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<title data-rh="true">Verifying RLN Proofs in Light Clients with Subtrees | Vac Research</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://vac.dev/rlog/rln-light-verifiers"><meta data-rh="true" property="og:locale" content="en"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docusaurus_tag" content="default"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docsearch:docusaurus_tag" content="default"><meta data-rh="true" name="image" content="https://vac.dev/_og/36ae1883b1ed0b2a15ae158f9cd93aaa9f5d7c69.png"><meta data-rh="true" property="og:title" content="Verifying RLN Proofs in Light Clients with Subtrees | Vac Research"><meta data-rh="true" name="description" content="How resource-restricted devices can verify RLN proofs fast and efficiently."><meta data-rh="true" property="og:description" content="How resource-restricted devices can verify RLN proofs fast and efficiently."><meta data-rh="true" property="og:type" content="article"><meta data-rh="true" property="article:published_time" content="2024-05-03T12:00:00.000Z"><link data-rh="true" rel="icon" href="/theme/image/favicon.ico"><link data-rh="true" rel="canonical" href="https://vac.dev/rlog/rln-light-verifiers"><link data-rh="true" rel="alternate" href="https://vac.dev/rlog/rln-light-verifiers" hreflang="en"><link data-rh="true" rel="alternate" href="https://vac.dev/rlog/rln-light-verifiers" hreflang="x-default"><script data-rh="true" type="application/ld+json">{"@context":"https://schema.org","@type":"BlogPosting","@id":"https://vac.dev/rlog/rln-light-verifiers","mainEntityOfPage":"https://vac.dev/rlog/rln-light-verifiers","url":"https://vac.dev/rlog/rln-light-verifiers","headline":"Verifying RLN Proofs in Light Clients with Subtrees","name":"Verifying RLN Proofs in Light Clients with Subtrees","description":"How resource-restricted devices can verify RLN proofs fast and efficiently.","datePublished":"2024-05-03T12:00:00.000Z","author":{"@type":"Person","name":"Aaryamann"},"keywords":[],"isPartOf":{"@type":"Blog","@id":"https://vac.dev/rlog","name":"Research Blog"}}</script><link rel="alternate icon" type="image/png" href="/theme/image/favicon.png">
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class="col col--9 col--offset-1" itemscope="" itemtype="http://schema.org/Blog"><article class=""><header><nav class="theme-doc-breadcrumbs breadcrumbsContainer_RLvU" aria-label="Breadcrumbs"><ul class="breadcrumbs" itemscope="" itemtype="https://schema.org/BreadcrumbList"><a class="breadcrumbs__item" href="/blog"><span class="lsd-typography lsd-typography--body3">All posts</span></a><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item breadcrumbs__item--active"><span class="lsd-typography lsd-typography--body3 breadcrumbs__link" itemprop="name">Verifying RLN Proofs in Light Clients with Subtrees</span><meta itemprop="position" content="1"></li></ul></nav><h1 class="title_f1Hy">Verifying RLN Proofs in Light Clients with Subtrees</h1><div class="container_PkUo margin-vert--md"><time datetime="2024-05-03T12:00:00.000Z" itemprop="datePublished">May 3, 2024</time><div class="authors_dZ4g"><span class="lsd-typography lsd-typography--body2">by </span><div class="col col--6 authorCol_y4tx"><div class="avatar margin-bottom--sm"><div class="avatar__intro" itemprop="author" itemscope="" itemtype="https://schema.org/Person"><div class="avatar__name"><span class="lsd-typography lsd-typography--body2" itemprop="name">Aaryamann</span></div></div></div></div></div><div>5 min read</div></div><hr class="blog-divider"></header><div id="__blog-post-container" class="markdown"><p>How resource-restricted devices can verify RLN proofs fast and efficiently.</p>
<h2 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="introduction">Introduction<a href="#introduction" class="hash-link" aria-label="Direct link to Introduction" title="Direct link to Introduction">​</a></h2>
<p>Recommended previous reading: <a href="https://research.logos.co/rlog/rln-anonymous-dos-prevention" target="_blank" rel="noopener noreferrer">Strengthening Anonymous DoS Prevention with Rate Limiting Nullifiers in Waku</a>.</p>
<p>This post expands upon ideas described in the previous post,
focusing on how resource-restricted devices can verify RLN proofs fast and efficiently.</p>
<p>Previously, it was required to fetch all the memberships from the smart contract,
construct the merkle tree locally,
and derive the merkle root,
which is subsequently used to verify RLN proofs.</p>
<p>This process is not feasible for resource-restricted devices since it involves a lot of RPC calls, computation and fault tolerance.
