Gen html, small md-LaTeX fixes

1 year ago · e766aa9016
97 changed files with 46759 additions and 150 deletions
--- a/blogo-input/posts/kzg-batch-proof.md
+++ b/blogo-input/posts/kzg-batch-proof.md
@ -20,6 +20,7 @@ where \space\space\space l_j(x) = \prod\_{0\leq m \leq k} \frac{x-x_m}{x_j - x_m
 $$

 And the $x-z$, which was to ensure that $q(x)$ had a root at $z$, now, as we want to ensure that $q(x)$ has roots at all the points of the commitment, we will use the *zero polynomial*:

 $$
 Z(x) = \prod_{i=0}^{k} x-z_i =\newline
 =(x-z_0)(x-z_1)...(x-z_k)
@ -30,16 +31,19 @@ This polynomial ensures that when $x=z_i$ ($z_i$ being one of our points), the p
 Now we can put $I(x)$ and $Z(x)$ in place, obtaining $q(x)=\frac{p(x)-I(x)}{Z(x)}$. And the batch proof evaluation is obtained by $\pi=[q(\tau)]_1$.

 The verification is quite similar than what we did for single proofs, but using the mentioned $z(x)$ and $I(x)$:

 $$
 \hat{e}(\pi, [Z(\tau)]_2) == \hat{e}(c - [I(\tau)]_1, H)
 $$

 Which, as we did with the single proofs in the previous post, we can unroll it and see that:

 $$
 \hat{e}(\pi, [Z(\tau)]_2) == \hat{e}(c - [I(\tau)]_1, H)\newline
 \Rightarrow \hat{e}([q(\tau)]_1, [Z(\tau)]_2) == \hat{e}([p(\tau)]_1 - [I(\tau)]_1, H)\newline
 \Rightarrow [q(\tau) \cdot Z(\tau)]_T == [p(\tau) - I(\tau)]_T
 $$

 From where we see that is the equation $q(x)\cdot Z(x)=p(x)-I(x)$, which can be expressed as $q(x) = \frac{p(x) - I(x)}{Z(x)}$, evaluated at $\tau$ from the trusted setup, which is not known: $q(\tau) = \frac{p(\tau) - I(\tau)}{Z(\tau)}$.

 #### Vector commitments
--- a/blogo-input/posts/kzg-commitments.md
+++ b/blogo-input/posts/kzg-commitments.md
@ -32,6 +32,7 @@ $$
 $$

 Which in additive representation is:

 $$
 (G, \tau G, \tau^2 G, ..., \tau^{n-1} G) \in \mathbb{G}_1\newline
 (H, \tau H, \tau^2 H, ..., \tau^{n-1} H) \in \mathbb{G}_2
--- a/blogo-input/posts/shamir-secret-sharing.md
+++ b/blogo-input/posts/shamir-secret-sharing.md
@ -53,6 +53,7 @@ We will work over a finite field of size $p$, where $p$ is a prime number. For o
 Let our secret be $s=14$. We now generate our polynomial of degree $n-1=2$, where $s$ will be the constant coefficient: $p(x)= s + \alpha_1 x^1 + \alpha_2 x^2$. We can set $\alpha_1$ and $\alpha_2$ into any random value, as example $\alpha_1=4$ and $\alpha_2=6$. So we have our polynomial: $p(x) = 14 + 4 x + 6 x^2$.

 Now that we have the polynomial, we can pick $k$ points from it, using incremental indexes for the $x$ coordinate: $P_1=(1, p(1)), P_2=(2, p(2)), \space\ldots\space, P_k=(k, p(k))$. With the numbers of our example this is (remember, we work over $\mathbb{F}\_{19}$):

 $$
 p(x) = 14 + 4 x + 6 x^2,\newline
 p(1)=14 + 4 \cdot 1 + 6 \cdot 1^2 = 24 \space (mod \space 19) = 5\newline
@ -61,6 +62,7 @@ p(3)=14 + 4 \cdot 3 + 6 \cdot 3^2 = 80 \space (mod \space 19) = 4\newline
 p(4)=14 + 4 \cdot 4 + 6 \cdot 4^2 = 126 \space (mod \space 19) = 12\newline
 p(5)=14 + 4 \cdot 5 + 6 \cdot 5^2 = 184 \space (mod \space 19) = 13
 $$

 So our $k$ points are: $(1,5), (2,8), (3,4), (4,12), (5,13)$. We can distribute these points as our 'secret parts'.
 In order to recover the secret, we need at least $n=3$ points, for example $P_1$, $P_3$, $P_5$, and we compute the *Lagrange polynomial interpolation* to recover the original polynomial (remember, we work over $\mathbb{F}\_{19}$):

@ -68,11 +70,13 @@ $$
 I(x) = \sum_{i=0}^n y_i l_i(x) \space\space
 where \space\space\space l_i(x) = \prod\_{0 \leq j \leq n \\ j\neq i} \frac{x-x_j}{x_i - x_j}
 $$

 $$
 l_1(x) = \frac{x-3}{1-3} \cdot \frac{x-5}{1-5} = \frac{x-3}{17} \cdot \frac{x-5}{15}=\frac{x^2+11x+15}{8}\newline
 l_3(x) = \frac{x-1}{3-1} \cdot \frac{x-5}{3-5} = \frac{x-1}{2} \cdot \frac{x-5}{17} =\frac{x^2+13x+5}{15}\newline
 l_5(x) = \frac{x-1}{5-1} \cdot \frac{x-3}{5-3} = \frac{x-1}{4} \cdot \frac{x-3}{2} = \frac{x^2 + 15x + 3}{8}\newline
 $$

 $$
 I(x) = y_2 \cdot l_2(x) + y_4 \cdot l_4(x) + y_5 \cdot l_5(x)\newline
 = 5 \cdot (\frac{x^2+11x+15}{8}) + 4 \cdot (\frac{x^2+13x+5}{15}) + 13 \cdot (\frac{x^2 +15x + 3}{8})\newline
--- a/public/blind-signatures-ec.html
+++ b/public/blind-signatures-ec.html
@ -21,7 +21,7 @@

    <meta name="viewport" content="width=device-width, initial-scale=1">

    <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.0.2/dist/css/bootstrap.min.css" rel="stylesheet" integrity="sha384-EVSTQN3/azprG1Anm3QDgpJLIm9Nao0Yz1ztcQTwFspd3yD65VohhpuuCOmLASjC" crossorigin="anonymous">
    <link href="css/bootstrap.min.css" rel="stylesheet">
    <link rel="stylesheet" href="css/style.css">

    <!-- highlightjs -->
@ -30,7 +30,7 @@
    <script src="js/highlightjs/highlight.pack.js"></script>

    <!-- katex -->
    <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.11/dist/katex.min.css" integrity="sha384-Um5gpz1odJg5Z4HAmzPtgZKdTBHZdw8S29IecapCSB31ligYPhHQZMIlWLYQGVoc" crossorigin="anonymous">
    <link rel="stylesheet" href="js/katex/katex.min.css">
  </head>

  <body>
@ -82,18 +82,18 @@ The main idea is that one party has a message and blinds it, then sends the blin

 <p>In this notes, we will cover the scheme proposed at <em>&ldquo;<a href="https://sci-hub.do/10.1109/ICCKE.2013.6682844">New Blind Signature Schemes Based on the (Elliptic Curve) Discrete Logarithm Problem</a>&rdquo;</em> paper by Hamid Mala &amp; Nafiseh Nezhadansari (thanks to <a href="https://twitter.com/feministPLT">Daira Hopwood</a> who mentioned this paper in a Telegram group).</p>

 <p>First of all, the <em>signer</em> generates their key pair by generating a random scalar $d \in \mathbb{Z}_n$ (where $\mathbb{Z}_n$ is the elliptic curve field), which will be the <em>private key</em>. From $d$ they can compute the <em>public key</em> by $Q = dG$, where $G$ is the generator point of $\mathbb{G}$ (the elliptic curve group).</p>
 <p>First of all, the <em>signer</em> generates their key pair by generating a random scalar <span class="math inline">\(d \in \mathbb{Z}_n\)</span> (where <span class="math inline">\(\mathbb{Z}_n\)</span> is the elliptic curve field), which will be the <em>private key</em>. From <span class="math inline">\(d\)</span> they can compute the <em>public key</em> by <span class="math inline">\(Q = dG\)</span>, where <span class="math inline">\(G\)</span> is the generator point of <span class="math inline">\(\mathbb{G}\)</span> (the elliptic curve group).</p>

 <p>Appart from their key pair, the <em>signer</em> will generate for each request of signature another random value $k \in \mathbb{Z}_n$, and its respective $R&rsquo;=kG$.</p>
 <p>Appart from their key pair, the <em>signer</em> will generate for each request of signature another random value <span class="math inline">\(k \in \mathbb{Z}_n\)</span>, and its respective <span class="math inline">\(R'=kG\)</span>.</p>

 <p>The <em>user</em> has a message <em>m</em> that which they want to get signed by the <em>signer</em> (without the <em>signer</em> knowing the content of <em>m</em>). In order to achieve that, the user will generate a coupe of random values $a, b \in \mathbb{Z}_n$, and from these parameters will compute the <em>blinding factor</em> $R=aR&rsquo; + bG = (ak + b)G$, and as $R$ is a point we can get $R = (x, y)$.
 The user can <em>blind</em> the message by computing $m&rsquo; = a^{-1} \cdot x \cdot h(m)$, where $h(m)$ is the hash of the message.</p>
 <p>The <em>user</em> has a message <em>m</em> that which they want to get signed by the <em>signer</em> (without the <em>signer</em> knowing the content of <em>m</em>). In order to achieve that, the user will generate a coupe of random values <span class="math inline">\(a, b \in \mathbb{Z}_n\)</span>, and from these parameters will compute the <em>blinding factor</em> <span class="math inline">\(R=aR' + bG = (ak + b)G\)</span>, and as <span class="math inline">\(R\)</span> is a point we can get <span class="math inline">\(R = (x, y)\)</span>.
 The user can <em>blind</em> the message by computing <span class="math inline">\(m' = a^{-1} \cdot x \cdot h(m)\)</span>, where <span class="math inline">\(h(m)\)</span> is the hash of the message.</p>

