_3_3_homomorphic_encryption.html"><strong aria-hidden="true">6.3.3.</strong> Homomorphic Encryption</a></li></ol></li></ol></li><li class="chapter-item expanded "><a href="6_0_tutorial.html"><strong aria-hidden="true">7.</strong> Tutorial</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="6_1_plonk_pallet.html"><strong aria-hidden="true">7.1.</strong> pallet-plonk</a></li><li class="chapter-item expanded "><a href="6_2_confidential_transfer.html"><strong aria-hidden="true">7.2.</strong> confidential_transfer</a></li></ol></li><li class="chapter-item expanded "><a href="7_0_frequent_errors.html"><strong aria-hidden="true">8.</strong> Frequent Errors</a></li></ol>
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<h1 class="menu-title">Zero Network Documentation</h1>
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<h1 id="qap-quadratic-arithmetic-programs"><a class="header" href="#qap-quadratic-arithmetic-programs">QAP (Quadratic Arithmetic Programs)</a></h1>
<h2 id="abstract"><a class="header" href="#abstract">Abstract</a></h2>
<p>The <code>QAP</code> is the technology which converts <code>computation</code> to <code>polynomial groups</code>. With this, we can check whether the <code>computation</code> was executed correctly just factor the polynomial without execute <code>computation</code> again.</p>
<h2 id="details"><a class="header" href="#details">Details</a></h2>
<p>Let's take a look at details. I give a example. Let's prove that following computation was executed correctly.</p>
<p>$$a^2 \cdot b^2 = c.$$</p>
<p>Assume that <code>c</code> is public input. <code>a</code> and <code>b</code> are private input. Prove that knowledge of <code>a</code> and <code>b</code> satisfying above equation.</p>
<h3 id="flattening"><a class="header" href="#flattening">Flattening</a></h3>
<p>First of all, let's disassemble the <code>computation</code> to minimum form using multicative.</p>
<p>$$ 1: a * a = a^2 $$
$$ 2: b * b = b^2 $$
$$ 3: a^2 * b^2 = c $$</p>
<p>Now the computation was disassembled to three gate.</p>
<h3 id="r1cs"><a class="header" href="#r1cs">R1Cs</a></h3>
<p>As described, we have three multicative computation and want to check whether each steps are executed correctly to set constraint. Before that, we permute above characters as following.</p>
<p>$$ [a, a^2, b, b^2, c] -> [v, w, x, y, z] $$</p>
<p>And now, our computation can be expressed as following table. Left, Right and Output.</p>
<div class="table-wrapper"><table><thead><tr><th style="text-align: left">Gate</th><th style="text-align: left">L</th><th style="text-align: left">R</th><th style="text-align: left">O</th></tr></thead><tbody>
<tr><td style="text-align: left">1</td><td style="text-align: left">v</td><td style="text-align: left">v</td><td style="text-align: left">w</td></tr>
<tr><td style="text-align: left">2</td><td style="text-align: left">x</td><td style="text-align: left">x</td><td style="text-align: left">y</td></tr>
<tr><td style="text-align: left">3</td><td style="text-align: left">y</td><td style="text-align: left">w</td><td style="text-align: left">z</td></tr>
</tbody></table>
</div>
<h3 id="qap"><a class="header" href="#qap">QAP</a></h3>
<p>Let's express above table as polynomail groups. As example, express <code>v</code> polynomial on <code>L</code> column. <code>x</code> cordinate is <code>Gate</code> number and <code>y</code> cordinate is if that variable is used, it's going to be 1 and oserwise 0. In <code>L</code> column, <code>v</code> is only used <code>Gate</code> 1 so express as <code>(1, 1) (2, 0) (3, 0)</code>. Find polynomial using <a href="https://math.iitm.ac.in/public_html/sryedida/caimna/interpolation/lagrange.html"><code>Lagrange interpolation formula</code></a> for each variables.</p>
<p><em>L column polynomial</em></p>
<div class="table-wrapper"><table><thead><tr><th style="text-align: left">Variable</th><th style="text-align: left">Cordinate</th><th style="text-align: left">Polynomial</th><th style="text-align: left">Name</th></tr></thead><tbody>
<tr><td style="text-align: left">v</td><td style="text-align: left">(1, 1) (2, 0) (3, 0)</td><td style="text-align: left">$$ \frac{x^2}{2} - \frac{5x}{2} + 3 \ $$</td><td style="text-align: left">Lv</td></tr>
