zec
/
orchard
mirror of https://github.com/zcash/orchard.git
1
0
Fork 0
Code Issues Projects Releases Wiki Activity
orchard/design/nullifiers.html

339 lines
196 KiB
HTML
Raw Normal View History

deploy: a4fb24b724aba340595b71da9b8936b0fd4cace1
2021-01-08 09:28:54 -08:00
<!DOCTYPE HTML>
<html lang="en" class="sidebar-visible no-js light">
<head>
<!-- Book generated using mdBook -->
<meta charset="UTF-8">
<title>Nullifiers - The Orchard Book</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" />
<link rel="icon" href="../favicon.svg">
<link rel="shortcut icon" href="../favicon.png">
<link rel="stylesheet" href="../css/variables.css">
<link rel="stylesheet" href="../css/general.css">
<link rel="stylesheet" href="../css/chrome.css">
<link rel="stylesheet" href="../css/print.css" media="print">
<!-- Fonts -->
<link rel="stylesheet" href="../FontAwesome/css/font-awesome.css">
<link rel="stylesheet" href="../fonts/fonts.css">
<!-- Highlight.js Stylesheets -->
<link rel="stylesheet" href="../highlight.css">
<link rel="stylesheet" href="../tomorrow-night.css">
<link rel="stylesheet" href="../ayu-highlight.css">
<!-- Custom theme stylesheets -->
</head>
<body>
<!-- Provide site root to javascript -->
<script type="text/javascript">
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">
try {
var theme = localStorage.getItem('mdbook-theme');
var sidebar = localStorage.getItem('mdbook-sidebar');
if (theme.startsWith('"') && theme.endsWith('"')) {
localStorage.setItem('mdbook-theme', theme.slice(1, theme.length - 1));
}
if (sidebar.startsWith('"') && sidebar.endsWith('"')) {
localStorage.setItem('mdbook-sidebar', sidebar.slice(1, sidebar.length - 1));
}
} catch (e) { }
</script>
<!-- Set the theme before any content is loaded, prevents flash -->
<script type="text/javascript">
var theme;
try { theme = localStorage.getItem('mdbook-theme'); } catch(e) { }
if (theme === null || theme === undefined) { theme = default_theme; }
var html = document.querySelector('html');
html.classList.remove('no-js')
html.classList.remove('light')
html.classList.add(theme);
html.classList.add('js');
</script>
<!-- Hide / unhide sidebar before it is displayed -->
<script type="text/javascript">
var html = document.querySelector('html');
var sidebar = 'hidden';
if (document.body.clientWidth >= 1080) {
try { sidebar = localStorage.getItem('mdbook-sidebar'); } catch(e) { }
sidebar = sidebar || 'visible';
}
html.classList.remove('sidebar-visible');
html.classList.add("sidebar-" + sidebar);
</script>
<nav id="sidebar" class="sidebar" aria-label="Table of contents">
<div class="sidebar-scrollbox">
<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">Orchard</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/preliminaries.html"><strong aria-hidden="true">1.1.</strong> Preliminaries</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/keys.html"><strong aria-hidden="true">2.1.</strong> Creating keys and addresses</a></li><li class="chapter-item expanded "><a href="../user/creating-notes.html"><strong aria-hidden="true">2.2.</strong> Creating notes</a></li><li class="chapter-item expanded "><a href="../user/spending-notes.html"><strong aria-hidden="true">2.3.</strong> Spending notes</a></li><li class="chapter-item expanded "><a href="../user/integration.html"><strong aria-hidden="true">2.4.</strong> Integration into an existing chain</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/actions.html"><strong aria-hidden="true">3.1.</strong> Actions</a></li><li class="chapter-item expanded "><a href="../design/commitments.html"><strong aria-hidden="true">3.2.</strong> Commitments</a></li><li class="chapter-item expanded "><a href="../design/commitment-tree.html"><strong aria-hidden="true">3.3.</strong> Commitment tree</a></li><li class="chapter-item expanded "><a href="../design/nullifiers.html" class="active"><strong aria-hidden="true">3.4.</strong> Nullifiers</a></li><li class="chapter-item expanded "><a href="../design/signatures.html"><strong aria-hidden="true">3.5.</strong> Signatures</a></li><li class="chapter-item expanded "><a href="../design/circuit.html"><strong aria-hidden="true">3.6.</strong> Circuit</a></li></ol></li></ol>
</div>
<div id="sidebar-resize-handle" class="sidebar-resize-handle"></div>
</nav>
<div id="page-wrapper" class="page-wrapper">
<div class="page">
<div id="menu-bar-hover-placeholder"></div>
<div id="menu-bar" class="menu-bar sticky bordered">
<div class="left-buttons">
<button id="sidebar-toggle" class="icon-button" type="button" title="Toggle Table of Contents" aria-label="Toggle Table of Contents" aria-controls="sidebar">
<i class="fa fa-bars"></i>
</button>
<button id="theme-toggle" class="icon-button" type="button" title="Change theme" aria-label="Change theme" aria-haspopup="true" aria-expanded="false" aria-controls="theme-list">
<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="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>
<li role="none"><button role="menuitem" class="theme" id="ayu">Ayu</button></li>
</ul>
<button id="search-toggle" class="icon-button" type="button" title="Search. (Shortkey: s)" aria-label="Toggle Searchbar" aria-expanded="false" aria-keyshortcuts="S" aria-controls="searchbar">
<i class="fa fa-search"></i>
</button>
</div>
<h1 class="menu-title">The Orchard Book</h1>
<div class="right-buttons">
<a href="../print.html" title="Print this book" aria-label="Print this book">
<i id="print-button" class="fa fa-print"></i>
</a>
</div>
</div>
<div id="search-wrapper" class="hidden">
<form id="searchbar-outer" class="searchbar-outer">
<input type="search" name="search" id="searchbar" name="searchbar" placeholder="Search this book ..." aria-controls="searchresults-outer" aria-describedby="searchresults-header">
</form>
<div id="searchresults-outer" class="searchresults-outer hidden">
<div id="searchresults-header" class="searchresults-header"></div>
<ul id="searchresults">
</ul>
</div>
</div>