 <p>Then, the <em>user</em> sends the <em>blinded message</em> ($m&rsquo;$) to the <em>signer</em>, who will perform the <em>blind signature</em> by computing $s&rsquo; = d m&rsquo; + k$, which is sent back to the <em>user</em>.</p>
 <p>Then, the <em>user</em> sends the <em>blinded message</em> (<span class="math inline">\(m'\)</span>) to the <em>signer</em>, who will perform the <em>blind signature</em> by computing <span class="math inline">\(s' = d m' + k\)</span>, which is sent back to the <em>user</em>.</p>

 <p>The <em>user</em> can unblind the signature by $s = a s&rsquo; + b$, and the complete signature will be $(R, s)$.</p>
 <p>The <em>user</em> can unblind the signature by <span class="math inline">\(s = a s' + b\)</span>, and the complete signature will be <span class="math inline">\((R, s)\)</span>.</p>

 <p>And now, we are in a point where the signature can be verified by a third party for the <em>signer</em>&rsquo;s public key by checking $sG == R + x h(m) Q$.</p>
 <p>And now, we are in a point where the signature can be verified by a third party for the <em>signer</em>&rsquo;s public key by checking <span class="math inline">\(sG == R + x h(m) Q\)</span>.</p>

 <p><div style="text-align:center; font-size:80%;">
    <img style="padding:50px;max-width:100%;" src="img/posts/blind-signatures-ec/flow1.png" />
@ -101,15 +101,13 @@ The user can blind the message by computing $m’ = a^{-1} \cdot x
 </div>
 <br></p>

 <p>From the verification $sG == R + x h(m) Q$, we can unroll it and check that:</p>

 <p>$$
 \fbox{sG} = (a s&rsquo; + b) G = (a (d m&rsquo; + k) + b) G\newline
 = (a d m&rsquo; + ak + b) G = ((a d (a^{-1} x h(m))) + ak + b) G\newline
 <p>From the verification <span class="math inline">\(sG == R + x h(m) Q\)</span>, we can unroll it and check that:</p>
 <p><span class="math display">\[
 \fbox{sG} = (a s' + b) G = (a (d m' + k) + b) G\newline
 = (a d m' + ak + b) G = ((a d (a^{-1} x h(m))) + ak + b) G\newline
 = (d x h(m) + ak + b) G\newline
 = dG x h(m) + (ak + b)G = \fbox{R + x h(m) Q}
 $$</p>

 \]</span></p>
 <h4>Code</h4>

 <p>Here is an example of how this scheme on the <a href="https://en.bitcoin.it/wiki/Secp256k1">secp256k1</a> curve could be used using the implementation from <a href="https://github.com/arnaucube/go-blindsecp256k1">go-blindsecp256k1</a>.</p>
@ -189,8 +187,8 @@ func main() {
    </script>
    <script src="js/external-links.js"></script>
    <script>hljs.initHighlightingOnLoad();</script>
    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.13.11/dist/katex.min.js" integrity="sha384-YNHdsYkH6gMx9y3mRkmcJ2mFUjTd0qNQQvY9VYZgQd7DcN7env35GzlmFaZ23JGp" crossorigin="anonymous"></script>
    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.13.11/dist/contrib/auto-render.min.js" integrity="sha384-vZTG03m+2yp6N6BNi5iM4rW4oIwk5DfcNdFfxkk9ZWpDriOkXX8voJBFrAO7MpVl" crossorigin="anonymous"></script>
    <script defer src="js/katex/katex.min.js"></script>
    <script defer src="js/katex/auto-render.min.js"></script>
    <script>
      document.addEventListener("DOMContentLoaded", function() {
        renderMathInElement(document.body, {
@ -200,6 +198,8 @@ func main() {
          delimiters: [
            {left: '$$', right: '$$', display: true},
            {left: '$', right: '$', display: false},
            {left: "\\[", right: "\\]", display: true},
            {left: "\\(", right: "\\)", display: false},
          ],
          // • rendering keys, e.g.:
          throwOnError : true
@ -249,7 +249,7 @@ func main() {
    tagLinks("h4");
    tagLinks("h5");
    </script>
    <script src="https://cdn.jsdelivr.net/npm/mermaid/dist/mermaid.min.js"></script>
    <script src="js/mermaid.min.js"></script>


  </body>
--- a/public/blogo.html
+++ b/public/blogo.html
@ -21,7 +21,7 @@

    <meta name="viewport" content="width=device-width, initial-scale=1">

    <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.0.2/dist/css/bootstrap.min.css" rel="stylesheet" integrity="sha384-EVSTQN3/azprG1Anm3QDgpJLIm9Nao0Yz1ztcQTwFspd3yD65VohhpuuCOmLASjC" crossorigin="anonymous">
    <link href="css/bootstrap.min.css" rel="stylesheet">
    <link rel="stylesheet" href="css/style.css">

    <!-- highlightjs -->
@ -30,7 +30,7 @@
    <script src="js/highlightjs/highlight.pack.js"></script>

    <!-- katex -->
    <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.11/dist/katex.min.css" integrity="sha384-Um5gpz1odJg5Z4HAmzPtgZKdTBHZdw8S29IecapCSB31ligYPhHQZMIlWLYQGVoc" crossorigin="anonymous">
    <link rel="stylesheet" href="js/katex/katex.min.css">
  </head>

  <body>
@ -436,8 +436,8 @@ func putHTMLToTemplate(template string, m map[string]string) string {
    </script>
    <script src="js/external-links.js"></script>
    <script>hljs.initHighlightingOnLoad();</script>
    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.13.11/dist/katex.min.js" integrity="sha384-YNHdsYkH6gMx9y3mRkmcJ2mFUjTd0qNQQvY9VYZgQd7DcN7env35GzlmFaZ23JGp" crossorigin="anonymous"></script>
    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.13.11/dist/contrib/auto-render.min.js" integrity="sha384-vZTG03m+2yp6N6BNi5iM4rW4oIwk5DfcNdFfxkk9ZWpDriOkXX8voJBFrAO7MpVl" crossorigin="anonymous"></script>
    <script defer src="js/katex/katex.min.js"></script>
    <script defer src="js/katex/auto-render.min.js"></script>
    <script>
      document.addEventListener("DOMContentLoaded", function() {
        renderMathInElement(document.body, {
@ -447,6 +447,8 @@ func putHTMLToTemplate(template string, m map[string]string) string {
          delimiters: [
            {left: '$$', right: '$$', display: true},
            {left: '$', right: '$', display: false},
            {left: "\\[", right: "\\]", display: true},
            {left: "\\(", right: "\\)", display: false},
          ],
          // • rendering keys, e.g.:
          throwOnError : true
@ -496,7 +498,7 @@ func putHTMLToTemplate(template string, m map[string]string) string {
    tagLinks("h4");
    tagLinks("h5");
    </script>
    <script src="https://cdn.jsdelivr.net/npm/mermaid/dist/mermaid.min.js"></script>
    <script src="js/mermaid.min.js"></script>


  </body>
--- a/public/coffeeminer-hacking-wifi-cryptocurrency-miner.html
+++ b/public/coffeeminer-hacking-wifi-cryptocurrency-miner.html
@ -21,7 +21,7 @@

    <meta name="viewport" content="width=device-width, initial-scale=1">

    <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.0.2/dist/css/bootstrap.min.css" rel="stylesheet" integrity="sha384-EVSTQN3/azprG1Anm3QDgpJLIm9Nao0Yz1ztcQTwFspd3yD65VohhpuuCOmLASjC" crossorigin="anonymous">
    <link href="css/bootstrap.min.css" rel="stylesheet">
    <link rel="stylesheet" href="css/style.css">

    <!-- highlightjs -->
@ -30,7 +30,7 @@
    <script src="js/highlightjs/highlight.pack.js"></script>

    <!-- katex -->
    <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.11/dist/katex.min.css" integrity="sha384-Um5gpz1odJg5Z4HAmzPtgZKdTBHZdw8S29IecapCSB31ligYPhHQZMIlWLYQGVoc" crossorigin="anonymous">
    <link rel="stylesheet" href="js/katex/katex.min.css">
  </head>

  <body>
@ -575,8 +575,8 @@ def start():
    </script>
    <script src="js/external-links.js"></script>
    <script>hljs.initHighlightingOnLoad();</script>
    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.13.11/dist/katex.min.js" integrity="sha384-YNHdsYkH6gMx9y3mRkmcJ2mFUjTd0qNQQvY9VYZgQd7DcN7env35GzlmFaZ23JGp" crossorigin="anonymous"></script>
    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.13.11/dist/contrib/auto-render.min.js" integrity="sha384-vZTG03m+2yp6N6BNi5iM4rW4oIwk5DfcNdFfxkk9ZWpDriOkXX8voJBFrAO7MpVl" crossorigin="anonymous"></script>
    <script defer src="js/katex/katex.min.js"></script>
    <script defer src="js/katex/auto-render.min.js"></script>
    <script>
      document.addEventListener("DOMContentLoaded", function() {
        renderMathInElement(document.body, {
@ -586,6 +586,8 @@ def start():
          delimiters: [
            {left: '$$', right: '$$', display: true},
            {left: '$', right: '$', display: false},
            {left: "\\[", right: "\\]", display: true},
            {left: "\\(", right: "\\)", display: false},
          ],
          // • rendering keys, e.g.:
          throwOnError : true
@ -635,7 +637,7 @@ def start():
    tagLinks("h4");
    tagLinks("h5");
    </script>
    <script src="https://cdn.jsdelivr.net/npm/mermaid/dist/mermaid.min.js"></script>
    <script src="js/mermaid.min.js"></script>


