From 4589335456d15c2d03afaff9ec7f7a14ac471bc3 Mon Sep 17 00:00:00 2001
From: PulpSpy <PulpSpy@users.noreply.github.com>
Date: Mon, 3 Jun 2024 17:51:56 +0000
Subject: [PATCH] Publishing Site MadibaGroup/2024-Gadgets at
 978afe979a3418acc90704a05fde2dca36a8de6b on Mon Jun  3 17:51:56 UTC 2024

---
 404.html                                      |  2 +-
 categories/index.html                         |  4 +--
 docs/background/index.html                    |  4 +--
 docs/background/poly-iop/index.html           |  7 ++---
 docs/gadgets/index.html                       |  4 +--
 docs/gadgets/index.xml                        |  7 ++---
 docs/gadgets/mult2/index.html                 | 29 +++++++++++--------
 docs/gadgets/productgadget/index.html         |  7 -----
 docs/gadgets/zero/index.html                  | 10 +++++++
 docs/index.html                               |  4 +--
 ...2f3b42865159bb58f2dbffc4a21acd4f958b8.json |  1 +
 ...b3cdb168d80fc9891a6e90b634767a453e188.json |  1 -
 ...dbdde95d267573c1f5b04dd2cb337c07f21791f.js |  2 +-
 index.html                                    |  4 +--
 index.xml                                     |  7 ++---
 sitemap.xml                                   |  2 +-
 tags/index.html                               |  4 +--
 17 files changed, 52 insertions(+), 47 deletions(-)
 delete mode 100644 docs/gadgets/productgadget/index.html
 create mode 100644 docs/gadgets/zero/index.html
 create mode 100644 en.search-data.min.6931ed3b61c13b8de0bdba543242f3b42865159bb58f2dbffc4a21acd4f958b8.json
 delete mode 100644 en.search-data.min.b6838b554e8ebd2c57c60f6f31db3cdb168d80fc9891a6e90b634767a453e188.json
 rename en.search.min.bb5d9164f3309a75ef9097bce777d0ab67b4a8025ae95f3105b06cec629ec033.js => en.search.min.93281b7c13f5b9b0a9578b33cdbdde95d267573c1f5b04dd2cb337c07f21791f.js (90%)
diff --git a/404.html b/404.html
index aaa2620..7c1ffc5 100644
--- a/404.html
+++ b/404.html
@@ -1,2 +1,2 @@
 <!doctype html><html lang=en><head><meta charset=UTF-8><meta name=viewport content="width=device-width,initial-scale=1"><meta name=description content><meta name=theme-color media="(prefers-color-scheme: light)" content="#ffffff"><meta name=theme-color media="(prefers-color-scheme: dark)" content="#343a40"><meta name=color-scheme content="light dark"><meta property="og:url" content="https://madibagroup.github.io/2024-Gadgets/404.html"><meta property="og:site_name" content="Handbook"><meta property="og:title" content="404 Page not found"><meta property="og:locale" content="en_us"><meta property="og:type" content="website"><title>404 Page not found | Handbook</title>
-<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/404.html><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.bb5d9164f3309a75ef9097bce777d0ab67b4a8025ae95f3105b06cec629ec033.js integrity="sha256-u12RZPMwmnXvkJe853fQq2e0qAJa6V8xBbBs7GKewDM=" crossorigin=anonymous></script><style>.not-found{text-align:center}.not-found h1{margin:.25em 0 0;opacity:.25;font-size:40vmin}</style></head><body><main class="flex justify-center not-found"><div><h1>404</h1><h2>Page Not Found</h2><h3><a href=/2024-Gadgets/>Handbook</a></h3></div></main></body></html>
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+<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/404.html><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.93281b7c13f5b9b0a9578b33cdbdde95d267573c1f5b04dd2cb337c07f21791f.js integrity="sha256-kygbfBP1ubCpV4szzb3eldJnVzwfWwTdLLM3wH8heR8=" crossorigin=anonymous></script><style>.not-found{text-align:center}.not-found h1{margin:.25em 0 0;opacity:.25;font-size:40vmin}</style></head><body><main class="flex justify-center not-found"><div><h1>404</h1><h2>Page Not Found</h2><h3><a href=/2024-Gadgets/>Handbook</a></h3></div></main></body></html>
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diff --git a/categories/index.html b/categories/index.html
index 3527d35..b00f106 100644
--- a/categories/index.html
+++ b/categories/index.html
@@ -1,5 +1,5 @@
 <!doctype html><html lang=en-us dir=ltr><head><meta charset=UTF-8><meta name=viewport content="width=device-width,initial-scale=1"><meta name=description content><meta name=theme-color media="(prefers-color-scheme: light)" content="#ffffff"><meta name=theme-color media="(prefers-color-scheme: dark)" content="#343a40"><meta name=color-scheme content="light dark"><meta property="og:url" content="https://madibagroup.github.io/2024-Gadgets/categories/"><meta property="og:site_name" content="Handbook"><meta property="og:title" content="Categories"><meta property="og:locale" content="en_us"><meta property="og:type" content="website"><title>Categories | Handbook</title>
-<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/categories/><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.bb5d9164f3309a75ef9097bce777d0ab67b4a8025ae95f3105b06cec629ec033.js integrity="sha256-u12RZPMwmnXvkJe853fQq2e0qAJa6V8xBbBs7GKewDM=" crossorigin=anonymous></script><link rel=alternate type=application/rss+xml href=https://madibagroup.github.io/2024-Gadgets/categories/index.xml title=Handbook><script id=MathJax-script async src=https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js></script><script>MathJax={tex:{displayMath:[["\\[","\\]"],["$$","$$"]],inlineMath:[["\\(","\\)"],["$","$"]]}}</script></head><body dir=ltr><input type=checkbox class="hidden toggle" id=menu-control>
-<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><a href=/2024-Gadgets/docs/background/><strong>Background</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/>Poly IOP</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/><strong>Gadgets</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/>Mult2</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/productgadget/>ProdGadget</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
+<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/categories/><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.93281b7c13f5b9b0a9578b33cdbdde95d267573c1f5b04dd2cb337c07f21791f.js integrity="sha256-kygbfBP1ubCpV4szzb3eldJnVzwfWwTdLLM3wH8heR8=" crossorigin=anonymous></script><link rel=alternate type=application/rss+xml href=https://madibagroup.github.io/2024-Gadgets/categories/index.xml title=Handbook><script id=MathJax-script async src=https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js></script><script>MathJax={tex:{displayMath:[["\\[","\\]"],["$$","$$"]],inlineMath:[["\\(","\\)"],["$","$"]]}}</script></head><body dir=ltr><input type=checkbox class="hidden toggle" id=menu-control>
+<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><strong>Background</strong></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/>Poly IOP</a></p></li><li><p><strong>Gadgets</strong></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/>Mult2</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
 </label><strong>Categories</strong>
 <label for=toc-control><img src=/2024-Gadgets/svg/toc.svg class=book-icon alt="Table of Contents"></label></div><aside class="hidden clearfix"><nav><ul><li class=book-section-flat><strong>Categories</strong><ul></ul></li><li class=book-section-flat><strong>Tags</strong><ul></ul></li></ul></nav></aside></header><footer class=book-footer><div class="flex flex-wrap justify-between"></div><script>(function(){function e(e){const t=window.getSelection(),n=document.createRange();n.selectNodeContents(e),t.removeAllRanges(),t.addRange(n)}document.querySelectorAll("pre code").forEach(t=>{t.addEventListener("click",function(){if(window.getSelection().toString())return;e(t.parentElement),navigator.clipboard&&navigator.clipboard.writeText(t.parentElement.textContent)})})})()</script></footer><label for=menu-control class="hidden book-menu-overlay"></label></div><aside class=book-toc><div class=book-toc-content><nav><ul><li class=book-section-flat><strong>Categories</strong><ul></ul></li><li class=book-section-flat><strong>Tags</strong><ul></ul></li></ul></nav></div></aside></main></body></html>
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diff --git a/docs/background/index.html b/docs/background/index.html
index 0b5e31a..b4ea8dd 100644
--- a/docs/background/index.html
+++ b/docs/background/index.html
@@ -1,5 +1,5 @@
 <!doctype html><html lang=en-us dir=ltr><head><meta charset=UTF-8><meta name=viewport content="width=device-width,initial-scale=1"><meta name=description content><meta name=theme-color media="(prefers-color-scheme: light)" content="#ffffff"><meta name=theme-color media="(prefers-color-scheme: dark)" content="#343a40"><meta name=color-scheme content="light dark"><meta property="og:url" content="https://madibagroup.github.io/2024-Gadgets/docs/background/"><meta property="og:site_name" content="Handbook"><meta property="og:title" content="Handbook"><meta property="og:locale" content="en_us"><meta property="og:type" content="website"><title>Background | Handbook</title>
-<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/docs/background/><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.bb5d9164f3309a75ef9097bce777d0ab67b4a8025ae95f3105b06cec629ec033.js integrity="sha256-u12RZPMwmnXvkJe853fQq2e0qAJa6V8xBbBs7GKewDM=" crossorigin=anonymous></script><link rel=alternate type=application/rss+xml href=https://madibagroup.github.io/2024-Gadgets/docs/background/index.xml title=Handbook><script id=MathJax-script async src=https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js></script><script>MathJax={tex:{displayMath:[["\\[","\\]"],["$$","$$"]],inlineMath:[["\\(","\\)"],["$","$"]]}}</script></head><body dir=ltr><input type=checkbox class="hidden toggle" id=menu-control>
-<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><a href=/2024-Gadgets/docs/background/ class=active><strong>Background</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/>Poly IOP</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/><strong>Gadgets</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/>Mult2</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/productgadget/>ProdGadget</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
+<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/docs/background/><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.93281b7c13f5b9b0a9578b33cdbdde95d267573c1f5b04dd2cb337c07f21791f.js integrity="sha256-kygbfBP1ubCpV4szzb3eldJnVzwfWwTdLLM3wH8heR8=" crossorigin=anonymous></script><link rel=alternate type=application/rss+xml href=https://madibagroup.github.io/2024-Gadgets/docs/background/index.xml title=Handbook><script id=MathJax-script async src=https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js></script><script>MathJax={tex:{displayMath:[["\\[","\\]"],["$$","$$"]],inlineMath:[["\\(","\\)"],["$","$"]]}}</script></head><body dir=ltr><input type=checkbox class="hidden toggle" id=menu-control>
+<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><strong>Background</strong></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/>Poly IOP</a></p></li><li><p><strong>Gadgets</strong></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/>Mult2</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
 </label><strong>Background</strong>
 <label for=toc-control><img src=/2024-Gadgets/svg/toc.svg class=book-icon alt="Table of Contents"></label></div><aside class="hidden clearfix"><nav id=TableOfContents></nav></aside></header><article class="markdown book-article"></article><footer class=book-footer><div class="flex flex-wrap justify-between"></div><script>(function(){function e(e){const t=window.getSelection(),n=document.createRange();n.selectNodeContents(e),t.removeAllRanges(),t.addRange(n)}document.querySelectorAll("pre code").forEach(t=>{t.addEventListener("click",function(){if(window.getSelection().toString())return;e(t.parentElement),navigator.clipboard&&navigator.clipboard.writeText(t.parentElement.textContent)})})})()</script></footer><label for=menu-control class="hidden book-menu-overlay"></label></div><aside class=book-toc><div class=book-toc-content><nav id=TableOfContents></nav></div></aside></main></body></html>
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diff --git a/docs/background/poly-iop/index.html b/docs/background/poly-iop/index.html
index 19302c3..1e5ab95 100644
--- a/docs/background/poly-iop/index.html
+++ b/docs/background/poly-iop/index.html
@@ -1,8 +1,8 @@
 <!doctype html><html lang=en-us dir=ltr><head><meta charset=UTF-8><meta name=viewport content="width=device-width,initial-scale=1"><meta name=description content="Notes on Polynomial Interactive Oracle Proofs (Draft) # Encoding Arrays of Data into Polynomials # Data starts as a array of integers where integers in each slot are between 0 and $q-1$, where $q$ is a large (generally 256 bit) prime number. Recall that we call this set of integers $\mathbb{Z}_q$.
 $\mathsf{data}_0$ $\mathsf{data}_1$ $\mathsf{data}_2$ $\mathsf{data}_3$ $\mathsf{data}_4$ It is common to denote a polynomial like $P(X)$ where $X$ is the variable of the polynomial."><meta name=theme-color media="(prefers-color-scheme: light)" content="#ffffff"><meta name=theme-color media="(prefers-color-scheme: dark)" content="#343a40"><meta name=color-scheme content="light dark"><meta property="og:url" content="https://madibagroup.github.io/2024-Gadgets/docs/background/poly-iop/"><meta property="og:site_name" content="Handbook"><meta property="og:title" content="Handbook"><meta property="og:description" content="Notes on Polynomial Interactive Oracle Proofs (Draft) # Encoding Arrays of Data into Polynomials # Data starts as a array of integers where integers in each slot are between 0 and $q-1$, where $q$ is a large (generally 256 bit) prime number. Recall that we call this set of integers $\mathbb{Z}_q$.
 $\mathsf{data}_0$ $\mathsf{data}_1$ $\mathsf{data}_2$ $\mathsf{data}_3$ $\mathsf{data}_4$ It is common to denote a polynomial like $P(X)$ where $X$ is the variable of the polynomial."><meta property="og:locale" content="en_us"><meta property="og:type" content="article"><meta property="article:section" content="docs"><title>Poly Iop | Handbook</title>
-<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/docs/background/poly-iop/><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.bb5d9164f3309a75ef9097bce777d0ab67b4a8025ae95f3105b06cec629ec033.js integrity="sha256-u12RZPMwmnXvkJe853fQq2e0qAJa6V8xBbBs7GKewDM=" crossorigin=anonymous></script><script id=MathJax-script async src=https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js></script><script>MathJax={tex:{displayMath:[["\\[","\\]"],["$$","$$"]],inlineMath:[["\\(","\\)"],["$","$"]]}}</script></head><body dir=ltr><input type=checkbox class="hidden toggle" id=menu-control>
-<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><a href=/2024-Gadgets/docs/background/><strong>Background</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/ class=active>Poly IOP</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/><strong>Gadgets</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/>Mult2</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/productgadget/>ProdGadget</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
+<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/docs/background/poly-iop/><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.93281b7c13f5b9b0a9578b33cdbdde95d267573c1f5b04dd2cb337c07f21791f.js integrity="sha256-kygbfBP1ubCpV4szzb3eldJnVzwfWwTdLLM3wH8heR8=" crossorigin=anonymous></script><script id=MathJax-script async src=https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js></script><script>MathJax={tex:{displayMath:[["\\[","\\]"],["$$","$$"]],inlineMath:[["\\(","\\)"],["$","$"]]}}</script></head><body dir=ltr><input type=checkbox class="hidden toggle" id=menu-control>
+<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><strong>Background</strong></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/ class=active>Poly IOP</a></p></li><li><p><strong>Gadgets</strong></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/>Mult2</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
 </label><strong>Poly Iop</strong>
 <label for=toc-control><img src=/2024-Gadgets/svg/toc.svg class=book-icon alt="Table of Contents"></label></div><aside class="hidden clearfix"><nav id=TableOfContents><ul><li><a href=#encoding-arrays-of-data-into-polynomials>Encoding Arrays of Data into Polynomials</a><ul><li><a href=#encoding-1-coefficients>Encoding 1: Coefficients</a></li><li><a href=#encoding-2-evaluation-points>Encoding 2: Evaluation Points</a></li><li><a href=#encoding-3-roots>Encoding 3: Roots</a></li><li><a href=#decision-tree-for-encoding>Decision Tree for Encoding</a></li></ul></li><li><a href=#optimizing-interpolation-roots-of-unity--fft>Optimizing Interpolation (Roots of Unity + FFT)</a></li><li><a href=#zeroing-parts-of-an-array>Zeroing Parts of an Array</a></li></ul></nav></aside></header><article class="markdown book-article"><h1 id=notes-on-polynomial-interactive-oracle-proofs-draft>Notes on Polynomial Interactive Oracle Proofs (Draft)
 <a class=anchor href=#notes-on-polynomial-interactive-oracle-proofs-draft>#</a></h1><h2 id=encoding-arrays-of-data-into-polynomials>Encoding Arrays of Data into Polynomials
@@ -17,5 +17,4 @@
 D -- Yes --> E[Evaluation Points];
 D -- No --> F[Coefficients];</p><p>The short answer is to start with evaluation point encoding until you realize you need something different.</p><h2 id=optimizing-interpolation-roots-of-unity--fft>Optimizing Interpolation (Roots of Unity + FFT)
 <a class=anchor href=#optimizing-interpolation-roots-of-unity--fft>#</a></h2><p>Moving forward, we will assume we are using Encoding 2: Evaluation Points. In short, this means placing the elements of our array into the $y$-coordinates ($\mathsf{data}_i=P(\boxed{x_i})$) of points on the polynomial. Before commiting to $P(\square)$, we need to use interpolation to find the coefficients of the polynomial that is fitted to these points. General interpolation algorithms are $O(n^2)$ work for $n$ evaluation points but this can be reduced to $O(n\log n)$ with an optimization.</p><p>The optimization we will explore enables interpolation via the fast Fourier transform (FFT). It concerns how to choose the $x$-coordinates, which will serve as the index for accessing the data: evaluating $P(X)$ at $x_i$ will reveal $\mathsf{data}_i$. First note, $x$-coordinates are from the exponent group ($Z_q$) and the choices exceed what is feasible to use ($2^{255}$ values in bls). Any subset can be used and interpolated. The optimization is to chose them with a mathematical structure. Specifically, instead an additive sequence (e.g., $0,1,2,3,\ldots$), we use a multiplicative sequence $1,\omega,\omega\cdot\omega,\omega\cdot\omega\cdot\omega,\ldots$ or equivalently: $\omega^0,\omega^1,\omega^2,\ldots,\omega^{\kappa-1}$. Further, the sequence is closed under multiplication which means that next index after $\omega^{\kappa-1}$ wraps back to the first index: $\omega^{k-1} \cdot \omega = \omega^\kappa = \omega^0=1$ (this property is also useful in proving relationships between data in the array and its neighbouring values).</p><p><img src=https://hackmd.io/_uploads/SyQdoMDxC.png alt="Roots of Unity"></p><p>For terminology, we say $\omega$ is a generator with multiplicative order $\kappa$ in $\mathbb{Z}_q$ (or $\omega \in \mathbb{G}_\kappa$). This implies $\omega^\kappa=1$. Rearranging, $\omega=\sqrt[\kappa]{1}$. Thus we can equivalently describe $\omega$ as a $\kappa$-th root of 1. Finally, as 1 is the unity element in $Z_q$, $\omega$ is commonly called a $\kappa$-th root of unity.</p><p>For practical purposes, $\kappa$ represents the length of the longest array of data we can use in our protocol. Where does $\kappa$ come from? Different elements of $Z_q$ will have different multiplicative orders but every order must be a divisor of $q-1$. Thus $\kappa$ is the largest divisor of the exact value of $q$ used in an elliptic curve standard. BLS12-384 has $\kappa=2^{32}$ (for terminology, this called a $2$-adicity of $32$). In summary, we can only encode data arrays of length up to $2^{32}=4,294,967,296$.</p><h2 id=zeroing-parts-of-an-array>Zeroing Parts of an Array
