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curryrasul
2022-12-07 15:31:44 +00:00
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33
404.html
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@@ -6,7 +6,6 @@
<title>Page not found - Rate-Limiting Nullifier</title>
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@@ -27,17 +26,17 @@
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@@ -53,7 +52,7 @@
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var theme;
try { theme = localStorage.getItem('mdbook-theme'); } catch(e) { }
if (theme === null || theme === undefined) { theme = default_theme; }
@@ -65,7 +64,7 @@
</script>
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<script type="text/javascript">
<script>
var html = document.querySelector('html');
var sidebar = 'hidden';
if (document.body.clientWidth >= 1080) {
@@ -96,7 +95,7 @@
<i class="fa fa-paint-brush"></i>
</button>
<ul id="theme-list" class="theme-popup" aria-label="Themes" role="menu">
<li role="none"><button role="menuitem" class="theme" id="light">Light (default)</button></li>
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<li role="none"><button role="menuitem" class="theme" id="coal">Coal</button></li>
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@@ -127,7 +126,7 @@
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@@ -154,18 +153,18 @@
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</body>
</html>

33
appendix.html
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@@ -5,7 +5,6 @@
<meta charset="UTF-8">
<title>Appendix - Rate-Limiting Nullifier</title>
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<meta content="text/html; charset=utf-8" http-equiv="Content-Type">
<meta name="description" content="">
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="theme-color" content="#ffffff" />
@@ -26,17 +25,17 @@
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<script async type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
</head>
<body>
<!-- Provide site root to javascript -->
<script type="text/javascript">
<script>
var path_to_root = "";
var default_theme = window.matchMedia("(prefers-color-scheme: dark)").matches ? "navy" : "light";
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<script type="text/javascript">
<script>
try {
var theme = localStorage.getItem('mdbook-theme');
var sidebar = localStorage.getItem('mdbook-sidebar');
@@ -52,7 +51,7 @@
</script>
<!-- Set the theme before any content is loaded, prevents flash -->
<script type="text/javascript">
<script>
var theme;
try { theme = localStorage.getItem('mdbook-theme'); } catch(e) { }
if (theme === null || theme === undefined) { theme = default_theme; }
@@ -64,7 +63,7 @@
</script>
<!-- Hide / unhide sidebar before it is displayed -->
<script type="text/javascript">
<script>
var html = document.querySelector('html');
var sidebar = 'hidden';
if (document.body.clientWidth >= 1080) {
@@ -95,7 +94,7 @@
<i class="fa fa-paint-brush"></i>
</button>
<ul id="theme-list" class="theme-popup" aria-label="Themes" role="menu">
<li role="none"><button role="menuitem" class="theme" id="light">Light (default)</button></li>
<li role="none"><button role="menuitem" class="theme" id="light">Light</button></li>
<li role="none"><button role="menuitem" class="theme" id="rust">Rust</button></li>
<li role="none"><button role="menuitem" class="theme" id="coal">Coal</button></li>
<li role="none"><button role="menuitem" class="theme" id="navy">Navy</button></li>
@@ -126,7 +125,7 @@
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</div>
<!-- Apply ARIA attributes after the sidebar and the sidebar toggle button are added to the DOM -->
<script type="text/javascript">
<script>
document.getElementById('sidebar-toggle').setAttribute('aria-expanded', sidebar === 'visible');
document.getElementById('sidebar').setAttribute('aria-hidden', sidebar !== 'visible');
Array.from(document.querySelectorAll('#sidebar a')).forEach(function(link) {
@@ -169,18 +168,18 @@
</div>
<script type="text/javascript">
<script>
window.playground_copyable = true;
</script>
<script src="elasticlunr.min.js" type="text/javascript" charset="utf-8"></script>
<script src="mark.min.js" type="text/javascript" charset="utf-8"></script>
<script src="searcher.js" type="text/javascript" charset="utf-8"></script>
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<script src="elasticlunr.min.js"></script>
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<script src="clipboard.min.js"></script>
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<script src="book.js"></script>
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<script src="mermaid.min.js"></script>
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</body>
</html>

10
book.js
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@@ -11,7 +11,7 @@ function playground_text(playground) {
let editor = window.ace.edit(code_block);
return editor.getValue();
} else {
return code_block.textContent;