One cannot expect a mobile phone to fetch all the memberships from the smart contract and construct the merkle tree locally.</p>
<h2 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="constraints-and-requirements">Constraints and requirements<a href="#constraints-and-requirements" class="hash-link" aria-label="Direct link to Constraints and requirements" title="Direct link to Constraints and requirements">​</a></h2>
<p>An alternative solution to the one proposed in this post is to construct the merkle tree on-chain,
and have the root accessible with a single RPC call.
However, this approach increases gas costs for inserting new memberships and <em>may</em> not be feasible until it is optimized further with batching mechanisms, etc.</p>
<p>The other methods have been explored in more depth <a href="https://hackmd.io/@rymnc/rln-tree-storages" target="_blank" rel="noopener noreferrer">here</a>.</p>
<p>Following are the requirements and constraints for the solution proposed in this post:</p>
<ol>
<li>Cheap membership insertions.</li>
<li>As few RPC calls as possible to reduce startup time.</li>
<li>Merkle root of the tree is available on-chain.</li>
<li>No centralized services to sequence membership insertions.</li>
<li>Map inserted commitments to the block in which they were inserted.</li>
</ol>
<h2 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="metrics-on-sync-time-for-a-tree-with-2653-leaves">Metrics on sync time for a tree with 2,653 leaves<a href="#metrics-on-sync-time-for-a-tree-with-2653-leaves" class="hash-link" aria-label="Direct link to Metrics on sync time for a tree with 2,653 leaves" title="Direct link to Metrics on sync time for a tree with 2,653 leaves">​</a></h2>
<p>The following metrics are based on the current implementation of RLN in the Waku gen0 network.</p>
<h3 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="test-bench">Test bench<a href="#test-bench" class="hash-link" aria-label="Direct link to Test bench" title="Direct link to Test bench">​</a></h3>
<ul>
<li>Hardware: Macbook Air M2, 16GB RAM</li>
<li>Network: 120 Megabits/sec</li>
<li>Nwaku commit: <a href="https://github.com/waku-org/nwaku/tree/e61e4ff90a235657a7dc4248f5be41b6e031e98c" target="_blank" rel="noopener noreferrer">e61e4ff</a></li>
<li>RLN membership set contract: <a href="https://sepolia.etherscan.io/address/0xF471d71E9b1455bBF4b85d475afb9BB0954A29c4#code" target="_blank" rel="noopener noreferrer">0xF471d71E9b1455bBF4b85d475afb9BB0954A29c4</a></li>
<li>Deployed block number: 4,230,716</li>
<li>RLN Membership set depth: 20</li>
<li>Hash function: PoseidonT3 (which is a gas guzzler)</li>
<li>Max size of the membership set: 2^20 = 1,048,576 leaves</li>
</ul>
<h3 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="metrics">Metrics<a href="#metrics" class="hash-link" aria-label="Direct link to Metrics" title="Direct link to Metrics">​</a></h3>
<ul>
<li>Time to sync the whole tree: 4 minutes</li>
<li>RPC calls: 702</li>
<li>Number of leaves: 2,653</li>
</ul>
<p>One can argue that the time to sync the tree at the current state is not <em>that</em> bad.