-<a class=anchor href=#zeroing-parts-of-an-array>#</a></h2><p>Assuming an input array of size $n$: $\langle \mathsf{data}_0,\mathsf{data}_1,\ldots,\mathsf{data}_{n-1}\rangle$ and input array encoded into the polynomial. This uses &ldquo;Encoding 2&rdquo; from above (evaluation points) and uses &ldquo;Roots of Unity + FFT&rdquo; from above where $\omega\in\mathbb{G}_\kappa$ is a generator for the x-coordinates of the points.</p><p>$\bot$ is an arbitrary non-zero integer.</p><table><thead><tr><th>Operation</th><th>Input Array</th><th>Input Polynomial</th><th>Output Array</th><th>Output Polynomial</th></tr></thead><tbody><tr><td>Zero all</td><td>$\langle 3,1,3,3,7 \rangle$</td><td>$P(X)$</td><td>$\langle 0,0,0,0,0 \rangle$</td><td>$P(X)\cdot(X^n-1)$</td></tr><tr><td>Zero first</td><td>$\langle 3,1,3,3,7 \rangle$</td><td>$P(X)$</td><td>$\langle 0,\bot,\bot,\bot,\bot \rangle$</td><td>$P(X)\cdot(X-\omega^0)$</td></tr><tr><td>Zero last</td><td>$\langle 3,1,3,3,7 \rangle$</td><td>$P(X)$</td><td>$\langle \bot,\bot,\bot,\bot,0 \rangle$</td><td>$P(X)\cdot(X-\omega^{n-1})$</td></tr><tr><td>Zero all but first</td><td>$\langle 3,1,3,3,7 \rangle$</td><td>$P(X)$</td><td>$\langle \bot,0,0,0,0 \rangle$</td><td>$P(X)\cdot\frac{(X^n-1)}{(X-\omega^0)}$</td></tr><tr><td>Zero all but last</td><td>$\langle 3,1,3,3,7 \rangle$</td><td>$P(X)$</td><td>$\langle 0,0,0,0,\bot \rangle$</td><td>$P(X)\cdot\frac{(X^n-1)}{(X-\omega^{n-1})}$</td></tr></tbody></table><p>Why are these useful? Consider a <a href=https://hackmd.io/@dabo/B1U4kx8XI>range proof</a> from Boneh, Fisch, Gabizon, and Williamson. At a certain point in the protocol, we reach the following:</p><p><img src=https://hackmd.io/_uploads/H1Xg2IvgR.png alt="Screenshot 2024-04-12 at 9.11.45 PM"></p><p>First note that $w$ and $\omega$ look the same but are different: $w$ are three new polynomials we are creating, while $\omega$ is the generator of $\mathbb{G}_\kappa$ we are using to pick x-coordinates for the polynomials.</p><ul><li>In the first constraint, we want to prove the value of f(1) and g(1) are the same. So we subtract (g-f) which leaves a zero in the first element of the array but the rest of the array will contain other stuff. By applying &ldquo;zero all but first&rdquo;, we can zero out the rest of the array. We now have an array that is all zero (in polynomial form, this is called a vanshing polynomial and we can prove a polynomial vanishes easily and in a batch).</li><li>In the second constraint, we prove the last element in g is a binary bit (0 or 1) and then we apply the &ldquo;zero all but last&rdquo; to make an array that is all zero (and vanishing polynomial).</li><li>In the third constraint, the first two terms of the multiplication are proving $g(X)$ has a certain form that does not need to be understood here. What is important is that there is a relationship between each integer in the array ($g(X)$) and the integer right beside it ($g(X\omega)$) in the array. When the relationship holds, the result is a zero in $w_3$. Unfortunately, there is a corner case: for the last element in the array, the &ldquo;next&rdquo; integer wraps back to the first integer and the relationship does not hold across this boundary. So we use &ldquo;zero last&rdquo; to manually zero out the last integer in the array, leaving us with a zero array (and vanishing polynomial).</li></ul><h1 id=kzg-commitments>KZG Commitments
-<a class=anchor href=#kzg-commitments>#</a></h1><p>KZG is a commitment scheme, designed specifically for polynomials. While any commitment scheme can be used to commit to a polynomial (e.g., hash the polynomial) and later reveal it, KZG has homomorphic properties that are useful for polynomials. They are as follows:</p><ol><li>Addition. $\mathsf{Comm}$</li><li>Multiplication Check. $P_1(z)\cdot P_2(z) - P_3(z)=0$</li><li>Evaluation Check.</li></ol><ul><li><p>g^P(X) -> g^(c_0+c_1X+c_2X^2)</p></li><li><p>Evaluate at tau to commit</p></li><li><p>Trusted setup (Powers of Tau) -> $g,g^{\tau}, g^{\tau^2}$</p></li><li><p>Array: [4, 6, 7, 2, 346, 24&mldr;]</p></li><li><p>Array -> Polynomial -> Polynomial Commitment (Poly-IOP; polynomial interactive oracle proof)</p></li><li><p>Array -> Polynomial -> How?</p><ul><li>Evaluation points -> [{1,4},{2,6},{3,7}&mldr;] -> Lagrange / FFT interpolation</li><li>Roots -> [{4,0},{6,0},{7,0},..] -> Interpolation is sufficient, not necessary -> Expansion -> (X-4)(X-6)(X-7)</li><li>Coefficients -> 4+6X+7X^2 -> Nothing</li></ul></li><li><p>KZG: $g^{c_0+c_1*\tau+c_2*\tau^2+\ldots}$</p></li><li><p>Comm(P1)*Comm(P2) = Comm (P1+P2)</p></li><li><p>Comm(P1)^P2=Comm(P1*P2)</p></li><li><p>Comm(P1)^a=Comm(a*P1)</p></li><li><p>Open Root (z=P1(0))</p><ul><li>Open a root at 7</li><li>Q=(X-4)(X-6)(X-7) / (X-7) -> evenly divide iff 7 is a root</li><li>Q(X-7) = P -> check with pairing</li></ul></li><li><p>Open point (z=P1(x))</p><ul><li>Subtract off z and now x is a root, see above</li></ul></li><li><p>Batch opening many points:</p><ul><li>Open many points on same polynomial P</li><li>Open the same point on many polynomials</li><li>Open many points on many polynomials -> not sure?</li></ul></li></ul><p>$P_1(X)=c_0+c_1*X+\ldots$</p><p>$P_3(\square)=(\square-\mathsf{data}_0)(\square-\mathsf{data}_1)$</p><p>$P_3(\square)=((\square-\mathsf{data}_2)(\square-\mathsf{data}_3)(\square-\mathsf{data}_4)$</p><p>$P_3(\square)=(\square-\mathsf{data}_0)(\square-\mathsf{data}_1)(\square-\mathsf{data}_2)(\square-\mathsf{data}_3)(\square-\mathsf{data}_4)$</p></article><footer class=book-footer><div class="flex flex-wrap justify-between"></div><script>(function(){function e(e){const t=window.getSelection(),n=document.createRange();n.selectNodeContents(e),t.removeAllRanges(),t.addRange(n)}document.querySelectorAll("pre code").forEach(t=>{t.addEventListener("click",function(){if(window.getSelection().toString())return;e(t.parentElement),navigator.clipboard&&navigator.clipboard.writeText(t.parentElement.textContent)})})})()</script></footer><div class=book-comments></div><label for=menu-control class="hidden book-menu-overlay"></label></div><aside class=book-toc><div class=book-toc-content><nav id=TableOfContents><ul><li><a href=#encoding-arrays-of-data-into-polynomials>Encoding Arrays of Data into Polynomials</a><ul><li><a href=#encoding-1-coefficients>Encoding 1: Coefficients</a></li><li><a href=#encoding-2-evaluation-points>Encoding 2: Evaluation Points</a></li><li><a href=#encoding-3-roots>Encoding 3: Roots</a></li><li><a href=#decision-tree-for-encoding>Decision Tree for Encoding</a></li></ul></li><li><a href=#optimizing-interpolation-roots-of-unity--fft>Optimizing Interpolation (Roots of Unity + FFT)</a></li><li><a href=#zeroing-parts-of-an-array>Zeroing Parts of an Array</a></li></ul></nav></div></aside></main></body></html>
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+<a class=anchor href=#zeroing-parts-of-an-array>#</a></h2><p>Assuming an input array of size $n$: $\langle \mathsf{data}_0,\mathsf{data}_1,\ldots,\mathsf{data}_{n-1}\rangle$ and input array encoded into the polynomial. This uses &ldquo;Encoding 2&rdquo; from above (evaluation points) and uses &ldquo;Roots of Unity + FFT&rdquo; from above where $\omega\in\mathbb{G}_\kappa$ is a generator for the x-coordinates of the points.</p><p>$\bot$ is an arbitrary non-zero integer.</p><table><thead><tr><th>Operation</th><th>Input Array</th><th>Input Polynomial</th><th>Output Array</th><th>Output Polynomial</th></tr></thead><tbody><tr><td>Zero all</td><td>$\langle 3,1,3,3,7 \rangle$</td><td>$P(X)$</td><td>$\langle 0,0,0,0,0 \rangle$</td><td>$P(X)\cdot(X^n-1)$</td></tr><tr><td>Zero first</td><td>$\langle 3,1,3,3,7 \rangle$</td><td>$P(X)$</td><td>$\langle 0,\bot,\bot,\bot,\bot \rangle$</td><td>$P(X)\cdot(X-\omega^0)$</td></tr><tr><td>Zero last</td><td>$\langle 3,1,3,3,7 \rangle$</td><td>$P(X)$</td><td>$\langle \bot,\bot,\bot,\bot,0 \rangle$</td><td>$P(X)\cdot(X-\omega^{n-1})$</td></tr><tr><td>Zero all but first</td><td>$\langle 3,1,3,3,7 \rangle$</td><td>$P(X)$</td><td>$\langle \bot,0,0,0,0 \rangle$</td><td>$P(X)\cdot\frac{(X^n-1)}{(X-\omega^0)}$</td></tr><tr><td>Zero all but last</td><td>$\langle 3,1,3,3,7 \rangle$</td><td>$P(X)$</td><td>$\langle 0,0,0,0,\bot \rangle$</td><td>$P(X)\cdot\frac{(X^n-1)}{(X-\omega^{n-1})}$</td></tr></tbody></table><p>Why are these useful? Consider a <a href=https://hackmd.io/@dabo/B1U4kx8XI>range proof</a> from Boneh, Fisch, Gabizon, and Williamson. At a certain point in the protocol, we reach the following:</p><p><img src=https://hackmd.io/_uploads/H1Xg2IvgR.png alt="Screenshot 2024-04-12 at 9.11.45 PM"></p><p>First note that $w$ and $\omega$ look the same but are different: $w$ are three new polynomials we are creating, while $\omega$ is the generator of $\mathbb{G}_\kappa$ we are using to pick x-coordinates for the polynomials.</p><ul><li>In the first constraint, we want to prove the value of f(1) and g(1) are the same. So we subtract (g-f) which leaves a zero in the first element of the array but the rest of the array will contain other stuff. By applying &ldquo;zero all but first&rdquo;, we can zero out the rest of the array. We now have an array that is all zero (in polynomial form, this is called a vanshing polynomial and we can prove a polynomial vanishes easily and in a batch).</li><li>In the second constraint, we prove the last element in g is a binary bit (0 or 1) and then we apply the &ldquo;zero all but last&rdquo; to make an array that is all zero (and vanishing polynomial).</li><li>In the third constraint, the first two terms of the multiplication are proving $g(X)$ has a certain form that does not need to be understood here. What is important is that there is a relationship between each integer in the array ($g(X)$) and the integer right beside it ($g(X\omega)$) in the array. When the relationship holds, the result is a zero in $w_3$. Unfortunately, there is a corner case: for the last element in the array, the &ldquo;next&rdquo; integer wraps back to the first integer and the relationship does not hold across this boundary. So we use &ldquo;zero last&rdquo; to manually zero out the last integer in the array, leaving us with a zero array (and vanishing polynomial).</li></ul></article><footer class=book-footer><div class="flex flex-wrap justify-between"></div><script>(function(){function e(e){const t=window.getSelection(),n=document.createRange();n.selectNodeContents(e),t.removeAllRanges(),t.addRange(n)}document.querySelectorAll("pre code").forEach(t=>{t.addEventListener("click",function(){if(window.getSelection().toString())return;e(t.parentElement),navigator.clipboard&&navigator.clipboard.writeText(t.parentElement.textContent)})})})()</script></footer><div class=book-comments></div><label for=menu-control class="hidden book-menu-overlay"></label></div><aside class=book-toc><div class=book-toc-content><nav id=TableOfContents><ul><li><a href=#encoding-arrays-of-data-into-polynomials>Encoding Arrays of Data into Polynomials</a><ul><li><a href=#encoding-1-coefficients>Encoding 1: Coefficients</a></li><li><a href=#encoding-2-evaluation-points>Encoding 2: Evaluation Points</a></li><li><a href=#encoding-3-roots>Encoding 3: Roots</a></li><li><a href=#decision-tree-for-encoding>Decision Tree for Encoding</a></li></ul></li><li><a href=#optimizing-interpolation-roots-of-unity--fft>Optimizing Interpolation (Roots of Unity + FFT)</a></li><li><a href=#zeroing-parts-of-an-array>Zeroing Parts of an Array</a></li></ul></nav></div></aside></main></body></html>
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-<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/docs/gadgets/><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.bb5d9164f3309a75ef9097bce777d0ab67b4a8025ae95f3105b06cec629ec033.js integrity="sha256-u12RZPMwmnXvkJe853fQq2e0qAJa6V8xBbBs7GKewDM=" crossorigin=anonymous></script><link rel=alternate type=application/rss+xml href=https://madibagroup.github.io/2024-Gadgets/docs/gadgets/index.xml title=Handbook><script id=MathJax-script async src=https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js></script><script>MathJax={tex:{displayMath:[["\\[","\\]"],["$$","$$"]],inlineMath:[["\\(","\\)"],["$","$"]]}}</script></head><body dir=ltr><input type=checkbox class="hidden toggle" id=menu-control>
-<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><a href=/2024-Gadgets/docs/background/><strong>Background</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/>Poly IOP</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/ class=active><strong>Gadgets</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/>Mult2</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/productgadget/>ProdGadget</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
+<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/docs/gadgets/><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.93281b7c13f5b9b0a9578b33cdbdde95d267573c1f5b04dd2cb337c07f21791f.js integrity="sha256-kygbfBP1ubCpV4szzb3eldJnVzwfWwTdLLM3wH8heR8=" crossorigin=anonymous></script><link rel=alternate type=application/rss+xml href=https://madibagroup.github.io/2024-Gadgets/docs/gadgets/index.xml title=Handbook><script id=MathJax-script async src=https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js></script><script>MathJax={tex:{displayMath:[["\\[","\\]"],["$$","$$"]],inlineMath:[["\\(","\\)"],["$","$"]]}}</script></head><body dir=ltr><input type=checkbox class="hidden toggle" id=menu-control>
+<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><strong>Background</strong></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/>Poly IOP</a></p></li><li><p><strong>Gadgets</strong></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/>Mult2</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
 </label><strong>Gadgets</strong>
 <label for=toc-control><img src=/2024-Gadgets/svg/toc.svg class=book-icon alt="Table of Contents"></label></div><aside class="hidden clearfix"><nav id=TableOfContents></nav></aside></header><article class="markdown book-article"></article><footer class=book-footer><div class="flex flex-wrap justify-between"></div><script>(function(){function e(e){const t=window.getSelection(),n=document.createRange();n.selectNodeContents(e),t.removeAllRanges(),t.addRange(n)}document.querySelectorAll("pre code").forEach(t=>{t.addEventListener("click",function(){if(window.getSelection().toString())return;e(t.parentElement),navigator.clipboard&&navigator.clipboard.writeText(t.parentElement.textContent)})})})()</script></footer><label for=menu-control class="hidden book-menu-overlay"></label></div><aside class=book-toc><div class=book-toc-content><nav id=TableOfContents></nav></div></aside></main></body></html>
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diff --git a/docs/gadgets/index.xml b/docs/gadgets/index.xml
index 60cdc53..37bf693 100644
--- a/docs/gadgets/index.xml
+++ b/docs/gadgets/index.xml
@@ -1,4 +1,3 @@
-<?xml version="1.0" encoding="utf-8" standalone="yes"?><rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom"><channel><title>Handbook</title><link>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/</link><description>Recent content on Handbook</description><generator>Hugo</generator><language>en-us</language><atom:link href="https://madibagroup.github.io/2024-Gadgets/docs/gadgets/index.xml" rel="self" type="application/rss+xml"/><item><title/><link>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/mult2/</link><pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate><guid>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/mult2/</guid><description> Multiplication (within polynomial) # Array-level # Given array $\mathsf{Arr}_1$ of length $\ell$ and integer $\mathsf{Prod}_1$, demonstrate that $\mathsf{Prod}=\prod_{i=0}^{\ell-1}\mathsf{Arr}_1[i]$.
-Build accumulator array $\mathsf{Acc}_1$ from $\mathsf{Arr}_1$ as follows: $\mathsf{Acc}_1[\ell-1]=\mathsf{Arr}_1[\ell-1]$ $\mathsf{Acc}_1[i]=\mathsf{Arr}_1[i]\cdot \mathsf{Acc}_1[i+1]$​ If built correctly, $\mathsf{Acc}_1[0]=\mathsf{Prod}_1$ Example:
-Checked input: $\mathsf{Prod}_1=66$ Unchecked input: $\mathsf{Arr}_1=\langle 84,67,11,92,36,67 \rangle$ Execution trace: $\mathsf{Acc}_1= \langle \bot,\bot,\bot,\bot,\bot,67 \rangle$​ $\mathsf{Acc}_1= \langle \bot,\bot,\bot,\bot,(36\cdot 67),67 \rangle$ $\mathsf{Acc}_1= \langle \bot,\bot,\bot,\bot,84,67 \rangle$​ $\mathsf{Acc}_1= \langle \bot,\bot,\bot,(92\cdot 84),84,67 \rangle$ Polynomial-level # Commitment-level #</description></item><item><title/><link>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/productgadget/</link><pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate><guid>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/productgadget/</guid><description>Array Level
-The prover has a list of value $A = [a_0, a_1, a_2, \dots, a_{n-1}]$ and wishes to commit to and prove that product of all entries in $A$ equals some value $z$. In other words, that $\prod_{i = 0}^{n-1} a_i= z$. First, the prover will compute an accumulator, $B$ according to the following rules: $i) B_i = A_i \cdot B_{i+1} \space 0 \leq i \lt n-1$ and $ii)B_{n-1} = A_{n-1}$.</description></item></channel></rss>
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+<?xml version="1.0" encoding="utf-8" standalone="yes"?><rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom"><channel><title>Handbook</title><link>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/</link><description>Recent content on Handbook</description><generator>Hugo</generator><language>en-us</language><atom:link href="https://madibagroup.github.io/2024-Gadgets/docs/gadgets/index.xml" rel="self" type="application/rss+xml"/><item><title/><link>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/mult2/</link><pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate><guid>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/mult2/</guid><description>Multiplication (Type 2) # Recap of types # Type Description Recap This Type 1 $\mathsf{Arr}_3=\mathsf{Arr}_1 \cdot \mathsf{Arr}_2$ $\mathsf{Arr}_3$ is the element-wise multiplication of $\mathsf{Arr}_1$ and $\mathsf{Arr}_2$. Type 2 $\mathsf{Prod}_\mathsf{Arr}=\prod_{i = 0}^{n-1} \mathsf{Arr}[i]$ $\mathsf{Prod}_\mathsf{Arr}$ is the disclosed product of all the elements in $\mathsf{Arr}$. ✅ Type 3 $\prod_{i = 0}^{n-1} \mathsf{Arr}_1[i]=\prod_{i = 0}^{n-1} \mathsf{Arr}_2[i]$ $\mathsf{Arr}_1$ and $\mathsf{Arr}_2$ have the same undisclosed product. Relation # $ \mathcal{R}_{\mathtt{mult2}} := \left\{ \begin{array}{l} (K_\mathsf{Arr},\mathsf{Prod}_\mathsf{Arr}) \end{array} \middle | \begin{array}{l} \mathsf{Arr} = [a_0, a_1, a_2, \dots, a_{n-1}], \\ \mathsf{Prod}_\mathsf{Arr} = \prod_{i = 0}^{n-1} a_i, \\ \mathsf{Poly}_\mathsf{Arr}=\mathsf{FFT.</description></item><item><title/><link>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/zero/</link><pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate><guid>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/zero/</guid><description>Zeroing Parts of an Array # Assuming an input array of size $n$: $\langle \mathsf{data}_0,\mathsf{data}_1,\ldots,\mathsf{data}_{n-1}\rangle$ and input array encoded into the polynomial. This uses &amp;ldquo;Encoding 2&amp;rdquo; from above (evaluation points) and uses &amp;ldquo;Roots of Unity + FFT&amp;rdquo; from above where $\omega\in\mathbb{G}_\kappa$ is a generator for the x-coordinates of the points.
+$\bot$ is an arbitrary non-zero integer.