return code_block.innerText;
}
}
@@ -300,6 +300,13 @@ function playground_text(playground) {
themePopup.querySelector("button#" + get_theme()).focus();
}
function updateThemeSelected() {
themePopup.querySelectorAll('.theme-selected').forEach(function (el) {
el.classList.remove('theme-selected');
});
themePopup.querySelector("button#" + get_theme()).classList.add('theme-selected');
}
function hideThemes() {
themePopup.style.display = 'none';
themeToggleButton.setAttribute('aria-expanded', false);
@@ -355,6 +362,7 @@ function playground_text(playground) {
html.classList.remove(previousTheme);
html.classList.add(theme);
updateThemeSelected();
}
// Set theme

73
circuits.html
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@@ -5,7 +5,6 @@
<meta charset="UTF-8">
<title>Circuits - Rate-Limiting Nullifier</title>
<!-- Custom HTML head -->
<meta content="text/html; charset=utf-8" http-equiv="Content-Type">
<meta name="description" content="">
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="theme-color" content="#ffffff" />
@@ -26,17 +25,17 @@
<!-- Custom theme stylesheets -->
<!-- MathJax -->
<script async type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
</head>
<body>
<!-- Provide site root to javascript -->
<script type="text/javascript">
<script>
var path_to_root = "";
var default_theme = window.matchMedia("(prefers-color-scheme: dark)").matches ? "navy" : "light";
</script>
<!-- Work around some values being stored in localStorage wrapped in quotes -->
<script type="text/javascript">
<script>
try {
var theme = localStorage.getItem('mdbook-theme');
var sidebar = localStorage.getItem('mdbook-sidebar');
@@ -52,7 +51,7 @@
</script>
<!-- Set the theme before any content is loaded, prevents flash -->
<script type="text/javascript">
<script>
var theme;
try { theme = localStorage.getItem('mdbook-theme'); } catch(e) { }
if (theme === null || theme === undefined) { theme = default_theme; }
@@ -64,7 +63,7 @@
</script>
<!-- Hide / unhide sidebar before it is displayed -->
<script type="text/javascript">
<script>
var html = document.querySelector('html');
var sidebar = 'hidden';
if (document.body.clientWidth >= 1080) {
@@ -95,7 +94,7 @@
<i class="fa fa-paint-brush"></i>
</button>
<ul id="theme-list" class="theme-popup" aria-label="Themes" role="menu">
<li role="none"><button role="menuitem" class="theme" id="light">Light (default)</button></li>
<li role="none"><button role="menuitem" class="theme" id="light">Light</button></li>
<li role="none"><button role="menuitem" class="theme" id="rust">Rust</button></li>
<li role="none"><button role="menuitem" class="theme" id="coal">Coal</button></li>
<li role="none"><button role="menuitem" class="theme" id="navy">Navy</button></li>
@@ -126,7 +125,7 @@
</div>
</div>
<!-- Apply ARIA attributes after the sidebar and the sidebar toggle button are added to the DOM -->
<script type="text/javascript">
<script>
document.getElementById('sidebar-toggle').setAttribute('aria-expanded', sidebar === 'visible');
document.getElementById('sidebar').setAttribute('aria-hidden', sidebar !== 'visible');
Array.from(document.querySelectorAll('#sidebar a')).forEach(function(link) {
@@ -213,32 +212,43 @@ template HashLeftRight() {
out &lt;== hasher.out;
}
template CalculateExternalNullifier() {
signal input epoch;
signal input rln_identifier;
signal output out;
component hasher = Poseidon(2);
hasher.inputs[0] &lt;== epoch;
hasher.inputs[1] &lt;== rln_identifier;
out &lt;== hasher.out;
}
template CalculateA1() {
signal input a_0;
signal input epoch;
signal input external_nullifier;
signal output out;
component hasher = Poseidon(2);
hasher.inputs[0] &lt;== a_0;
hasher.inputs[1] &lt;== epoch;
hasher.inputs[1] &lt;== external_nullifier;
out &lt;== hasher.out;
}
template CalculateNullifier() {
template CalculateInternalNullifier() {
signal input a_1;
signal input rln_identifier;
signal output out;
component hasher = Poseidon(2);
component hasher = Poseidon(1);
hasher.inputs[0] &lt;== a_1;
hasher.inputs[1] &lt;== rln_identifier;
out &lt;== hasher.out;
}
</code></pre>
<p>It's easy to understand these samples: <code>CalculateIdentityCommitment()</code> is used to calculate the identity commitment. It takes secret and outputs the commitment. <code>CalculateA1()</code> and <code>CalculateNullifier()</code> are used to calculate <code>a_1</code> and <code>nullifier</code> (internal nullifier); they are implemented as it's described in <a href="./protocol_spec.html">previous topic</a>.</p>
<p>It's easy to understand these samples: <code>CalculateIdentityCommitment()</code>, <code>CalculateA1()</code>, <code>CalculateInternalNullifier()</code>, <code>CalculateExternalNullifier()</code> - they do exactly what their name says; they are implemented as it's described in <a href="./protocol_spec.html">previous topic</a>.</p>
<p>Now, let's look at the core logic of the <strong>RLN</strong> circuit. </p>
<pre><code class="language-swift">...
@@ -256,7 +266,7 @@ template CalculateNullifier() {
...
