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<ol class="chapter"><li class="chapter-item expanded "><a href="rln.html"><strong aria-hidden="true">1.</strong> RLN</a></li><li class="chapter-item expanded "><a href="overview.html"><strong aria-hidden="true">2.</strong> Overview</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="what_is_rln.html"><strong aria-hidden="true">2.1.</strong> What is RLN</a></li><li class="chapter-item expanded "><a href="under_the_hood.html"><strong aria-hidden="true">2.2.</strong> Under the hood</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="protocol_spec.html"><strong aria-hidden="true">2.2.1.</strong> Protocol spec V1</a></li><li class="chapter-item expanded "><a href="protocol_spec_v2.html"><strong aria-hidden="true">2.2.2.</strong> Protocol spec V2</a></li><li class="chapter-item expanded "><a href="formal_spec.html" class="active"><strong aria-hidden="true">2.2.3.</strong> Formal spec</a></li></ol></li><li class="chapter-item expanded "><a href="uses.html"><strong aria-hidden="true">2.3.</strong> Uses</a></li></ol></li><li class="chapter-item expanded "><a href="how_to_use.html"><strong aria-hidden="true">3.</strong> How to use</a></li><li><ol class="section"><li class="chapter-item expanded "><div><strong aria-hidden="true">3.1.</strong> JavaScript RLN</div></li><li class="chapter-item expanded "><div><strong aria-hidden="true">3.2.</strong> Rust RLN</div></li></ol></li><li class="chapter-item expanded "><a href="theory.html"><strong aria-hidden="true">4.</strong> Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="sss.html"><strong aria-hidden="true">4.1.</strong> Shamir's Secret Sharing</a></li></ol></li><li class="chapter-item expanded "><a href="appendix.html"><strong aria-hidden="true">5.</strong> Appendix</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="terminology.html"><strong aria-hidden="true">5.1.</strong> A - Terminology</a></li><li class="chapter-item expanded "><a href="references.html"><strong aria-hidden="true">5.2.</strong> B - References</a></li></ol></li></ol>
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<h1 class="menu-title">Rate-Limiting Nullifier</h1>
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<h1 id="formal-spec"><a class="header" href="#formal-spec">Formal spec</a></h1>
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<p><em><a href="https://rfc.vac.dev/spec/58/">RFC for RLN-V2</a></em></p>
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<ul>
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<li><a href="#utils-templates">Utils</a>
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<ul>
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<li><a href="#merkletreeinclusionproof">MerkleTreeInclusionProof</a></li>
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<li><a href="#isininterval">IsInInterval</a></li>
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</ul>
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</li>
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<li><a href="#rln-same-templates">RLN-same</a></li>
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<li><a href="#rln-diff-templates">RLN-diff</a></li>
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<li><a href="#withdrawal">Withdrawal</a></li>
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</ul>
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<hr />
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<h2 id="utils"><a class="header" href="#utils">Utils</a></h2>
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<p><a href="https://github.com/Rate-Limiting-Nullifier/rln-circuits-v2/blob/main/circuits/utils.circom">utils.circom</a> is a set of templates/gadgets that the RLN circuits uses.</p>
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<p>These are: </p>
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<ul>
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<li>MerkleTreeInclusionProof - Merkle Tree inclusion check, used like set membership check;</li>
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<li>IsInInterval - used for range check.</li>
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</ul>
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<p>Their description is given below.</p>
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<h3 id="merkletreeinclusionproof"><a class="header" href="#merkletreeinclusionproof">MerkleTreeInclusionProof</a></h3>
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<p><code>MerkleTreeInclusionProof(DEPTH)</code> template used for verification of inclusion in full binary incremental merkle tree. The implementation is a fork of https://github.com/privacy-scaling-explorations/incrementalquintree, and changed to <em>binary</em> tree and refactored to <em>Circom 2.1.0</em>.</p>
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<p><strong>Parameters</strong>:</p>
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<ul>
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<li><code>DEPTH</code> - depth of the Merkle Tree.</li>
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</ul>
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<p><strong>Inputs</strong>:</p>
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<ul>
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<li><code>leaf</code> - <code>Poseidon(elem)</code>, where <code>elem</code> is the element that's checked for inclusion;</li>
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<li><code>pathIndex[DEPTH]</code> - array of length = <code>DEPTH</code>, consists of <code>0 | 1</code>, represents Merkle proof path.