+Operation Input Array Input Polynomial Output Array Output Polynomial Zero all $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle 0,0,0,0,0 \rangle$ $P(X)\cdot(X^\kappa-1)$ Zero first $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle 0,\bot,\bot,\bot,\bot \rangle$ $P(X)\cdot(X-\omega^0)$ Zero last $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle \bot,\bot,\bot,\bot,0 \rangle$ $P(X)\cdot(X-\omega^{\kappa-1})$ Zero all but first $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle \bot,0,0,0,0 \rangle$ $P(X)\cdot\frac{(X^\kappa-1)}{(X-\omega^0)}$ Zero all but last $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle 0,0,0,0,\bot \rangle$ $P(X)\cdot\frac{(X^\kappa-1)}{(X-\omega^{\kappa-1})}$ Why are these useful?</description></item></channel></rss>
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diff --git a/docs/gadgets/mult2/index.html b/docs/gadgets/mult2/index.html
index 06f1512..0c9978d 100644
--- a/docs/gadgets/mult2/index.html
+++ b/docs/gadgets/mult2/index.html
@@ -1,13 +1,18 @@
-<!doctype html><html lang=en-us dir=ltr><head><meta charset=UTF-8><meta name=viewport content="width=device-width,initial-scale=1"><meta name=description content=" Multiplication (within polynomial) # Array-level # Given array $\mathsf{Arr}_1$ of length $\ell$ and integer $\mathsf{Prod}_1$, demonstrate that $\mathsf{Prod}=\prod_{i=0}^{\ell-1}\mathsf{Arr}_1[i]$.
-Build accumulator array $\mathsf{Acc}_1$ from $\mathsf{Arr}_1$ as follows: $\mathsf{Acc}_1[\ell-1]=\mathsf{Arr}_1[\ell-1]$ $\mathsf{Acc}_1[i]=\mathsf{Arr}_1[i]\cdot \mathsf{Acc}_1[i+1]$​ If built correctly, $\mathsf{Acc}_1[0]=\mathsf{Prod}_1$ Example:
-Checked input: $\mathsf{Prod}_1=66$ Unchecked input: $\mathsf{Arr}_1=\langle 84,67,11,92,36,67 \rangle$ Execution trace: $\mathsf{Acc}_1= \langle \bot,\bot,\bot,\bot,\bot,67 \rangle$​ $\mathsf{Acc}_1= \langle \bot,\bot,\bot,\bot,(36\cdot 67),67 \rangle$ $\mathsf{Acc}_1= \langle \bot,\bot,\bot,\bot,84,67 \rangle$​ $\mathsf{Acc}_1= \langle \bot,\bot,\bot,(92\cdot 84),84,67 \rangle$ Polynomial-level # Commitment-level # "><meta name=theme-color media="(prefers-color-scheme: light)" content="#ffffff"><meta name=theme-color media="(prefers-color-scheme: dark)" content="#343a40"><meta name=color-scheme content="light dark"><meta property="og:url" content="https://madibagroup.github.io/2024-Gadgets/docs/gadgets/mult2/"><meta property="og:site_name" content="Handbook"><meta property="og:title" content="Handbook"><meta property="og:description" content=" Multiplication (within polynomial) # Array-level # Given array $\mathsf{Arr}_1$ of length $\ell$ and integer $\mathsf{Prod}_1$, demonstrate that $\mathsf{Prod}=\prod_{i=0}^{\ell-1}\mathsf{Arr}_1[i]$.
-Build accumulator array $\mathsf{Acc}_1$ from $\mathsf{Arr}_1$ as follows: $\mathsf{Acc}_1[\ell-1]=\mathsf{Arr}_1[\ell-1]$ $\mathsf{Acc}_1[i]=\mathsf{Arr}_1[i]\cdot \mathsf{Acc}_1[i+1]$​ If built correctly, $\mathsf{Acc}_1[0]=\mathsf{Prod}_1$ Example:
-Checked input: $\mathsf{Prod}_1=66$ Unchecked input: $\mathsf{Arr}_1=\langle 84,67,11,92,36,67 \rangle$ Execution trace: $\mathsf{Acc}_1= \langle \bot,\bot,\bot,\bot,\bot,67 \rangle$​ $\mathsf{Acc}_1= \langle \bot,\bot,\bot,\bot,(36\cdot 67),67 \rangle$ $\mathsf{Acc}_1= \langle \bot,\bot,\bot,\bot,84,67 \rangle$​ $\mathsf{Acc}_1= \langle \bot,\bot,\bot,(92\cdot 84),84,67 \rangle$ Polynomial-level # Commitment-level # "><meta property="og:locale" content="en_us"><meta property="og:type" content="article"><meta property="article:section" content="docs"><title>Mult2 | Handbook</title>
-<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/docs/gadgets/mult2/><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.bb5d9164f3309a75ef9097bce777d0ab67b4a8025ae95f3105b06cec629ec033.js integrity="sha256-u12RZPMwmnXvkJe853fQq2e0qAJa6V8xBbBs7GKewDM=" crossorigin=anonymous></script><script id=MathJax-script async src=https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js></script><script>MathJax={tex:{displayMath:[["\\[","\\]"],["$$","$$"]],inlineMath:[["\\(","\\)"],["$","$"]]}}</script></head><body dir=ltr><input type=checkbox class="hidden toggle" id=menu-control>
-<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><a href=/2024-Gadgets/docs/background/><strong>Background</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/>Poly IOP</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/><strong>Gadgets</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/ class=active>Mult2</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/productgadget/>ProdGadget</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
+<!doctype html><html lang=en-us dir=ltr><head><meta charset=UTF-8><meta name=viewport content="width=device-width,initial-scale=1"><meta name=description content="Multiplication (Type 2) # Recap of types # Type Description Recap This Type 1 $\mathsf{Arr}_3=\mathsf{Arr}_1 \cdot \mathsf{Arr}_2$ $\mathsf{Arr}_3$ is the element-wise multiplication of $\mathsf{Arr}_1$ and $\mathsf{Arr}_2$. Type 2 $\mathsf{Prod}_\mathsf{Arr}=\prod_{i = 0}^{n-1} \mathsf{Arr}[i]$ $\mathsf{Prod}_\mathsf{Arr}$ is the disclosed product of all the elements in $\mathsf{Arr}$. ✅ Type 3 $\prod_{i = 0}^{n-1} \mathsf{Arr}_1[i]=\prod_{i = 0}^{n-1} \mathsf{Arr}_2[i]$ $\mathsf{Arr}_1$ and $\mathsf{Arr}_2$ have the same undisclosed product. Relation # $ \mathcal{R}_{\mathtt{mult2}} := \left\{ \begin{array}{l} (K_\mathsf{Arr},\mathsf{Prod}_\mathsf{Arr}) \end{array} \middle | \begin{array}{l} \mathsf{Arr} = [a_0, a_1, a_2, \dots, a_{n-1}], \\ \mathsf{Prod}_\mathsf{Arr} = \prod_{i = 0}^{n-1} a_i, \\ \mathsf{Poly}_\mathsf{Arr}=\mathsf{FFT."><meta name=theme-color media="(prefers-color-scheme: light)" content="#ffffff"><meta name=theme-color media="(prefers-color-scheme: dark)" content="#343a40"><meta name=color-scheme content="light dark"><meta property="og:url" content="https://madibagroup.github.io/2024-Gadgets/docs/gadgets/mult2/"><meta property="og:site_name" content="Handbook"><meta property="og:title" content="Handbook"><meta property="og:description" content="Multiplication (Type 2) # Recap of types # Type Description Recap This Type 1 $\mathsf{Arr}_3=\mathsf{Arr}_1 \cdot \mathsf{Arr}_2$ $\mathsf{Arr}_3$ is the element-wise multiplication of $\mathsf{Arr}_1$ and $\mathsf{Arr}_2$. Type 2 $\mathsf{Prod}_\mathsf{Arr}=\prod_{i = 0}^{n-1} \mathsf{Arr}[i]$ $\mathsf{Prod}_\mathsf{Arr}$ is the disclosed product of all the elements in $\mathsf{Arr}$. ✅ Type 3 $\prod_{i = 0}^{n-1} \mathsf{Arr}_1[i]=\prod_{i = 0}^{n-1} \mathsf{Arr}_2[i]$ $\mathsf{Arr}_1$ and $\mathsf{Arr}_2$ have the same undisclosed product. Relation # $ \mathcal{R}_{\mathtt{mult2}} := \left\{ \begin{array}{l} (K_\mathsf{Arr},\mathsf{Prod}_\mathsf{Arr}) \end{array} \middle | \begin{array}{l} \mathsf{Arr} = [a_0, a_1, a_2, \dots, a_{n-1}], \\ \mathsf{Prod}_\mathsf{Arr} = \prod_{i = 0}^{n-1} a_i, \\ \mathsf{Poly}_\mathsf{Arr}=\mathsf{FFT."><meta property="og:locale" content="en_us"><meta property="og:type" content="article"><meta property="article:section" content="docs"><title>Mult2 | Handbook</title>
+<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/docs/gadgets/mult2/><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.93281b7c13f5b9b0a9578b33cdbdde95d267573c1f5b04dd2cb337c07f21791f.js integrity="sha256-kygbfBP1ubCpV4szzb3eldJnVzwfWwTdLLM3wH8heR8=" crossorigin=anonymous></script><script id=MathJax-script async src=https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js></script><script>MathJax={tex:{displayMath:[["\\[","\\]"],["$$","$$"]],inlineMath:[["\\(","\\)"],["$","$"]]}}</script></head><body dir=ltr><input type=checkbox class="hidden toggle" id=menu-control>
+<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><strong>Background</strong></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/>Poly IOP</a></p></li><li><p><strong>Gadgets</strong></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/ class=active>Mult2</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
 </label><strong>Mult2</strong>
-<label for=toc-control><img src=/2024-Gadgets/svg/toc.svg class=book-icon alt="Table of Contents"></label></div><aside class="hidden clearfix"><nav id=TableOfContents><ul><li><a href=#array-level>Array-level</a></li><li><a href=#polynomial-level>Polynomial-level</a></li><li><a href=#commitment-level>Commitment-level</a></li></ul></nav></aside></header><article class="markdown book-article"><h1 id=multiplication-within-polynomial>Multiplication (within polynomial)
-<a class=anchor href=#multiplication-within-polynomial>#</a></h1><h2 id=array-level>Array-level
-<a class=anchor href=#array-level>#</a></h2><p>Given array $\mathsf{Arr}_1$ of length $\ell$ and integer $\mathsf{Prod}_1$, demonstrate that $\mathsf{Prod}=\prod_{i=0}^{\ell-1}\mathsf{Arr}_1[i]$.</p><ul><li>Build accumulator array $\mathsf{Acc}_1$ from $\mathsf{Arr}_1$ as follows:<ul><li>$\mathsf{Acc}_1[\ell-1]=\mathsf{Arr}_1[\ell-1]$</li><li>$\mathsf{Acc}_1[i]=\mathsf{Arr}_1[i]\cdot \mathsf{Acc}_1[i+1]$​</li></ul></li><li>If built correctly, $\mathsf{Acc}_1[0]=\mathsf{Prod}_1$</li></ul><p>Example:</p><ul><li>Checked input:<ul><li>$\mathsf{Prod}_1=66$</li></ul></li><li>Unchecked input:<ul><li>$\mathsf{Arr}_1=\langle 84,67,11,92,36,67 \rangle$</li></ul></li><li>Execution trace:<ul><li>$\mathsf{Acc}_1= \langle \bot,\bot,\bot,\bot,\bot,67 \rangle$​</li><li>$\mathsf{Acc}_1= \langle \bot,\bot,\bot,\bot,(36\cdot 67),67 \rangle$</li><li>$\mathsf{Acc}_1= \langle \bot,\bot,\bot,\bot,84,67 \rangle$​</li><li>$\mathsf{Acc}_1= \langle \bot,\bot,\bot,(92\cdot 84),84,67 \rangle$</li></ul></li></ul><h2 id=polynomial-level>Polynomial-level
-<a class=anchor href=#polynomial-level>#</a></h2><h2 id=commitment-level>Commitment-level
-<a class=anchor href=#commitment-level>#</a></h2></article><footer class=book-footer><div class="flex flex-wrap justify-between"></div><script>(function(){function e(e){const t=window.getSelection(),n=document.createRange();n.selectNodeContents(e),t.removeAllRanges(),t.addRange(n)}document.querySelectorAll("pre code").forEach(t=>{t.addEventListener("click",function(){if(window.getSelection().toString())return;e(t.parentElement),navigator.clipboard&&navigator.clipboard.writeText(t.parentElement.textContent)})})})()</script></footer><div class=book-comments></div><label for=menu-control class="hidden book-menu-overlay"></label></div><aside class=book-toc><div class=book-toc-content><nav id=TableOfContents><ul><li><a href=#array-level>Array-level</a></li><li><a href=#polynomial-level>Polynomial-level</a></li><li><a href=#commitment-level>Commitment-level</a></li></ul></nav></div></aside></main></body></html>
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+<label for=toc-control><img src=/2024-Gadgets/svg/toc.svg class=book-icon alt="Table of Contents"></label></div><aside class="hidden clearfix"><nav id=TableOfContents><ul><li><a href=#recap-of-types>Recap of types</a></li><li><a href=#relation>Relation</a></li><li><a href=#intuition>Intuition</a></li><li><a href=#array-level>Array Level</a></li><li><a href=#polynomial-level>Polynomial Level</a><ul><li><a href=#optimized>Optimized</a></li></ul></li><li><a href=#commitment-level>Commitment Level</a></li><li><a href=#implementations>Implementations</a></li><li><a href=#security-proof>Security Proof</a><ul><li><a href=#completeness>Completeness</a></li><li><a href=#soundness>Soundness</a></li><li><a href=#zero-knowledge>Zero-Knowledge</a></li></ul></li></ul></nav></aside></header><article class="markdown book-article"><h1 id=multiplication-type-2>Multiplication (Type 2)
+<a class=anchor href=#multiplication-type-2>#</a></h1><h2 id=recap-of-types>Recap of types
+<a class=anchor href=#recap-of-types>#</a></h2><table><thead><tr><th>Type</th><th>Description</th><th>Recap</th><th>This</th></tr></thead><tbody><tr><td>Type 1</td><td>$\mathsf{Arr}_3=\mathsf{Arr}_1 \cdot \mathsf{Arr}_2$</td><td>$\mathsf{Arr}_3$ is the element-wise multiplication of $\mathsf{Arr}_1$ and $\mathsf{Arr}_2$.</td><td></td></tr><tr><td>Type 2</td><td>$\mathsf{Prod}_\mathsf{Arr}=\prod_{i = 0}^{n-1} \mathsf{Arr}[i]$</td><td>$\mathsf{Prod}_\mathsf{Arr}$ is the disclosed product of all the elements in $\mathsf{Arr}$.</td><td>✅</td></tr><tr><td>Type 3</td><td>$\prod_{i = 0}^{n-1} \mathsf{Arr}_1[i]=\prod_{i = 0}^{n-1} \mathsf{Arr}_2[i]$</td><td>$\mathsf{Arr}_1$ and $\mathsf{Arr}_2$ have the same undisclosed product.</td><td></td></tr></tbody></table><h2 id=relation>Relation
+<a class=anchor href=#relation>#</a></h2><p>$ \mathcal{R}_{\mathtt{mult2}} := \left\{ \begin{array}{l} (K_\mathsf{Arr},\mathsf{Prod}_\mathsf{Arr}) \end{array} \middle | \begin{array}{l} \mathsf{Arr} = [a_0, a_1, a_2, \dots, a_{n-1}], \\ \mathsf{Prod}_\mathsf{Arr} = \prod_{i = 0}^{n-1} a_i, \\ \mathsf{Poly}_\mathsf{Arr}=\mathsf{FFT.Interp}(\omega,\mathsf{Arr}), \\ K_\mathsf{Arr}=\mathsf{KZG.Commit}(\mathsf{Poly}_\mathsf{Arr}) \end{array} \right\} $</p><h2 id=intuition>Intuition
+<a class=anchor href=#intuition>#</a></h2><p>The prover ($\mathcal{P}$) holds an array $\mathsf{Arr} = [a_0, a_1, a_2, \dots, a_{n-1}]$ and a public value $\mathsf{Prod}_\mathsf{Arr}$. It will produce a succinct (independent of $n$) proof that $\mathsf{Prod}_\mathsf{Arr}$ is the product of all the elements in the array. The prover will encode the array into a polynomial $\mathsf{Poly}_\mathsf{Arr}$ (using evaluation points) and commit to the polynomial $K_\mathsf{Arr}$. The verifier ($\mathcal{V}$) cannot check $\textsf{Arr}$ or $\mathsf{Poly}_\mathsf{Arr}$ directly (they may contain secret information, and even if they do not, it is too long to check) so the verifier only sees $K_\mathsf{Arr}$ and the asserted value $\mathsf{Prod_\mathsf{Arr}}$.</p><p>In order to prove $K_\mathsf{Arr}$ and $\mathsf{Prod}_\mathsf{Arr}$ are consistent, the prover will build a helper array $\mathsf{Acc}_\mathsf{Arr}$ called an accumulator (or accumulating array or incremental array). This should not be confused with accumulators from cryptography, which are a concept related to succinct proofs but are distinct. As with $\mathsf{Arr}$, the prover will also encode $\mathsf{Acc}$ as a polynomial and provide a commitment of it to the verifier. The idea is that the prover will prove a relation between $\mathsf{Arr}$ and $\mathsf{Acc}$; and a relation between $\mathsf{Acc}$ and $\mathsf{Prod_\mathsf{Arr}}$. Put together, it will imply the correct relation between $\mathsf{Arr}$ and $\mathsf{Prod_\mathsf{Arr}}$​.</p><p>Consider a small numeric example in $\mathbb{Z}_{97}$ where $\mathsf{Arr}= [84,67,11,92,36,67]$ and $\mathsf{Prod}_\mathsf{Arr}=72$. The first idea is to get $\mathsf{Prod}_\mathsf{Arr}$ into an array. Say, we just append it: $\mathsf{Arr}''= [84,67,11,92,36,67,72] $. How does the prover show $\mathsf{Arr}''$ is correct? The last value of the array depends on every single element that precedes it, which will be a complex constraint to prove.</p><p>An alternative idea is to create a new array that starts the same as $\mathsf{Arr}$ and ends up at $\mathsf{Prod}_\mathsf{Arr}$ by folding in the integers from $\mathsf{Arr}$ one-by-one with multiplication. Then each value in the new array will depend on only two values, as below.</p><p>The first value in $\mathsf{Acc}$ will be a copy of the first value from $\mathsf{Arr}$:</p><ul><li><p>$\mathsf{Arr}= [84,67,11,92,36,67]$</p></li><li><p>$\mathsf{Acc}= [84, \bot,\bot,\bot,\bot,\bot] $​</p></li></ul><p>The next value will be the multiplication (mod 97) of: 67 (the value at the same index in $\mathsf{Arr}$) and 84 (the previous value in $\mathsf{Acc}$​):</p><ul><li><p>$\mathsf{Arr}= [84,67,11,92,36,67]$</p></li><li><p>$ \mathsf{Acc} = [84, (67\cdot84),\bot,\bot,\bot,\bot] = [84, 2,\bot,\bot,\bot,\bot]$</p></li></ul><p>The next value will be the multiplication of: 11 (the value at the same index in $\mathsf{Arr}$) and 2 (the previous value in $\mathsf{Acc}$):</p><ul><li>$\mathsf{Arr}= [84,67,11,92,36,67]$</li><li>$ \mathsf{Acc} = [84, 2,(11\cdot2),\bot,\bot,\bot] = [84,2,22,\bot,\bot,\bot]$​​</li><li>$ \mathsf{Acc} = [84, 2,22,(92\cdot22),\bot,\bot] = [84,2,22,84,\bot,\bot]$</li><li>$ \mathsf{Acc} = [84, 2,22,84,(36\cdot84),\bot] = [84,2,22,84,17,\bot]$​</li><li>$ \mathsf{Acc} = [84,2,22,84,17,(67\cdot17)] = [84, 2, 22, 84, 17, 72]$​</li><li>$\mathsf{Prod}_\mathsf{Arr}=72$</li></ul><p>Notice the last value in $\mathsf{Acc}$ is $\mathsf{Prod_\mathsf{Arr}}$. The prover wants to show three constraints:</p><ol><li>The first value in $\mathsf{Acc}$ matches the first value in $\mathsf{Arr}$,</li><li>The rest of the values in $\mathsf{Acc}$ are of the form $\mathsf{Acc}[i]=\mathsf{Arr}[i]\cdot\mathsf{Acc}[i-1]$,</li><li>The last value in $\mathsf{Acc}$ matches $\mathsf{Prod}_\mathsf{Arr}$​​.</li></ol><p>If all three constraints are true, then $\mathsf{Prod}_\mathsf{Arr}$ is the product of the elements of $\mathsf{Arr}$.</p><p>Last, while it is not necessary to do, it is often convenient to hold the the value $\mathsf{Prod}_\mathsf{Arr}$ at the start of the array $\mathsf{Acc}$ instead of the end. For this reason, the mathematical explaination below will construct $\mathsf{Acc}$ &ldquo;backwards&rdquo; (or right-to-left) from the above example, where the last value of $\mathsf{Acc}$ matches the last value of $\mathsf{Arr}$, the values are folded in from right to left, and the first (leftmost) value of $\mathsf{Acc}$ is $\mathsf{Prod}_\mathsf{Arr}$:</p><ul><li>$\mathsf{Arr}= [84,67,11,92,36,67]$​</li><li>$ \mathsf{Acc} = [\bot, \bot, \bot, \bot, \bot, 67]$</li><li>$ \mathsf{Acc} = [\bot, \bot, \bot, \bot, 84, 67]$</li><li>$\ldots$</li><li>$ \mathsf{Acc} = [72, 84, 36, 65, 84, 67]$​</li><li>$\mathsf{Prod}_\mathsf{Arr}=72$</li></ul><h2 id=array-level>Array Level