</code></pre>
<p>So, here we have many inputs. Private inputs are: <code>identity_secret</code> (basically <code>a_0</code> from the polynomial), <code>path_elements[][]</code>, <code>identity_path_index[]</code>. Public inputs are: <code>x</code> (actually just the hash of a signal), <code>epoch,</code> <code>rln_identifier.</code> Outputs are: <code>y' (share of the secret), </code>root<code>of a Merkle Tree, and</code>nullifier.`</p>
<p>So, here we have many inputs. Private inputs are: <code>identity_secret</code> (basically <code>a_0</code> from the polynomial), <code>path_elements[][]</code>, <code>identity_path_index[]</code>. Public inputs are: <code>x</code> (actually just the hash of a signal), <code>epoch,</code> <code>rln_identifier</code>. Outputs are: <code>y</code> (polynomial share/secret share), <code>root</code> of a Merkle Tree, and <code>nullifier</code> (which is basically <code>internal_nullifier</code>).</p>
<p><strong>RLN</strong> circuit consists of two checks:</p>
<ul>
<li>Membership in Merkle Tree</li>
@@ -287,12 +297,16 @@ template CalculateNullifier() {
<pre><code class="language-swift">root &lt;== inclusionProof.root;
</code></pre>
<h3 id="correctness-of-secret-share"><a class="header" href="#correctness-of-secret-share">Correctness of secret share</a></h3>
<p>As we use linear polynomial we need to check that <code>y = a_1 * x + a_0</code> (<code>a_0</code> is identity secret). For that, we need these constraints:</p>
<p>As we use linear polynomial we need to check that <code>y = a_1 * x + a_0</code> (<code>a_0</code> is identity secret). For that, we need to calculate <code>external_nullifier</code> and constraints on <code>a_1</code> and secret share:</p>
<pre><code class="language-swift">...
component external_nullifier = CalculateExternalNullifier();
external_nullifier.epoch &lt;== epoch;
external_nullifier.rln_identifier &lt;== rln_identifier;
component a_1 = CalculateA1();
a_1.a_0 &lt;== identity_secret;
a_1.epoch &lt;== epoch;
a_1.external_nullifier &lt;== external_nullifier.out;
y &lt;== identity_secret + a_1.out * x;
@@ -301,9 +315,8 @@ template CalculateNullifier() {
<p>To calculate and reveal the <code>nullifier</code>:</p>
<pre><code class="language-swift">...
component calculateNullifier = CalculateNullifier();
component calculateNullifier = CalculateInternalNullifier();
calculateNullifier.a_1 &lt;== a_1.out;
calculateNullifier.rln_identifier &lt;== rln_identifier;
nullifier &lt;== calculateNullifier.out;
@@ -316,7 +329,7 @@ template CalculateNullifier() {
include &quot;./rln-base.circom&quot;;
component main {public [x, epoch, rln_identifier ]} = RLN(15);
component main { public [x, epoch, rln_identifier] } = RLN(15);
</code></pre>
<p>That's the whole <strong>RLN</strong> Circom Circuit :) Here we just need to list all public inputs (<code>x,</code> <code>epoch,</code> <code>rln_identifier</code>; the rest of the inputs are private). Also, we set the depth of the Merkle Tree = 15 (max of 32768 members).</p>
@@ -346,18 +359,18 @@ component main {public [x, epoch, rln_identifier ]} = RLN(15);
</div>
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<script>
window.playground_copyable = true;
</script>
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<script src="mermaid-init.js"></script>
</body>
</html>

14
css/chrome.css
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@@ -507,6 +507,8 @@ ul#searchresults span.teaser em {
padding: 0;
list-style: none;
display: none;
/* Don't let the children's background extend past the rounded corners. */
overflow: hidden;
}
.theme-popup .default {
color: var(--icons);
@@ -515,7 +517,7 @@ ul#searchresults span.teaser em {
width: 100%;
border: 0;
margin: 0;
padding: 2px 10px;
padding: 2px 20px;
line-height: 25px;
white-space: nowrap;
text-align: left;
@@ -527,8 +529,10 @@ ul#searchresults span.teaser em {
.theme-popup .theme:hover {
background-color: var(--theme-hover);
}
.theme-popup .theme:hover:first-child,
.theme-popup .theme:hover:last-child {
border-top-left-radius: inherit;
border-top-right-radius: inherit;
.theme-selected::before {
display: inline-block;
content: "✓";
margin-left: -14px;
width: 14px;
}

16
css/general.css
View File

@@ -22,8 +22,8 @@ body {
}
code {
font-family: "Source Code Pro", Consolas, "Ubuntu Mono", Menlo, "DejaVu Sans Mono", monospace, monospace !important;
font-size: 0.875em; /* please adjust the ace font size accordingly in editor.js */
font-family: var(--mono-font) !important;
font-size: var(--code-font-size);
}
/* make long words/inline code not x overflow */
@@ -148,6 +148,18 @@ blockquote {
border-bottom: .1em solid var(--quote-border);
}
kbd {
background-color: var(--table-border-color);
border-radius: 4px;
border: solid 1px var(--theme-popup-border);
box-shadow: inset 0 -1px 0 var(--theme-hover);
display: inline-block;
font-size: var(--code-font-size);