However, the number of RPC calls is a concern,
which scales linearly with the number of blocks since the contract was deployed
This is because the implementation fetches all events from the contract,
chunking 2,000 blocks at a time.
This is done to avoid hitting the block limit of 10,000 events per call,
which is a limitation of popular RPC providers.</p>
<h2 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="proposed-solution">Proposed solution<a href="#proposed-solution" class="hash-link" aria-label="Direct link to Proposed solution" title="Direct link to Proposed solution">​</a></h2>
<p>From a theoretical perspective,
one could construct the merkle tree on-chain,
in a view call, in-memory.
However, this is not feasible due to the gas costs associated with it.</p>
<p>To compute the root of a Merkle tree with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>20</mn></msup></mrow><annotation encoding="application/x-tex">2^{20}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">20</span></span></span></span></span></span></span></span></span></span></span></span> leaves it costs approximately 2 billion gas.
With Infura and Alchemy capping the gas limit to 350M and 550M gas respectively,
it is not possible to compute the root of the tree in a single call.</p>
<p>Acknowledging that <a href="https://polygon.technology/blog/polygon-miden-state-model" target="_blank" rel="noopener noreferrer">Polygon Miden</a> and <a href="https://penumbra.zone/blog/tiered-commitment-tree/" target="_blank" rel="noopener noreferrer">Penumbra</a> both make use of a tiered commitment tree,
we propose a similar approach for RLN.</p>
<p>A tiered commitment tree is a tree which is sharded into multiple smaller subtrees,
each of which is a tree in itself.
This allows scaling in terms of the number of leaves,
as well as reducing state bloat by just storing the root of a subtree when it is full instead of all its leaves.</p>
<p>Here, the question arises:
What is the maximum number of leaves in a subtree with which the root can be computed in a single call?</p>
<p>It costs approximately 217M gas to compute the root of a Merkle tree with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>10</mn></msup></mrow><annotation encoding="application/x-tex">2^{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span></span></span></span></span></span></span></span> leaves.</p>
<p>This is a feasible number for a single call,
and hence we propose a tiered commitment tree with a maximum of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>10</mn></msup></mrow><annotation encoding="application/x-tex">2^{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span></span></span></span></span></span></span></span> leaves in a subtree and the number of subtrees is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>10</mn></msup></mrow><annotation encoding="application/x-tex">2^{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span></span></span></span></span></span></span></span>.
Therefore, the maximum number of leaves in the tree is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>20</mn></msup></mrow><annotation encoding="application/x-tex">2^{20}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">20</span></span></span></span></span></span></span></span></span></span></span></span> (the same as the current implementation).</p>
<p></p><div class="wrapper_SWrM active_qZD5"><img decoding="async" loading="lazy" alt="img" src="/assets/images/light-rln-verifiers-f801999160884be6a1223ee7d76cebcf.png" width="631" height="381" class="img_ev3q"><button class="fullscreenButton_Bocn lsd-icon-button lsd-icon-button--medium lsd-icon-button--outlined"><div class="icon_S7Kx m_thRi"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="14" fill="none" viewBox="0 0 14 14"><path fill="#fff" d="M1.75 2.917V5.25h1.167V2.917H5.25V1.75H2.917A1.17 1.17 0 0 0 1.75 2.917M2.917 8.75H1.75v2.333a1.17 1.17 0 0 0 1.167 1.167H5.25v-1.167H2.917zm8.166 2.333H8.75v1.167h2.333a1.17 1.17 0 0 0 1.167-1.167V8.75h-1.167zm0-9.333H8.75v1.167h2.333V5.25h1.167V2.917a1.17 1.17 0 0 0-1.167-1.167"></path></svg></div></button></div><p></p>
<h3 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="insertion">Insertion<a href="#insertion" class="hash-link" aria-label="Direct link to Insertion" title="Direct link to Insertion">​</a></h3>
<p>When a commitment is inserted into the tree it is first inserted into the first subtree.