+<a class=anchor href=#array-level>#</a></h2><p>The prover has a list of value $A = [a_0, a_1, a_2, \dots, a_{n-1}]$ and wishes to commit to and prove that product of all entries in $A$ equals some value $z$. In other words, that $\prod_{i = 0}^{n-1} a_i= z$. First, the prover will compute an accumulator, $B$ according to the following rules: $i) B_i = A_i \cdot B_{i+1} \space 0 \leq i \lt n-1$ and $ii)B_{n-1} = A_{n-1}$. Then $B_0 = z$. To prove that $\prod_{i = 0}^{n-1} a_i= z$, it suffices for the prover to demonstrate that $B_0 = z$ and that $i)$ and $ii)$ hold, and that $B_0 = z$​.</p><h2 id=polynomial-level>Polynomial Level
+<a class=anchor href=#polynomial-level>#</a></h2><p>We assume arrays are encoded as y-coordinates into a univariant polynomial where the x-coordinates are chosen as the multiplicative group of order $\kappa$ with generator $\omega\in\mathbb{G}_\kappa$ (see <a href=../background/poly-iop.md>Background</a> for more). In short, $\omega^0$ is the first element and $\omega^{\kappa-1}$ is the last element of this domain $\mathcal{H}$.</p><p>In polynomial form, the three constraints are:</p><ol><li>For $X=w^\kappa$: $\mathsf{Poly}_\mathsf{Acc}(X)=\mathsf{Poly}_\mathsf{Arr}(X)$,</li><li>For all $X$ except $X=\omega^{\kappa-1}$: $\mathsf{Poly}_\mathsf{Acc}(X)=\mathsf{Poly}_\mathsf{Arr}(X)\cdot\mathsf{Poly}_\mathsf{Acc}(\omega\cdot X)$</li><li>For $X=w^0$: $\mathsf{Poly}_\mathsf{Acc}(X)=\mathsf{Prod}_\mathsf{Arr}$</li></ol><p>We adjust each of these constraints to show an equality with 0:</p><ol><li>For $X=w^\kappa$: $\mathsf{Poly}_\mathsf{Acc}(X)-\mathsf{Poly}_\mathsf{Arr}(X)=0$,</li><li>For all $X$ except $X=\omega^{\kappa-1}$: $\mathsf{Poly}_\mathsf{Acc}(X)-\mathsf{Poly}_\mathsf{Arr}(X)\cdot\mathsf{Poly}_\mathsf{Acc}(\omega\cdot X)=0$</li><li>For $X=w^0$: $\mathsf{Poly}_\mathsf{Acc}(X)-\mathsf{Prod}_\mathsf{Arr}=0$​</li></ol><p>Next we take care of the &ldquo;for $X$​&rdquo; conditions by zeroing out the rest of the polynomial that is not zero. See the gadget: <a href=./zero.md>Zero-ing</a> for more on why this works.</p><ol><li>$\mathsf{Poly}_\mathsf{Vanish1}=(\mathsf{Poly}_\mathsf{Acc}(X)-\mathsf{Poly}_\mathsf{Arr}(X))\cdot\frac{(X^\kappa-1)}{(X-\omega^{\kappa-1})}=0$,</li><li>$\mathsf{Poly}_\mathsf{Vanish2}=(\mathsf{Poly}_\mathsf{Acc}(X)-\mathsf{Poly}_\mathsf{Arr}(X)\cdot\mathsf{Poly}_\mathsf{Acc}(\omega\cdot X))\cdot(X-\omega^{\kappa-1})=0$</li><li>$\mathsf{Poly}_\mathsf{Vanish3}=(\mathsf{Poly}_\mathsf{Acc}(X)-\mathsf{Prod}_\mathsf{Arr})\cdot\frac{(X^\kappa-1)}{(X-\omega^0)}=0$</li></ol><p>These equations are true for every value of $X \in \mathbb{G}_\kappa$ (but not necessarily true outside of these values) show we show each is a vanishing polynomial.</p><hr><p>The prover interpolates $A$ and $B$ to get $P_A(x)$ and $P_B(x)$. To do so, an $n$th root of unity (which we will call $\omega$) is used, and the values of $A$ and $B$ are paired with the powers of this root of unity as follows: $(\omega^0, a_0), (\omega^1, a_1), \dots, (\omega^{n-1}, a_{n-1})$ and similarly for $B$. Then the interpolation (using the fast fourier transform (FFT)) creates a polynomial that passes through the points represented by these tuples. We also define $H$ as the set $[\omega^0, \omega^1, \dots, \omega^{n-1}]$. We are assuming that $n$ is a power of 2, so that we can use FFT. In practice, if it is not, then a power of 2 greater than $n$, say $2^m$, is used and the last $2^m - n$ entries of $A$ are padded with 1s.</p><p>The prover then compute $W_1(x) = [P_B(x) - P_B(x\omega) \cdot P_a(x))] \cdot (x - \omega^{n-1})$ and $W_2(x) = [P_A(x) - P_B(x)] \cdot \frac{x^n - 1} {x - w^{n-1}}$. If $W_1$ is vanishing on $H$, then $i)$ holds for $0 \leq i \lt n$. We note that the $(x - \omega^{n-1})$ is included in the equation to add a root at $x = n -1$, since $i)$ does not hold for $i = n -1 $ (unless $z = 1$, in which case the ($x - \omega^{n-1}$) is unnecessary). Similarly, If $W_2$ is vanishing on $H$, then $ii)$ holds. We note that multiplying [$P_A(x) - P_B(x)]$ by $\frac{x^n - 1} {x - w^{n-1}}$ zeroes $W_2$ on all of $H$ except $\omega^{n-1}$, thus $W_2$ vanishing on $H$ is testing only the equality of $A_{n-1}$ and $B_{n-1}$​.</p><p>The prover then computes $Q_1(x) = \frac{W_1(x)}{x^n - 1}$ and $Q_2(x) = \frac{W_2(x)}{x^n - 1}$. Since $x^n - 1$ is the vanishing polynomial for $H$, if $Q_1$ and $Q_2$ are polynomials (and not rational functions), then $W_1$ and $W_2$ must be vanishing on $H$.</p><p>The prover then commits to $P_A$, $P_B$, $Q_1$, and $Q_2$. They then hash the four commitments to get a random challenge, $\tau$. They open $P_A(\tau), P_B(\tau), Q_1(\tau), Q_2(\tau),$ and $P_B(\tau \cdot \omega)$ to demonstrate that $Q_1$ and $Q_2$ are polynomials, as defined above $-$ thus $i)$ and $ii)$ hold. They also open $P_B(1)$, showing that $B_0 = z$. Thus they have demontrated that $\prod_{i = 0}^{n-1} a_i= z$​​.</p><h3 id=optimized>Optimized
+<a class=anchor href=#optimized>#</a></h3><p>Here, rework each constraint so it is a vanishing polynomial.</p><h2 id=commitment-level>Commitment Level
+<a class=anchor href=#commitment-level>#</a></h2><p>The verifier first checks that $\tau$​ is correct by hashing the four commitments which have been sent to them by the prover.</p><p>Next, the verifier checks that the evaluations $P_A(\tau), P_B(\tau), Q_1(\tau), Q_2(\tau),$ and $P_B(\tau \cdot \omega)$​ sent to them by the prover are correct. This can be done (assuming we are working with KZG) as a single batch opening.</p><p>The verifier will then check the following two equalities, so make sure that $Q_1$ and $Q_2$​ were defined as they should be:</p><p>​ $a) \space [P_B(\tau) - P_B(\tau \cdot \omega)\cdot P_A(\tau)] \cdot (\tau - \omega^{n-1}) = (\tau^n - 1) \cdot Q_1(\tau)$​</p><p>​ $b) \space \frac{P_A(\tau) - P_B(\tau)}{\tau - \omega^{n-1}} = Q_2(\tau)$</p><p>Finally, the verifier will check the opening of $P_B(1)$. This value is $z$, and they have verified that it is equal to the product of the entries in $A$​.</p><h2 id=implementations>Implementations
+<a class=anchor href=#implementations>#</a></h2><ul><li><a href>Rust</a></li><li><a href>Mathematica</a> (Toy Example)</li></ul><h2 id=security-proof>Security Proof
+<a class=anchor href=#security-proof>#</a></h2><h3 id=completeness>Completeness
+<a class=anchor href=#completeness>#</a></h3><h3 id=soundness>Soundness
+<a class=anchor href=#soundness>#</a></h3><h3 id=zero-knowledge>Zero-Knowledge
+<a class=anchor href=#zero-knowledge>#</a></h3></article><footer class=book-footer><div class="flex flex-wrap justify-between"></div><script>(function(){function e(e){const t=window.getSelection(),n=document.createRange();n.selectNodeContents(e),t.removeAllRanges(),t.addRange(n)}document.querySelectorAll("pre code").forEach(t=>{t.addEventListener("click",function(){if(window.getSelection().toString())return;e(t.parentElement),navigator.clipboard&&navigator.clipboard.writeText(t.parentElement.textContent)})})})()</script></footer><div class=book-comments></div><label for=menu-control class="hidden book-menu-overlay"></label></div><aside class=book-toc><div class=book-toc-content><nav id=TableOfContents><ul><li><a href=#recap-of-types>Recap of types</a></li><li><a href=#relation>Relation</a></li><li><a href=#intuition>Intuition</a></li><li><a href=#array-level>Array Level</a></li><li><a href=#polynomial-level>Polynomial Level</a><ul><li><a href=#optimized>Optimized</a></li></ul></li><li><a href=#commitment-level>Commitment Level</a></li><li><a href=#implementations>Implementations</a></li><li><a href=#security-proof>Security Proof</a><ul><li><a href=#completeness>Completeness</a></li><li><a href=#soundness>Soundness</a></li><li><a href=#zero-knowledge>Zero-Knowledge</a></li></ul></li></ul></nav></div></aside></main></body></html>
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-<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><a href=/2024-Gadgets/docs/background/><strong>Background</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/>Poly IOP</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/><strong>Gadgets</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/>Mult2</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/productgadget/ class=active>ProdGadget</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
-</label><strong>Product Gadget</strong>
-<label for=toc-control><img src=/2024-Gadgets/svg/toc.svg class=book-icon alt="Table of Contents"></label></div><aside class="hidden clearfix"><nav id=TableOfContents></nav></aside></header><article class="markdown book-article"><p><strong>Array Level</strong></p><p>The prover has a list of value $A = [a_0, a_1, a_2, \dots, a_{n-1}]$ and wishes to commit to and prove that product of all entries in $A$ equals some value $z$. In other words, that $\prod_{i = 0}^{n-1} a_i= z$. First, the prover will compute an accumulator, $B$ according to the following rules: $i) B_i = A_i \cdot B_{i+1} \space 0 \leq i \lt n-1$ and $ii)B_{n-1} = A_{n-1}$. Then $B_0 = z$. To prove that $\prod_{i = 0}^{n-1} a_i= z$, it suffices for the prover to demonstrate that $B_0 = z$ and that $i)$ and $ii)$ hold, and that $B_0 = z$.</p><p><strong>Polynomial Level</strong></p><p>The prover interpolates $A$ and $B$ to get $P_A(x)$ and $P_B(x)$. To do so, an $n$th root of unity (which we will call $\omega$) is used, and the values of $A$ and $B$ are paired with the powers of this root of unity as follows: $(\omega^0, a_0), (\omega^1, a_1), \dots, (\omega^{n-1}, a_{n-1})$ and similarly for $B$. Then the interpolation (using the fast fourier transform (FFT)) creates a polynomial that passes through the points represented by these tuples. We also define $H$ as the set $[\omega^0, \omega^1, \dots, \omega^{n-1}]$. We are assuming that $n$ is a power of 2, so that we can use FFT. In practice, if it is not, then a power of 2 greater than $n$, say $2^m$, is used and the last $2^m - n$ entries of $A$ are padded with 1s.</p><p>The prover then compute $W_1(x) = [P_B(x) - P_B(x\omega) \cdot P_a(x))] \cdot (x - \omega^{n-1})$ and $W_2(x) = [P_A(x) - P_B(x)] \cdot \frac{x^n - 1} {x - w^{n-1}}$. If $W_1$ is vanishing on $H$, then $i)$ holds for $0 \leq i \lt n$. We note that the $(x - \omega^{n-1})$ is included in the equation to add a root at $x = n -1$, since $i)$ does not hold for $i = n -1 $ (unless $z = 1$, in which case the ($x - \omega^{n-1}$) is unnecessary). Similarly, If $W_2$ is vanishing on $H$, then $ii)$ holds. We note that multiplying [$P_A(x) - P_B(x)]$ by $\frac{x^n - 1} {x - w^{n-1}}$ zeroes $W_2$ on all of $H$ except $\omega^{n-1}$, thus $W_2$ vanishing on $H$ is testing only the equality of $A_{n-1}$ and $B_{n-1}$​.</p><p>The prover then computes $Q_1(x) = \frac{W_1(x)}{x^n - 1}$ and $Q_2(x) = \frac{W_2(x)}{x^n - 1}$. Since $x^n - 1$ is the vanishing polynomial for $H$, if $Q_1$ and $Q_2$ are polynomials (and not rational functions), then $W_1$ and $W_2$ must be vanishing on $H$.</p><p>The prover then commits to $P_A$, $P_B$, $Q_1$, and $Q_2$. They then hash the four commitments to get a random challenge, $\tau$. They open $P_A(\tau), P_B(\tau), Q_1(\tau), Q_2(\tau),$ and $P_B(\tau \cdot \omega)$ to demonstrate that $Q_1$ and $Q_2$ are polynomials, as defined above $-$ thus $i)$ and $ii)$ hold. They also open $P_B(1)$, showing that $B_0 = z$. Thus they have demontrated that $\prod_{i = 0}^{n-1} a_i= z$.</p><p><strong>Verifier Level</strong></p><p>The verifier first checks that $\tau$​ is correct by hashing the four commitments which have been sent to them by the prover.</p><p>Next, the verifier checks that the evaluations $P_A(\tau), P_B(\tau), Q_1(\tau), Q_2(\tau),$ and $P_B(\tau \cdot \omega)$​ sent to them by the prover are correct. This can be done (assuming we are working with KZG) as a single batch opening.</p><p>The verifier will then check the following two equalities, so make sure that $Q_1$ and $Q_2$​ were defined as they should be:</p><p>​ $a) \space [P_B(\tau) - P_B(\tau \cdot \omega)\cdot P_A(\tau)] \cdot (\tau - \omega^{n-1}) = (\tau^n - 1) \cdot Q_1(\tau)$​</p><p>​ $b) \space \frac{P_A(\tau) - P_B(\tau)}{\tau - \omega^{n-1}} = Q_2(\tau)$</p><p>Finally, the verifier will check the opening of $P_B(1)$. This value is $z$, and they have verified that it is equal to the product of the entries in $A$.</p></article><footer class=book-footer><div class="flex flex-wrap justify-between"></div><script>(function(){function e(e){const t=window.getSelection(),n=document.createRange();n.selectNodeContents(e),t.removeAllRanges(),t.addRange(n)}document.querySelectorAll("pre code").forEach(t=>{t.addEventListener("click",function(){if(window.getSelection().toString())return;e(t.parentElement),navigator.clipboard&&navigator.clipboard.writeText(t.parentElement.textContent)})})})()</script></footer><div class=book-comments></div><label for=menu-control class="hidden book-menu-overlay"></label></div><aside class=book-toc><div class=book-toc-content><nav id=TableOfContents></nav></div></aside></main></body></html>
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+<!doctype html><html lang=en-us dir=ltr><head><meta charset=UTF-8><meta name=viewport content="width=device-width,initial-scale=1"><meta name=description content="Zeroing Parts of an Array # Assuming an input array of size $n$: $\langle \mathsf{data}_0,\mathsf{data}_1,\ldots,\mathsf{data}_{n-1}\rangle$ and input array encoded into the polynomial. This uses &ldquo;Encoding 2&rdquo; from above (evaluation points) and uses &ldquo;Roots of Unity + FFT&rdquo; from above where $\omega\in\mathbb{G}_\kappa$ is a generator for the x-coordinates of the points.
+$\bot$ is an arbitrary non-zero integer.
+Operation Input Array Input Polynomial Output Array Output Polynomial Zero all $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle 0,0,0,0,0 \rangle$ $P(X)\cdot(X^\kappa-1)$ Zero first $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle 0,\bot,\bot,\bot,\bot \rangle$ $P(X)\cdot(X-\omega^0)$ Zero last $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle \bot,\bot,\bot,\bot,0 \rangle$ $P(X)\cdot(X-\omega^{\kappa-1})$ Zero all but first $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle \bot,0,0,0,0 \rangle$ $P(X)\cdot\frac{(X^\kappa-1)}{(X-\omega^0)}$ Zero all but last $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle 0,0,0,0,\bot \rangle$ $P(X)\cdot\frac{(X^\kappa-1)}{(X-\omega^{\kappa-1})}$ Why are these useful?"><meta name=theme-color media="(prefers-color-scheme: light)" content="#ffffff"><meta name=theme-color media="(prefers-color-scheme: dark)" content="#343a40"><meta name=color-scheme content="light dark"><meta property="og:url" content="https://madibagroup.github.io/2024-Gadgets/docs/gadgets/zero/"><meta property="og:site_name" content="Handbook"><meta property="og:title" content="Handbook"><meta property="og:description" content="Zeroing Parts of an Array # Assuming an input array of size $n$: $\langle \mathsf{data}_0,\mathsf{data}_1,\ldots,\mathsf{data}_{n-1}\rangle$ and input array encoded into the polynomial. This uses “Encoding 2” from above (evaluation points) and uses “Roots of Unity + FFT” from above where $\omega\in\mathbb{G}_\kappa$ is a generator for the x-coordinates of the points.
+$\bot$ is an arbitrary non-zero integer.