font-family: var(--mono-font);
line-height: 10px;
padding: 4px 5px;
vertical-align: middle;
}
:not(.footnote-definition) + .footnote-definition,
.footnote-definition + :not(.footnote-definition) {

2
css/variables.css
View File

@@ -6,6 +6,8 @@
--page-padding: 15px;
--content-max-width: 750px;
--menu-bar-height: 50px;
--mono-font: "Source Code Pro", Consolas, "Ubuntu Mono", Menlo, "DejaVu Sans Mono", monospace, monospace;
--code-font-size: 0.875em /* please adjust the ace font size accordingly in editor.js */
}
/* Themes */

33
how_to_use.html
View File

@@ -5,7 +5,6 @@
<meta charset="UTF-8">
<title>How to use - Rate-Limiting Nullifier</title>
<!-- Custom HTML head -->
<meta content="text/html; charset=utf-8" http-equiv="Content-Type">
<meta name="description" content="">
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="theme-color" content="#ffffff" />
@@ -26,17 +25,17 @@
<!-- Custom theme stylesheets -->
<!-- MathJax -->
<script async type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
</head>
<body>
<!-- Provide site root to javascript -->
<script type="text/javascript">
<script>
var path_to_root = "";
var default_theme = window.matchMedia("(prefers-color-scheme: dark)").matches ? "navy" : "light";
</script>
<!-- Work around some values being stored in localStorage wrapped in quotes -->
<script type="text/javascript">
<script>
try {
var theme = localStorage.getItem('mdbook-theme');
var sidebar = localStorage.getItem('mdbook-sidebar');
@@ -52,7 +51,7 @@
</script>
<!-- Set the theme before any content is loaded, prevents flash -->
<script type="text/javascript">
<script>
var theme;
try { theme = localStorage.getItem('mdbook-theme'); } catch(e) { }
if (theme === null || theme === undefined) { theme = default_theme; }
@@ -64,7 +63,7 @@
</script>
<!-- Hide / unhide sidebar before it is displayed -->
<script type="text/javascript">
<script>
var html = document.querySelector('html');
var sidebar = 'hidden';
if (document.body.clientWidth >= 1080) {
@@ -95,7 +94,7 @@
<i class="fa fa-paint-brush"></i>
</button>
<ul id="theme-list" class="theme-popup" aria-label="Themes" role="menu">
<li role="none"><button role="menuitem" class="theme" id="light">Light (default)</button></li>
<li role="none"><button role="menuitem" class="theme" id="light">Light</button></li>
<li role="none"><button role="menuitem" class="theme" id="rust">Rust</button></li>
<li role="none"><button role="menuitem" class="theme" id="coal">Coal</button></li>
<li role="none"><button role="menuitem" class="theme" id="navy">Navy</button></li>
@@ -126,7 +125,7 @@
</div>
</div>
<!-- Apply ARIA attributes after the sidebar and the sidebar toggle button are added to the DOM -->
<script type="text/javascript">
<script>
document.getElementById('sidebar-toggle').setAttribute('aria-expanded', sidebar === 'visible');
document.getElementById('sidebar').setAttribute('aria-hidden', sidebar !== 'visible');
Array.from(document.querySelectorAll('#sidebar a')).forEach(function(link) {
@@ -169,18 +168,18 @@
</div>
<script type="text/javascript">
<script>
window.playground_copyable = true;
</script>
<script src="elasticlunr.min.js" type="text/javascript" charset="utf-8"></script>
<script src="mark.min.js" type="text/javascript" charset="utf-8"></script>
<script src="searcher.js" type="text/javascript" charset="utf-8"></script>
<script src="clipboard.min.js" type="text/javascript" charset="utf-8"></script>
<script src="highlight.js" type="text/javascript" charset="utf-8"></script>
<script src="book.js" type="text/javascript" charset="utf-8"></script>
<script src="elasticlunr.min.js"></script>
<script src="mark.min.js"></script>
<script src="searcher.js"></script>
<script src="clipboard.min.js"></script>
<script src="highlight.js"></script>
<script src="book.js"></script>
<!-- Custom JS scripts -->
<script type="text/javascript" src="mermaid.min.js"></script>
<script type="text/javascript" src="mermaid-init.js"></script>
<script src="mermaid.min.js"></script>
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@@ -150,8 +149,6 @@
<li><strong>RLN</strong> uses</li>
</ul>
<div style="break-before: page; page-break-before: always;"></div><h1 id="what-is-rate-limiting-nullifier"><a class="header" href="#what-is-rate-limiting-nullifier">What is Rate-Limiting Nullifier?</a></h1>
<p><em>This topic is a part of <a href="https://medium.com/privacy-scaling-explorations/rate-limiting-nullifier-a-spam-protection-mechanism-for-anonymous-environments-bbe4006a57d">complete overview</a> by Blagoj</em>.</p>
<hr />
<p><strong>RLN</strong> is a zero-knowledge gadget that enables spam prevention for decentralized, anonymous environments.</p>