<!-- Apply ARIA attributes after the sidebar and the sidebar toggle button are added to the DOM -->
<script type="text/javascript">
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) {
link.setAttribute('tabIndex', sidebar === 'visible' ? 0 : -1);
});
</script>
<div id="content" class="content">
<main>
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
<h1><a class="header" href="#nullifiers" id="nullifiers">Nullifiers</a></h1>
<p>The nullifier design we use for Orchard is</p>
deploy: f3a51fa6f420ec0d21bb92e6eddab84316432055
2021-01-08 17:01:47 -08:00
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;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"><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight">k</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mclose">)</span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mclose">]</span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span><span class="mpunct">,</span></span></span></span></span></p>
deploy: a4fb24b724aba340595b71da9b8936b0fd4cace1
2021-01-08 09:28:54 -08:00
<p>where:</p>
<ul>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span></span></span></span> is a keyed circuit-efficient hash (such as Rescue).</li>
deploy: f3a51fa6f420ec0d21bb92e6eddab84316432055
2021-01-08 17:01:47 -08:00
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">G</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is a cryptographic hash into the group (such as BLAKE2s with simplified SWU).</li>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span></span></span></span> is a fixed base, independent of any others returned by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">G</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</li>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is a base unique to this output.
deploy: a4fb24b724aba340595b71da9b8936b0fd4cace1
2021-01-08 09:28:54 -08:00
<ul>
deploy: f3a51fa6f420ec0d21bb92e6eddab84316432055
2021-01-08 17:01:47 -08:00
<li>For non-zero-valued notes, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">G</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mclose">)</span></span></span></span>. As with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;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 mathsf mtight">S</span><span class="mord mathsf mtight">i</span><span class="mord mathsf mtight" style="margin-right:0.01389em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span> in Sprout,
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">ρ</span></span></span></span> includes the nullifiers of any Orchard notes being spent in the same action.
Given that an action consists of a single spend and a single output, we set <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">ρ</span></span></span></span> to
be the nullifier of the spent note.</li>
<li>For zero-valued notes, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is constrained by the circuit to a fixed base independent
of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span></span></span></span> and any others returned by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">G</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</li>
deploy: a4fb24b724aba340595b71da9b8936b0fd4cace1
2021-01-08 09:28:54 -08:00
</ul>
</li>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span></span></span></span> is sender-controlled randomness. It is not required to be unique, and in practice
deploy: f3a51fa6f420ec0d21bb92e6eddab84316432055
2021-01-08 17:01:47 -08:00
is derived from both <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">ρ</span></span></span></span> and a sender-selected random value <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">s</span><span class="mord mathsf">e</span><span class="mord mathsf">e</span><span class="mord mathsf">d</span></span></span></span></span>:
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0991079999999998em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><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 mathnormal mtight" style="margin-right:0.03588em;">ψ</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">s</span><span class="mord mathsf">e</span><span class="mord mathsf">e</span><span class="mord mathsf">d</span></span><span class="mclose">)</span></span></span></span>.</li>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span></span></span></span> is a blinding scalar, similarly generated as
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><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 mathsf mtight" style="margin-right:0.01389em;">r</span><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight" style="margin-right:0.06944em;">f</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">s</span><span class="mord mathsf">e</span><span class="mord mathsf">e</span><span class="mord mathsf">d</span></span><span class="mclose">)</span></span></span></span>.</li>
deploy: a4fb24b724aba340595b71da9b8936b0fd4cace1
2021-01-08 09:28:54 -08:00
</ul>
<p>This gives a note structure of</p>
deploy: f3a51fa6f420ec0d21bb92e6eddab84316432055
2021-01-08 17:01:47 -08:00
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">d</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span><span class="mclose">)</span><span class="mord">.</span></span></span></span></span></p>
<p>The note plaintext includes <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">s</span><span class="mord mathsf">e</span><span class="mord mathsf">e</span><span class="mord mathsf">d</span></span></span></span></span> in place of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span></span></span></span>, and
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span></span></span></span>. <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> is omitted entirely from the action:</p>
<ul>
