2020-12-22 12:47:31 -08:00
<!DOCTYPE HTML>
< html lang = "en" class = "sidebar-visible no-js light" >
< head >
<!-- Book generated using mdBook -->
< meta charset = "UTF-8" >
< title > Multipoint opening argument - The halo2 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" / >
2021-01-11 13:36:59 -08:00
< link rel = "icon" href = "../../favicon.svg" >
2020-12-22 12:47:31 -08:00
2021-01-11 13:36:59 -08:00
< link rel = "shortcut icon" href = "../../favicon.png" >
2020-12-22 12:47:31 -08:00
2021-01-11 13:36:59 -08:00
< link rel = "stylesheet" href = "../../css/variables.css" >
< link rel = "stylesheet" href = "../../css/general.css" >
< link rel = "stylesheet" href = "../../css/chrome.css" >
2020-12-22 12:47:31 -08:00
2021-01-11 13:36:59 -08:00
< link rel = "stylesheet" href = "../../css/print.css" media = "print" >
2020-12-22 12:47:31 -08:00
<!-- Fonts -->
2021-01-11 13:36:59 -08:00
< link rel = "stylesheet" href = "../../FontAwesome/css/font-awesome.css" >
2020-12-22 12:47:31 -08:00
2021-01-11 13:36:59 -08:00
< link rel = "stylesheet" href = "../../fonts/fonts.css" >
2020-12-22 12:47:31 -08:00
<!-- Highlight.js Stylesheets -->
2021-01-11 13:36:59 -08:00
< link rel = "stylesheet" href = "../../highlight.css" >
< link rel = "stylesheet" href = "../../tomorrow-night.css" >
< link rel = "stylesheet" href = "../../ayu-highlight.css" >
2020-12-22 12:47:31 -08:00
<!-- Custom theme stylesheets -->
< / head >
< body >
<!-- Provide site root to javascript -->
< script type = "text/javascript" >
2021-01-11 13:36:59 -08:00
var path_to_root = "../../";
2020-12-22 12:47:31 -08:00
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" >
2021-06-01 09:31:34 -07:00
< ol class = "chapter" > < li class = "chapter-item expanded affix " > < a href = "../../index.html" > halo2< / 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/proofs.html" > < strong aria-hidden = "true" > 1.1.< / strong > Proof systems< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../concepts/arithmetization.html" > < strong aria-hidden = "true" > 1.2.< / strong > UltraPLONK Arithmetization< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../concepts/chips.html" > < strong aria-hidden = "true" > 1.3.< / strong > Chips< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../concepts/gadgets.html" > < strong aria-hidden = "true" > 1.4.< / strong > Gadgets< / 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/dev-tools.html" > < strong aria-hidden = "true" > 2.1.< / strong > Developer tools< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../user/simple-example.html" > < strong aria-hidden = "true" > 2.2.< / strong > A simple example< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../user/lookup-tables.html" > < strong aria-hidden = "true" > 2.3.< / strong > Lookup tables< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../user/gadgets.html" > < strong aria-hidden = "true" > 2.4.< / strong > Gadgets< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../user/tips-and-tricks.html" > < strong aria-hidden = "true" > 2.5.< / strong > Tips and tricks< / 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/proving-system.html" > < strong aria-hidden = "true" > 3.1.< / strong > Proving system< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../../design/proving-system/lookup.html" > < strong aria-hidden = "true" > 3.1.1.< / strong > Lookup argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../design/proving-system/permutation.html" > < strong aria-hidden = "true" > 3.1.2.< / strong > Permutation argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../design/proving-system/circuit-commitments.html" > < strong aria-hidden = "true" > 3.1.3.< / strong > Circuit commitments< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../design/proving-system/vanishing.html" > < strong aria-hidden = "true" > 3.1.4.< / strong > Vanishing argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../design/proving-system/multipoint-opening.html" class = "active" > < strong aria-hidden = "true" > 3.1.5.< / strong > Multipoint opening argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../design/proving-system/inner-product.html" > < strong aria-hidden = "true" > 3.1.6.< / strong > Inner product argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../design/proving-system/comparison.html" > < strong aria-hidden = "true" > 3.1.7.< / strong > Comparison to other work< / a > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../../design/implementation.html" > < strong aria-hidden = "true" > 3.2.< / strong > Implementation< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../../design/implementation/proofs.html" > < strong aria-hidden = "true" > 3.2.1.< / strong > Proofs< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../design/implementation/fields.html" > < strong aria-hidden = "true" > 3.2.2.< / strong > Fields< / a > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../../design/gadgets.html" > < strong aria-hidden = "true" > 3.3.< / strong > Gadgets< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../../design/gadgets/sha256.html" > < strong aria-hidden = "true" > 3.3.1.< / strong > SHA-256< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../../design/gadgets/sha256/table16.html" > < strong aria-hidden = "true" > 3.3.1.1.< / strong > 16-bit table chip< / a > < / li > < / ol > < / li > < / ol > < /
2020-12-22 12:47:31 -08:00
< / 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 halo2 Book< / h1 >
< div class = "right-buttons" >
2021-01-11 13:36:59 -08:00
< a href = "../../print.html" title = "Print this book" aria-label = "Print this book" >
2020-12-22 12:47:31 -08:00
< i id = "print-button" class = "fa fa-print" > < / i >
< / a >
2021-06-05 03:42:23 -07:00
2020-12-22 12:47:31 -08:00
< / div >
< / div >
< div id = "search-wrapper" class = "hidden" >
< form id = "searchbar-outer" class = "searchbar-outer" >
2021-06-05 03:42:23 -07:00
< input type = "search" id = "searchbar" name = "searchbar" placeholder = "Search this book ..." aria-controls = "searchresults-outer" aria-describedby = "searchresults-header" >
2020-12-22 12:47:31 -08:00
< / 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" >
2021-06-05 03:42:23 -07:00