<p>The anonymity property opens up the possibility for spam and Sybil attack vectors for certain applications, which could seriously degrade the user experience and the overall functioning of the application. For example, imagine a chat application where users are anonymous. Now, everyone can write an unlimited number of spam messages, but we don't have the ability to kick this member because the spammer is anonymous. </p>
<p><strong>RLN</strong> helps us identify and &quot;kick&quot; the spammer.</p>
@@ -203,8 +200,8 @@
<p>So, each member generates a secret key, denoted by \(a_0\). Identity commitment \(q\) is the hash (Poseidon) of the secret key: \(q = Poseidon(a_0)\).</p>
<p><strong>RLN</strong> wouldn't work if there were no punishment for spam; that's why to become a member, a user has to register and provide something at stake. So, whoever has our \(a_0\) can &quot;slash&quot; us. </p>
<p>The slight difference is that we must enable a <em>secret sharing</em> scheme (to split the <code>commitment</code> into parts). We need to come up with a polynomial. For simplicity we use linear polynomial (e.g. \(f(x) = kx + b\). Therefore, with two points, we can reconstruct the polynomial and recover the secret. </p>
<p>Our polynomial will be: \(A(x) = a_1 * x + a_0\), where \(a_1 = Poseidon(a_0, epoch)\).</p>
<p><code>epoch</code> is a simple identifier (also called <em>external nullifier</em>). And each epoch, there is a polynomial with new \(a_1\) and the same \(a_0\). </p>
<p>Our polynomial will be: \(A(x) = a_1 * x + a_0\), where \(a_1 = Poseidon(a_0, external\_nullifier)\).
The meaning of \(external\_nullifier\) is described below.</p>
<h2 id="signalling"><a class="header" href="#signalling">Signalling</a></h2>
<p>Now that the user is registered, he wants to interact with the system. Imagine that the system is an <em>anonymous chat</em> and the interaction is the sending of messages.
So, to send a message user have to come up with <em>share</em> - the point \((x, y)\) on her polynomial.
@@ -213,10 +210,12 @@ We denote: \(x = Poseidon(message), y = A(x)\). </p>
<p>Of course, we somehow must prove that our <em>share</em> = \((x, y)\) is valid (that this is really a point on our <code>polynomial = A(x)</code>), as well as we must prove other things are valid too, that's why we use zkSNARK. An explanation of the zk-circuits can be found in the next topic.</p>
<h2 id="slashing"><a class="header" href="#slashing">Slashing</a></h2>
<p>As it's been said, if a user sends more than one message, everyone else will be able to recover his secret, slash them and take their stake.</p>
<h2 id="nullifiers"><a class="header" href="#nullifiers">Nullifiers</a></h2>
<p>There are also \(internal\_nullifier\) and \(external\_nullifier\), which can be found in the <strong>RLN</strong> protocol/circuits.</p>
<p>\(external\_nullifier = Poseidon(epoch, rln\_identifier)\), where \(rln\_identifier\) is a random finite field value, unique per RLN app.</p>
<p>The \(external\_nullifier\) is required so that the user can securely use the same private key \(a_0\) across different <strong>RLN</strong> apps - in different applications (and in different eras) with the same secret key, the user will have different values ​​of the coefficient \(a_1\).</p>
<p>Now, imagine there are a lot of users sending messages, and after each received message, we need to check if any member can be slashed. To do this, we can use all combinations of received <em>shares</em> and try to recover the polynomial, but this is a naive and non-optimal approach. Suppose we have a mechanism that will tell us about the connection between a person and their messages while not revealing their identity. In that case, we can solve this without brute-forcing all possibilities by using a public \(internal\_nullifier = Poseidon(a_1)\), so if a user sends more than one message, it will be immediately visible to everyone.</p>
<h2 id="some-important-notes"><a class="header" href="#some-important-notes">Some important notes</a></h2>
<p>There are also <code>nullifier</code> and <code>rln-identifier</code>, which can be found in the <strong>RLN</strong> protocol/circuits.</p>
<p>So, <code>rln-identifier</code> is just a random value that's unique per <strong>RLN</strong> app. It's used for additional cross-application security - to protect the user secrets from being compromised if they use the same credentials across different <strong>RLN</strong> apps. If <code>rln-identifier</code> is not present, the user uses the same credentials and sends a message in two different <strong>RLN</strong> apps using the same epoch, then their secret key can be revealed. Adding the <code>rln-identifier</code> field, we obscure the nullifier, so this kind of attack cannot happen. The only kind of attack that is possible is if we have an entity with a global view of all messages, and they try to brute-force different combinations of x and y shares for different nullifiers.</p>