<li>Consensus nodes directly derive <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">G</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mclose">)</span></span></span></span> and provide it as a public input to the
circuit (which ignores it for zero-valued notes, as with the commitment tree anchor).</li>
<li>The recipient can recompute the correct <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> given their additional knowledge of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span>.</li>
</ul>
deploy: a4fb24b724aba340595b71da9b8936b0fd4cace1
2021-01-08 09:28:54 -08:00
<h2><a class="header" href="#security-properties" id="security-properties">Security properties</a></h2>
<p>We care about several security properties for our nullifiers:</p>
<ul>
<li>
<p><strong>Balance:</strong> can I forge money?</p>
</li>
<li>
<p><strong>Note Privacy:</strong> can I gain information about notes only from the public block chain?</p>
<ul>
<li>This describes notes sent in-band.</li>
</ul>
</li>
<li>
<p><strong>Note Privacy (OOB):</strong> can I gain information about notes sent out-of-band, only from
the public block chain?</p>
<ul>
<li>In this case, we assume privacy of the channel over which the note is sent, and that
the adversary does not have access to any notes sent to the same address which are
then spent (so that the nullifier is on the block chain somewhere).</li>
</ul>
</li>
<li>
<p><strong>Spend Unlinkability:</strong> given the incoming viewing key for an address, and not the full
viewing key, can I (possibly the sender) detect spends of any notes sent to that address?</p>
<ul>
<li>We're giving <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">i</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> to the attacker and allowing it to be the sender in order to make
this property as strong as possible: they will have <em>all</em> the notes sent to that
address.</li>
</ul>
</li>
<li>
<p><strong>Faerie Resistance:</strong> can I perform a Faerie Gold attack (i.e. cause notes to be
accepted that are unspendable)?</p>
deploy: f3a51fa6f420ec0d21bb92e6eddab84316432055
2021-01-08 17:01:47 -08:00
<ul>
<li>We're giving the full viewing key to the attacker and allowing it to be the sender in
order to make this property as strong as possible: they will have <em>all</em> the notes sent
to that address, and be able to derive <em>every</em> nullifier.</li>
</ul>
deploy: a4fb24b724aba340595b71da9b8936b0fd4cace1
2021-01-08 09:28:54 -08:00
</li>
</ul>
<p>We assume (and instantiate elsewhere) the following primitives:</p>
<ul>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span></span></span></span> is an elliptic curve (such as Pallas).</li>
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> is the note encryption key derivation function.</li>
</ul>
<p>For our chosen design, our desired security properties rely on the following assumptions:</p>
deploy: f3a51fa6f420ec0d21bb92e6eddab84316432055
2021-01-08 17:01:47 -08:00
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.128347em;vertical-align:-2.8141735em;"></span><span class="mord"><span class="mtable"><span class="vertical-separator" style="height:6.128347em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-2.8141735em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.3141735000000003em;"><span style="top:-5.4741735em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Balance</span></span></span></span><span style="top:-4.272842499999999em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Note Privacy</span></span></span></span><span style="top:-3.0728424999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Note Privacy (OOB)</span></span></span></span><span style="top:-1.7458265000000004em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Spend Unlinkability</span></span></span></span><span style="top:-0.5458265000000002em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">Faerie Resistance</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.8141735em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:6.128347em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-2.8141735em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.3141735000000003em;"><span style="top:-5.4741735em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.272842499999999em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.424669em;margin-left:-0.08125em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><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 mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mor
deploy: a4fb24b724aba340595b71da9b8936b0fd4cace1
2021-01-08 09:28:54 -08:00
<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1166619999999998em;vertical-align:-0.275331em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.424669em;margin-left:-0.08125em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span><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 mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em;"><span></span></span></span></span></span></span></span></span></span> is computational Diffie-Hellman using <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> for the key derivation, with
one-time ephemeral keys. This assumption is heuristically weaker than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> but stronger
than <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>.</p>
<blockquote>
deploy: f3a51fa6f420ec0d21bb92e6eddab84316432055
2021-01-08 17:01:47 -08:00
<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">†</span></span></span></span> We additionally assume that for any input <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span></span></span></span>,
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;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"><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight">k</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord"><span class="mord mathsf">n</span><span class="mord mathsf">k</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mclose">}</span></span></span></span> gives a scalar in an adequate range for
