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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" class="active"><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"><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="pedersen-commitment"><a class="header" href="#pedersen-commitment">Pedersen Commitment</a></h1>
<h2 id="abstract"><a class="header" href="#abstract">Abstract</a></h2>
<p>The <code>Pedersen Commitment</code> is the technology which allows us to check the transfer amount is valid without revealing actual amount.</p>
<h2 id="details"><a class="header" href="#details">Details</a></h2>
<p>This technology uses additivity of elliptic curve. Hiding the transfer amount as scalar of elliptic curve point and mixing random value refer as to <code>blinding factor</code>. Let's take a look the squence.</p>
<ol>
<li>Setup the parameters</li>
<li>Hide the transfer amount</li>
<li>Verify the transfer amount</li>
</ol>
<p>Above sequence used for confidential transfer to keep the transfer amount secret.</p>
<h2 id="setup-the-parameters"><a class="header" href="#setup-the-parameters">Setup The Parameters</a></h2>
<p>First of all, we'd like to setup the parameters we are going to use with the <code>Pedersen Commitment</code>. Selecting generator <code>G</code> over prime order elliptic curve group and randomness <code>a</code> less than order prime. Calculating <code>H = aG</code> and making <code>H</code> and <code>G</code> public. We can't predict the <code>a</code> value from <code>H</code> and <code>G</code> because of discrete logarithm.</p>
<div class="table-wrapper"><table><thead><tr><th style="text-align: left">Variable</th><th style="text-align: left">Explanation</th><th style="text-align: left">Derivation</th></tr></thead><tbody>
<tr><td style="text-align: left">p</td><td style="text-align: left">prime number</td><td style="text-align: left">-</td></tr>
<tr><td style="text-align: left">a</td><td style="text-align: left">random number</td><td style="text-align: left">a ∈ Fp</td></tr>
<tr><td style="text-align: left">C(x)</td><td style="text-align: left">elliptic curve function</td><td style="text-align: left">-</td></tr>
<tr><td style="text-align: left">G</td><td style="text-align: left">generator of elliptic curve point</td><td style="text-align: left">G ∈ C(Fp)</td></tr>
<tr><td style="text-align: left">H</td><td style="text-align: left">generator made by random a</td><td style="text-align: left">H = aG</td></tr>
</tbody></table>
</div>
<h2 id="hide-the-transfer-amount"><a class="header" href="#hide-the-transfer-amount">Hide The Transfer Amount</a></h2>
<p>Let's assume that Alice has <code>10</code> balance and send Bob to <code>3</code>. We need to check <code>{Alice balance} - {transfer amount} = {Alice after balance}</code> without revealing any information about Alice balance. Alice knows <code>H</code>, <code>G</code> and her balance so she computes following value.</p>
<ul>
<li>Alice balance commitment
$$ C(10) = 10H + x_1G $$</li>
<li>Transfer amount commitment
$$ C(3) = 3H + x_2G $$</li>
<li>Alice after balance commitment
$$ C(10 - 3) = 7H + x_3G $$</li>
</ul>
<p>In above equation, <code>x_1 ~ x_3</code> are called <code>blinding factor</code> and each blinding factor need to be set to hold <code>{Alice balance} - {transfer amount} = {Alice after balance}</code> equation. Let's say <code>x_1 = 330</code>, <code>x_2 = 30</code> and <code>x_3 = 300</code>.</p>
<h2 id="verify-the-transfer-amount"><a class="header" href="#verify-the-transfer-amount">Verify The Transfer Amount</a></h2>
<p>In previous section, Alice computes the following commitment.</p>
<p>$$ C(10, 330) = 10H + 330G $$
$$ C(3, 30) = 3H + 30G $$
$$ C(10 - 3, 300) = 7H + 300G $$</p>
<p>Let's check if Alice transfer amount is valid. The elliptic curve arithmetic supports additive so we can check <code>{Alice balance} - {transfer amount} = {Alice after balance}</code> as following.</p>
<p>$$ C(10, 330) - C(3, 30) - C(10 - 3, 300) = C(10 - 3 - 7, 330 - 30 - 300) = 0H + 0G = 0 $$</p>
<p>If above equation holds up, we can know the transfer amount is valid. Through this process, <code>balance</code> and <code>transfer amount</code> are encrypted by elliptic curve so no one can predict actual value from public information. This is how we conceal the transfer transaction and verify the validity.</p>
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