<p>Now, imagine there are a lot of users sending messages, and after each received message, we need to check if any member can be slashed. To do this, we can use all combinations of received <em>shares</em> and try to recover the polynomial, but this is a naive and non-optimal approach. Suppose we have a mechanism that will tell us about the connection between a person and their messages while not revealing their identity. In that case, we can solve this without brute-forcing all possibilities by using a public <code>nullifier</code> (\(Poseidon(a_1, rln-identifier)\)), so if a user sends more than one message, it will be immediately visible to everyone.</p>
<p>Also, in our example (and <a href="https://github.com/njofce/zk-chat">zk-chat</a> implementation), we use linear polynomial, but <a href="sss.html">SSS</a> allows us to use various degree polynomials; therefore we can implement a protocol, where more than one signal (message) can be sent in per epoch. </p>
<p>To learn more, check out the <a href="https://hackmd.io/7GR5Vi28Rz2EpEmLK0E0Aw?view">specification</a>; there are also <a href="https://github.com/privacy-scaling-explorations/rln/tree/master/circuits">circuits</a> implemented for various degree polynomials too.</p>
<div style="break-before: page; page-break-before: always;"></div><h1 id="circuits"><a class="header" href="#circuits">Circuits</a></h1>
@@ -296,32 +295,43 @@ template HashLeftRight() {
out &lt;== hasher.out;
}
template CalculateExternalNullifier() {
signal input epoch;
signal input rln_identifier;
signal output out;
component hasher = Poseidon(2);
hasher.inputs[0] &lt;== epoch;
hasher.inputs[1] &lt;== rln_identifier;
out &lt;== hasher.out;
}
template CalculateA1() {
signal input a_0;
signal input epoch;
signal input external_nullifier;
signal output out;
component hasher = Poseidon(2);
hasher.inputs[0] &lt;== a_0;
hasher.inputs[1] &lt;== epoch;
hasher.inputs[1] &lt;== external_nullifier;
out &lt;== hasher.out;
}
template CalculateNullifier() {
template CalculateInternalNullifier() {
signal input a_1;
signal input rln_identifier;
signal output out;
component hasher = Poseidon(2);
component hasher = Poseidon(1);
hasher.inputs[0] &lt;== a_1;
hasher.inputs[1] &lt;== rln_identifier;
out &lt;== hasher.out;
}
</code></pre>
<p>It's easy to understand these samples: <code>CalculateIdentityCommitment()</code> is used to calculate the identity commitment. It takes secret and outputs the commitment. <code>CalculateA1()</code> and <code>CalculateNullifier()</code> are used to calculate <code>a_1</code> and <code>nullifier</code> (internal nullifier); they are implemented as it's described in <a href="./protocol_spec.html">previous topic</a>.</p>
<p>It's easy to understand these samples: <code>CalculateIdentityCommitment()</code>, <code>CalculateA1()</code>, <code>CalculateInternalNullifier()</code>, <code>CalculateExternalNullifier()</code> - they do exactly what their name says; they are implemented as it's described in <a href="./protocol_spec.html">previous topic</a>.</p>
<p>Now, let's look at the core logic of the <strong>RLN</strong> circuit. </p>
<pre><code class="language-swift">...
@@ -339,7 +349,7 @@ template CalculateNullifier() {
...
</code></pre>
<p>So, here we have many inputs. Private inputs are: <code>identity_secret</code> (basically <code>a_0</code> from the polynomial), <code>path_elements[][]</code>, <code>identity_path_index[]</code>. Public inputs are: <code>x</code> (actually just the hash of a signal), <code>epoch,</code> <code>rln_identifier.</code> Outputs are: <code>y' (share of the secret), </code>root<code>of a Merkle Tree, and</code>nullifier.`</p>
<p>So, here we have many inputs. Private inputs are: <code>identity_secret</code> (basically <code>a_0</code> from the polynomial), <code>path_elements[][]</code>, <code>identity_path_index[]</code>. Public inputs are: <code>x</code> (actually just the hash of a signal), <code>epoch,</code> <code>rln_identifier</code>. Outputs are: <code>y</code> (polynomial share/secret share), <code>root</code> of a Merkle Tree, and <code>nullifier</code> (which is basically <code>internal_nullifier</code>).</p>
<p><strong>RLN</strong> circuit consists of two checks:</p>
<ul>
<li>Membership in Merkle Tree</li>
@@ -370,12 +380,16 @@ template CalculateNullifier() {
<pre><code class="language-swift">root &lt;== inclusionProof.root;
</code></pre>
<h3 id="correctness-of-secret-share"><a class="header" href="#correctness-of-secret-share">Correctness of secret share</a></h3>
<p>As we use linear polynomial we need to check that <code>y = a_1 * x + a_0</code> (<code>a_0</code> is identity secret). For that, we need these constraints:</p>
<p>As we use linear polynomial we need to check that <code>y = a_1 * x + a_0</code> (<code>a_0</code> is identity secret). For that, we need to calculate <code>external_nullifier</code> and constraints on <code>a_1</code> and secret share:</p>
<pre><code class="language-swift">...