< h1 id = "multipoint-opening-argument" > < a class = "header" href = "#multipoint-opening-argument" > Multipoint opening argument< / a > < / h1 >
2020-12-22 12:47:31 -08:00
< p > Consider the commitments < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.8777699999999999em;vertical-align:-0.19444em;" > < / span > < span class = "mord mathnormal" > A< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.05017em;" > B< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.07153em;" > C< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.02778em;" > D< / span > < / span > < / span > < / span > to polynomials < 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" > a< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" > b< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" > c< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" > d< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > .
Let's say that < 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" > a< / span > < / span > < / span > < / span > and < 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" > b< / span > < / span > < / span > < / span > were queried at the point < 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 > , while < 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" > c< / span > < / span > < / span > < / span > and < 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" > d< / span > < / span > < / span > < / span >
were queried at both points < 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 > and < 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;" > ω< / span > < span class = "mord mathnormal" > x< / span > < / span > < / span > < / span > . (Here, < 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;" > ω< / span > < / span > < / span > < / span > is the primitive
root of unity in the multiplicative subgroup over which we constructed the
polynomials).< / p >
2021-02-12 08:17:18 -08:00
< p > To open these commitments, we could create a polynomial < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.8777699999999999em;vertical-align:-0.19444em;" > < / span > < span class = "mord mathnormal" > Q< / span > < / span > < / span > < / span > for each point that we queried
at (corresponding to each relative rotation used in the circuit). But this would not be
efficient in the circuit; for example, < 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" > c< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > would appear in multiple polynomials.< / p >
< p > Instead, we can group the commitments by the sets of points at which they were queried:
2020-12-22 12:47:31 -08:00
< span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:3.6000000000000005em;vertical-align:-1.5500000000000007em;" > < / span > < span class = "mord" > < span class = "mtable" > < 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:2.05em;" > < span style = "top:-4.05em;" > < span class = "pstrut" style = "height:2.84em;" > < / span > < span class = "mord" > < / span > < / span > < span style = "top:-2.849999999999999em;" > < span class = "pstrut" style = "height:2.84em;" > < / span > < span class = "mord" > < / span > < / span > < span style = "top:-1.6499999999999992em;" > < span class = "pstrut" style = "height:2.84em;" > < / span > < span class = "mord" > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:1.5500000000000007em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / 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:2.05em;" > < span style = "top:-4.21em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mopen" > {< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > }< / span > < / span > < / span > < span style = "top:-3.0099999999999993em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > A< / span > < / span > < / span > < span style = "top:-1.8099999999999994em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.05017em;" > B< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:1.5500000000000007em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / 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:2.05em;" > < span style = "top:-4.05em;" > < span class = "pstrut" style = "height:2.84em;" > < / span > < span class = "mord" > < / span > < / span > < span style = "top:-2.849999999999999em;" > < span class = "pstrut" style = "height:2.84em;" > < / span > < span class = "mord" > < / span > < / span > < span style = "top:-1.6499999999999992em;" > < span class = "pstrut" style = "height:2.84em;" > < / span > < span class = "mord" > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:1.5500000000000007em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / 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:2.05em;" > < span style = "top:-4.21em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mopen" > {< / span > < span class = "mord mathnormal" > x< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > ω< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > }< / span > < / span > < / span > < span style = "top:-3.0099999999999993em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.07153em;" > C< / span > < / span > < / span > < span style = "top:-1.8099999999999994em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.02778em;" > D< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:1.5500000000000007em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / 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:2.05em;" > < span style
2021-02-12 08:17:18 -08:00
< p > For each of these groups, we combine them into a polynomial set, and create a single < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.8777699999999999em;vertical-align:-0.19444em;" > < / span > < span class = "mord mathnormal" > Q< / span > < / span > < / span > < / span >
for that set, which we open at each rotation.< / p >
2021-06-05 03:42:23 -07:00
< h2 id = "optimisation-steps" > < a class = "header" href = "#optimisation-steps" > Optimisation steps< / a > < / h2 >
2021-02-12 08:17:18 -08:00
< p > The multipoint opening optimisation takes as input:< / p >
< ul >
< li > A random < 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 > sampled by the verifier, at which we evaluate < 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" > a< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" > b< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" > c< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" > d< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > .< / li >