  </body>
--- a/public/css/bootstrap.min.css
+++ b/public/css/bootstrap.min.css
--- a/public/flock-botnet.html
+++ b/public/flock-botnet.html
@ -21,7 +21,7 @@

    <meta name="viewport" content="width=device-width, initial-scale=1">

    <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.0.2/dist/css/bootstrap.min.css" rel="stylesheet" integrity="sha384-EVSTQN3/azprG1Anm3QDgpJLIm9Nao0Yz1ztcQTwFspd3yD65VohhpuuCOmLASjC" crossorigin="anonymous">
    <link href="css/bootstrap.min.css" rel="stylesheet">
    <link rel="stylesheet" href="css/style.css">

    <!-- highlightjs -->
@ -30,7 +30,7 @@
    <script src="js/highlightjs/highlight.pack.js"></script>

    <!-- katex -->
    <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.11/dist/katex.min.css" integrity="sha384-Um5gpz1odJg5Z4HAmzPtgZKdTBHZdw8S29IecapCSB31ligYPhHQZMIlWLYQGVoc" crossorigin="anonymous">
    <link rel="stylesheet" href="js/katex/katex.min.css">
  </head>

  <body>
@ -361,8 +361,8 @@
    </script>
    <script src="js/external-links.js"></script>
    <script>hljs.initHighlightingOnLoad();</script>
    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.13.11/dist/katex.min.js" integrity="sha384-YNHdsYkH6gMx9y3mRkmcJ2mFUjTd0qNQQvY9VYZgQd7DcN7env35GzlmFaZ23JGp" crossorigin="anonymous"></script>
    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.13.11/dist/contrib/auto-render.min.js" integrity="sha384-vZTG03m+2yp6N6BNi5iM4rW4oIwk5DfcNdFfxkk9ZWpDriOkXX8voJBFrAO7MpVl" crossorigin="anonymous"></script>
    <script defer src="js/katex/katex.min.js"></script>
    <script defer src="js/katex/auto-render.min.js"></script>
    <script>
      document.addEventListener("DOMContentLoaded", function() {
        renderMathInElement(document.body, {
@ -372,6 +372,8 @@
          delimiters: [
            {left: '$$', right: '$$', display: true},
            {left: '$', right: '$', display: false},
            {left: "\\[", right: "\\]", display: true},
            {left: "\\(", right: "\\)", display: false},
          ],
          // • rendering keys, e.g.:
          throwOnError : true
@ -421,7 +423,7 @@
    tagLinks("h4");
    tagLinks("h5");
    </script>
    <script src="https://cdn.jsdelivr.net/npm/mermaid/dist/mermaid.min.js"></script>
    <script src="js/mermaid.min.js"></script>


  </body>
--- a/public/index.html
+++ b/public/index.html
@ -21,7 +21,7 @@

    <meta name="viewport" content="width=device-width, initial-scale=1">

    <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.0.2/dist/css/bootstrap.min.css" rel="stylesheet" integrity="sha384-EVSTQN3/azprG1Anm3QDgpJLIm9Nao0Yz1ztcQTwFspd3yD65VohhpuuCOmLASjC" crossorigin="anonymous">
    <link href="css/bootstrap.min.css" rel="stylesheet">
    <link rel="stylesheet" href="css/style.css">

    <!-- highlightjs -->
@ -30,7 +30,7 @@
    <script src="js/highlightjs/highlight.pack.js"></script>

    <!-- katex -->
    <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.11/dist/katex.min.css" integrity="sha384-Um5gpz1odJg5Z4HAmzPtgZKdTBHZdw8S29IecapCSB31ligYPhHQZMIlWLYQGVoc" crossorigin="anonymous">
    <link rel="stylesheet" href="js/katex/katex.min.css">
  </head>

  <body>
@ -67,7 +67,7 @@
 </a><a href='/blog/kzg-batch-proof.html'><div class="row postThumb">
        <h3>Batch proof in KZG Commitments</h3>

 <p>The benefit of <em>batch proof</em> is that allows us to proof multiple points while the proof size remains constant to one $\mathbb{G}_1$ point.</p>
 <p>The benefit of <em>batch proof</em> is that allows us to proof multiple points while the proof size remains constant to one <span class="math inline">\(\mathbb{G}_1\)</span> point.</p>

 <p><em>2021-08-14</em></p>

@ -146,8 +146,8 @@
    </script>
    <script src="js/external-links.js"></script>
    <script>hljs.initHighlightingOnLoad();</script>
    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.13.11/dist/katex.min.js" integrity="sha384-YNHdsYkH6gMx9y3mRkmcJ2mFUjTd0qNQQvY9VYZgQd7DcN7env35GzlmFaZ23JGp" crossorigin="anonymous"></script>
    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.13.11/dist/contrib/auto-render.min.js" integrity="sha384-vZTG03m+2yp6N6BNi5iM4rW4oIwk5DfcNdFfxkk9ZWpDriOkXX8voJBFrAO7MpVl" crossorigin="anonymous"></script>
    <script defer src="js/katex/katex.min.js"></script>
    <script defer src="js/katex/auto-render.min.js"></script>
    <script>
      document.addEventListener("DOMContentLoaded", function() {
        renderMathInElement(document.body, {
@ -157,6 +157,8 @@
          delimiters: [
            {left: '$$', right: '$$', display: true},
            {left: '$', right: '$', display: false},
            {left: "\\[", right: "\\]", display: true},
            {left: "\\(", right: "\\)", display: false},
          ],
          // • rendering keys, e.g.:
          throwOnError : true
@ -206,7 +208,7 @@
    tagLinks("h4");
    tagLinks("h5");
    </script>
    <script src="https://cdn.jsdelivr.net/npm/mermaid/dist/mermaid.min.js"></script>
    <script src="js/mermaid.min.js"></script>


  </body>
--- a/public/js/katex/README.md
+++ b/public/js/katex/README.md
@ -0,0 +1,119 @@
 # [<img src="https://katex.org/img/katex-logo-black.svg" width="130" alt="KaTeX">](https://katex.org/)
 [![npm](https://img.shields.io/npm/v/katex.svg)](https://www.npmjs.com/package/katex)
 [![semantic-release](https://img.shields.io/badge/%20%20%F0%9F%93%A6%F0%9F%9A%80-semantic--release-e10079.svg)](https://github.com/semantic-release/semantic-release)
 [![CI](https://github.com/KaTeX/KaTeX/workflows/CI/badge.svg?branch=main&event=push)](https://github.com/KaTeX/KaTeX/actions?query=workflow%3ACI)
 [![codecov](https://codecov.io/gh/KaTeX/KaTeX/branch/main/graph/badge.svg)](https://codecov.io/gh/KaTeX/KaTeX)
 [![Discussions](https://img.shields.io/badge/Discussions-join-brightgreen)](https://github.com/KaTeX/KaTeX/discussions)
 [![jsDelivr](https://data.jsdelivr.com/v1/package/npm/katex/badge?style=rounded)](https://www.jsdelivr.com/package/npm/katex)
 ![katex.min.js size](https://img.badgesize.io/https://unpkg.com/katex/dist/katex.min.js?compression=gzip)
 [![Gitpod ready-to-code](https://img.shields.io/badge/Gitpod-ready--to--code-blue?logo=gitpod)](https://gitpod.io/#https://github.com/KaTeX/KaTeX)
 [![Financial Contributors on Open Collective](https://opencollective.com/katex/all/badge.svg?label=financial+contributors)](https://opencollective.com/katex)

 KaTeX is a fast, easy-to-use JavaScript library for TeX math rendering on the web.

 * **Fast:** KaTeX renders its math synchronously and doesn't need to reflow the page. See how it compares to a competitor in [this speed test](http://www.intmath.com/cg5/katex-mathjax-comparison.php).
 * **Print quality:** KaTeX's layout is based on Donald Knuth's TeX, the gold standard for math typesetting.
 * **Self contained:** KaTeX has no dependencies and can easily be bundled with your website resources.
 * **Server side rendering:** KaTeX produces the same output regardless of browser or environment, so you can pre-render expressions using Node.js and send them as plain HTML.

 KaTeX is compatible with all major browsers, including Chrome, Safari, Firefox, Opera, Edge, and IE 11.

 KaTeX supports much (but not all) of LaTeX and many LaTeX packages. See the [list of supported functions](https://katex.org/docs/supported.html).

 Try out KaTeX [on the demo page](https://katex.org/#demo)!

 ## Getting started

 ### Starter template

 ```html
 <!DOCTYPE html>
 <!-- KaTeX requires the use of the HTML5 doctype. Without it, KaTeX may not render properly -->
 <html>
  <head>
    <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.15.3/dist/katex.min.css" integrity="sha384-KiWOvVjnN8qwAZbuQyWDIbfCLFhLXNETzBQjA/92pIowpC0d2O3nppDGQVgwd2nB" crossorigin="anonymous">

    <!-- The loading of KaTeX is deferred to speed up page rendering -->
    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.15.3/dist/katex.min.js" integrity="sha384-0fdwu/T/EQMsQlrHCCHoH10pkPLlKA1jL5dFyUOvB3lfeT2540/2g6YgSi2BL14p" crossorigin="anonymous"></script>

    <!-- To automatically render math in text elements, include the auto-render extension: -->
    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.15.3/dist/contrib/auto-render.min.js" integrity="sha384-+XBljXPPiv+OzfbB3cVmLHf4hdUFHlWNZN5spNQ7rmHTXpd7WvJum6fIACpNNfIR" crossorigin="anonymous"
        onload="renderMathInElement(document.body);"></script>
  </head>
  ...
 </html>
 ```

 You can also [download KaTeX](https://github.com/KaTeX/KaTeX/releases) and host it yourself.