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Basically, it says how to calculate Poseidon hash, e.g. for two inputs <code>input1</code>, <code>input2</code>, if the <code>pathIndex[i] = 0</code> it shoud be calculated as <code>Poseidon(input1, input2)</code>, otherwise <code>Poseidon(input2, input1)</code>;</li>
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<li><code>pathElements[DEPTH]</code> - array of length = <code>DEPTH</code>, represents elements of the Merkle proof.</li>
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</ul>
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<p><strong>Outputs</strong>:</p>
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<ul>
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<li><code>root</code> - Root of the merkle tree.</li>
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</ul>
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<p><strong>Templates used</strong>:</p>
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<ul>
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<li><a href="https://github.com/iden3/circomlib/blob/master/circuits/mux1.circom">mux1.circom</a> from circomlib;</li>
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<li><a href="https://github.com/iden3/circomlib/blob/master/circuits/poseidon.circom">poseidon.circom</a> from circomlib.</li>
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</ul>
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<h3 id="isininterval"><a class="header" href="#isininterval">IsInInterval</a></h3>
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<p><code>IsInInterval(LIMIT_BIT_SIZE)</code> template used for range check, e.g. (x <= y <= z).</p>
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<p><strong>Parameters</strong>:</p>
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<ul>
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<li><code>LIMIT_BIT_SIZE</code> - maximum bit size of numbers that are used in range check, f.e. for the <code>LIMIT_BIT_SIZE</code> = 16, input numbers allowed to be in the interval <code>[0, 65536)</code>.</li>
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</ul>
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<p><strong>Inputs</strong>:</p>
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<ul>
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<li><code>in[3]</code> - array of 3 elements.</li>
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</ul>
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<p><strong>Outputs</strong>:</p>
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<ul>
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<li><code>out</code> - bool value (<code>0 | 1</code>). Outputs 1 when the circuit is satisfied, otherwise - 0.</li>
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</ul>
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<p><strong>Templates used</strong>:</p>
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<ul>
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<li><a href="https://github.com/iden3/circomlib/blob/master/circuits/comparators.circom#L105"><code>LessEqThan(n)</code></a> from circomlib.</li>
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</ul>
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<p><strong>Logic/Constraints</strong>:
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Checked that <code>in[0] <= in[1] <= in[2]</code>. That's done by combining two <code>LessEqThan</code> checks.
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<code>out</code> value is calculated as a multiplication of two <code>LessEqThan</code> outputs.</p>
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<hr />
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<h2 id="rln-same"><a class="header" href="#rln-same">RLN-same</a></h2>
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<p><a href="https://github.com/Rate-Limiting-Nullifier/rln-circuits-v2/blob/main/circuits/rln-same.circom">rln-same.circom</a> is a template that's used for <a href="https://rfc.vac.dev/spec/58/#rln-same-flow">RLN-same protocol</a>. </p>
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<p><strong>Parameters</strong>:</p>
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<ul>
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<li><code>DEPTH</code> - depth of a Merkle Tree. Described <a href="#merkletreeinclusionproof">here</a>;</li>
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<li><code>LIMIT_BIT_SIZE</code> - maximum bit size of numbers that are used in range check. Described <a href="#isininterval">here</a>.</li>
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</ul>
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<p><strong>Private inputs</strong>:</p>
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<ul>
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<li><code>identitySecret</code> - randomly generated number in <code>F_p</code>, used as private key;</li>
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<li><code>messageId</code> - id of the message;</li>
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<li><code>pathElements[DEPTH]</code> - pathElements[DEPTH], described <a href="#merkletreeinclusionproof">here</a>;</li>
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<li><code>identityPathIndex[DEPTH]</code> - pathIndex[DEPTH], described <a href="#merkletreeinclusionproof">here</a>.</li>
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</ul>
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<p><strong>Public inputs</strong>:</p>
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<ul>
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<li><code>x</code> - <code>Hash(signal)</code>, where <code>signal</code> is for example message, that was sent by user;</li>
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<li><code>externalNullifier</code> - <code>Hash(epoch, rln_identifier)</code>;</li>
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<li><code>messageLimit</code> - message limit of an RLN app.</li>
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</ul>
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<p><strong>Outputs</strong>:</p>
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<ul>
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<li><code>y</code> - calculated first-degree linear polynomial (y = kx + b);</li>
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<li><code>root</code> - root of the Merkle Tree;</li>
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<li><code>nullifier</code> - internal nullifier/pseudonym of the user in anonyomus environment.</li>
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</ul>
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<p><strong>Logic/Constraints</strong>:</p>
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<ol>
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<li>Merkle tree membership check:
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<ul>
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<li><code>identityCommitment</code> = <code>Poseidon(identitySecret)</code> calculation;</li>
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<li><a href="#merkletreeinclusionproof">Merkle tree inclusion check</a> for the <code>identityCommitment</code>.</li>
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</ul>
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</li>
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<li>Range check:
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<ul>
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<li><a href="#isininterval">Range check</a> that <code>1 <= messageId <= messageLimit</code>.</li>
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</ul>
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</li>
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<li>Polynomial share calculation:
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<ul>
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<li><code>a1</code> = <code>Poseidon(identitySecret, externalNullifier, messageId)</code>;</li>
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<li><code>y</code> = <code>identitySecret + a1 * x</code>.</li>
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</ul>
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</li>
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<li>Output of calculated <code>root</code>, <code>share</code> and <code>nullifier</code> = <code>Poseidon(a_1)</code> values.</li>
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</ol>
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<hr />
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<h2 id="rln-diff"><a class="header" href="#rln-diff">RLN-diff</a></h2>
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<p><a href="https://github.com/Rate-Limiting-Nullifier/rln-circuits-v2/blob/main/circuits/rln-diff.circom">rln-diff.circom</a> is a template that's used for <a href="https://rfc.vac.dev/spec/58/#rln-diff-flow">RLN-diff protocol</a>. </p>
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<p><strong>Parameters</strong>:</p>
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<ul>
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<li><code>DEPTH</code> - depth of a Merkle Tree. Described <a href="#merkletreeinclusionproof">here</a>;</li>
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<li><code>LIMIT_BIT_SIZE</code> - maximum bit size of numbers that are used in range check. Described <a href="#isininterval">here</a>.</li>
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</ul>
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<p><strong>Private inputs</strong>:</p>
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<ul>
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<li><code>identitySecret</code> - randomly generated number in <code>F_p</code>, used as a private key;</li>
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<li><code>userMessageLimit</code> - message limit of the user;</li>
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<li><code>messageId</code> - id of the message;</li>
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<li><code>pathElements[DEPTH]</code> - pathElements[DEPTH], described <a href="#merkletreeinclusionproof">here</a>;</li>
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<li><code>identityPathIndex[DEPTH]</code> - pathIndex[DEPTH], described <a href="#merkletreeinclusionproof">here</a>.</li>
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</ul>
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<p><strong>Public inputs</strong>:</p>
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<ul>
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<li><code>x</code> - <code>Hash(signal)</code>, where <code>signal</code> is for example message, that was sent by user;</li>
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<li><code>externalNullifier</code> - <code>Hash(epoch, rln_identifier)</code>.</li>
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</ul>
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<p><strong>Outputs</strong>:</p>
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<ul>
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<li><code>y</code> - calculated first-degree linear polynomial (y = kx + b);</li>
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<li><code>root</code> - root of the Merkle Tree;</li>
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<li><code>nullifier</code> - internal nullifier/pseudonym of the user in anonyomus environment.</li>
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</ul>
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<p><strong>Logic/Constraints</strong>:</p>
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<ol>
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<li>Merkle tree membership check:
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<ul>
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<li><code>identityCommitment</code> = <code>Poseidon(identitySecret, )</code> calculation;</li>
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<li><code>rateCommitment</code> = <code>Poseidon(identityCommitment, userMessageLimit)</code> calculation;</li>
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<li><a href="#merkletreeinclusionproof">Merkle tree inclusion check</a> for the <code>rateCommitment</code>.</li>
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</ul>
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</li>
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<li>Range check:
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<ul>
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<li><a href="#isininterval">Range check</a> that <code>1 <= messageId <= userMessageLimit</code>.</li>
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</ul>
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</li>
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<li>Polynomial share calculation:
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<ul>
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<li><code>a1</code> = <code>Poseidon(identitySecret, externalNullifier, messageId)</code>;</li>
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<li><code>y</code> = <code>identitySecret + a1 * x</code>.</li>
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</ul>
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</li>
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<li>Output of calculated <code>root</code>, <code>share</code> and <code>nullifier</code> = <code>Poseidon(a_1)</code> values.</li>
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</ol>
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<hr />
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<h3 id="withdrawal"><a class="header" href="#withdrawal">Withdrawal</a></h3>
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|
<p><a href="https://github.com/Rate-Limiting-Nullifier/rln-circuits-v2/blob/main/circuits/withdraw.circom">withdraw.circom</a> is a template that's used for the withdrawal/slashing and is needed to prevent front run while withdrawing the stake from the smart-contract/registry. </p>
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|
<p><strong>Private inputs</strong>:</p>
|
|
<ul>
|
|
<li><code>identitySecret</code> - randomly generated number in <code>F_p</code>, used as private key.</li>
|
|
</ul>
|
|
<p><strong>Public inputs</strong>:</p>
|
|
<ul>
|
|
<li><code>addressHash</code> - <code>F_p</code> scalar field element. <code>addressHash</code> = <code>Hash(address)</code>, where <code>address</code> is ETH address that'll receive stake. </li>
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</ul>
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|
<p><strong>Outputs</strong>:</p>
|
|
<ul>
|
|
<li><code>identityCommitment</code> = <code>Poseidon(identitySecret)</code>.</li>
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</ul>
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