< li > Evaluations of each polynomial at each point of interest, provided by the prover:
< 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" > a< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" > b< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" > c< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" > d< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" > c< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > ω< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" > d< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > ω< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < / li >
< / ul >
< p > These are the outputs of the < a href = "vanishing.html#evaluating-the-polynomials" > vanishing argument< / a > .< / p >
2020-12-22 12:47:31 -08:00
< p > The multipoint opening optimisation proceeds as such:< / p >
< ol >
< li >
2021-02-12 08:17:18 -08:00
< p > Sample random < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.58056em;vertical-align:-0.15em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < 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" > 1< / 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 > , to keep < 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" > a< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" > b< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" > c< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" > d< / span > < / span > < / span > < / span > linearly independent.< / p >
2020-12-22 12:47:31 -08:00
< / li >
< li >
< p > Accumulate polynomials and their corresponding evaluations according
to the point set at which they were queried:
< code > q_polys< / code > :
2021-02-12 08:17:18 -08:00
< span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:2.4000000000000004em;vertical-align:-0.9500000000000004em;" > < / span > < span class = "mord" > < span class = "mtable" > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "col-align-r" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:1.45em;" > < span style = "top:-3.61em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > q< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 1< / 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:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < span style = "top:-2.4099999999999997em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > q< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / 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:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.9500000000000004em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / 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:1.45em;" > < span style = "top:-3.61em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mrel" > =< / span > < / span > < / span > < span style = "top:-2.4099999999999997em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mrel" > =< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.9500000000000004em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / 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:1.45em;" > < span style = "top:-3.61em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > a< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < span style = "top:-2.4099999999999997em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > c< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.9500000000000004em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "col-align-l" > < span class = "vlist-t vlist-t2
2020-12-22 12:47:31 -08:00
< code > q_eval_sets< / code > :< / p >
< pre > < code class = "language-math" > [
2021-02-12 08:17:18 -08:00
[a(x) + x_1 b(x)],
2020-12-22 12:47:31 -08:00
[
2021-02-12 08:17:18 -08:00
c(x) + x_1 d(x),
c(\omega x) + x_1 d(\omega x)
2020-12-22 12:47:31 -08:00
]
]
< / code > < / pre >
< p > NB: < code > q_eval_sets< / code > is a vector of sets of evaluations, where the outer vector
goes over the point sets, and the inner vector goes over the points in each set.< / p >
< / li >
< li >
< p > Interpolate each set of values in < code > q_eval_sets< / code > :
< code > r_polys< / code > :
2021-02-12 08:17:18 -08:00
< span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:6.000000000000001em;vertical-align:-2.7500000000000004em;" > < / span > < span class = "mord" > < span class = "mtable" > < 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:3.2500000000000004em;" > < span style = "top:-5.410000000000001em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.02778em;" > r< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 1< / 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:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mord mathnormal" > s< / span > < span class = "mord" > .< / span > < span class = "mord mathnormal" > t< / span > < span class = "mord" > .< / span > < / span > < / span > < span style = "top:-4.21em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < / span > < / span > < span style = "top:-3.01em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.02778em;" > r< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / 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:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mord mathnormal" > s< / span > < span class = "mord" > .< / span > < span class = "mord mathnormal" > t< / span > < span class = "mord" > .< / span > < / span > < / span > < span style = "top:-1.8099999999999998em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < / span > < / span > < span style = "top:-0.6099999999999997em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:2.7500000000000004em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / 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:3.2500000000000004em;" > < span style = "top:-5.410000000000001em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < / span > < / span > < span style = "top:-4.21em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.02778em;" > r< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 1< / 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 > < /spa
2020-12-22 12:47:31 -08:00
< / li >
< li >