 For details on how to configure auto-render extension, refer to [the documentation](https://katex.org/docs/autorender.html).

 ### API

 Call `katex.render` to render a TeX expression directly into a DOM element.
 For example:

 ```js
 katex.render("c = \\pm\\sqrt{a^2 + b^2}", element, {
    throwOnError: false
 });
 ```

 Call `katex.renderToString` to generate an HTML string of the rendered math,
 e.g., for server-side rendering.  For example:

 ```js
 var html = katex.renderToString("c = \\pm\\sqrt{a^2 + b^2}", {
    throwOnError: false
 });
 // '<span class="katex">...</span>'
 ```

 Make sure to include the CSS and font files in both cases.
 If you are doing all rendering on the server, there is no need to include the
 JavaScript on the client.

 The examples above use the `throwOnError: false` option, which renders invalid
 inputs as the TeX source code in red (by default), with the error message as
 hover text.  For other available options, see the
 [API documentation](https://katex.org/docs/api.html),
 [options documentation](https://katex.org/docs/options.html), and
 [handling errors documentation](https://katex.org/docs/error.html).

 ## Demo and Documentation

 Learn more about using KaTeX [on the website](https://katex.org)!

 ## Contributors

 ### Code Contributors

 This project exists thanks to all the people who contribute code. If you'd like to help, see [our guide to contributing code](CONTRIBUTING.md).
 <a href="https://github.com/KaTeX/KaTeX/graphs/contributors"><img src="https://contributors-svg.opencollective.com/katex/contributors.svg?width=890&button=false" alt="Code contributors" /></a>

 ### Financial Contributors

 Become a financial contributor and help us sustain our community.

 #### Individuals

 <a href="https://opencollective.com/katex"><img src="https://opencollective.com/katex/individuals.svg?width=890" alt="Contribute on Open Collective"></a>

 #### Organizations

 Support this project with your organization. Your logo will show up here with a link to your website.

 <a href="https://opencollective.com/katex/organization/0/website"><img src="https://opencollective.com/katex/organization/0/avatar.svg" alt="Organization 1"></a>
 <a href="https://opencollective.com/katex/organization/1/website"><img src="https://opencollective.com/katex/organization/1/avatar.svg" alt="Organization 2"></a>
 <a href="https://opencollective.com/katex/organization/2/website"><img src="https://opencollective.com/katex/organization/2/avatar.svg" alt="Organization 3"></a>
 <a href="https://opencollective.com/katex/organization/3/website"><img src="https://opencollective.com/katex/organization/3/avatar.svg" alt="Organization 4"></a>
 <a href="https://opencollective.com/katex/organization/4/website"><img src="https://opencollective.com/katex/organization/4/avatar.svg" alt="Organization 5"></a>
 <a href="https://opencollective.com/katex/organization/5/website"><img src="https://opencollective.com/katex/organization/5/avatar.svg" alt="Organization 6"></a>
 <a href="https://opencollective.com/katex/organization/6/website"><img src="https://opencollective.com/katex/organization/6/avatar.svg" alt="Organization 7"></a>
 <a href="https://opencollective.com/katex/organization/7/website"><img src="https://opencollective.com/katex/organization/7/avatar.svg" alt="Organization 8"></a>
 <a href="https://opencollective.com/katex/organization/8/website"><img src="https://opencollective.com/katex/organization/8/avatar.svg" alt="Organization 9"></a>
 <a href="https://opencollective.com/katex/organization/9/website"><img src="https://opencollective.com/katex/organization/9/avatar.svg" alt="Organization 10"></a>

 ## License

 KaTeX is licensed under the [MIT License](http://opensource.org/licenses/MIT).
--- a/public/js/katex/auto-render.min.js
+++ b/public/js/katex/auto-render.min.js
@ -0,0 +1 @@
 !function(e,t){"object"==typeof exports&&"object"==typeof module?module.exports=t(require("katex")):"function"==typeof define&&define.amd?define(["katex"],t):"object"==typeof exports?exports.renderMathInElement=t(require("katex")):e.renderMathInElement=t(e.katex)}("undefined"!=typeof self?self:this,(function(e){return function(){"use strict";var t={974:function(t){t.exports=e}},r={};function n(e){var a=r[e];if(void 0!==a)return a.exports;var i=r[e]={exports:{}};return t[e](i,i.exports,n),i.exports}n.n=function(e){var t=e&&e.__esModule?function(){return e.default}:function(){return e};return n.d(t,{a:t}),t},n.d=function(e,t){for(var r in t)n.o(t,r)&&!n.o(e,r)&&Object.defineProperty(e,r,{enumerable:!0,get:t[r]})},n.o=function(e,t){return Object.prototype.hasOwnProperty.call(e,t)};var a={};return function(){n.d(a,{default:function(){return s}});var e=n(974),t=n.n(e),r=function(e,t,r){for(var n=r,a=0,i=e.length;n<t.length;){var o=t[n];if(a<=0&&t.slice(n,n+i)===e)return n;"\\"===o?n++:"{"===o?a++:"}"===o&&a--,n++}return-1},i=/^\\begin{/,o=function(e,t){for(var n,a=[],o=new RegExp("("+t.map((function(e){return e.left.replace(/[-/\\^$*+?.()|[\]{}]/g,"\\$&")})).join("|")+")");-1!==(n=e.search(o));){n>0&&(a.push({type:"text",data:e.slice(0,n)}),e=e.slice(n));var l=t.findIndex((function(t){return e.startsWith(t.left)}));if(-1===(n=r(t[l].right,e,t[l].left.length)))break;var d=e.slice(0,n+t[l].right.length),s=i.test(d)?d:e.slice(t[l].left.length,n);a.push({type:"math",data:s,rawData:d,display:t[l].display}),e=e.slice(n+t[l].right.length)}return""!==e&&a.push({type:"text",data:e}),a},l=function(e,r){var n=o(e,r.delimiters);if(1===n.length&&"text"===n[0].type)return null;for(var a=document.createDocumentFragment(),i=0;i<n.length;i++)if("text"===n[i].type)a.appendChild(document.createTextNode(n[i].data));else{var l=document.createElement("span"),d=n[i].data;r.displayMode=n[i].display;try{r.preProcess&&(d=r.preProcess(d)),t().render(d,l,r)}catch(e){if(!(e instanceof t().ParseError))throw e;r.errorCallback("KaTeX auto-render: Failed to parse `"+n[i].data+"` with ",e),a.appendChild(document.createTextNode(n[i].rawData));continue}a.appendChild(l)}return a},d=function e(t,r){for(var n=0;n<t.childNodes.length;n++){var a=t.childNodes[n];if(3===a.nodeType){var i=l(a.textContent,r);i&&(n+=i.childNodes.length-1,t.replaceChild(i,a))}else 1===a.nodeType&&function(){var t=" "+a.className+" ";-1===r.ignoredTags.indexOf(a.nodeName.toLowerCase())&&r.ignoredClasses.every((function(e){return-1===t.indexOf(" "+e+" ")}))&&e(a,r)}()}},s=function(e,t){if(!e)throw new Error("No element provided to render");var r={};for(var n in t)t.hasOwnProperty(n)&&(r[n]=t[n]);r.delimiters=r.delimiters||[{left:"$$",right:"$$",display:!0},{left:"\\(",right:"\\)",display:!1},{left:"\\begin{equation}",right:"\\end{equation}",display:!0},{left:"\\begin{align}",right:"\\end{align}",display:!0},{left:"\\begin{alignat}",right:"\\end{alignat}",display:!0},{left:"\\begin{gather}",right:"\\end{gather}",display:!0},{left:"\\begin{CD}",right:"\\end{CD}",display:!0},{left:"\\[",right:"\\]",display:!0}],r.ignoredTags=r.ignoredTags||["script","noscript","style","textarea","pre","code","option"],r.ignoredClasses=r.ignoredClasses||[],r.errorCallback=r.errorCallback||console.error,r.macros=r.macros||{},d(e,r)}}(),a=a.default}()}));
--- a/public/js/katex/contrib/auto-render.js
+++ b/public/js/katex/contrib/auto-render.js
@ -0,0 +1,327 @@
 (function webpackUniversalModuleDefinition(root, factory) {
 	if(typeof exports === 'object' && typeof module === 'object')
 		module.exports = factory(require("katex"));
 	else if(typeof define === 'function' && define.amd)
 		define(["katex"], factory);
 	else if(typeof exports === 'object')
 		exports["renderMathInElement"] = factory(require("katex"));
 	else
 		root["renderMathInElement"] = factory(root["katex"]);
 })((typeof self !== 'undefined' ? self : this), function(__WEBPACK_EXTERNAL_MODULE__771__) {
 return /******/ (function() { // webpackBootstrap
 /******/ 	"use strict";
 /******/ 	var __webpack_modules__ = ({