<tr><td style="text-align: left">w</td><td style="text-align: left">(1, 0) (2, 0) (3, 0)</td><td style="text-align: left">0</td><td style="text-align: left">Lw</td></tr>
<tr><td style="text-align: left">x</td><td style="text-align: left">(1, 0) (2, 1) (3, 0)</td><td style="text-align: left">$$ -x^2 + 4x - 3 $$</td><td style="text-align: left">Lx</td></tr>
<tr><td style="text-align: left">y</td><td style="text-align: left">(1, 0) (2, 0) (3, 1)</td><td style="text-align: left">$$ \frac{x^2}{2} - \frac{3x}{2} + 1 \ $$</td><td style="text-align: left">Ly</td></tr>
<tr><td style="text-align: left">z</td><td style="text-align: left">(1, 0) (2, 0) (3, 0)</td><td style="text-align: left">0</td><td style="text-align: left">Lz</td></tr>
</tbody></table>
</div>
<p>Above polynomial expresses the gate that variable uses. When we pass gate number to polynomial <code>v</code> $$ \frac{x^2}{2} - \frac{5x}{2} + 3 \ $$, we can know which gate the <code>v</code> is used. For example, we pass <code>1</code> to polynomial <code>v</code>, it returns <code>1</code> so the variable <code>v</code> is used on gate <code>1</code> but it returns <code>0</code> when we pass <code>2</code> and <code>3</code> so it's not used on these gate. When we add all polynomial <code>Lv + Lw + Lx + Ly + Lz = L(x)</code>, it returns <code>1</code> when we pass <code>1</code>, <code>2</code> and <code>3</code>. We do the same operation for each column and get polynomials as well.</p>
<ul>
<li>L(x)<br />
<code>Lv + Lw + Lx + Ly + Lz</code></li>
<li>R(x)<br />
<code>Rv + Rw + Rx + Ry + Rz</code></li>
<li>O(x)<br />
<code>Ov + Ow + Ox + Oy + Oz</code></li>
</ul>
<p>We can intruduce above polynomials when we decide the <code>computation</code>.</p>
<h3 id="proof"><a class="header" href="#proof">Proof</a></h3>
<p>From now on, we are going to prove the state ment. In here, we use <code>a = 2</code>, <code>b = 3</code> and <code>c = 36</code>. We can get actual value as following.</p>
<p>$$ [v, w, x, y, z] -> [2, 4, 3, 9, 36] $$</p>
<p>And we multiply above variables by for each polynomial. For example, <code>L(x)</code> is following.</p>
<p>$$ L(x) = v * Lv + w * Lw + x * Lx + y * Ly + z * Lz $$</p>
<p>When we pass the <code>1</code> to <code>L(x)</code>, we can get <code>v</code> because only <code>Lv</code> returns <code>1</code> and others return <code>0</code>. <code>R(x)</code> as well and <code>O(x)</code> returns <code>w</code> so following equation holds.</p>
<p>$$ L(1) * R(1) - O(1) = v * v - w = 2 * 2 - 4 = 0 $$</p>
<p>It corresponds the table we saw in <code>R1Cs</code> and above also holds the case <code>x = 2</code> and <code>x = 3</code>. When we want to prove the statement, we are going to make above polynomial with secret <code>[v, w, x, y, z] -> [2, 4, 3, 9, 36]</code> so polynomial would be integrated as one as following.</p>
<p>$$ L(x) = 2 * Lv + 4 * Lw + 3 * Lx + 9 * Ly + 36 * Lz $$
$$ R(x) = 2 * Rv + 4 * Rw + 3 * Rx + 9 * Ry + 36 * Rz $$
$$ O(x) = 2 * Ov + 4 * Ow + 3 * Ox + 9 * Oy + 36 * Oz $$
$$ L(x) * R(x) - O(x) = P(x) $$</p>
<p>We make the <code>P(x)</code> to prove the statement.</p>
<h3 id="verification"><a class="header" href="#verification">Verification</a></h3>
<p>We can know whether <code>computation</code> was executed correctly to devide <code>P(x)</code> with <code>(x - 1) * (x - 2) * (x - 3)</code>. If it's devided as following prover knows the secret <code>a</code> and <code>b</code> leading <code>c</code>.</p>
<p>$$ P(x) = (x - 1) * (x - 2) * (x - 3) * T(x) $$</p>
<h3 id="next"><a class="header" href="#next">Next</a></h3>
<p>In this section, we understood how to convert <code>computation</code> to <code>polynomial groups</code> but there are some possibility that <code>P(x)</code> was made without using secret. In addition to this, we can know the secret to factor the polynomial. Zk SNARKs addresses the former problem with <code>Polynomial Commitment</code> and latter problem with <code>homomorphic encryption</code>.</p>
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