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>. (Otherwise, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span></span></span></span> could be trivial, e.g. independent of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf">n</span><span class="mord mathsf">k</span></span></span></span></span>.)</p>
deploy: a4fb24b724aba340595b71da9b8936b0fd4cace1
2021-01-08 09:28:54 -08:00
</blockquote>
<h2><a class="header" href="#considered-alternatives" id="considered-alternatives">Considered alternatives</a></h2>
<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord" style="color:red;"><span class="mord text" style="color:red;"><span class="mord" style="color:red;">⚠</span><span class="mord textsf" style="color:red;"> Caution</span></span></span></span></span></span>: be skeptical of the claims in this table about what
problem(s) each security property depends on. They may not be accurate and are definitely
not fully rigorous.</p>
deploy: f3a51fa6f420ec0d21bb92e6eddab84316432055
2021-01-08 17:01:47 -08:00
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:17.98846799999999em;vertical-align:-8.724234em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.264233999999991em;"><span style="top:-11.224234em;"><span class="pstrut" style="height:11.224234em;"></span><span class="mtable"><span class="vertical-separator" style="height:17.948467999999995em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-8.724233999999997em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.224234em;"><span style="top:-11.384233999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span></span></span><span style="top:-10.182902999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathsf">n</span><span class="mord mathsf">k</span></span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span><span style="top:-8.981571999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathsf">n</span><span class="mord mathsf">k</span></span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mclose">]</span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span></span></span><span style="top:-7.780240999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mopen">[</span><span class="mord"><span class="mord mathsf">n</span><span class="mord mathsf">k</span></span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span></span></span><span style="top:-6.578909999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mord mathnormal">a</span><span class="mord mathnormal">s</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mopen">[</span><span class="mord"><span class="mord mathsf">n</span><span class="mord mathsf">k</span></span><span class="mclose">]</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathsf" style="margin-right:0.01389em;">r</span><span class="mord mathsf">n</span><span class="mord mathsf" style="margin-right:0.06944em;">f</span></span><span class="mclose">]</span><span class="mord"><span class="mord mathcal" style=
deploy: a4fb24b724aba340595b71da9b8936b0fd4cace1
2021-01-08 09:28:54 -08:00
<p>In the above alternatives:</p>
<ul>
deploy: f3a51fa6f420ec0d21bb92e6eddab84316432055
2021-01-08 17:01:47 -08:00
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78055em;vertical-align:-0.09722em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.0593em;">G</span></span></span></span></span> is an fixed independent base, independent of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.07382em;">I</span></span></span></span></span> and any others
deploy: a4fb24b724aba340595b71da9b8936b0fd4cace1
2021-01-08 09:28:54 -08:00
returned by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">G</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>.</li>
</ul>
<p>The <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.849108em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal">m</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><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"><span class="mord mathsf mtight">n</span><span class="mord mathsf mtight" style="margin-right:0.06944em;">f</span></span></span></span></span></span></span></span></span></span></span></span></span> variants enabled nullifier domain separation based on note
value, without directly depending on <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span></span></span></span> (which in its native type is a base
field element, not a group element). We decided instead to follow Sapling by defining an
intermediate representation of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathsf">c</span><span class="mord mathsf">m</span></span></span></span></span> as a group element, that is only used in
nullifier computation.</p>
</main>
<nav class="nav-wrapper" aria-label="Page navigation">
<!-- Mobile navigation buttons -->
<a rel="prev" href="../design/commitment-tree.html" class="mobile-nav-chapters previous" title="Previous chapter" aria-label="Previous chapter" aria-keyshortcuts="Left">
<i class="fa fa-angle-left"></i>
</a>
<a rel="next" href="../design/signatures.html" class="mobile-nav-chapters next" title="Next chapter" aria-label="Next chapter" aria-keyshortcuts="Right">
<i class="fa fa-angle-right"></i>
</a>
<div style="clear: both"></div>
</nav>
</div>
</div>
<nav class="nav-wide-wrapper" aria-label="Page navigation">
<a rel="prev" href="../design/commitment-tree.html" class="nav-chapters previous" title="Previous chapter" aria-label="Previous chapter" aria-keyshortcuts="Left">
<i class="fa fa-angle-left"></i>
</a>
<a rel="next" href="../design/signatures.html" class="nav-chapters next" title="Next chapter" aria-label="Next chapter" aria-keyshortcuts="Right">
<i class="fa fa-angle-right"></i>
</a>
</nav>
</div>
<script type="text/javascript">
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>
<!-- Custom JS scripts -->
</body>
</html>