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 /***/ (function(module) {

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 /******/ 		var module = __webpack_module_cache__[moduleId] = {
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 /******/ 			exports: {}
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 /******/ 	
 /******/ 		// Execute the module function
 /******/ 		__webpack_modules__[moduleId](module, module.exports, __webpack_require__);
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 /******/ 		// Return the exports of the module
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 /******/ 	}
 /******/ 	
 /************************************************************************/
 /******/ 	/* webpack/runtime/compat get default export */
 /******/ 	!function() {
 /******/ 		// getDefaultExport function for compatibility with non-harmony modules
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 /******/ 			var getter = module && module.__esModule ?
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 var __webpack_exports__ = {};
 // This entry need to be wrapped in an IIFE because it need to be isolated against other modules in the chunk.
 !function() {

 // EXPORTS
 __webpack_require__.d(__webpack_exports__, {
  "default": function() { return /* binding */ auto_render; }
 });

 // EXTERNAL MODULE: external "katex"
 var external_katex_ = __webpack_require__(771);
 var external_katex_default = /*#__PURE__*/__webpack_require__.n(external_katex_);
 ;// CONCATENATED MODULE: ./contrib/auto-render/splitAtDelimiters.js
 /* eslint no-constant-condition:0 */
 var findEndOfMath = function findEndOfMath(delimiter, text, startIndex) {
  // Adapted from
  // https://github.com/Khan/perseus/blob/master/src/perseus-markdown.jsx
  var index = startIndex;
  var braceLevel = 0;
  var delimLength = delimiter.length;

  while (index < text.length) {
    var character = text[index];

    if (braceLevel <= 0 && text.slice(index, index + delimLength) === delimiter) {
      return index;
    } else if (character === "\\") {
      index++;
    } else if (character === "{") {
      braceLevel++;
    } else if (character === "}") {
      braceLevel--;
    }

    index++;
  }

  return -1;
 };

 var escapeRegex = function escapeRegex(string) {
  return string.replace(/[-/\\^$*+?.()|[\]{}]/g, "\\$&");
 };

 var amsRegex = /^\\begin{/;

 var splitAtDelimiters = function splitAtDelimiters(text, delimiters) {
  var index;
  var data = [];
  var regexLeft = new RegExp("(" + delimiters.map(function (x) {
    return escapeRegex(x.left);
  }).join("|") + ")");

  while (true) {
    index = text.search(regexLeft);

    if (index === -1) {
      break;
    }

    if (index > 0) {
      data.push({
        type: "text",
        data: text.slice(0, index)
      });
      text = text.slice(index); // now text starts with delimiter
    } // ... so this always succeeds:


    var i = delimiters.findIndex(function (delim) {
      return text.startsWith(delim.left);
    });
    index = findEndOfMath(delimiters[i].right, text, delimiters[i].left.length);

    if (index === -1) {
      break;
    }

    var rawData = text.slice(0, index + delimiters[i].right.length);
    var math = amsRegex.test(rawData) ? rawData : text.slice(delimiters[i].left.length, index);
    data.push({
      type: "math",
      data: math,
      rawData: rawData,
      display: delimiters[i].display
    });
    text = text.slice(index + delimiters[i].right.length);
  }

  if (text !== "") {
    data.push({
      type: "text",
      data: text
    });
  }

  return data;
 };

 /* harmony default export */ var auto_render_splitAtDelimiters = (splitAtDelimiters);
 ;// CONCATENATED MODULE: ./contrib/auto-render/auto-render.js
 /* eslint no-console:0 */


 /* Note: optionsCopy is mutated by this method. If it is ever exposed in the
 * API, we should copy it before mutating.
 */

 var renderMathInText = function renderMathInText(text, optionsCopy) {
  var data = auto_render_splitAtDelimiters(text, optionsCopy.delimiters);

  if (data.length === 1 && data[0].type === 'text') {
    // There is no formula in the text.
    // Let's return null which means there is no need to replace
    // the current text node with a new one.
    return null;
  }

  var fragment = document.createDocumentFragment();

  for (var i = 0; i < data.length; i++) {
    if (data[i].type === "text") {
      fragment.appendChild(document.createTextNode(data[i].data));
    } else {
      var span = document.createElement("span");
      var math = data[i].data; // Override any display mode defined in the settings with that
      // defined by the text itself

      optionsCopy.displayMode = data[i].display;

      try {
        if (optionsCopy.preProcess) {
          math = optionsCopy.preProcess(math);
        }

        external_katex_default().render(math, span, optionsCopy);
      } catch (e) {
        if (!(e instanceof (external_katex_default()).ParseError)) {
          throw e;
        }

        optionsCopy.errorCallback("KaTeX auto-render: Failed to parse `" + data[i].data + "` with ", e);
        fragment.appendChild(document.createTextNode(data[i].rawData));
        continue;
      }

      fragment.appendChild(span);
    }
  }

  return fragment;
 };

 var renderElem = function renderElem(elem, optionsCopy) {
  for (var i = 0; i < elem.childNodes.length; i++) {
    var childNode = elem.childNodes[i];

    if (childNode.nodeType === 3) {
      // Text node
      var frag = renderMathInText(childNode.textContent, optionsCopy);

      if (frag) {
        i += frag.childNodes.length - 1;
        elem.replaceChild(frag, childNode);
      }
    } else if (childNode.nodeType === 1) {
      (function () {
        // Element node
        var className = ' ' + childNode.className + ' ';
        var shouldRender = optionsCopy.ignoredTags.indexOf(childNode.nodeName.toLowerCase()) === -1 && optionsCopy.ignoredClasses.every(function (x) {
          return className.indexOf(' ' + x + ' ') === -1;
        });

        if (shouldRender) {
          renderElem(childNode, optionsCopy);
        }
      })();
    } // Otherwise, it's something else, and ignore it.

  }
 };

 var renderMathInElement = function renderMathInElement(elem, options) {
  if (!elem) {
    throw new Error("No element provided to render");
  }

  var optionsCopy = {}; // Object.assign(optionsCopy, option)

  for (var option in options) {
    if (options.hasOwnProperty(option)) {
      optionsCopy[option] = options[option];
    }
  } // default options


  optionsCopy.delimiters = optionsCopy.delimiters || [{
    left: "$$",
    right: "$$",
    display: true
  }, {
    left: "\\(",
    right: "\\)",
    display: false
  }, // LaTeX uses $…$, but it ruins the display of normal `$` in text:
  // {left: "$", right: "$", display: false},
  // $ must come after $$
  // Render AMS environments even if outside $$…$$ delimiters.
  {
    left: "\\begin{equation}",
    right: "\\end{equation}",
    display: true
  }, {
    left: "\\begin{align}",
    right: "\\end{align}",
    display: true
  }, {
    left: "\\begin{alignat}",
    right: "\\end{alignat}",
    display: true
  }, {
    left: "\\begin{gather}",
    right: "\\end{gather}",
    display: true
  }, {
    left: "\\begin{CD}",
    right: "\\end{CD}",
    display: true
  }, {
    left: "\\[",
    right: "\\]",
    display: true
  }];
  optionsCopy.ignoredTags = optionsCopy.ignoredTags || ["script", "noscript", "style", "textarea", "pre", "code", "option"];
  optionsCopy.ignoredClasses = optionsCopy.ignoredClasses || [];
  optionsCopy.errorCallback = optionsCopy.errorCallback || console.error; // Enable sharing of global macros defined via `\gdef` between different
  // math elements within a single call to `renderMathInElement`.

  optionsCopy.macros = optionsCopy.macros || {};
  renderElem(elem, optionsCopy);
 };

 /* harmony default export */ var auto_render = (renderMathInElement);
 }();
 __webpack_exports__ = __webpack_exports__["default"];
 /******/ 	return __webpack_exports__;
 /******/ })()
 ;
 });
--- a/public/js/katex/contrib/auto-render.min.js
+++ b/public/js/katex/contrib/auto-render.min.js
@ -0,0 +1 @@
 !function(e,t){"object"==typeof exports&&"object"==typeof module?module.exports=t(require("katex")):"function"==typeof define&&define.amd?define(["katex"],t):"object"==typeof exports?exports.renderMathInElement=t(require("katex")):e.renderMathInElement=t(e.katex)}("undefined"!=typeof self?self:this,(function(e){return function(){"use strict";var t={771:function(t){t.exports=e}},r={};function n(e){var a=r[e];if(void 0!==a)return a.exports;var i=r[e]={exports:{}};return t[e](i,i.exports,n),i.exports}n.n=function(e){var t=e&&e.__esModule?function(){return e.default}:function(){return e};return n.d(t,{a:t}),t},n.d=function(e,t){for(var r in t)n.o(t,r)&&!n.o(e,r)&&Object.defineProperty(e,r,{enumerable:!0,get:t[r]})},n.o=function(e,t){return Object.prototype.hasOwnProperty.call(e,t)};var a={};return function(){n.d(a,{default:function(){return s}});var e=n(771),t=n.n(e),r=function(e,t,r){for(var n=r,a=0,i=e.length;n<t.length;){var o=t[n];if(a<=0&&t.slice(n,n+i)===e)return n;"\\"===o?n++:"{"===o?a++:"}"===o&&a--,n++}return-1},i=/^\\begin{/,o=function(e,t){for(var n,a=[],o=new RegExp("("+t.map((function(e){return e.left.replace(/[-/\\^$*+?.()|[\]{}]/g,"\\$&")})).join("|")+")");-1!==(n=e.search(o));){n>0&&(a.push({type:"text",data:e.slice(0,n)}),e=e.slice(n));var l=t.findIndex((function(t){return e.startsWith(t.left)}));if(-1===(n=r(t[l].right,e,t[l].left.length)))break;var d=e.slice(0,n+t[l].right.length),s=i.test(d)?d:e.slice(t[l].left.length,n);a.push({type:"math",data:s,rawData:d,display:t[l].display}),e=e.slice(n+t[l].right.length)}return""!==e&&a.push({type:"text",data:e}),a},l=function(e,r){var n=o(e,r.delimiters);if(1===n.length&&"text"===n[0].type)return null;for(var a=document.createDocumentFragment(),i=0;i<n.length;i++)if("text"===n[i].type)a.appendChild(document.createTextNode(n[i].data));else{var l=document.createElement("span"),d=n[i].data;r.displayMode=n[i].display;try{r.preProcess&&(d=r.preProcess(d)),t().render(d,l,r)}catch(e){if(!(e instanceof t().ParseError))throw e;r.errorCallback("KaTeX auto-render: Failed to parse `"+n[i].data+"` with ",e),a.appendChild(document.createTextNode(n[i].rawData));continue}a.appendChild(l)}return a},d=function e(t,r){for(var n=0;n<t.childNodes.length;n++){var a=t.childNodes[n];if(3===a.nodeType){var i=l(a.textContent,r);i&&(n+=i.childNodes.length-1,t.replaceChild(i,a))}else 1===a.nodeType&&function(){var t=" "+a.className+" ";-1===r.ignoredTags.indexOf(a.nodeName.toLowerCase())&&r.ignoredClasses.every((function(e){return-1===t.indexOf(" "+e+" ")}))&&e(a,r)}()}},s=function(e,t){if(!e)throw new Error("No element provided to render");var r={};for(var n in t)t.hasOwnProperty(n)&&(r[n]=t[n]);r.delimiters=r.delimiters||[{left:"$$",right:"$$",display:!0},{left:"\\(",right:"\\)",display:!1},{left:"\\begin{equation}",right:"\\end{equation}",display:!0},{left:"\\begin{align}",right:"\\end{align}",display:!0},{left:"\\begin{alignat}",right:"\\end{alignat}",display:!0},{left:"\\begin{gather}",right:"\\end{gather}",display:!0},{left:"\\begin{CD}",right:"\\end{CD}",display:!0},{left:"\\[",right:"\\]",display:!0}],r.ignoredTags=r.ignoredTags||["script","noscript","style","textarea","pre","code","option"],r.ignoredClasses=r.ignoredClasses||[],r.errorCallback=r.errorCallback||console.error,r.macros=r.macros||{},d(e,r)}}(),a=a.default}()}));
--- a/public/js/katex/contrib/auto-render.mjs
+++ b/public/js/katex/contrib/auto-render.mjs
@ -0,0 +1,222 @@
 import katex from '../katex.mjs';