+Operation Input Array Input Polynomial Output Array Output Polynomial Zero all $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle 0,0,0,0,0 \rangle$ $P(X)\cdot(X^\kappa-1)$ Zero first $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle 0,\bot,\bot,\bot,\bot \rangle$ $P(X)\cdot(X-\omega^0)$ Zero last $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle \bot,\bot,\bot,\bot,0 \rangle$ $P(X)\cdot(X-\omega^{\kappa-1})$ Zero all but first $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle \bot,0,0,0,0 \rangle$ $P(X)\cdot\frac{(X^\kappa-1)}{(X-\omega^0)}$ Zero all but last $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle 0,0,0,0,\bot \rangle$ $P(X)\cdot\frac{(X^\kappa-1)}{(X-\omega^{\kappa-1})}$ Why are these useful?"><meta property="og:locale" content="en_us"><meta property="og:type" content="article"><meta property="article:section" content="docs"><title>Zero | Handbook</title>
+<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/docs/gadgets/zero/><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.93281b7c13f5b9b0a9578b33cdbdde95d267573c1f5b04dd2cb337c07f21791f.js integrity="sha256-kygbfBP1ubCpV4szzb3eldJnVzwfWwTdLLM3wH8heR8=" crossorigin=anonymous></script><script id=MathJax-script async src=https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js></script><script>MathJax={tex:{displayMath:[["\\[","\\]"],["$$","$$"]],inlineMath:[["\\(","\\)"],["$","$"]]}}</script></head><body dir=ltr><input type=checkbox class="hidden toggle" id=menu-control>
+<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><strong>Background</strong></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/>Poly IOP</a></p></li><li><p><strong>Gadgets</strong></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/>Mult2</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
+</label><strong>Zero</strong>
+<label for=toc-control><img src=/2024-Gadgets/svg/toc.svg class=book-icon alt="Table of Contents"></label></div><aside class="hidden clearfix"><nav id=TableOfContents><ul><li><a href=#zeroing-parts-of-an-array>Zeroing Parts of an Array</a></li></ul></nav></aside></header><article class="markdown book-article"><h2 id=zeroing-parts-of-an-array>Zeroing Parts of an Array
+<a class=anchor href=#zeroing-parts-of-an-array>#</a></h2><p>Assuming an input array of size $n$: $\langle \mathsf{data}_0,\mathsf{data}_1,\ldots,\mathsf{data}_{n-1}\rangle$ and input array encoded into the polynomial. This uses &ldquo;Encoding 2&rdquo; from above (evaluation points) and uses &ldquo;Roots of Unity + FFT&rdquo; from above where $\omega\in\mathbb{G}_\kappa$ is a generator for the x-coordinates of the points.</p><p>$\bot$ is an arbitrary non-zero integer.</p><table><thead><tr><th>Operation</th><th>Input Array</th><th>Input Polynomial</th><th>Output Array</th><th>Output Polynomial</th></tr></thead><tbody><tr><td>Zero all</td><td>$\langle 3,1,3,3,7 \rangle$</td><td>$P(X)$</td><td>$\langle 0,0,0,0,0 \rangle$</td><td>$P(X)\cdot(X^\kappa-1)$</td></tr><tr><td>Zero first</td><td>$\langle 3,1,3,3,7 \rangle$</td><td>$P(X)$</td><td>$\langle 0,\bot,\bot,\bot,\bot \rangle$</td><td>$P(X)\cdot(X-\omega^0)$</td></tr><tr><td>Zero last</td><td>$\langle 3,1,3,3,7 \rangle$</td><td>$P(X)$</td><td>$\langle \bot,\bot,\bot,\bot,0 \rangle$</td><td>$P(X)\cdot(X-\omega^{\kappa-1})$</td></tr><tr><td>Zero all but first</td><td>$\langle 3,1,3,3,7 \rangle$</td><td>$P(X)$</td><td>$\langle \bot,0,0,0,0 \rangle$</td><td>$P(X)\cdot\frac{(X^\kappa-1)}{(X-\omega^0)}$</td></tr><tr><td>Zero all but last</td><td>$\langle 3,1,3,3,7 \rangle$</td><td>$P(X)$</td><td>$\langle 0,0,0,0,\bot \rangle$</td><td>$P(X)\cdot\frac{(X^\kappa-1)}{(X-\omega^{\kappa-1})}$</td></tr></tbody></table><p>Why are these useful? Consider a <a href=https://hackmd.io/@dabo/B1U4kx8XI>range proof</a> from Boneh, Fisch, Gabizon, and Williamson. At a certain point in the protocol, we reach the following:</p><p><img src=https://hackmd.io/_uploads/H1Xg2IvgR.png alt="Screenshot 2024-04-12 at 9.11.45 PM"></p><p>First note that $w$ and $\omega$ look the same but are different: $w$ are three new polynomials we are creating, while $\omega$ is the generator of $\mathbb{G}_\kappa$ we are using to pick x-coordinates for the polynomials.</p><ul><li>In the first constraint, we want to prove the value of f(1) and g(1) are the same. So we subtract (g-f) which leaves a zero in the first element of the array but the rest of the array will contain other stuff. By applying &ldquo;zero all but first&rdquo;, we can zero out the rest of the array. We now have an array that is all zero (in polynomial form, this is called a vanshing polynomial and we can prove a polynomial vanishes easily and in a batch).</li><li>In the second constraint, we prove the last element in g is a binary bit (0 or 1) and then we apply the &ldquo;zero all but last&rdquo; to make an array that is all zero (and vanishing polynomial).</li><li>In the third constraint, the first two terms of the multiplication are proving $g(X)$ has a certain form that does not need to be understood here. What is important is that there is a relationship between each integer in the array ($g(X)$) and the integer right beside it ($g(X\omega)$) in the array. When the relationship holds, the result is a zero in $w_3$. Unfortunately, there is a corner case: for the last element in the array, the &ldquo;next&rdquo; integer wraps back to the first integer and the relationship does not hold across this boundary. So we use &ldquo;zero last&rdquo; to manually zero out the last integer in the array, leaving us with a zero array (and vanishing polynomial).</li></ul></article><footer class=book-footer><div class="flex flex-wrap justify-between"></div><script>(function(){function e(e){const t=window.getSelection(),n=document.createRange();n.selectNodeContents(e),t.removeAllRanges(),t.addRange(n)}document.querySelectorAll("pre code").forEach(t=>{t.addEventListener("click",function(){if(window.getSelection().toString())return;e(t.parentElement),navigator.clipboard&&navigator.clipboard.writeText(t.parentElement.textContent)})})})()</script></footer><div class=book-comments></div><label for=menu-control class="hidden book-menu-overlay"></label></div><aside class=book-toc><div class=book-toc-content><nav id=TableOfContents><ul><li><a href=#zeroing-parts-of-an-array>Zeroing Parts of an Array</a></li></ul></nav></div></aside></main></body></html>
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 <!doctype html><html lang=en-us dir=ltr><head><meta charset=UTF-8><meta name=viewport content="width=device-width,initial-scale=1"><meta name=description content><meta name=theme-color media="(prefers-color-scheme: light)" content="#ffffff"><meta name=theme-color media="(prefers-color-scheme: dark)" content="#343a40"><meta name=color-scheme content="light dark"><meta property="og:url" content="https://madibagroup.github.io/2024-Gadgets/docs/"><meta property="og:site_name" content="Handbook"><meta property="og:title" content="Docs"><meta property="og:locale" content="en_us"><meta property="og:type" content="website"><title>Docs | Handbook</title>
-<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/docs/><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.bb5d9164f3309a75ef9097bce777d0ab67b4a8025ae95f3105b06cec629ec033.js integrity="sha256-u12RZPMwmnXvkJe853fQq2e0qAJa6V8xBbBs7GKewDM=" crossorigin=anonymous></script><link rel=alternate type=application/rss+xml href=https://madibagroup.github.io/2024-Gadgets/docs/index.xml title=Handbook><script id=MathJax-script async src=https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js></script><script>MathJax={tex:{displayMath:[["\\[","\\]"],["$$","$$"]],inlineMath:[["\\(","\\)"],["$","$"]]}}</script></head><body dir=ltr><input type=checkbox class="hidden toggle" id=menu-control>
-<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><a href=/2024-Gadgets/docs/background/><strong>Background</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/>Poly IOP</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/><strong>Gadgets</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/>Mult2</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/productgadget/>ProdGadget</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
+<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/docs/><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.93281b7c13f5b9b0a9578b33cdbdde95d267573c1f5b04dd2cb337c07f21791f.js integrity="sha256-kygbfBP1ubCpV4szzb3eldJnVzwfWwTdLLM3wH8heR8=" crossorigin=anonymous></script><link rel=alternate type=application/rss+xml href=https://madibagroup.github.io/2024-Gadgets/docs/index.xml title=Handbook><script id=MathJax-script async src=https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js></script><script>MathJax={tex:{displayMath:[["\\[","\\]"],["$$","$$"]],inlineMath:[["\\(","\\)"],["$","$"]]}}</script></head><body dir=ltr><input type=checkbox class="hidden toggle" id=menu-control>
+<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><strong>Background</strong></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/>Poly IOP</a></p></li><li><p><strong>Gadgets</strong></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/>Mult2</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
 </label><strong>Docs</strong>
 <label for=toc-control><img src=/2024-Gadgets/svg/toc.svg class=book-icon alt="Table of Contents"></label></div><aside class="hidden clearfix"><nav id=TableOfContents></nav></aside></header><article class="markdown book-article"></article><footer class=book-footer><div class="flex flex-wrap justify-between"></div><script>(function(){function e(e){const t=window.getSelection(),n=document.createRange();n.selectNodeContents(e),t.removeAllRanges(),t.addRange(n)}document.querySelectorAll("pre code").forEach(t=>{t.addEventListener("click",function(){if(window.getSelection().toString())return;e(t.parentElement),navigator.clipboard&&navigator.clipboard.writeText(t.parentElement.textContent)})})})()</script></footer><label for=menu-control class="hidden book-menu-overlay"></label></div><aside class=book-toc><div class=book-toc-content><nav id=TableOfContents></nav></div></aside></main></body></html>
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+[{"id":0,"href":"/2024-Gadgets/docs/background/poly-iop/","title":"Poly Iop","section":"Background","content":" Notes on Polynomial Interactive Oracle Proofs (Draft) # Encoding Arrays of Data into Polynomials # Data starts as a array of integers where integers in each slot are between 0 and $q-1$, where $q$ is a large (generally 256 bit) prime number. Recall that we call this set of integers $\\mathbb{Z}_q$.\n$\\mathsf{data}_0$ $\\mathsf{data}_1$ $\\mathsf{data}_2$ $\\mathsf{data}_3$ $\\mathsf{data}_4$ It is common to denote a polynomial like $P(X)$ where $X$ is the variable of the polynomial. We are going denote the variable with an empty box $\\square$ which can be interpreted as a place where you can put any integer in $\\mathbb{Z}_q$ you want evaluated (or equivalent, where you place an x-coordinate to learn what the y-coordinate is).\nA polynomial in this notation looks like:\n$P(\\square)=c_0+c_1\\cdot\\square+c_2\\cdot\\square^2+c_3\\cdot\\square^3+c_4\\cdot\\square^4=\\sum_{i=0}^d c_i\\cdot\\square^i$ The values $c_i$ are called coefficients. Different arrays of data will (depending on how data is encoded, next) result in different coefficients and thus different polynomials. The degree of the polynomial is the largest exponent. So the polynomial above has degree 4 and thus will have 5 coefficients and 5 terms of the form $c_i\\cdot\\square^i$ (including $i=0$). Sometimes the coefficient will be zero: the term is thus not written down but in a list of coefficients, it will be included as a 0.\nThe main question to tackle is how to \u0026ldquo;encode\u0026rdquo; a array of integers into a polynomial. This is generally done one of three ways:\nCoefficients Evaluation Points Roots Each has its advantages and disadvantages, which we discuss next.\nFast forwarding a bit, once the polynomial is created, it is not shared directly with anyone. Instead a commitment to it is shared. The commitment does two things: (1) it makes it succinct: e.g., constant size regardless of how long the array is; and (2) it can hide the data in the array as necessary. We will discuss one specific polynomial commitment scheme called KZG. KZG needs the polynomial in the format of a list of its coefficients. If we have the polynomial in a different form, we will have to convert to coefficients. Thus this needs to be considered when weighing the pros/cons of the three encoding methods.\nEncoding 1: Coefficients # Create polynomial as: $P_1(\\square)=\\mathsf{data}_0+\\square+\\mathsf{data}_2\\cdot\\square^2+\\mathsf{data}_3\\cdot\\square^3+\\mathsf{data}_4\\cdot\\square^4=\\sum_{i=0}^d \\mathsf{data}_i\\cdot\\square^i$\nProperties:\nFast 👍: fastest path to commitment as the output is already in coefficient form. Addition 👍: two arrays can be added together (slot-by-slot) by simply adding the polynomials together Multiplication 👎: no support for multiplication of arrays (Remark: multiplying the polynomials does not multiply the coefficients. It results in a cross multiplication of every term in the first polynomial with every term in the second polynomial. Further the degree of the resulting polynomial will double that of the starting polynomials). Opening 🤷🏻: proving the value of the $i$th element in the array is $\\mathsf{data}_i$ doing polynomial math on $P_1(\\square)$ is not possible. However $\\Sigma$-protocols done directly on KZG may enable this kind of proof. In any case, this is not particularly well explored. Other useful properties 👍: the sum of all values in a array can be computed by evaluating the polynomial at $P_1(\\boxed{1})$! $\\sum_{i=0}^d \\mathsf{data}_i\\cdot\\boxed{1}^i=\\sum_{i=0}^d \\mathsf{data}_i$ . You can also show two arrays have the same sum (called a \u0026ldquo;sum check\u0026rdquo;) by subtracting them and showing $P(\\boxed{1})=0$. Other useful properties 👍: when all coefficients are 0, the polynomial will be the zero polynomial ($P_1(\\square)=0$). Coefficients can be entire polynomials, not just integers. A common optimization in Poly-IOP systems is taking a set of equations of polynomials that should equal 0, placing each into the coefficient of a super-polynomial, and showing the super-polynomial is the zero polynomial (which can be proven overwhelmingly by showing it is 0 at a randomly selected point). Encoding 2: Evaluation Points # Create a list of points $\\{x,y\\}$ for the data: $\\langle\\{0,\\mathsf{data}_0\\},\\{1,\\mathsf{data}_1\\},\\{2,\\mathsf{data}_2\\},\\{3,\\mathsf{data}_3\\},\\{4,\\mathsf{data}_4\\}\\rangle$ and interpolate a polynomial $P_2(\\square)$ through these points.\nProperties:\nSlow (or moderate) 👎: converting a set of points into a set of coefficients is called interpolation and is $O(n^2)$ time generally. A certain optimization allows $O(n\\log n)$ time by chosing $x$ coordinates with a mathematical relationship (more on this later). Addition 👍: two arrays can be added together (slot-by-slot) by simply adding the polynomials together Multiplication 👍: two arrays can be multiplied together (slot-by-slot) by simply multiplying the polynomials together Opening 👍: proving the value of the $i$th element in the array is $\\mathsf{data}_i$ is possible with polynomial math by showing $P_2(\\boxed{i})=\\mathsf{data}_i$ and KZG has a precise algorithm for this. Encoding 3: Roots # Create polynomial as: $P_3(\\square)=(\\square-\\mathsf{data}_0)(\\square-\\mathsf{data}_1)(\\square-\\mathsf{data}_2)(\\square-\\mathsf{data}_3)(\\square-\\mathsf{data}_4)$\nAlternatively, create a list of roots $\\{x,0\\}$ for the data: $\\langle\\{\\mathsf{data}_0,0\\},\\{\\mathsf{data}_1,0\\},\\{\\mathsf{data}_2,0\\},\\{\\mathsf{data}_3,0\\},\\{\\mathsf{data}_4,0\\}\\rangle$ and interpolate a polynomial $P_3(\\square)$ through these points.\nProperties:\nSlow 👎 (or moderate): multiplying out naively requires $O(n^2)$ time. Treating as a set of points and interpolating also requires $O(n^2)$ time (because the x-coordinates are the data, they cannot be chosen freely to optimize interpolation). Applying divide and conquer can provide $O(n \\log^2 n)$.^[1] Addition 👎: two arrays cannot be added from adding (or otherwise manipulating) the polynomials. Multiplication 👎: two arrays cannot be multiplied from multiplying (or otherwise manipulating) the polynomials (but you can do a \u0026ldquo;union\u0026rdquo; operation below). Opening 👍: proving that a value is in the array somewhere is easy and KZG as a precise algorithm for this (opening a root is the same as opening a point, where the y-coordinate is 0). However you cannot show a value is specifically the $i$th value in the array because the polynomial loses the order of the data in the array (see next property). Other useful properties 👍: the order of the data in the array does not matter. The same polynomial will be produced even if the order is changed. This is useful when the array represents a \u0026ldquo;bag\u0026rdquo; of unordered data. You can easily prove two \u0026ldquo;bags\u0026rdquo; of data are the same because the polynomials will be the same. One use-case of this is proving the output of a shuffle/permutation is the same data as the input (just in a different order). Other useful properties 👍: multiplying two polynomials results in a concatenation of the data in the arrays (or conjunction/union of the data in both bags). This might be useful in some protocols. ^[1]: Hat tip Pratyush Mishra.\nDecision Tree for Encoding # Basically we decide if we specifically need unordered \u0026ldquo;bags\u0026rdquo; of data. If so, encoding as roots is the only option. If not, we consider if we need to ever get the data back from the polynomial. Generally we do and encoding as evaluation points is the most common encoding technique. When do we encode the data and never want it back? Usually when (1) the coefficients are all supposed to be zero so we are just showing that property, or (2) we want back the sum of the data and not the data itself. In these cases, you can still work with evaluation point encoding but it will be faster to just do coefficient encoding.\nflowchart LR; A[Array to Polynomial] --\u003e B{Is the data unordered?}; B -- Unordered --\u003e C[Roots]; B -- Ordered or don't care --\u003e D{Open data from polynomial later?}; D -- Yes --\u003e E[Evaluation Points]; D -- No --\u003e F[Coefficients]; The short answer is to start with evaluation point encoding until you realize you need something different.