< p > Construct < code > f_polys< / code > which check the correctness of < code > q_polys< / code > :
< code > f_polys< / code >
2021-02-12 08:17:18 -08:00
< span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:2.943331em;vertical-align:-1.2216655000000003em;" > < / span > < span class = "mord" > < span class = "mtable" > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "col-align-r" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:1.7216654999999998em;" > < span style = "top:-3.7216655em;" > < span class = "pstrut" style = "height:3.01em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 1< / 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:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < span style = "top:-2.3083344999999995em;" > < span class = "pstrut" style = "height:3.01em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / 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:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:1.2216655000000003em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / 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:1.7216654999999998em;" > < span style = "top:-3.7216655em;" > < span class = "pstrut" style = "height:3.01em;" > < / span > < span class = "mord" > < span class = "mrel" > =< / span > < / span > < / span > < span style = "top:-2.3083344999999995em;" > < span class = "pstrut" style = "height:3.01em;" > < / span > < span class = "mord" > < span class = "mrel" > =< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:1.2216655000000003em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / 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:1.7216654999999998em;" > < span style = "top:-3.7216655em;" > < span class = "pstrut" style = "height:3.01em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mopen nulldelimiter" > < / span > < span class = "mfrac" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:1.01em;" > < span style = "top:-2.655em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mathnormal mtight" style = "margin-right:0.07847em;" > X< / span > < span class = "mbin mtight" > − < / span > < span class = "mord mathnormal mtight" > x< / span > < / span > < / span > < / span > < span style = "top:-3.23em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "frac-line" style = "border-bottom-width:0.04em;" > < / span > < / span > < span style = "top:-3.485em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span
< p > If < 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" > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > q< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 1< / 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:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.02778em;" > r< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 1< / 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 > < / span > < / span > , then < 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" > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 1< / 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:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > should be a polynomial.
If < 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" > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > q< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / 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:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.02778em;" > r< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / 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 > < / span > < / span > and < 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" > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > q< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / 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;" > ω< / 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:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.02778em;" > r< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / 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;" > ω< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span >
2020-12-22 12:47:31 -08:00
then < 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" > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / 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:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > should be a polynomial.< / p >
< / li >
< li >
2021-02-12 08:17:18 -08:00
< p > Sample random < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.58056em;vertical-align:-0.15em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < 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" > 2< / 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 > to keep the < code > f_polys< / code > linearly independent.< / p >
2020-12-22 12:47:31 -08:00
< / li >
< li >
2021-02-12 08:17:18 -08:00
< p > Construct < 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" style = "margin-right:0.10764em;" > f< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > 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:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 1< / 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:0.07847em;" > X< / span > < span class = "mclose" > )< / 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 = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < 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" > 2< / 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 = "mord" > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / 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:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > .< / p >
2020-12-22 12:47:31 -08:00
< / li >
< li >
2021-02-12 08:17:18 -08:00
< p > Sample random < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.58056em;vertical-align:-0.15em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < 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" > 3< / 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 > , at which we evaluate < 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" style = "margin-right:0.10764em;" > f< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > :
< span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:2.73em;vertical-align:-1.115em;" > < / span > < span class = "mord" > < span class = "mtable" > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "col-align-r" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:1.615em;" > < span style = "top:-3.7849999999999997em;" > < span class = "pstrut" style = "height:3.01em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < span class = "mopen" > (< / span > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < 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" > 3< / 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 = "mclose" > )< / span > < / span > < / span > < span style = "top:-2.415em;" > < span class = "pstrut" style = "height:3.01em;" > < / span > < span class = "mord" > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:1.115em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / 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:1.615em;" > < span style = "top:-3.7849999999999997em;" > < span class = "pstrut" style = "height:3.01em;" > < / span > < span class = "mord" > < span class = "mrel" > =< / span > < / span > < / span > < span style = "top:-2.415em;" > < span class = "pstrut" style = "height:3.01em;" > < / span > < span class = "mord" > < span class = "mrel" > =< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:1.115em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / 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:1.615em;" > < span style = "top:-3.7849999999999997em;" > < span class = "pstrut" style = "height:3.01em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 1< / 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" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < 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" > 3< / 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 = "mclose" > )< / span > < / span > < / span > < span style = "top:-2.415em;" > < span class = "pstrut" style = "height:3.01em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mopen nulldelimiter" > < / span > < span class = "mfrac" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:1.01em;" > < span style = "top:-2.655em;" > < span class = "pstrut" style = "height:3e
2020-12-22 12:47:31 -08:00
< / li >
< li >
2021-02-12 08:17:18 -08:00
< p > Sample random < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.58056em;vertical-align:-0.15em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < 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" > 4< / 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 > to keep < 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" style = "margin-right:0.10764em;" > f< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > and < code > q_polys< / code > linearly independent.< / p >
2020-12-22 12:47:31 -08:00
< / li >
< li >
2021-05-03 15:55:00 -07:00
< p > Construct < code > final_poly< / code > , < span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1.06em;vertical-align:-0.31em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < span class = "mord mathnormal" > ina< / span > < span class = "mord mathnormal" style = "margin-right:0.01968em;" > l< / span > < span class = "mord" style = "margin-right:0.02778em;" > _< / span > < span class = "mord mathnormal" > p< / span > < span class = "mord mathnormal" > o< / span > < span class = "mord mathnormal" style = "margin-right:0.01968em;" > l< / span > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > y< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > 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:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / 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 = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < 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" > 4< / 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 = "mord" > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > q< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 1< / 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:0.07847em;" > X< / span > < span class = "mclose" > )< / 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:1.1141079999999999em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.8641079999999999em;" > < span style = "top:-2.4530000000000003em;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" > 4< / span > < / span > < / span > < span style = "top:-3.113em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.247em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > q< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999
2020-12-22 12:47:31 -08:00
which is the polynomial we commit to in the inner product argument.< / p >
< / li >
< / ol >
< / main >
< nav class = "nav-wrapper" aria-label = "Page navigation" >
<!-- Mobile navigation buttons -->
2021-02-12 08:17:18 -08:00
< a rel = "prev" href = "../../design/proving-system/vanishing.html" class = "mobile-nav-chapters previous" title = "Previous chapter" aria-label = "Previous chapter" aria-keyshortcuts = "Left" >
2020-12-22 12:47:31 -08:00
< i class = "fa fa-angle-left" > < / i >
< / a >
2021-02-12 08:17:18 -08:00
< a rel = "next" href = "../../design/proving-system/inner-product.html" class = "mobile-nav-chapters next" title = "Next chapter" aria-label = "Next chapter" aria-keyshortcuts = "Right" >
2020-12-22 12:47:31 -08:00
< 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" >
2021-02-12 08:17:18 -08:00
< a rel = "prev" href = "../../design/proving-system/vanishing.html" class = "nav-chapters previous" title = "Previous chapter" aria-label = "Previous chapter" aria-keyshortcuts = "Left" >
2020-12-22 12:47:31 -08:00
< i class = "fa fa-angle-left" > < / i >
< / a >
2021-02-12 08:17:18 -08:00
< a rel = "next" href = "../../design/proving-system/inner-product.html" class = "nav-chapters next" title = "Next chapter" aria-label = "Next chapter" aria-keyshortcuts = "Right" >
2020-12-22 12:47:31 -08:00
< i class = "fa fa-angle-right" > < / i >
< / a >
< / nav >
< / div >
< script type = "text/javascript" >
window.playground_copyable = true;
< / script >
2021-01-11 13:36:59 -08:00
< 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 >
2020-12-22 12:47:31 -08:00
2021-01-11 13:36:59 -08:00
< 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 >
2020-12-22 12:47:31 -08:00
<!-- Custom JS scripts -->
< / body >
< / html >