 /* eslint no-constant-condition:0 */
 var findEndOfMath = function findEndOfMath(delimiter, text, startIndex) {
  // Adapted from
  // https://github.com/Khan/perseus/blob/master/src/perseus-markdown.jsx
  var index = startIndex;
  var braceLevel = 0;
  var delimLength = delimiter.length;

  while (index < text.length) {
    var character = text[index];

    if (braceLevel <= 0 && text.slice(index, index + delimLength) === delimiter) {
      return index;
    } else if (character === "\\") {
      index++;
    } else if (character === "{") {
      braceLevel++;
    } else if (character === "}") {
      braceLevel--;
    }

    index++;
  }

  return -1;
 };

 var escapeRegex = function escapeRegex(string) {
  return string.replace(/[-/\\^$*+?.()|[\]{}]/g, "\\$&");
 };

 var amsRegex = /^\\begin{/;

 var splitAtDelimiters = function splitAtDelimiters(text, delimiters) {
  var index;
  var data = [];
  var regexLeft = new RegExp("(" + delimiters.map(x => escapeRegex(x.left)).join("|") + ")");

  while (true) {
    index = text.search(regexLeft);

    if (index === -1) {
      break;
    }

    if (index > 0) {
      data.push({
        type: "text",
        data: text.slice(0, index)
      });
      text = text.slice(index); // now text starts with delimiter
    } // ... so this always succeeds:


    var i = delimiters.findIndex(delim => text.startsWith(delim.left));
    index = findEndOfMath(delimiters[i].right, text, delimiters[i].left.length);

    if (index === -1) {
      break;
    }

    var rawData = text.slice(0, index + delimiters[i].right.length);
    var math = amsRegex.test(rawData) ? rawData : text.slice(delimiters[i].left.length, index);
    data.push({
      type: "math",
      data: math,
      rawData,
      display: delimiters[i].display
    });
    text = text.slice(index + delimiters[i].right.length);
  }

  if (text !== "") {
    data.push({
      type: "text",
      data: text
    });
  }

  return data;
 };

 /* eslint no-console:0 */
 /* Note: optionsCopy is mutated by this method. If it is ever exposed in the
 * API, we should copy it before mutating.
 */

 var renderMathInText = function renderMathInText(text, optionsCopy) {
  var data = splitAtDelimiters(text, optionsCopy.delimiters);

  if (data.length === 1 && data[0].type === 'text') {
    // There is no formula in the text.
    // Let's return null which means there is no need to replace
    // the current text node with a new one.
    return null;
  }

  var fragment = document.createDocumentFragment();

  for (var i = 0; i < data.length; i++) {
    if (data[i].type === "text") {
      fragment.appendChild(document.createTextNode(data[i].data));
    } else {
      var span = document.createElement("span");
      var math = data[i].data; // Override any display mode defined in the settings with that
      // defined by the text itself

      optionsCopy.displayMode = data[i].display;

      try {
        if (optionsCopy.preProcess) {
          math = optionsCopy.preProcess(math);
        }

        katex.render(math, span, optionsCopy);
      } catch (e) {
        if (!(e instanceof katex.ParseError)) {
          throw e;
        }

        optionsCopy.errorCallback("KaTeX auto-render: Failed to parse `" + data[i].data + "` with ", e);
        fragment.appendChild(document.createTextNode(data[i].rawData));
        continue;
      }

      fragment.appendChild(span);
    }
  }

  return fragment;
 };

 var renderElem = function renderElem(elem, optionsCopy) {
  for (var i = 0; i < elem.childNodes.length; i++) {
    var childNode = elem.childNodes[i];

    if (childNode.nodeType === 3) {
      // Text node
      var frag = renderMathInText(childNode.textContent, optionsCopy);

      if (frag) {
        i += frag.childNodes.length - 1;
        elem.replaceChild(frag, childNode);
      }
    } else if (childNode.nodeType === 1) {
      (function () {
        // Element node
        var className = ' ' + childNode.className + ' ';
        var shouldRender = optionsCopy.ignoredTags.indexOf(childNode.nodeName.toLowerCase()) === -1 && optionsCopy.ignoredClasses.every(x => className.indexOf(' ' + x + ' ') === -1);

        if (shouldRender) {
          renderElem(childNode, optionsCopy);
        }
      })();
    } // Otherwise, it's something else, and ignore it.

  }
 };

 var renderMathInElement = function renderMathInElement(elem, options) {
  if (!elem) {
    throw new Error("No element provided to render");
  }

  var optionsCopy = {}; // Object.assign(optionsCopy, option)

  for (var option in options) {
    if (options.hasOwnProperty(option)) {
      optionsCopy[option] = options[option];
    }
  } // default options


  optionsCopy.delimiters = optionsCopy.delimiters || [{
    left: "$$",
    right: "$$",
    display: true
  }, {
    left: "\\(",
    right: "\\)",
    display: false
  }, // LaTeX uses $…$, but it ruins the display of normal `$` in text:
  // {left: "$", right: "$", display: false},
  // $ must come after $$
  // Render AMS environments even if outside $$…$$ delimiters.
  {
    left: "\\begin{equation}",
    right: "\\end{equation}",
    display: true
  }, {
    left: "\\begin{align}",
    right: "\\end{align}",
    display: true
  }, {
    left: "\\begin{alignat}",
    right: "\\end{alignat}",
    display: true
  }, {
    left: "\\begin{gather}",
    right: "\\end{gather}",
    display: true
  }, {
    left: "\\begin{CD}",
    right: "\\end{CD}",
    display: true
  }, {
    left: "\\[",
    right: "\\]",
    display: true
  }];
  optionsCopy.ignoredTags = optionsCopy.ignoredTags || ["script", "noscript", "style", "textarea", "pre", "code", "option"];
  optionsCopy.ignoredClasses = optionsCopy.ignoredClasses || [];
  optionsCopy.errorCallback = optionsCopy.errorCallback || console.error; // Enable sharing of global macros defined via `\gdef` between different
  // math elements within a single call to `renderMathInElement`.

  optionsCopy.macros = optionsCopy.macros || {};
  renderElem(elem, optionsCopy);
 };

 export { renderMathInElement as default };
--- a/public/js/katex/contrib/copy-tex.css
+++ b/public/js/katex/contrib/copy-tex.css
@ -0,0 +1,13 @@
 /* Force selection of entire .katex/.katex-display blocks, so that we can
 * copy/paste the entire source code.  If you omit this CSS, partial
 * selections of a formula will work, but will copy the ugly HTML
 * representation instead of the LaTeX source code.  (Full selections will
 * still produce the LaTeX source code.)
 */
 .katex,
 .katex-display {
    -webkit-user-select: all;
       -moz-user-select: all;
            user-select: all;
 }

--- a/public/js/katex/contrib/copy-tex.js
+++ b/public/js/katex/contrib/copy-tex.js
@ -0,0 +1,116 @@
 (function webpackUniversalModuleDefinition(root, factory) {
 	if(typeof exports === 'object' && typeof module === 'object')
 		module.exports = factory();
 	else if(typeof define === 'function' && define.amd)
 		define([], factory);
 	else {
 		var a = factory();
 		for(var i in a) (typeof exports === 'object' ? exports : root)[i] = a[i];
 	}
 })((typeof self !== 'undefined' ? self : this), function() {
 return /******/ (function() { // webpackBootstrap
 /******/ 	"use strict";
 var __webpack_exports__ = {};