\nOptimizing Interpolation (Roots of Unity + FFT) # Moving forward, we will assume we are using Encoding 2: Evaluation Points. In short, this means placing the elements of our array into the $y$-coordinates ($\\mathsf{data}_i=P(\\boxed{x_i})$) of points on the polynomial. Before commiting to $P(\\square)$, we need to use interpolation to find the coefficients of the polynomial that is fitted to these points. General interpolation algorithms are $O(n^2)$ work for $n$ evaluation points but this can be reduced to $O(n\\log n)$ with an optimization.\nThe optimization we will explore enables interpolation via the fast Fourier transform (FFT). It concerns how to choose the $x$-coordinates, which will serve as the index for accessing the data: evaluating $P(X)$ at $x_i$ will reveal $\\mathsf{data}_i$. First note, $x$-coordinates are from the exponent group ($Z_q$) and the choices exceed what is feasible to use ($2^{255}$ values in bls). Any subset can be used and interpolated. The optimization is to chose them with a mathematical structure. Specifically, instead an additive sequence (e.g., $0,1,2,3,\\ldots$), we use a multiplicative sequence $1,\\omega,\\omega\\cdot\\omega,\\omega\\cdot\\omega\\cdot\\omega,\\ldots$ or equivalently: $\\omega^0,\\omega^1,\\omega^2,\\ldots,\\omega^{\\kappa-1}$. Further, the sequence is closed under multiplication which means that next index after $\\omega^{\\kappa-1}$ wraps back to the first index: $\\omega^{k-1} \\cdot \\omega = \\omega^\\kappa = \\omega^0=1$ (this property is also useful in proving relationships between data in the array and its neighbouring values).\nFor terminology, we say $\\omega$ is a generator with multiplicative order $\\kappa$ in $\\mathbb{Z}_q$ (or $\\omega \\in \\mathbb{G}_\\kappa$). This implies $\\omega^\\kappa=1$. Rearranging, $\\omega=\\sqrt[\\kappa]{1}$. Thus we can equivalently describe $\\omega$ as a $\\kappa$-th root of 1. Finally, as 1 is the unity element in $Z_q$, $\\omega$ is commonly called a $\\kappa$-th root of unity.\nFor practical purposes, $\\kappa$ represents the length of the longest array of data we can use in our protocol. Where does $\\kappa$ come from? Different elements of $Z_q$ will have different multiplicative orders but every order must be a divisor of $q-1$. Thus $\\kappa$ is the largest divisor of the exact value of $q$ used in an elliptic curve standard. BLS12-384 has $\\kappa=2^{32}$ (for terminology, this called a $2$-adicity of $32$). In summary, we can only encode data arrays of length up to $2^{32}=4,294,967,296$.\nZeroing Parts of an Array # Assuming an input array of size $n$: $\\langle \\mathsf{data}_0,\\mathsf{data}_1,\\ldots,\\mathsf{data}_{n-1}\\rangle$ and input array encoded into the polynomial. This uses \u0026ldquo;Encoding 2\u0026rdquo; from above (evaluation points) and uses \u0026ldquo;Roots of Unity + FFT\u0026rdquo; from above where $\\omega\\in\\mathbb{G}_\\kappa$ is a generator for the x-coordinates of the points.\n$\\bot$ is an arbitrary non-zero integer.\nOperation Input Array Input Polynomial Output Array Output Polynomial Zero all $\\langle 3,1,3,3,7 \\rangle$ $P(X)$ $\\langle 0,0,0,0,0 \\rangle$ $P(X)\\cdot(X^n-1)$ Zero first $\\langle 3,1,3,3,7 \\rangle$ $P(X)$ $\\langle 0,\\bot,\\bot,\\bot,\\bot \\rangle$ $P(X)\\cdot(X-\\omega^0)$ Zero last $\\langle 3,1,3,3,7 \\rangle$ $P(X)$ $\\langle \\bot,\\bot,\\bot,\\bot,0 \\rangle$ $P(X)\\cdot(X-\\omega^{n-1})$ Zero all but first $\\langle 3,1,3,3,7 \\rangle$ $P(X)$ $\\langle \\bot,0,0,0,0 \\rangle$ $P(X)\\cdot\\frac{(X^n-1)}{(X-\\omega^0)}$ Zero all but last $\\langle 3,1,3,3,7 \\rangle$ $P(X)$ $\\langle 0,0,0,0,\\bot \\rangle$ $P(X)\\cdot\\frac{(X^n-1)}{(X-\\omega^{n-1})}$ Why are these useful? Consider a range proof from Boneh, Fisch, Gabizon, and Williamson. At a certain point in the protocol, we reach the following:\nFirst note that $w$ and $\\omega$ look the same but are different: $w$ are three new polynomials we are creating, while $\\omega$ is the generator of $\\mathbb{G}_\\kappa$ we are using to pick x-coordinates for the polynomials.\nIn the first constraint, we want to prove the value of f(1) and g(1) are the same. So we subtract (g-f) which leaves a zero in the first element of the array but the rest of the array will contain other stuff. By applying \u0026ldquo;zero all but first\u0026rdquo;, we can zero out the rest of the array. We now have an array that is all zero (in polynomial form, this is called a vanshing polynomial and we can prove a polynomial vanishes easily and in a batch). In the second constraint, we prove the last element in g is a binary bit (0 or 1) and then we apply the \u0026ldquo;zero all but last\u0026rdquo; to make an array that is all zero (and vanishing polynomial). In the third constraint, the first two terms of the multiplication are proving $g(X)$ has a certain form that does not need to be understood here. What is important is that there is a relationship between each integer in the array ($g(X)$) and the integer right beside it ($g(X\\omega)$) in the array. When the relationship holds, the result is a zero in $w_3$. Unfortunately, there is a corner case: for the last element in the array, the \u0026ldquo;next\u0026rdquo; integer wraps back to the first integer and the relationship does not hold across this boundary. So we use \u0026ldquo;zero last\u0026rdquo; to manually zero out the last integer in the array, leaving us with a zero array (and vanishing polynomial). "},{"id":1,"href":"/2024-Gadgets/docs/gadgets/mult2/","title":"Mult2","section":"Gadgets","content":" Multiplication (Type 2) # Recap of types # Type Description Recap This Type 1 $\\mathsf{Arr}_3=\\mathsf{Arr}_1 \\cdot \\mathsf{Arr}_2$ $\\mathsf{Arr}_3$ is the element-wise multiplication of $\\mathsf{Arr}_1$ and $\\mathsf{Arr}_2$. Type 2 $\\mathsf{Prod}_\\mathsf{Arr}=\\prod_{i = 0}^{n-1} \\mathsf{Arr}[i]$ $\\mathsf{Prod}_\\mathsf{Arr}$ is the disclosed product of all the elements in $\\mathsf{Arr}$. ✅ Type 3 $\\prod_{i = 0}^{n-1} \\mathsf{Arr}_1[i]=\\prod_{i = 0}^{n-1} \\mathsf{Arr}_2[i]$ $\\mathsf{Arr}_1$ and $\\mathsf{Arr}_2$ have the same undisclosed product. Relation # $ \\mathcal{R}_{\\mathtt{mult2}} := \\left\\{ \\begin{array}{l} (K_\\mathsf{Arr},\\mathsf{Prod}_\\mathsf{Arr}) \\end{array} \\middle | \\begin{array}{l} \\mathsf{Arr} = [a_0, a_1, a_2, \\dots, a_{n-1}], \\\\ \\mathsf{Prod}_\\mathsf{Arr} = \\prod_{i = 0}^{n-1} a_i, \\\\ \\mathsf{Poly}_\\mathsf{Arr}=\\mathsf{FFT.Interp}(\\omega,\\mathsf{Arr}), \\\\ K_\\mathsf{Arr}=\\mathsf{KZG.Commit}(\\mathsf{Poly}_\\mathsf{Arr}) \\end{array} \\right\\} $\nIntuition # The prover ($\\mathcal{P}$) holds an array $\\mathsf{Arr} = [a_0, a_1, a_2, \\dots, a_{n-1}]$ and a public value $\\mathsf{Prod}_\\mathsf{Arr}$. It will produce a succinct (independent of $n$) proof that $\\mathsf{Prod}_\\mathsf{Arr}$ is the product of all the elements in the array. The prover will encode the array into a polynomial $\\mathsf{Poly}_\\mathsf{Arr}$ (using evaluation points) and commit to the polynomial $K_\\mathsf{Arr}$. The verifier ($\\mathcal{V}$) cannot check $\\textsf{Arr}$ or $\\mathsf{Poly}_\\mathsf{Arr}$ directly (they may contain secret information, and even if they do not, it is too long to check) so the verifier only sees $K_\\mathsf{Arr}$ and the asserted value $\\mathsf{Prod_\\mathsf{Arr}}$.\nIn order to prove $K_\\mathsf{Arr}$ and $\\mathsf{Prod}_\\mathsf{Arr}$ are consistent, the prover will build a helper array $\\mathsf{Acc}_\\mathsf{Arr}$ called an accumulator (or accumulating array or incremental array). This should not be confused with accumulators from cryptography, which are a concept related to succinct proofs but are distinct. As with $\\mathsf{Arr}$, the prover will also encode $\\mathsf{Acc}$ as a polynomial and provide a commitment of it to the verifier. The idea is that the prover will prove a relation between $\\mathsf{Arr}$ and $\\mathsf{Acc}$; and a relation between $\\mathsf{Acc}$ and $\\mathsf{Prod_\\mathsf{Arr}}$. Put together, it will imply the correct relation between $\\mathsf{Arr}$ and $\\mathsf{Prod_\\mathsf{Arr}}$​.\nConsider a small numeric example in $\\mathbb{Z}_{97}$ where $\\mathsf{Arr}= [84,67,11,92,36,67]$ and $\\mathsf{Prod}_\\mathsf{Arr}=72$. The first idea is to get $\\mathsf{Prod}_\\mathsf{Arr}$ into an array. Say, we just append it: $\\mathsf{Arr}''= [84,67,11,92,36,67,72] $. How does the prover show $\\mathsf{Arr}''$ is correct? The last value of the array depends on every single element that precedes it, which will be a complex constraint to prove.\nAn alternative idea is to create a new array that starts the same as $\\mathsf{Arr}$ and ends up at $\\mathsf{Prod}_\\mathsf{Arr}$ by folding in the integers from $\\mathsf{Arr}$ one-by-one with multiplication. Then each value in the new array will depend on only two values, as below.\nThe first value in $\\mathsf{Acc}$ will be a copy of the first value from $\\mathsf{Arr}$:\n$\\mathsf{Arr}= [84,67,11,92,36,67]$\n$\\mathsf{Acc}= [84, \\bot,\\bot,\\bot,\\bot,\\bot] $​\nThe next value will be the multiplication (mod 97) of: 67 (the value at the same index in $\\mathsf{Arr}$) and 84 (the previous value in $\\mathsf{Acc}$​):\n$\\mathsf{Arr}= [84,67,11,92,36,67]$\n$ \\mathsf{Acc} = [84, (67\\cdot84),\\bot,\\bot,\\bot,\\bot] = [84, 2,\\bot,\\bot,\\bot,\\bot]$\nThe next value will be the multiplication of: 11 (the value at the same index in $\\mathsf{Arr}$) and 2 (the previous value in $\\mathsf{Acc}$):\n$\\mathsf{Arr}= [84,67,11,92,36,67]$ $ \\mathsf{Acc} = [84, 2,(11\\cdot2),\\bot,\\bot,\\bot] = [84,2,22,\\bot,\\bot,\\bot]$​​ $ \\mathsf{Acc} = [84, 2,22,(92\\cdot22),\\bot,\\bot] = [84,2,22,84,\\bot,\\bot]$ $ \\mathsf{Acc} = [84, 2,22,84,(36\\cdot84),\\bot] = [84,2,22,84,17,\\bot]$​ $ \\mathsf{Acc} = [84,2,22,84,17,(67\\cdot17)] = [84, 2, 22, 84, 17, 72]$​ $\\mathsf{Prod}_\\mathsf{Arr}=72$ Notice the last value in $\\mathsf{Acc}$ is $\\mathsf{Prod_\\mathsf{Arr}}$. The prover wants to show three constraints:\nThe first value in $\\mathsf{Acc}$ matches the first value in $\\mathsf{Arr}$, The rest of the values in $\\mathsf{Acc}$ are of the form $\\mathsf{Acc}[i]=\\mathsf{Arr}[i]\\cdot\\mathsf{Acc}[i-1]$, The last value in $\\mathsf{Acc}$ matches $\\mathsf{Prod}_\\mathsf{Arr}$​​. If all three constraints are true, then $\\mathsf{Prod}_\\mathsf{Arr}$ is the product of the elements of $\\mathsf{Arr}$.\nLast, while it is not necessary to do, it is often convenient to hold the the value $\\mathsf{Prod}_\\mathsf{Arr}$ at the start of the array $\\mathsf{Acc}$ instead of the end. For this reason, the mathematical explaination below will construct $\\mathsf{Acc}$ \u0026ldquo;backwards\u0026rdquo; (or right-to-left) from the above example, where the last value of $\\mathsf{Acc}$ matches the last value of $\\mathsf{Arr}$, the values are folded in from right to left, and the first (leftmost) value of $\\mathsf{Acc}$ is $\\mathsf{Prod}_\\mathsf{Arr}$:\n$\\mathsf{Arr}= [84,67,11,92,36,67]$​ $ \\mathsf{Acc} = [\\bot, \\bot, \\bot, \\bot, \\bot, 67]$ $ \\mathsf{Acc} = [\\bot, \\bot, \\bot, \\bot, 84, 67]$ $\\ldots$ $ \\mathsf{Acc} = [72, 84, 36, 65, 84, 67]$​ $\\mathsf{Prod}_\\mathsf{Arr}=72$ Array Level # The prover has a list of value $A = [a_0, a_1, a_2, \\dots, a_{n-1}]$ and wishes to commit to and prove that product of all entries in $A$ equals some value $z$. In other words, that $\\prod_{i = 0}^{n-1} a_i= z$. First, the prover will compute an accumulator, $B$ according to the following rules: $i) B_i = A_i \\cdot B_{i+1} \\space 0 \\leq i \\lt n-1$ and $ii)B_{n-1} = A_{n-1}$. Then $B_0 = z$. To prove that $\\prod_{i = 0}^{n-1} a_i= z$, it suffices for the prover to demonstrate that $B_0 = z$ and that $i)$ and $ii)$ hold, and that $B_0 = z$​.\nPolynomial Level # We assume arrays are encoded as y-coordinates into a univariant polynomial where the x-coordinates are chosen as the multiplicative group of order $\\kappa$ with generator $\\omega\\in\\mathbb{G}_\\kappa$ (see Background for more). In short, $\\omega^0$ is the first element and $\\omega^{\\kappa-1}$ is the last element of this domain $\\mathcal{H}$.\nIn polynomial form, the three constraints are:\nFor $X=w^\\kappa$: $\\mathsf{Poly}_\\mathsf{Acc}(X)=\\mathsf{Poly}_\\mathsf{Arr}(X)$, For all $X$ except $X=\\omega^{\\kappa-1}$: $\\mathsf{Poly}_\\mathsf{Acc}(X)=\\mathsf{Poly}_\\mathsf{Arr}(X)\\cdot\\mathsf{Poly}_\\mathsf{Acc}(\\omega\\cdot X)$ For $X=w^0$: $\\mathsf{Poly}_\\mathsf{Acc}(X)=\\mathsf{Prod}_\\mathsf{Arr}$ We adjust each of these constraints to show an equality with 0:\nFor $X=w^\\kappa$: $\\mathsf{Poly}_\\mathsf{Acc}(X)-\\mathsf{Poly}_\\mathsf{Arr}(X)=0$, For all $X$ except $X=\\omega^{\\kappa-1}$: $\\mathsf{Poly}_\\mathsf{Acc}(X)-\\mathsf{Poly}_\\mathsf{Arr}(X)\\cdot\\mathsf{Poly}_\\mathsf{Acc}(\\omega\\cdot X)=0$ For $X=w^0$: $\\mathsf{Poly}_\\mathsf{Acc}(X)-\\mathsf{Prod}_\\mathsf{Arr}=0$​ Next we take care of the \u0026ldquo;for $X$​\u0026rdquo; conditions by zeroing out the rest of the polynomial that is not zero. See the gadget: Zero-ing for more on why this works.\n$\\mathsf{Poly}_\\mathsf{Vanish1}=(\\mathsf{Poly}_\\mathsf{Acc}(X)-\\mathsf{Poly}_\\mathsf{Arr}(X))\\cdot\\frac{(X^\\kappa-1)}{(X-\\omega^{\\kappa-1})}=0$, $\\mathsf{Poly}_\\mathsf{Vanish2}=(\\mathsf{Poly}_\\mathsf{Acc}(X)-\\mathsf{Poly}_\\mathsf{Arr}(X)\\cdot\\mathsf{Poly}_\\mathsf{Acc}(\\omega\\cdot X))\\cdot(X-\\omega^{\\kappa-1})=0$ $\\mathsf{Poly}_\\mathsf{Vanish3}=(\\mathsf{Poly}_\\mathsf{Acc}(X)-\\mathsf{Prod}_\\mathsf{Arr})\\cdot\\frac{(X^\\kappa-1)}{(X-\\omega^0)}=0$ These equations are true for every value of $X \\in \\mathbb{G}_\\kappa$ (but not necessarily true outside of these values) show we show each is a vanishing polynomial.\nThe prover interpolates $A$ and $B$ to get $P_A(x)$ and $P_B(x)$. To do so, an $n$th root of unity (which we will call $\\omega$) is used, and the values of $A$ and $B$ are paired with the powers of this root of unity as follows: $(\\omega^0, a_0), (\\omega^1, a_1), \\dots, (\\omega^{n-1}, a_{n-1})$ and similarly for $B$. Then the interpolation (using the fast fourier transform (FFT)) creates a polynomial that passes through the points represented by these tuples. We also define $H$ as the set $[\\omega^0, \\omega^1, \\dots, \\omega^{n-1}]$. We are assuming that $n$ is a power of 2, so that we can use FFT. In practice, if it is not, then a power of 2 greater than $n$, say $2^m$, is used and the last $2^m - n$ entries of $A$ are padded with 1s.\nThe prover then compute $W_1(x) = [P_B(x) - P_B(x\\omega) \\cdot P_a(x))] \\cdot (x - \\omega^{n-1})$ and $W_2(x) = [P_A(x) - P_B(x)] \\cdot \\frac{x^n - 1} {x - w^{n-1}}$. If $W_1$ is vanishing on $H$, then $i)$ holds for $0 \\leq i \\lt n$. We note that the $(x - \\omega^{n-1})$ is included in the equation to add a root at $x = n -1$, since $i)$ does not hold for $i = n -1 $ (unless $z = 1$, in which case the ($x - \\omega^{n-1}$) is unnecessary). Similarly, If $W_2$ is vanishing on $H$, then $ii)$ holds. We note that multiplying [$P_A(x) - P_B(x)]$ by $\\frac{x^n - 1} {x - w^{n-1}}$ zeroes $W_2$ on all of $H$ except $\\omega^{n-1}$, thus $W_2$ vanishing on $H$ is testing only the equality of $A_{n-1}$ and $B_{n-1}$​.\nThe prover then computes $Q_1(x) = \\frac{W_1(x)}{x^n - 1}$ and $Q_2(x) = \\frac{W_2(x)}{x^n - 1}$. Since $x^n - 1$ is the vanishing polynomial for $H$, if $Q_1$ and $Q_2$ are polynomials (and not rational functions), then $W_1$ and $W_2$ must be vanishing on $H$.\nThe prover then commits to $P_A$, $P_B$, $Q_1$, and $Q_2$. They then hash the four commitments to get a random challenge, $\\tau$. They open $P_A(\\tau), P_B(\\tau), Q_1(\\tau), Q_2(\\tau),$ and $P_B(\\tau \\cdot \\omega)$ to demonstrate that $Q_1$ and $Q_2$ are polynomials, as defined above $-$ thus $i)$ and $ii)$ hold. They also open $P_B(1)$, showing that $B_0 = z$. Thus they have demontrated that $\\prod_{i = 0}^{n-1} a_i= z$​​.\nOptimized # Here, rework each constraint so it is a vanishing polynomial.\nCommitment Level # The verifier first checks that $\\tau$​ is correct by hashing the four commitments which have been sent to them by the prover.\nNext, the verifier checks that the evaluations $P_A(\\tau), P_B(\\tau), Q_1(\\tau), Q_2(\\tau),$ and $P_B(\\tau \\cdot \\omega)$​ sent to them by the prover are correct. This can be done (assuming we are working with KZG) as a single batch opening.\nThe verifier will then check the following two equalities, so make sure that $Q_1$ and $Q_2$​ were defined as they should be:\n​\t$a) \\space [P_B(\\tau) - P_B(\\tau \\cdot \\omega)\\cdot P_A(\\tau)] \\cdot (\\tau - \\omega^{n-1}) = (\\tau^n - 1) \\cdot Q_1(\\tau)$​\n​\t$b) \\space \\frac{P_A(\\tau) - P_B(\\tau)}{\\tau - \\omega^{n-1}} = Q_2(\\tau)$\nFinally, the verifier will check the opening of $P_B(1)$. This value is $z$, and they have verified that it is equal to the product of the entries in $A$​.\nImplementations # Rust Mathematica (Toy Example) Security Proof # Completeness # Soundness # Zero-Knowledge # "},{"id":2,"href":"/2024-Gadgets/docs/gadgets/zero/","title":"Zero","section":"Gadgets","content":" Zeroing Parts of an Array # Assuming an input array of size $n$: $\\langle \\mathsf{data}_0,\\mathsf{data}_1,\\ldots,\\mathsf{data}_{n-1}\\rangle$ and input array encoded into the polynomial. This uses \u0026ldquo;Encoding 2\u0026rdquo; from above (evaluation points) and uses \u0026ldquo;Roots of Unity + FFT\u0026rdquo; from above where $\\omega\\in\\mathbb{G}_\\kappa$ is a generator for the x-coordinates of the points.\n$\\bot$ is an arbitrary non-zero integer.\nOperation Input Array Input Polynomial Output Array Output Polynomial Zero all $\\langle 3,1,3,3,7 \\rangle$ $P(X)$ $\\langle 0,0,0,0,0 \\rangle$ $P(X)\\cdot(X^\\kappa-1)$ Zero first $\\langle 3,1,3,3,7 \\rangle$ $P(X)$ $\\langle 0,\\bot,\\bot,\\bot,\\bot \\rangle$ $P(X)\\cdot(X-\\omega^0)$ Zero last $\\langle 3,1,3,3,7 \\rangle$ $P(X)$ $\\langle \\bot,\\bot,\\bot,\\bot,0 \\rangle$ $P(X)\\cdot(X-\\omega^{\\kappa-1})$ Zero all but first $\\langle 3,1,3,3,7 \\rangle$ $P(X)$ $\\langle \\bot,0,0,0,0 \\rangle$ $P(X)\\cdot\\frac{(X^\\kappa-1)}{(X-\\omega^0)}$ Zero all but last $\\langle 3,1,3,3,7 \\rangle$ $P(X)$ $\\langle 0,0,0,0,\\bot \\rangle$ $P(X)\\cdot\\frac{(X^\\kappa-1)}{(X-\\omega^{\\kappa-1})}$ Why are these useful? Consider a range proof from Boneh, Fisch, Gabizon, and Williamson. At a certain point in the protocol, we reach the following:\nFirst note that $w$ and $\\omega$ look the same but are different: $w$ are three new polynomials we are creating, while $\\omega$ is the generator of $\\mathbb{G}_\\kappa$ we are using to pick x-coordinates for the polynomials.\nIn the first constraint, we want to prove the value of f(1) and g(1) are the same. So we subtract (g-f) which leaves a zero in the first element of the array but the rest of the array will contain other stuff. By applying \u0026ldquo;zero all but first\u0026rdquo;, we can zero out the rest of the array. We now have an array that is all zero (in polynomial form, this is called a vanshing polynomial and we can prove a polynomial vanishes easily and in a batch). In the second constraint, we prove the last element in g is a binary bit (0 or 1) and then we apply the \u0026ldquo;zero all but last\u0026rdquo; to make an array that is all zero (and vanishing polynomial). In the third constraint, the first two terms of the multiplication are proving $g(X)$ has a certain form that does not need to be understood here. What is important is that there is a relationship between each integer in the array ($g(X)$) and the integer right beside it ($g(X\\omega)$) in the array. When the relationship holds, the result is a zero in $w_3$. Unfortunately, there is a corner case: for the last element in the array, the \u0026ldquo;next\u0026rdquo; integer wraps back to the first integer and the relationship does not hold across this boundary. So we use \u0026ldquo;zero last\u0026rdquo; to manually zero out the last integer in the array, leaving us with a zero array (and vanishing polynomial). "}]