component external_nullifier = CalculateExternalNullifier();
external_nullifier.epoch &lt;== epoch;
external_nullifier.rln_identifier &lt;== rln_identifier;
component a_1 = CalculateA1();
a_1.a_0 &lt;== identity_secret;
a_1.epoch &lt;== epoch;
a_1.external_nullifier &lt;== external_nullifier.out;
y &lt;== identity_secret + a_1.out * x;
@@ -384,9 +398,8 @@ template CalculateNullifier() {
<p>To calculate and reveal the <code>nullifier</code>:</p>
<pre><code class="language-swift">...
component calculateNullifier = CalculateNullifier();
component calculateNullifier = CalculateInternalNullifier();
calculateNullifier.a_1 &lt;== a_1.out;
calculateNullifier.rln_identifier &lt;== rln_identifier;
nullifier &lt;== calculateNullifier.out;
@@ -399,7 +412,7 @@ template CalculateNullifier() {
include &quot;./rln-base.circom&quot;;
component main {public [x, epoch, rln_identifier ]} = RLN(15);
component main { public [x, epoch, rln_identifier] } = RLN(15);
</code></pre>
<p>That's the whole <strong>RLN</strong> Circom Circuit :) Here we just need to list all public inputs (<code>x,</code> <code>epoch,</code> <code>rln_identifier</code>; the rest of the inputs are private). Also, we set the depth of the Merkle Tree = 15 (max of 32768 members).</p>
<div style="break-before: page; page-break-before: always;"></div><h1 id="uses"><a class="header" href="#uses">Uses</a></h1>
@@ -534,20 +547,20 @@ So, the shares are: \((5, 55), (8, 70), (16, 110)\)</li>
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<p>So, each member generates a secret key, denoted by \(a_0\). Identity commitment \(q\) is the hash (Poseidon) of the secret key: \(q = Poseidon(a_0)\).</p>
<p><strong>RLN</strong> wouldn't work if there were no punishment for spam; that's why to become a member, a user has to register and provide something at stake. So, whoever has our \(a_0\) can &quot;slash&quot; us. </p>
<p>The slight difference is that we must enable a <em>secret sharing</em> scheme (to split the <code>commitment</code> into parts). We need to come up with a polynomial. For simplicity we use linear polynomial (e.g. \(f(x) = kx + b\). Therefore, with two points, we can reconstruct the polynomial and recover the secret. </p>
<p>Our polynomial will be: \(A(x) = a_1 * x + a_0\), where \(a_1 = Poseidon(a_0, epoch)\).</p>
<p><code>epoch</code> is a simple identifier (also called <em>external nullifier</em>). And each epoch, there is a polynomial with new \(a_1\) and the same \(a_0\). </p>
<p>Our polynomial will be: \(A(x) = a_1 * x + a_0\), where \(a_1 = Poseidon(a_0, external\_nullifier)\).
The meaning of \(external\_nullifier\) is described below.</p>
<h2 id="signalling"><a class="header" href="#signalling">Signalling</a></h2>
<p>Now that the user is registered, he wants to interact with the system. Imagine that the system is an <em>anonymous chat</em> and the interaction is the sending of messages.
So, to send a message user have to come up with <em>share</em> - the point \((x, y)\) on her polynomial.