 ;// CONCATENATED MODULE: ./contrib/copy-tex/katex2tex.js
 // Set these to how you want inline and display math to be delimited.
 var defaultCopyDelimiters = {
  inline: ['$', '$'],
  // alternative: ['\(', '\)']
  display: ['$$', '$$'] // alternative: ['\[', '\]']

 }; // Replace .katex elements with their TeX source (<annotation> element).
 // Modifies fragment in-place.  Useful for writing your own 'copy' handler,
 // as in copy-tex.js.

 var katexReplaceWithTex = function katexReplaceWithTex(fragment, copyDelimiters) {
  if (copyDelimiters === void 0) {
    copyDelimiters = defaultCopyDelimiters;
  }

  // Remove .katex-html blocks that are preceded by .katex-mathml blocks
  // (which will get replaced below).
  var katexHtml = fragment.querySelectorAll('.katex-mathml + .katex-html');

  for (var i = 0; i < katexHtml.length; i++) {
    var element = katexHtml[i];

    if (element.remove) {
      element.remove(null);
    } else {
      element.parentNode.removeChild(element);
    }
  } // Replace .katex-mathml elements with their annotation (TeX source)
  // descendant, with inline delimiters.


  var katexMathml = fragment.querySelectorAll('.katex-mathml');

  for (var _i = 0; _i < katexMathml.length; _i++) {
    var _element = katexMathml[_i];

    var texSource = _element.querySelector('annotation');

    if (texSource) {
      if (_element.replaceWith) {
        _element.replaceWith(texSource);
      } else {
        _element.parentNode.replaceChild(texSource, _element);
      }

      texSource.innerHTML = copyDelimiters.inline[0] + texSource.innerHTML + copyDelimiters.inline[1];
    }
  } // Switch display math to display delimiters.


  var displays = fragment.querySelectorAll('.katex-display annotation');

  for (var _i2 = 0; _i2 < displays.length; _i2++) {
    var _element2 = displays[_i2];
    _element2.innerHTML = copyDelimiters.display[0] + _element2.innerHTML.substr(copyDelimiters.inline[0].length, _element2.innerHTML.length - copyDelimiters.inline[0].length - copyDelimiters.inline[1].length) + copyDelimiters.display[1];
  }

  return fragment;
 };
 /* harmony default export */ var katex2tex = (katexReplaceWithTex);
 ;// CONCATENATED MODULE: ./contrib/copy-tex/copy-tex.js
 // Global copy handler to modify behavior on .katex elements.

 document.addEventListener('copy', function (event) {
  var selection = window.getSelection();

  if (selection.isCollapsed) {
    return; // default action OK if selection is empty
  }

  var fragment = selection.getRangeAt(0).cloneContents();

  if (!fragment.querySelector('.katex-mathml')) {
    return; // default action OK if no .katex-mathml elements
  } // Preserve usual HTML copy/paste behavior.


  var html = [];

  for (var i = 0; i < fragment.childNodes.length; i++) {
    html.push(fragment.childNodes[i].outerHTML);
  }

  event.clipboardData.setData('text/html', html.join('')); // Rewrite plain-text version.

  event.clipboardData.setData('text/plain', katex2tex(fragment).textContent); // Prevent normal copy handling.

  event.preventDefault();
 });
 ;// CONCATENATED MODULE: ./contrib/copy-tex/copy-tex.webpack.js
 /**
 * This is the webpack entry point for KaTeX. As ECMAScript doesn't support
 * CSS modules natively, a separate entry point is used.
 */


 __webpack_exports__ = __webpack_exports__["default"];
 /******/ 	return __webpack_exports__;
 /******/ })()
 ;
 });
--- a/public/js/katex/contrib/copy-tex.min.css
+++ b/public/js/katex/contrib/copy-tex.min.css
@ -0,0 +1 @@
 .katex,.katex-display{-webkit-user-select:all;-moz-user-select:all;user-select:all}
--- a/public/js/katex/contrib/copy-tex.min.js
+++ b/public/js/katex/contrib/copy-tex.min.js
@ -0,0 +1 @@
 !function(e,t){if("object"==typeof exports&&"object"==typeof module)module.exports=t();else if("function"==typeof define&&define.amd)define([],t);else{var n=t();for(var l in n)("object"==typeof exports?exports:e)[l]=n[l]}}("undefined"!=typeof self?self:this,(function(){return function(){"use strict";var e={},t={inline:["$","$"],display:["$$","$$"]},n=function(e,n){void 0===n&&(n=t);for(var l=e.querySelectorAll(".katex-mathml + .katex-html"),r=0;r<l.length;r++){var i=l[r];i.remove?i.remove(null):i.parentNode.removeChild(i)}for(var o=e.querySelectorAll(".katex-mathml"),a=0;a<o.length;a++){var d=o[a],f=d.querySelector("annotation");f&&(d.replaceWith?d.replaceWith(f):d.parentNode.replaceChild(f,d),f.innerHTML=n.inline[0]+f.innerHTML+n.inline[1])}for(var c=e.querySelectorAll(".katex-display annotation"),s=0;s<c.length;s++){var p=c[s];p.innerHTML=n.display[0]+p.innerHTML.substr(n.inline[0].length,p.innerHTML.length-n.inline[0].length-n.inline[1].length)+n.display[1]}return e};return document.addEventListener("copy",(function(e){var t=window.getSelection();if(!t.isCollapsed){var l=t.getRangeAt(0).cloneContents();if(l.querySelector(".katex-mathml")){for(var r=[],i=0;i<l.childNodes.length;i++)r.push(l.childNodes[i].outerHTML);e.clipboardData.setData("text/html",r.join("")),e.clipboardData.setData("text/plain",n(l).textContent),e.preventDefault()}}})),e=e.default}()}));
--- a/public/js/katex/contrib/copy-tex.mjs
+++ b/public/js/katex/contrib/copy-tex.mjs
@ -0,0 +1,86 @@
 // Set these to how you want inline and display math to be delimited.
 var defaultCopyDelimiters = {
  inline: ['$', '$'],
  // alternative: ['\(', '\)']
  display: ['$$', '$$'] // alternative: ['\[', '\]']

 }; // Replace .katex elements with their TeX source (<annotation> element).
 // Modifies fragment in-place.  Useful for writing your own 'copy' handler,
 // as in copy-tex.js.

 var katexReplaceWithTex = function katexReplaceWithTex(fragment, copyDelimiters) {
  if (copyDelimiters === void 0) {
    copyDelimiters = defaultCopyDelimiters;
  }

  // Remove .katex-html blocks that are preceded by .katex-mathml blocks
  // (which will get replaced below).
  var katexHtml = fragment.querySelectorAll('.katex-mathml + .katex-html');

  for (var i = 0; i < katexHtml.length; i++) {
    var element = katexHtml[i];

    if (element.remove) {
      element.remove(null);
    } else {
      element.parentNode.removeChild(element);
    }
  } // Replace .katex-mathml elements with their annotation (TeX source)
  // descendant, with inline delimiters.


  var katexMathml = fragment.querySelectorAll('.katex-mathml');

  for (var _i = 0; _i < katexMathml.length; _i++) {
    var _element = katexMathml[_i];

    var texSource = _element.querySelector('annotation');

    if (texSource) {
      if (_element.replaceWith) {
        _element.replaceWith(texSource);
      } else {
        _element.parentNode.replaceChild(texSource, _element);
      }

      texSource.innerHTML = copyDelimiters.inline[0] + texSource.innerHTML + copyDelimiters.inline[1];
    }
  } // Switch display math to display delimiters.


  var displays = fragment.querySelectorAll('.katex-display annotation');

  for (var _i2 = 0; _i2 < displays.length; _i2++) {
    var _element2 = displays[_i2];
    _element2.innerHTML = copyDelimiters.display[0] + _element2.innerHTML.substr(copyDelimiters.inline[0].length, _element2.innerHTML.length - copyDelimiters.inline[0].length - copyDelimiters.inline[1].length) + copyDelimiters.display[1];
  }

  return fragment;
 };

 document.addEventListener('copy', function (event) {
  var selection = window.getSelection();

  if (selection.isCollapsed) {
    return; // default action OK if selection is empty
  }

  var fragment = selection.getRangeAt(0).cloneContents();

  if (!fragment.querySelector('.katex-mathml')) {
    return; // default action OK if no .katex-mathml elements
  } // Preserve usual HTML copy/paste behavior.


  var html = [];

  for (var i = 0; i < fragment.childNodes.length; i++) {
    html.push(fragment.childNodes[i].outerHTML);
  }

  event.clipboardData.setData('text/html', html.join('')); // Rewrite plain-text version.

  event.clipboardData.setData('text/plain', katexReplaceWithTex(fragment).textContent); // Prevent normal copy handling.

  event.preventDefault();
 });
--- a/public/js/katex/contrib/mathtex-script-type.js
+++ b/public/js/katex/contrib/mathtex-script-type.js
@ -0,0 +1,112 @@
 (function webpackUniversalModuleDefinition(root, factory) {
 	if(typeof exports === 'object' && typeof module === 'object')
 		module.exports = factory(require("katex"));
 	else if(typeof define === 'function' && define.amd)
 		define(["katex"], factory);
 	else {
 		var a = typeof exports === 'object' ? factory(require("katex")) : factory(root["katex"]);
 		for(var i in a) (typeof exports === 'object' ? exports : root)[i] = a[i];
 	}
 })((typeof self !== 'undefined' ? self : this), function(__WEBPACK_EXTERNAL_MODULE__771__) {
 return /******/ (function() { // webpackBootstrap
 /******/ 	"use strict";
 /******/ 	var __webpack_modules__ = ({