When the first subtree is full the next insertions go into the second subtree and so on.</p>
<h3 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="syncing">Syncing<a href="#syncing" class="hash-link" aria-label="Direct link to Syncing" title="Direct link to Syncing">​</a></h3>
<p>When syncing the tree,
one only needs to fetch the roots of the subtrees.
The root of the full tree can be computed in-memory or on-chain.</p>
<p>This allows us to derive the following relation:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi mathvariant="normal">_</mi><mi>o</mi><mi>f</mi><mi mathvariant="normal">_</mi><mi>r</mi><mi>p</mi><mi>c</mi><mi mathvariant="normal">_</mi><mi>c</mi><mi>a</mi><mi>l</mi><mi>l</mi><mi>s</mi><mo>=</mo><mi>n</mi><mi>u</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi mathvariant="normal">_</mi><mi>o</mi><mi>f</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>i</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>d</mi><mi mathvariant="normal">_</mi><mi>s</mi><mi>u</mi><mi>b</mi><mi>t</mi><mi>r</mi><mi>e</mi><mi>e</mi><mi>s</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">number\_of\_rpc\_calls = number\_of\_filled\_subtrees + 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0044em;vertical-align:-0.31em"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">mb</span><span class="mord mathnormal" style="margin-right:0.02778em">er</span><span class="mord" style="margin-right:0.02778em">_</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mord" style="margin-right:0.02778em">_</span><span class="mord mathnormal" style="margin-right:0.02778em">r</span><span class="mord mathnormal">p</span><span class="mord mathnormal">c</span><span class="mord" style="margin-right:0.02778em">_</span><span class="mord mathnormal">c</span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.01968em">ll</span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1.0044em;vertical-align:-0.31em"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">mb</span><span class="mord mathnormal" style="margin-right:0.02778em">er</span><span class="mord" style="margin-right:0.02778em">_</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mord" style="margin-right:0.02778em">_</span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mord mathnormal">i</span><span class="mord mathnormal" style="margin-right:0.01968em">ll</span><span class="mord mathnormal">e</span><span class="mord mathnormal">d</span><span class="mord" style="margin-right:0.02778em">_</span><span class="mord mathnormal">s</span><span class="mord mathnormal">u</span><span class="mord mathnormal">b</span><span class="mord mathnormal">t</span><span class="mord mathnormal">rees</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">1</span></span></span></span></span>
<p>This is a significant improvement over the current implementation,
which requires fetching all the memberships from the smart contract.</p>
<h3 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="gas-costs">Gas costs<a href="#gas-costs" class="hash-link" aria-label="Direct link to Gas costs" title="Direct link to Gas costs">​</a></h3>
<p>The gas costs for inserting a commitment into the tree are the same as the current implementation except it consists of an extra SSTORE operation to store the <code>shardIndex</code> of the commitment.</p>
<h3 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="events">Events<a href="#events" class="hash-link" aria-label="Direct link to Events" title="Direct link to Events">​</a></h3>
<p>The events emitted by the contract are the same as the current implementation,
appending the <code>shardIndex</code> of the commitment.</p>
<h3 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="proof-of-concept">Proof of concept<a href="#proof-of-concept" class="hash-link" aria-label="Direct link to Proof of concept" title="Direct link to Proof of concept">​</a></h3>
<p>A proof of concept implementation of the tiered commitment tree is available <a href="https://github.com/vacp2p/rln-contract/pull/37" target="_blank" rel="noopener noreferrer">here</a>,
and is deployed on Sepolia at <a href="https://sepolia.etherscan.io/address/0xE7987c70B54Ff32f0D5CBbAA8c8Fc1cAf632b9A5" target="_blank" rel="noopener noreferrer">0xE7987c70B54Ff32f0D5CBbAA8c8Fc1cAf632b9A5</a>.</p>
<p>It is compatible with the current implementation of the RLN verifier.</p>
<h2 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="future-work">Future work<a href="#future-work" class="hash-link" aria-label="Direct link to Future work" title="Direct link to Future work">​</a></h2>
<ol>
<li>Optimize the gas costs of the tiered commitment tree.</li>
<li>Explore using different number of leaves under a given node in the tree (currently set to 2).</li>
</ol>
<h2 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="conclusion">Conclusion<a href="#conclusion" class="hash-link" aria-label="Direct link to Conclusion" title="Direct link to Conclusion">​</a></h2>
<p>The tiered commitment tree is a promising approach to reduce the number of RPC calls required to sync the tree and reduce the gas costs associated with computing the root of the tree.