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-[{"id":0,"href":"/2024-Gadgets/docs/background/poly-iop/","title":"Poly Iop","section":"Background","content":" Notes on Polynomial Interactive Oracle Proofs (Draft) # Encoding Arrays of Data into Polynomials # Data starts as a array of integers where integers in each slot are between 0 and $q-1$, where $q$ is a large (generally 256 bit) prime number. Recall that we call this set of integers $\\mathbb{Z}_q$.\n$\\mathsf{data}_0$ $\\mathsf{data}_1$ $\\mathsf{data}_2$ $\\mathsf{data}_3$ $\\mathsf{data}_4$ It is common to denote a polynomial like $P(X)$ where $X$ is the variable of the polynomial. We are going denote the variable with an empty box $\\square$ which can be interpreted as a place where you can put any integer in $\\mathbb{Z}_q$ you want evaluated (or equivalent, where you place an x-coordinate to learn what the y-coordinate is).\nA polynomial in this notation looks like:\n$P(\\square)=c_0+c_1\\cdot\\square+c_2\\cdot\\square^2+c_3\\cdot\\square^3+c_4\\cdot\\square^4=\\sum_{i=0}^d c_i\\cdot\\square^i$ The values $c_i$ are called coefficients. Different arrays of data will (depending on how data is encoded, next) result in different coefficients and thus different polynomials. The degree of the polynomial is the largest exponent. So the polynomial above has degree 4 and thus will have 5 coefficients and 5 terms of the form $c_i\\cdot\\square^i$ (including $i=0$). Sometimes the coefficient will be zero: the term is thus not written down but in a list of coefficients, it will be included as a 0.\nThe main question to tackle is how to \u0026ldquo;encode\u0026rdquo; a array of integers into a polynomial. This is generally done one of three ways:\nCoefficients Evaluation Points Roots Each has its advantages and disadvantages, which we discuss next.\nFast forwarding a bit, once the polynomial is created, it is not shared directly with anyone. Instead a commitment to it is shared. The commitment does two things: (1) it makes it succinct: e.g., constant size regardless of how long the array is; and (2) it can hide the data in the array as necessary. We will discuss one specific polynomial commitment scheme called KZG. KZG needs the polynomial in the format of a list of its coefficients. If we have the polynomial in a different form, we will have to convert to coefficients. Thus this needs to be considered when weighing the pros/cons of the three encoding methods.\nEncoding 1: Coefficients # Create polynomial as: $P_1(\\square)=\\mathsf{data}_0+\\square+\\mathsf{data}_2\\cdot\\square^2+\\mathsf{data}_3\\cdot\\square^3+\\mathsf{data}_4\\cdot\\square^4=\\sum_{i=0}^d \\mathsf{data}_i\\cdot\\square^i$\nProperties:\nFast 👍: fastest path to commitment as the output is already in coefficient form. Addition 👍: two arrays can be added together (slot-by-slot) by simply adding the polynomials together Multiplication 👎: no support for multiplication of arrays (Remark: multiplying the polynomials does not multiply the coefficients. It results in a cross multiplication of every term in the first polynomial with every term in the second polynomial. Further the degree of the resulting polynomial will double that of the starting polynomials). Opening 🤷🏻: proving the value of the $i$th element in the array is $\\mathsf{data}_i$ doing polynomial math on $P_1(\\square)$ is not possible. However $\\Sigma$-protocols done directly on KZG may enable this kind of proof. In any case, this is not particularly well explored. Other useful properties 👍: the sum of all values in a array can be computed by evaluating the polynomial at $P_1(\\boxed{1})$! $\\sum_{i=0}^d \\mathsf{data}_i\\cdot\\boxed{1}^i=\\sum_{i=0}^d \\mathsf{data}_i$ . You can also show two arrays have the same sum (called a \u0026ldquo;sum check\u0026rdquo;) by subtracting them and showing $P(\\boxed{1})=0$. Other useful properties 👍: when all coefficients are 0, the polynomial will be the zero polynomial ($P_1(\\square)=0$). Coefficients can be entire polynomials, not just integers. A common optimization in Poly-IOP systems is taking a set of equations of polynomials that should equal 0, placing each into the coefficient of a super-polynomial, and showing the super-polynomial is the zero polynomial (which can be proven overwhelmingly by showing it is 0 at a randomly selected point). Encoding 2: Evaluation Points # Create a list of points $\\{x,y\\}$ for the data: $\\langle\\{0,\\mathsf{data}_0\\},\\{1,\\mathsf{data}_1\\},\\{2,\\mathsf{data}_2\\},\\{3,\\mathsf{data}_3\\},\\{4,\\mathsf{data}_4\\}\\rangle$ and interpolate a polynomial $P_2(\\square)$ through these points.\nProperties:\nSlow (or moderate) 👎: converting a set of points into a set of coefficients is called interpolation and is $O(n^2)$ time generally. A certain optimization allows $O(n\\log n)$ time by chosing $x$ coordinates with a mathematical relationship (more on this later). Addition 👍: two arrays can be added together (slot-by-slot) by simply adding the polynomials together Multiplication 👍: two arrays can be multiplied together (slot-by-slot) by simply multiplying the polynomials together Opening 👍: proving the value of the $i$th element in the array is $\\mathsf{data}_i$ is possible with polynomial math by showing $P_2(\\boxed{i})=\\mathsf{data}_i$ and KZG has a precise algorithm for this. Encoding 3: Roots # Create polynomial as: $P_3(\\square)=(\\square-\\mathsf{data}_0)(\\square-\\mathsf{data}_1)(\\square-\\mathsf{data}_2)(\\square-\\mathsf{data}_3)(\\square-\\mathsf{data}_4)$\nAlternatively, create a list of roots $\\{x,0\\}$ for the data: $\\langle\\{\\mathsf{data}_0,0\\},\\{\\mathsf{data}_1,0\\},\\{\\mathsf{data}_2,0\\},\\{\\mathsf{data}_3,0\\},\\{\\mathsf{data}_4,0\\}\\rangle$ and interpolate a polynomial $P_3(\\square)$ through these points.\nProperties:\nSlow 👎 (or moderate): multiplying out naively requires $O(n^2)$ time. Treating as a set of points and interpolating also requires $O(n^2)$ time (because the x-coordinates are the data, they cannot be chosen freely to optimize interpolation). Applying divide and conquer can provide $O(n \\log^2 n)$.^[1] Addition 👎: two arrays cannot be added from adding (or otherwise manipulating) the polynomials. Multiplication 👎: two arrays cannot be multiplied from multiplying (or otherwise manipulating) the polynomials (but you can do a \u0026ldquo;union\u0026rdquo; operation below). Opening 👍: proving that a value is in the array somewhere is easy and KZG as a precise algorithm for this (opening a root is the same as opening a point, where the y-coordinate is 0). However you cannot show a value is specifically the $i$th value in the array because the polynomial loses the order of the data in the array (see next property). Other useful properties 👍: the order of the data in the array does not matter. The same polynomial will be produced even if the order is changed. This is useful when the array represents a \u0026ldquo;bag\u0026rdquo; of unordered data. You can easily prove two \u0026ldquo;bags\u0026rdquo; of data are the same because the polynomials will be the same. One use-case of this is proving the output of a shuffle/permutation is the same data as the input (just in a different order). Other useful properties 👍: multiplying two polynomials results in a concatenation of the data in the arrays (or conjunction/union of the data in both bags). This might be useful in some protocols. ^[1]: Hat tip Pratyush Mishra.\nDecision Tree for Encoding # Basically we decide if we specifically need unordered \u0026ldquo;bags\u0026rdquo; of data. If so, encoding as roots is the only option. If not, we consider if we need to ever get the data back from the polynomial. Generally we do and encoding as evaluation points is the most common encoding technique. When do we encode the data and never want it back? Usually when (1) the coefficients are all supposed to be zero so we are just showing that property, or (2) we want back the sum of the data and not the data itself. In these cases, you can still work with evaluation point encoding but it will be faster to just do coefficient encoding.\nflowchart LR; A[Array to Polynomial] --\u003e B{Is the data unordered?}; B -- Unordered --\u003e C[Roots]; B -- Ordered or don't care --\u003e D{Open data from polynomial later?}; D -- Yes --\u003e E[Evaluation Points]; D -- No --\u003e F[Coefficients]; The short answer is to start with evaluation point encoding until you realize you need something different.\nOptimizing Interpolation (Roots of Unity + FFT) # Moving forward, we will assume we are using Encoding 2: Evaluation Points. In short, this means placing the elements of our array into the $y$-coordinates ($\\mathsf{data}_i=P(\\boxed{x_i})$) of points on the polynomial. Before commiting to $P(\\square)$, we need to use interpolation to find the coefficients of the polynomial that is fitted to these points. General interpolation algorithms are $O(n^2)$ work for $n$ evaluation points but this can be reduced to $O(n\\log n)$ with an optimization.\nThe optimization we will explore enables interpolation via the fast Fourier transform (FFT). It concerns how to choose the $x$-coordinates, which will serve as the index for accessing the data: evaluating $P(X)$ at $x_i$ will reveal $\\mathsf{data}_i$. First note, $x$-coordinates are from the exponent group ($Z_q$) and the choices exceed what is feasible to use ($2^{255}$ values in bls). Any subset can be used and interpolated. The optimization is to chose them with a mathematical structure. Specifically, instead an additive sequence (e.g., $0,1,2,3,\\ldots$), we use a multiplicative sequence $1,\\omega,\\omega\\cdot\\omega,\\omega\\cdot\\omega\\cdot\\omega,\\ldots$ or equivalently: $\\omega^0,\\omega^1,\\omega^2,\\ldots,\\omega^{\\kappa-1}$. Further, the sequence is closed under multiplication which means that next index after $\\omega^{\\kappa-1}$ wraps back to the first index: $\\omega^{k-1} \\cdot \\omega = \\omega^\\kappa = \\omega^0=1$ (this property is also useful in proving relationships between data in the array and its neighbouring values).\nFor terminology, we say $\\omega$ is a generator with multiplicative order $\\kappa$ in $\\mathbb{Z}_q$ (or $\\omega \\in \\mathbb{G}_\\kappa$). This implies $\\omega^\\kappa=1$. Rearranging, $\\omega=\\sqrt[\\kappa]{1}$. Thus we can equivalently describe $\\omega$ as a $\\kappa$-th root of 1. Finally, as 1 is the unity element in $Z_q$, $\\omega$ is commonly called a $\\kappa$-th root of unity.\nFor practical purposes, $\\kappa$ represents the length of the longest array of data we can use in our protocol. Where does $\\kappa$ come from? Different elements of $Z_q$ will have different multiplicative orders but every order must be a divisor of $q-1$. Thus $\\kappa$ is the largest divisor of the exact value of $q$ used in an elliptic curve standard. BLS12-384 has $\\kappa=2^{32}$ (for terminology, this called a $2$-adicity of $32$). In summary, we can only encode data arrays of length up to $2^{32}=4,294,967,296$.\nZeroing Parts of an Array # Assuming an input array of size $n$: $\\langle \\mathsf{data}_0,\\mathsf{data}_1,\\ldots,\\mathsf{data}_{n-1}\\rangle$ and input array encoded into the polynomial. This uses \u0026ldquo;Encoding 2\u0026rdquo; from above (evaluation points) and uses \u0026ldquo;Roots of Unity + FFT\u0026rdquo; from above where $\\omega\\in\\mathbb{G}_\\kappa$ is a generator for the x-coordinates of the points.\n$\\bot$ is an arbitrary non-zero integer.\nOperation Input Array Input Polynomial Output Array Output Polynomial Zero all $\\langle 3,1,3,3,7 \\rangle$ $P(X)$ $\\langle 0,0,0,0,0 \\rangle$ $P(X)\\cdot(X^n-1)$ Zero first $\\langle 3,1,3,3,7 \\rangle$ $P(X)$ $\\langle 0,\\bot,\\bot,\\bot,\\bot \\rangle$ $P(X)\\cdot(X-\\omega^0)$ Zero last $\\langle 3,1,3,3,7 \\rangle$ $P(X)$ $\\langle \\bot,\\bot,\\bot,\\bot,0 \\rangle$ $P(X)\\cdot(X-\\omega^{n-1})$ Zero all but first $\\langle 3,1,3,3,7 \\rangle$ $P(X)$ $\\langle \\bot,0,0,0,0 \\rangle$ $P(X)\\cdot\\frac{(X^n-1)}{(X-\\omega^0)}$ Zero all but last $\\langle 3,1,3,3,7 \\rangle$ $P(X)$ $\\langle 0,0,0,0,\\bot \\rangle$ $P(X)\\cdot\\frac{(X^n-1)}{(X-\\omega^{n-1})}$ Why are these useful? Consider a range proof from Boneh, Fisch, Gabizon, and Williamson. At a certain point in the protocol, we reach the following:\nFirst note that $w$ and $\\omega$ look the same but are different: $w$ are three new polynomials we are creating, while $\\omega$ is the generator of $\\mathbb{G}_\\kappa$ we are using to pick x-coordinates for the polynomials.\nIn the first constraint, we want to prove the value of f(1) and g(1) are the same. So we subtract (g-f) which leaves a zero in the first element of the array but the rest of the array will contain other stuff. By applying \u0026ldquo;zero all but first\u0026rdquo;, we can zero out the rest of the array. We now have an array that is all zero (in polynomial form, this is called a vanshing polynomial and we can prove a polynomial vanishes easily and in a batch). In the second constraint, we prove the last element in g is a binary bit (0 or 1) and then we apply the \u0026ldquo;zero all but last\u0026rdquo; to make an array that is all zero (and vanishing polynomial). In the third constraint, the first two terms of the multiplication are proving $g(X)$ has a certain form that does not need to be understood here. What is important is that there is a relationship between each integer in the array ($g(X)$) and the integer right beside it ($g(X\\omega)$) in the array. When the relationship holds, the result is a zero in $w_3$. Unfortunately, there is a corner case: for the last element in the array, the \u0026ldquo;next\u0026rdquo; integer wraps back to the first integer and the relationship does not hold across this boundary. So we use \u0026ldquo;zero last\u0026rdquo; to manually zero out the last integer in the array, leaving us with a zero array (and vanishing polynomial). KZG Commitments # KZG is a commitment scheme, designed specifically for polynomials. While any commitment scheme can be used to commit to a polynomial (e.g., hash the polynomial) and later reveal it, KZG has homomorphic properties that are useful for polynomials. They are as follows:\nAddition. $\\mathsf{Comm}$ Multiplication Check. $P_1(z)\\cdot P_2(z) - P_3(z)=0$ Evaluation Check. g^P(X) -\u0026gt; g^(c_0+c_1X+c_2X^2)\nEvaluate at tau to commit\nTrusted setup (Powers of Tau) -\u0026gt; $g,g^{\\tau}, g^{\\tau^2}$\nArray: [4, 6, 7, 2, 346, 24\u0026hellip;]\nArray -\u0026gt; Polynomial -\u0026gt; Polynomial Commitment (Poly-IOP; polynomial interactive oracle proof)\nArray -\u0026gt; Polynomial -\u0026gt; How?\nEvaluation points -\u0026gt; [{1,4},{2,6},{3,7}\u0026hellip;] -\u0026gt; Lagrange / FFT interpolation Roots -\u0026gt; [{4,0},{6,0},{7,0},..] -\u0026gt; Interpolation is sufficient, not necessary -\u0026gt; Expansion -\u0026gt; (X-4)(X-6)(X-7) Coefficients -\u0026gt; 4+6X+7X^2 -\u0026gt; Nothing KZG: $g^{c_0+c_1*\\tau+c_2*\\tau^2+\\ldots}$\nComm(P1)*Comm(P2) = Comm (P1+P2)\nComm(P1)^P2=Comm(P1*P2)\nComm(P1)^a=Comm(a*P1)\nOpen Root (z=P1(0))\nOpen a root at 7 Q=(X-4)(X-6)(X-7) / (X-7) -\u0026gt; evenly divide iff 7 is a root Q(X-7) = P -\u0026gt; check with pairing Open point (z=P1(x))\nSubtract off z and now x is a root, see above Batch opening many points:\nOpen many points on same polynomial P Open the same point on many polynomials Open many points on many polynomials -\u0026gt; not sure? $P_1(X)=c_0+c_1*X+\\ldots$\n$P_3(\\square)=(\\square-\\mathsf{data}_0)(\\square-\\mathsf{data}_1)$\n$P_3(\\square)=((\\square-\\mathsf{data}_2)(\\square-\\mathsf{data}_3)(\\square-\\mathsf{data}_4)$\n$P_3(\\square)=(\\square-\\mathsf{data}_0)(\\square-\\mathsf{data}_1)(\\square-\\mathsf{data}_2)(\\square-\\mathsf{data}_3)(\\square-\\mathsf{data}_4)$\n"},{"id":1,"href":"/2024-Gadgets/docs/gadgets/mult2/","title":"Mult2","section":"Gadgets","content":" Multiplication (within polynomial) # Array-level # Given array $\\mathsf{Arr}_1$ of length $\\ell$ and integer $\\mathsf{Prod}_1$, demonstrate that $\\mathsf{Prod}=\\prod_{i=0}^{\\ell-1}\\mathsf{Arr}_1[i]$.\nBuild accumulator array $\\mathsf{Acc}_1$ from $\\mathsf{Arr}_1$ as follows: $\\mathsf{Acc}_1[\\ell-1]=\\mathsf{Arr}_1[\\ell-1]$ $\\mathsf{Acc}_1[i]=\\mathsf{Arr}_1[i]\\cdot \\mathsf{Acc}_1[i+1]$​ If built correctly, $\\mathsf{Acc}_1[0]=\\mathsf{Prod}_1$ Example:\nChecked input: $\\mathsf{Prod}_1=66$ Unchecked input: $\\mathsf{Arr}_1=\\langle 84,67,11,92,36,67 \\rangle$ Execution trace: $\\mathsf{Acc}_1= \\langle \\bot,\\bot,\\bot,\\bot,\\bot,67 \\rangle$​ $\\mathsf{Acc}_1= \\langle \\bot,\\bot,\\bot,\\bot,(36\\cdot 67),67 \\rangle$ $\\mathsf{Acc}_1= \\langle \\bot,\\bot,\\bot,\\bot,84,67 \\rangle$​ $\\mathsf{Acc}_1= \\langle \\bot,\\bot,\\bot,(92\\cdot 84),84,67 \\rangle$ Polynomial-level # Commitment-level # "},{"id":2,"href":"/2024-Gadgets/docs/gadgets/productgadget/","title":"Product Gadget","section":"Gadgets","content":"Array Level\nThe prover has a list of value $A = [a_0, a_1, a_2, \\dots, a_{n-1}]$ and wishes to commit to and prove that product of all entries in $A$ equals some value $z$. In other words, that $\\prod_{i = 0}^{n-1} a_i= z$. First, the prover will compute an accumulator, $B$ according to the following rules: $i) B_i = A_i \\cdot B_{i+1} \\space 0 \\leq i \\lt n-1$ and $ii)B_{n-1} = A_{n-1}$. Then $B_0 = z$. To prove that $\\prod_{i = 0}^{n-1} a_i= z$, it suffices for the prover to demonstrate that $B_0 = z$ and that $i)$ and $ii)$ hold, and that $B_0 = z$.