@@ -165,10 +164,12 @@ We denote: \(x = Poseidon(message), y = A(x)\). </p>
<p>Of course, we somehow must prove that our <em>share</em> = \((x, y)\) is valid (that this is really a point on our <code>polynomial = A(x)</code>), as well as we must prove other things are valid too, that's why we use zkSNARK. An explanation of the zk-circuits can be found in the next topic.</p>
<h2 id="slashing"><a class="header" href="#slashing">Slashing</a></h2>
<p>As it's been said, if a user sends more than one message, everyone else will be able to recover his secret, slash them and take their stake.</p>
<h2 id="nullifiers"><a class="header" href="#nullifiers">Nullifiers</a></h2>
<p>There are also \(internal\_nullifier\) and \(external\_nullifier\), which can be found in the <strong>RLN</strong> protocol/circuits.</p>
<p>\(external\_nullifier = Poseidon(epoch, rln\_identifier)\), where \(rln\_identifier\) is a random finite field value, unique per RLN app.</p>
<p>The \(external\_nullifier\) is required so that the user can securely use the same private key \(a_0\) across different <strong>RLN</strong> apps - in different applications (and in different eras) with the same secret key, the user will have different values ​​of the coefficient \(a_1\).</p>
<p>Now, imagine there are a lot of users sending messages, and after each received message, we need to check if any member can be slashed. To do this, we can use all combinations of received <em>shares</em> and try to recover the polynomial, but this is a naive and non-optimal approach. Suppose we have a mechanism that will tell us about the connection between a person and their messages while not revealing their identity. In that case, we can solve this without brute-forcing all possibilities by using a public \(internal\_nullifier = Poseidon(a_1)\), so if a user sends more than one message, it will be immediately visible to everyone.</p>
<h2 id="some-important-notes"><a class="header" href="#some-important-notes">Some important notes</a></h2>
<p>There are also <code>nullifier</code> and <code>rln-identifier</code>, which can be found in the <strong>RLN</strong> protocol/circuits.</p>
<p>So, <code>rln-identifier</code> is just a random value that's unique per <strong>RLN</strong> app. It's used for additional cross-application security - to protect the user secrets from being compromised if they use the same credentials across different <strong>RLN</strong> apps. If <code>rln-identifier</code> is not present, the user uses the same credentials and sends a message in two different <strong>RLN</strong> apps using the same epoch, then their secret key can be revealed. Adding the <code>rln-identifier</code> field, we obscure the nullifier, so this kind of attack cannot happen. The only kind of attack that is possible is if we have an entity with a global view of all messages, and they try to brute-force different combinations of x and y shares for different nullifiers.</p>
<p>Now, imagine there are a lot of users sending messages, and after each received message, we need to check if any member can be slashed. To do this, we can use all combinations of received <em>shares</em> and try to recover the polynomial, but this is a naive and non-optimal approach. Suppose we have a mechanism that will tell us about the connection between a person and their messages while not revealing their identity. In that case, we can solve this without brute-forcing all possibilities by using a public <code>nullifier</code> (\(Poseidon(a_1, rln-identifier)\)), so if a user sends more than one message, it will be immediately visible to everyone.</p>
<p>Also, in our example (and <a href="https://github.com/njofce/zk-chat">zk-chat</a> implementation), we use linear polynomial, but <a href="sss.html">SSS</a> allows us to use various degree polynomials; therefore we can implement a protocol, where more than one signal (message) can be sent in per epoch. </p>
<p>To learn more, check out the <a href="https://hackmd.io/7GR5Vi28Rz2EpEmLK0E0Aw?view">specification</a>; there are also <a href="https://github.com/privacy-scaling-explorations/rln/tree/master/circuits">circuits</a> implemented for various degree polynomials too.</p>
@@ -198,18 +199,18 @@ We denote: \(x = Poseidon(message), y = A(x)\). </p>
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sss.html
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@@ -5,7 +5,6 @@
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<title>Shamir&#x27;s Secret Sharing - Rate-Limiting Nullifier</title>
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@@ -211,18 +210,18 @@ So, the shares are: \((5, 55), (8, 70), (16, 110)\)</li>
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terminology.html
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33
under_the_hood.html
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@@ -5,7 +5,6 @@
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33
uses.html
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@@ -5,7 +5,6 @@
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35
what_is_rln.html
View File

@@ -5,7 +5,6 @@
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<title>What is RLN - Rate-Limiting Nullifier</title>
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<h1 id="what-is-rate-limiting-nullifier"><a class="header" href="#what-is-rate-limiting-nullifier">What is Rate-Limiting Nullifier?</a></h1>
<p><em>This topic is a part of <a href="https://medium.com/privacy-scaling-explorations/rate-limiting-nullifier-a-spam-protection-mechanism-for-anonymous-environments-bbe4006a57d">complete overview</a> by Blagoj</em>.</p>
<hr />
<p><strong>RLN</strong> is a zero-knowledge gadget that enables spam prevention for decentralized, anonymous environments.</p>
<p>The anonymity property opens up the possibility for spam and Sybil attack vectors for certain applications, which could seriously degrade the user experience and the overall functioning of the application. For example, imagine a chat application where users are anonymous. Now, everyone can write an unlimited number of spam messages, but we don't have the ability to kick this member because the spammer is anonymous. </p>
<p><strong>RLN</strong> helps us identify and &quot;kick&quot; the spammer.</p>
@@ -191,18 +188,18 @@
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