Consequently, it allows for a more scalable and efficient RLN verifier.</p>
<h2 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="references">References<a href="#references" class="hash-link" aria-label="Direct link to References" title="Direct link to References">​</a></h2>
<ul>
<li><a href="https://github.com/rate-limiting-nullifier/circom-rln" target="_blank" rel="noopener noreferrer">RLN Circuits</a></li>
<li><a href="https://github.com/vacp2p/zerokit" target="_blank" rel="noopener noreferrer">Zerokit</a></li>
<li><a href="https://rfc.vac.dev/vac/32/rln-v1" target="_blank" rel="noopener noreferrer">RLN-V1 RFC</a></li>
<li><a href="https://rfc.vac.dev/vac/raw/rln-v2" target="_blank" rel="noopener noreferrer">RLN-V2 RFC</a></li>
<li><a href="https://hackmd.io/7cBCMU5hS5OYv8PTaW2wAQ?view" target="_blank" rel="noopener noreferrer">RLN Implementers guide</a></li>
<li><a href="https://research.logos.co/rlog/rln-anonymous-dos-prevention" target="_blank" rel="noopener noreferrer">Strengthening Anonymous DoS Prevention with Rate Limiting Nullifiers in Waku</a></li>
</ul></div></article><nav class="pagination-nav docusaurus-mt-lg" aria-label="Blog post page navigation"><a class="pagination-nav__link pagination-nav__link--prev" href="/rlog/rln-v3"><div class="icon_S7Kx m_thRi"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="14" fill="none" viewBox="0 0 14 14"><path fill="#fff" d="M11.667 6.417h-7.1L7.83 3.156 7 2.333 2.334 7 7 11.667l.823-.823-3.255-3.26h7.099z"></path></svg></div><span class="lsd-typography lsd-typography--body2 pagination-nav__label">RLN-v3: Towards a Flexible and Cost-Efficient Implementation</span></a><a class="pagination-nav__link pagination-nav__link--next" href="/rlog/rln-anonymous-dos-prevention"><span class="lsd-typography lsd-typography--body2 pagination-nav__label">Strengthening Anonymous DoS Prevention with Rate Limiting Nullifiers in Waku</span><div class="icon_S7Kx m_thRi"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="14" fill="none" viewBox="0 0 14 14"><path fill="#fff" d="m7 2.334-.823.822 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toc-highlight">Insertion</a></li><li><a href="#syncing" class="table-of-contents__link toc-highlight">Syncing</a></li><li><a href="#gas-costs" class="table-of-contents__link toc-highlight">Gas costs</a></li><li><a href="#events" class="table-of-contents__link toc-highlight">Events</a></li><li><a href="#proof-of-concept" class="table-of-contents__link toc-highlight">Proof of concept</a></li></ul></li><li><a href="#future-work" class="table-of-contents__link toc-highlight">Future work</a></li><li><a href="#conclusion" class="table-of-contents__link toc-highlight">Conclusion</a></li><li><a href="#references" class="table-of-contents__link toc-highlight">References</a></li></ul></div></div></div></div></div><footer class="footer"><div class="container container-fluid firstRow_ar1q"><div class="footer__bottom text--center"><div class="margin-bottom--sm"><a class="footerLogoLink_BH7S" href="/"><img src="/theme/image/logo.svg" alt="Vac Research" class="themedImage_kfRS themedImage--light_BL8e 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