\nPolynomial Level\nThe prover interpolates $A$ and $B$ to get $P_A(x)$ and $P_B(x)$. To do so, an $n$th root of unity (which we will call $\\omega$) is used, and the values of $A$ and $B$ are paired with the powers of this root of unity as follows: $(\\omega^0, a_0), (\\omega^1, a_1), \\dots, (\\omega^{n-1}, a_{n-1})$ and similarly for $B$. Then the interpolation (using the fast fourier transform (FFT)) creates a polynomial that passes through the points represented by these tuples. We also define $H$ as the set $[\\omega^0, \\omega^1, \\dots, \\omega^{n-1}]$. We are assuming that $n$ is a power of 2, so that we can use FFT. In practice, if it is not, then a power of 2 greater than $n$, say $2^m$, is used and the last $2^m - n$ entries of $A$ are padded with 1s.\nThe prover then compute $W_1(x) = [P_B(x) - P_B(x\\omega) \\cdot P_a(x))] \\cdot (x - \\omega^{n-1})$ and $W_2(x) = [P_A(x) - P_B(x)] \\cdot \\frac{x^n - 1} {x - w^{n-1}}$. If $W_1$ is vanishing on $H$, then $i)$ holds for $0 \\leq i \\lt n$. We note that the $(x - \\omega^{n-1})$ is included in the equation to add a root at $x = n -1$, since $i)$ does not hold for $i = n -1 $ (unless $z = 1$, in which case the ($x - \\omega^{n-1}$) is unnecessary). Similarly, If $W_2$ is vanishing on $H$, then $ii)$ holds. We note that multiplying [$P_A(x) - P_B(x)]$ by $\\frac{x^n - 1} {x - w^{n-1}}$ zeroes $W_2$ on all of $H$ except $\\omega^{n-1}$, thus $W_2$ vanishing on $H$ is testing only the equality of $A_{n-1}$ and $B_{n-1}$​.\nThe prover then computes $Q_1(x) = \\frac{W_1(x)}{x^n - 1}$ and $Q_2(x) = \\frac{W_2(x)}{x^n - 1}$. Since $x^n - 1$ is the vanishing polynomial for $H$, if $Q_1$ and $Q_2$ are polynomials (and not rational functions), then $W_1$ and $W_2$ must be vanishing on $H$.\nThe prover then commits to $P_A$, $P_B$, $Q_1$, and $Q_2$. They then hash the four commitments to get a random challenge, $\\tau$. They open $P_A(\\tau), P_B(\\tau), Q_1(\\tau), Q_2(\\tau),$ and $P_B(\\tau \\cdot \\omega)$ to demonstrate that $Q_1$ and $Q_2$ are polynomials, as defined above $-$ thus $i)$ and $ii)$ hold. They also open $P_B(1)$, showing that $B_0 = z$. Thus they have demontrated that $\\prod_{i = 0}^{n-1} a_i= z$.\nVerifier Level\nThe verifier first checks that $\\tau$​ is correct by hashing the four commitments which have been sent to them by the prover.\nNext, the verifier checks that the evaluations $P_A(\\tau), P_B(\\tau), Q_1(\\tau), Q_2(\\tau),$ and $P_B(\\tau \\cdot \\omega)$​ sent to them by the prover are correct. This can be done (assuming we are working with KZG) as a single batch opening.\nThe verifier will then check the following two equalities, so make sure that $Q_1$ and $Q_2$​ were defined as they should be:\n​\t$a) \\space [P_B(\\tau) - P_B(\\tau \\cdot \\omega)\\cdot P_A(\\tau)] \\cdot (\\tau - \\omega^{n-1}) = (\\tau^n - 1) \\cdot Q_1(\\tau)$​\n​\t$b) \\space \\frac{P_A(\\tau) - P_B(\\tau)}{\\tau - \\omega^{n-1}} = Q_2(\\tau)$\nFinally, the verifier will check the opening of $P_B(1)$. This value is $z$, and they have verified that it is equal to the product of the entries in $A$.\n"}]
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diff --git a/en.search.min.bb5d9164f3309a75ef9097bce777d0ab67b4a8025ae95f3105b06cec629ec033.js b/en.search.min.93281b7c13f5b9b0a9578b33cdbdde95d267573c1f5b04dd2cb337c07f21791f.js
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rename from en.search.min.bb5d9164f3309a75ef9097bce777d0ab67b4a8025ae95f3105b06cec629ec033.js
rename to en.search.min.93281b7c13f5b9b0a9578b33cdbdde95d267573c1f5b04dd2cb337c07f21791f.js
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diff --git a/index.html b/index.html
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 <!doctype html><html lang=en-us dir=ltr><head><meta name=generator content="Hugo 0.126.1"><meta charset=UTF-8><meta name=viewport content="width=device-width,initial-scale=1"><meta name=description content="Acerbo datus maxime # Astris ipse furtiva # Est in vagis et Pittheus tu arge accipiter regia iram vocatur nurus. Omnes ut olivae sensit arma sorori deducit, inesset crudus, ego vetuere aliis, modo arsit? Utinam rapta fiducia valuere litora adicit cursu, ad facies
 Suis quot vota # Ea furtique risere fratres edidit terrae magis. Colla tam mihi tenebat: miseram excita suadent es pecudes iam. Concilio quam velatus posset ait quod nunc!"><meta name=theme-color media="(prefers-color-scheme: light)" content="#ffffff"><meta name=theme-color media="(prefers-color-scheme: dark)" content="#343a40"><meta name=color-scheme content="light dark"><meta property="og:url" content="https://madibagroup.github.io/2024-Gadgets/"><meta property="og:site_name" content="Handbook"><meta property="og:title" content="Introduction"><meta property="og:description" content="Acerbo datus maxime # Astris ipse furtiva # Est in vagis et Pittheus tu arge accipiter regia iram vocatur nurus. Omnes ut olivae sensit arma sorori deducit, inesset crudus, ego vetuere aliis, modo arsit? Utinam rapta fiducia valuere litora adicit cursu, ad facies
 Suis quot vota # Ea furtique risere fratres edidit terrae magis. Colla tam mihi tenebat: miseram excita suadent es pecudes iam. Concilio quam velatus posset ait quod nunc!"><meta property="og:locale" content="en_us"><meta property="og:type" content="website"><title>Introduction | Handbook</title>
-<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.bb5d9164f3309a75ef9097bce777d0ab67b4a8025ae95f3105b06cec629ec033.js integrity="sha256-u12RZPMwmnXvkJe853fQq2e0qAJa6V8xBbBs7GKewDM=" crossorigin=anonymous></script><link rel=alternate type=application/rss+xml href=https://madibagroup.github.io/2024-Gadgets/index.xml title=Handbook><script id=MathJax-script async src=https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js></script><script>MathJax={tex:{displayMath:[["\\[","\\]"],["$$","$$"]],inlineMath:[["\\(","\\)"],["$","$"]]}}</script></head><body dir=ltr><input type=checkbox class="hidden toggle" id=menu-control>
-<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><a href=/2024-Gadgets/docs/background/><strong>Background</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/>Poly IOP</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/><strong>Gadgets</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/>Mult2</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/productgadget/>ProdGadget</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
+<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.93281b7c13f5b9b0a9578b33cdbdde95d267573c1f5b04dd2cb337c07f21791f.js integrity="sha256-kygbfBP1ubCpV4szzb3eldJnVzwfWwTdLLM3wH8heR8=" crossorigin=anonymous></script><link rel=alternate type=application/rss+xml href=https://madibagroup.github.io/2024-Gadgets/index.xml title=Handbook><script id=MathJax-script async src=https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js></script><script>MathJax={tex:{displayMath:[["\\[","\\]"],["$$","$$"]],inlineMath:[["\\(","\\)"],["$","$"]]}}</script></head><body dir=ltr><input type=checkbox class="hidden toggle" id=menu-control>
+<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><strong>Background</strong></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/>Poly IOP</a></p></li><li><p><strong>Gadgets</strong></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/>Mult2</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
 </label><strong>Introduction</strong>
 <label for=toc-control><img src=/2024-Gadgets/svg/toc.svg class=book-icon alt="Table of Contents"></label></div><aside class="hidden clearfix"><nav id=TableOfContents><ul><li><a href=#tempora-nisi-nunc>Tempora nisi nunc</a></li><li><a href=#locis-suis-novi-cum-suoque-decidit-eadem>Locis suis novi cum suoque decidit eadem</a></li></ul></nav></aside></header><article class="markdown book-article"><h1 id=acerbo-datus-maxime>Acerbo datus maxime
 <a class=anchor href=#acerbo-datus-maxime>#</a></h1><div class="book-columns flex flex-wrap"><div class="flex-even markdown-inner"><h2 id=astris-ipse-furtiva>Astris ipse furtiva
diff --git a/index.xml b/index.xml
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@@ -1,5 +1,4 @@
 <?xml version="1.0" encoding="utf-8" standalone="yes"?><rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom"><channel><title>Introduction on Handbook</title><link>https://madibagroup.github.io/2024-Gadgets/</link><description>Recent content in Introduction on Handbook</description><generator>Hugo</generator><language>en-us</language><atom:link href="https://madibagroup.github.io/2024-Gadgets/index.xml" rel="self" type="application/rss+xml"/><item><title/><link>https://madibagroup.github.io/2024-Gadgets/docs/background/poly-iop/</link><pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate><guid>https://madibagroup.github.io/2024-Gadgets/docs/background/poly-iop/</guid><description>Notes on Polynomial Interactive Oracle Proofs (Draft) # Encoding Arrays of Data into Polynomials # Data starts as a array of integers where integers in each slot are between 0 and $q-1$, where $q$ is a large (generally 256 bit) prime number. Recall that we call this set of integers $\mathbb{Z}_q$.
-$\mathsf{data}_0$ $\mathsf{data}_1$ $\mathsf{data}_2$ $\mathsf{data}_3$ $\mathsf{data}_4$ It is common to denote a polynomial like $P(X)$ where $X$ is the variable of the polynomial.</description></item><item><title/><link>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/mult2/</link><pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate><guid>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/mult2/</guid><description> Multiplication (within polynomial) # Array-level # Given array $\mathsf{Arr}_1$ of length $\ell$ and integer $\mathsf{Prod}_1$, demonstrate that $\mathsf{Prod}=\prod_{i=0}^{\ell-1}\mathsf{Arr}_1[i]$.
-Build accumulator array $\mathsf{Acc}_1$ from $\mathsf{Arr}_1$ as follows: $\mathsf{Acc}_1[\ell-1]=\mathsf{Arr}_1[\ell-1]$ $\mathsf{Acc}_1[i]=\mathsf{Arr}_1[i]\cdot \mathsf{Acc}_1[i+1]$​ If built correctly, $\mathsf{Acc}_1[0]=\mathsf{Prod}_1$ Example:
-Checked input: $\mathsf{Prod}_1=66$ Unchecked input: $\mathsf{Arr}_1=\langle 84,67,11,92,36,67 \rangle$ Execution trace: $\mathsf{Acc}_1= \langle \bot,\bot,\bot,\bot,\bot,67 \rangle$​ $\mathsf{Acc}_1= \langle \bot,\bot,\bot,\bot,(36\cdot 67),67 \rangle$ $\mathsf{Acc}_1= \langle \bot,\bot,\bot,\bot,84,67 \rangle$​ $\mathsf{Acc}_1= \langle \bot,\bot,\bot,(92\cdot 84),84,67 \rangle$ Polynomial-level # Commitment-level #</description></item><item><title/><link>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/productgadget/</link><pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate><guid>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/productgadget/</guid><description>Array Level
-The prover has a list of value $A = [a_0, a_1, a_2, \dots, a_{n-1}]$ and wishes to commit to and prove that product of all entries in $A$ equals some value $z$. In other words, that $\prod_{i = 0}^{n-1} a_i= z$. First, the prover will compute an accumulator, $B$ according to the following rules: $i) B_i = A_i \cdot B_{i+1} \space 0 \leq i \lt n-1$ and $ii)B_{n-1} = A_{n-1}$.</description></item></channel></rss>
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+$\mathsf{data}_0$ $\mathsf{data}_1$ $\mathsf{data}_2$ $\mathsf{data}_3$ $\mathsf{data}_4$ It is common to denote a polynomial like $P(X)$ where $X$ is the variable of the polynomial.</description></item><item><title/><link>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/mult2/</link><pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate><guid>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/mult2/</guid><description>Multiplication (Type 2) # Recap of types # Type Description Recap This Type 1 $\mathsf{Arr}_3=\mathsf{Arr}_1 \cdot \mathsf{Arr}_2$ $\mathsf{Arr}_3$ is the element-wise multiplication of $\mathsf{Arr}_1$ and $\mathsf{Arr}_2$. Type 2 $\mathsf{Prod}_\mathsf{Arr}=\prod_{i = 0}^{n-1} \mathsf{Arr}[i]$ $\mathsf{Prod}_\mathsf{Arr}$ is the disclosed product of all the elements in $\mathsf{Arr}$. ✅ Type 3 $\prod_{i = 0}^{n-1} \mathsf{Arr}_1[i]=\prod_{i = 0}^{n-1} \mathsf{Arr}_2[i]$ $\mathsf{Arr}_1$ and $\mathsf{Arr}_2$ have the same undisclosed product. Relation # $ \mathcal{R}_{\mathtt{mult2}} := \left\{ \begin{array}{l} (K_\mathsf{Arr},\mathsf{Prod}_\mathsf{Arr}) \end{array} \middle | \begin{array}{l} \mathsf{Arr} = [a_0, a_1, a_2, \dots, a_{n-1}], \\ \mathsf{Prod}_\mathsf{Arr} = \prod_{i = 0}^{n-1} a_i, \\ \mathsf{Poly}_\mathsf{Arr}=\mathsf{FFT.</description></item><item><title/><link>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/zero/</link><pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate><guid>https://madibagroup.github.io/2024-Gadgets/docs/gadgets/zero/</guid><description>Zeroing Parts of an Array # Assuming an input array of size $n$: $\langle \mathsf{data}_0,\mathsf{data}_1,\ldots,\mathsf{data}_{n-1}\rangle$ and input array encoded into the polynomial. This uses &amp;ldquo;Encoding 2&amp;rdquo; from above (evaluation points) and uses &amp;ldquo;Roots of Unity + FFT&amp;rdquo; from above where $\omega\in\mathbb{G}_\kappa$ is a generator for the x-coordinates of the points.
+$\bot$ is an arbitrary non-zero integer.
+Operation Input Array Input Polynomial Output Array Output Polynomial Zero all $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle 0,0,0,0,0 \rangle$ $P(X)\cdot(X^\kappa-1)$ Zero first $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle 0,\bot,\bot,\bot,\bot \rangle$ $P(X)\cdot(X-\omega^0)$ Zero last $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle \bot,\bot,\bot,\bot,0 \rangle$ $P(X)\cdot(X-\omega^{\kappa-1})$ Zero all but first $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle \bot,0,0,0,0 \rangle$ $P(X)\cdot\frac{(X^\kappa-1)}{(X-\omega^0)}$ Zero all but last $\langle 3,1,3,3,7 \rangle$ $P(X)$ $\langle 0,0,0,0,\bot \rangle$ $P(X)\cdot\frac{(X^\kappa-1)}{(X-\omega^{\kappa-1})}$ Why are these useful?</description></item></channel></rss>
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diff --git a/sitemap.xml b/sitemap.xml
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--- a/sitemap.xml
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 <!doctype html><html lang=en-us dir=ltr><head><meta charset=UTF-8><meta name=viewport content="width=device-width,initial-scale=1"><meta name=description content><meta name=theme-color media="(prefers-color-scheme: light)" content="#ffffff"><meta name=theme-color media="(prefers-color-scheme: dark)" content="#343a40"><meta name=color-scheme content="light dark"><meta property="og:url" content="https://madibagroup.github.io/2024-Gadgets/tags/"><meta property="og:site_name" content="Handbook"><meta property="og:title" content="Tags"><meta property="og:locale" content="en_us"><meta property="og:type" content="website"><title>Tags | Handbook</title>
-<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/tags/><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.bb5d9164f3309a75ef9097bce777d0ab67b4a8025ae95f3105b06cec629ec033.js integrity="sha256-u12RZPMwmnXvkJe853fQq2e0qAJa6V8xBbBs7GKewDM=" crossorigin=anonymous></script><link rel=alternate type=application/rss+xml href=https://madibagroup.github.io/2024-Gadgets/tags/index.xml title=Handbook><script id=MathJax-script async src=https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js></script><script>MathJax={tex:{displayMath:[["\\[","\\]"],["$$","$$"]],inlineMath:[["\\(","\\)"],["$","$"]]}}</script></head><body dir=ltr><input type=checkbox class="hidden toggle" id=menu-control>
-<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><a href=/2024-Gadgets/docs/background/><strong>Background</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/>Poly IOP</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/><strong>Gadgets</strong></a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/>Mult2</a></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/productgadget/>ProdGadget</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
+<link rel=manifest href=/2024-Gadgets/manifest.json><link rel=icon href=/2024-Gadgets/favicon.png><link rel=canonical href=https://madibagroup.github.io/2024-Gadgets/tags/><link rel=stylesheet href=/2024-Gadgets/book.min.309b7ed028807cdb68d8d61e26d609f48369c098dbf5e4d8c0dcf4cdf49feafc.css integrity="sha256-MJt+0CiAfNto2NYeJtYJ9INpwJjb9eTYwNz0zfSf6vw=" crossorigin=anonymous><script defer src=/2024-Gadgets/fuse.min.js></script><script defer src=/2024-Gadgets/en.search.min.93281b7c13f5b9b0a9578b33cdbdde95d267573c1f5b04dd2cb337c07f21791f.js integrity="sha256-kygbfBP1ubCpV4szzb3eldJnVzwfWwTdLLM3wH8heR8=" crossorigin=anonymous></script><link rel=alternate type=application/rss+xml href=https://madibagroup.github.io/2024-Gadgets/tags/index.xml title=Handbook><script id=MathJax-script async src=https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js></script><script>MathJax={tex:{displayMath:[["\\[","\\]"],["$$","$$"]],inlineMath:[["\\(","\\)"],["$","$"]]}}</script></head><body dir=ltr><input type=checkbox class="hidden toggle" id=menu-control>
+<input type=checkbox class="hidden toggle" id=toc-control><main class="container flex"><aside class=book-menu><div class=book-menu-content><nav><h2 class=book-brand><a class="flex align-center" href=/2024-Gadgets/><span>Handbook</span></a></h2><div class="book-search hidden"><input type=text id=book-search-input placeholder=Search aria-label=Search maxlength=64 data-hotkeys=s/><div class="book-search-spinner hidden"></div><ul id=book-search-results></ul></div><script>document.querySelector(".book-search").classList.remove("hidden")</script><ul><li><p><strong>Background</strong></p></li><li><p><a href=/2024-Gadgets/docs/background/poly-iop/>Poly IOP</a></p></li><li><p><strong>Gadgets</strong></p></li><li><p><a href=/2024-Gadgets/docs/gadgets/mult2/>Mult2</a></p></li></ul></nav><script>(function(){var e=document.querySelector("aside .book-menu-content");addEventListener("beforeunload",function(){localStorage.setItem("menu.scrollTop",e.scrollTop)}),e.scrollTop=localStorage.getItem("menu.scrollTop")})()</script></div></aside><div class=book-page><header class=book-header><div class="flex align-center justify-between"><label for=menu-control><img src=/2024-Gadgets/svg/menu.svg class=book-icon alt=Menu>
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