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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/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>
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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/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.40dd2bd45cbc90e37cacf108fa39df9b3f18212420316d3ce5ac26b720d12005.js integrity="sha256-QN0r1Fy8kON8rPEI+jnfmz8YISQgMW085awmtyDRIAU=" 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/zero1/>zero1</a></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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diff --git a/docs/background/index.html b/docs/background/index.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="Select a background topic from the menu."><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:description" content="Select a background topic from the menu."><meta property="og:locale" content="en_us"><meta property="og:type" content="website"><title>Background | Handbook</title>
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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/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.40dd2bd45cbc90e37cacf108fa39df9b3f18212420316d3ce5ac26b720d12005.js integrity="sha256-QN0r1Fy8kON8rPEI+jnfmz8YISQgMW085awmtyDRIAU=" 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/zero1/>zero1</a></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"><p>Select a background topic from the menu.</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>
diff --git a/docs/background/poly-iop/index.html b/docs/background/poly-iop/index.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="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.d6a9c4a61d3bef64a3412749c21372c745bd218a35a963628a2f75095d0f4b78.js integrity="sha256-1qnEph0772SjQSdJwhNyx0W9IYo1qWNiii91CV0PS3g=" 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>
+<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.40dd2bd45cbc90e37cacf108fa39df9b3f18212420316d3ce5ac26b720d12005.js integrity="sha256-QN0r1Fy8kON8rPEI+jnfmz8YISQgMW085awmtyDRIAU=" 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/zero1/>zero1</a></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><li><a href=#additional-text-to-merge>Additional Text to Merge</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)