 /***/ 771:
 /***/ (function(module) {

 module.exports = __WEBPACK_EXTERNAL_MODULE__771__;

 /***/ })

 /******/ 	});
 /************************************************************************/
 /******/ 	// The module cache
 /******/ 	var __webpack_module_cache__ = {};
 /******/ 	
 /******/ 	// The require function
 /******/ 	function __webpack_require__(moduleId) {
 /******/ 		// Check if module is in cache
 /******/ 		var cachedModule = __webpack_module_cache__[moduleId];
 /******/ 		if (cachedModule !== undefined) {
 /******/ 			return cachedModule.exports;
 /******/ 		}
 /******/ 		// Create a new module (and put it into the cache)
 /******/ 		var module = __webpack_module_cache__[moduleId] = {
 /******/ 			// no module.id needed
 /******/ 			// no module.loaded needed
 /******/ 			exports: {}
 /******/ 		};
 /******/ 	
 /******/ 		// Execute the module function
 /******/ 		__webpack_modules__[moduleId](module, module.exports, __webpack_require__);
 /******/ 	
 /******/ 		// Return the exports of the module
 /******/ 		return module.exports;
 /******/ 	}
 /******/ 	
 /************************************************************************/
 /******/ 	/* webpack/runtime/compat get default export */
 /******/ 	!function() {
 /******/ 		// getDefaultExport function for compatibility with non-harmony modules
 /******/ 		__webpack_require__.n = function(module) {
 /******/ 			var getter = module && module.__esModule ?
 /******/ 				function() { return module['default']; } :
 /******/ 				function() { return module; };
 /******/ 			__webpack_require__.d(getter, { a: getter });
 /******/ 			return getter;
 /******/ 		};
 /******/ 	}();
 /******/ 	
 /******/ 	/* webpack/runtime/define property getters */
 /******/ 	!function() {
 /******/ 		// define getter functions for harmony exports
 /******/ 		__webpack_require__.d = function(exports, definition) {
 /******/ 			for(var key in definition) {
 /******/ 				if(__webpack_require__.o(definition, key) && !__webpack_require__.o(exports, key)) {
 /******/ 					Object.defineProperty(exports, key, { enumerable: true, get: definition[key] });
 /******/ 				}
 /******/ 			}
 /******/ 		};
 /******/ 	}();
 /******/ 	
 /******/ 	/* webpack/runtime/hasOwnProperty shorthand */
 /******/ 	!function() {
 /******/ 		__webpack_require__.o = function(obj, prop) { return Object.prototype.hasOwnProperty.call(obj, prop); }
 /******/ 	}();
 /******/ 	
 /************************************************************************/
 var __webpack_exports__ = {};
 // This entry need to be wrapped in an IIFE because it need to be isolated against other modules in the chunk.
 !function() {
 /* harmony import */ var katex__WEBPACK_IMPORTED_MODULE_0__ = __webpack_require__(771);
 /* harmony import */ var katex__WEBPACK_IMPORTED_MODULE_0___default = /*#__PURE__*/__webpack_require__.n(katex__WEBPACK_IMPORTED_MODULE_0__);

 var scripts = document.body.getElementsByTagName("script");
 scripts = Array.prototype.slice.call(scripts);
 scripts.forEach(function (script) {
  if (!script.type || !script.type.match(/math\/tex/i)) {
    return -1;
  }

  var display = script.type.match(/mode\s*=\s*display(;|\s|\n|$)/) != null;
  var katexElement = document.createElement(display ? "div" : "span");
  katexElement.setAttribute("class", display ? "equation" : "inline-equation");

  try {
    katex__WEBPACK_IMPORTED_MODULE_0___default().render(script.text, katexElement, {
      displayMode: display
    });
  } catch (err) {
    //console.error(err); linter doesn't like this
    katexElement.textContent = script.text;
  }

  script.parentNode.replaceChild(katexElement, script);
 });
 }();
 __webpack_exports__ = __webpack_exports__["default"];
 /******/ 	return __webpack_exports__;
 /******/ })()
 ;
 });
--- a/public/js/katex/contrib/mathtex-script-type.min.js
+++ b/public/js/katex/contrib/mathtex-script-type.min.js
@ -0,0 +1 @@
 !function(e,t){if("object"==typeof exports&&"object"==typeof module)module.exports=t(require("katex"));else if("function"==typeof define&&define.amd)define(["katex"],t);else{var r="object"==typeof exports?t(require("katex")):t(e.katex);for(var n in r)("object"==typeof exports?exports:e)[n]=r[n]}}("undefined"!=typeof self?self:this,(function(e){return function(){"use strict";var t={771:function(t){t.exports=e}},r={};function n(e){var o=r[e];if(void 0!==o)return o.exports;var i=r[e]={exports:{}};return t[e](i,i.exports,n),i.exports}n.n=function(e){var t=e&&e.__esModule?function(){return e.default}:function(){return e};return n.d(t,{a:t}),t},n.d=function(e,t){for(var r in t)n.o(t,r)&&!n.o(e,r)&&Object.defineProperty(e,r,{enumerable:!0,get:t[r]})},n.o=function(e,t){return Object.prototype.hasOwnProperty.call(e,t)};var o,i,a,u={};return o=n(771),i=n.n(o),a=document.body.getElementsByTagName("script"),(a=Array.prototype.slice.call(a)).forEach((function(e){if(!e.type||!e.type.match(/math\/tex/i))return-1;var t=null!=e.type.match(/mode\s*=\s*display(;|\s|\n|$)/),r=document.createElement(t?"div":"span");r.setAttribute("class",t?"equation":"inline-equation");try{i().render(e.text,r,{displayMode:t})}catch(t){r.textContent=e.text}e.parentNode.replaceChild(r,e)})),u=u.default}()}));
--- a/public/js/katex/contrib/mathtex-script-type.mjs
+++ b/public/js/katex/contrib/mathtex-script-type.mjs
@ -0,0 +1,24 @@
 import katex from '../katex.mjs';

 var scripts = document.body.getElementsByTagName("script");
 scripts = Array.prototype.slice.call(scripts);
 scripts.forEach(function (script) {
  if (!script.type || !script.type.match(/math\/tex/i)) {
    return -1;
  }

  var display = script.type.match(/mode\s*=\s*display(;|\s|\n|$)/) != null;
  var katexElement = document.createElement(display ? "div" : "span");
  katexElement.setAttribute("class", display ? "equation" : "inline-equation");

  try {
    katex.render(script.text, katexElement, {
      displayMode: display
    });
  } catch (err) {
    //console.error(err); linter doesn't like this
    katexElement.textContent = script.text;
  }

  script.parentNode.replaceChild(katexElement, script);
 });
--- a/public/js/katex/contrib/mhchem.js
+++ b/public/js/katex/contrib/mhchem.js
--- a/public/js/katex/contrib/mhchem.min.js
+++ b/public/js/katex/contrib/mhchem.min.js
--- a/public/js/katex/contrib/mhchem.mjs
+++ b/public/js/katex/contrib/mhchem.mjs
--- a/public/js/katex/contrib/render-a11y-string.js
+++ b/public/js/katex/contrib/render-a11y-string.js
@ -0,0 +1,875 @@
 (function webpackUniversalModuleDefinition(root, factory) {
 	if(typeof exports === 'object' && typeof module === 'object')
 		module.exports = factory(require("katex"));
 	else if(typeof define === 'function' && define.amd)
 		define(["katex"], factory);
 	else {
 		var a = typeof exports === 'object' ? factory(require("katex")) : factory(root["katex"]);
 		for(var i in a) (typeof exports === 'object' ? exports : root)[i] = a[i];
 	}
 })((typeof self !== 'undefined' ? self : this), function(__WEBPACK_EXTERNAL_MODULE__771__) {
 return /******/ (function() { // webpackBootstrap
 /******/ 	"use strict";
 /******/ 	var __webpack_modules__ = ({

 /***/ 771:
 /***/ (function(module) {

 module.exports = __WEBPACK_EXTERNAL_MODULE__771__;

 /***/ })

 /******/ 	});
 /************************************************************************/
 /******/ 	// The module cache
 /******/ 	var __webpack_module_cache__ = {};
 /******/ 	
 /******/ 	// The require function
 /******/ 	function __webpack_require__(moduleId) {
 /******/ 		// Check if module is in cache
 /******/ 		var cachedModule = __webpack_module_cache__[moduleId];
 /******/ 		if (cachedModule !== undefined) {
 /******/ 			return cachedModule.exports;
 /******/ 		}
 /******/ 		// Create a new module (and put it into the cache)
 /******/ 		var module = __webpack_module_cache__[moduleId] = {
 /******/ 			// no module.id needed
 /******/ 			// no module.loaded needed
 /******/ 			exports: {}
 /******/ 		};
 /******/ 	
 /******/ 		// Execute the module function
 /******/ 		__webpack_modules__[moduleId](module, module.exports, __webpack_require__);
 /******/ 	
 /******/ 		// Return the exports of the module
 /******/ 		return module.exports;
 /******/ 	}
 /******/ 	
 /************************************************************************/
 /******/ 	/* webpack/runtime/compat get default export */
 /******/ 	!function() {
 /******/ 		// getDefaultExport function for compatibility with non-harmony modules
 /******/ 		__webpack_require__.n = function(module) {
 /******/ 			var getter = module && module.__esModule ?
 /******/ 				function() { return module['default']; } :
 /******/ 				function() { return module; };
 /******/ 			__webpack_require__.d(getter, { a: getter });
 /******/ 			return getter;
 /******/ 		};
 /******/ 	}();
 /******/ 	
 /******/ 	/* webpack/runtime/define property getters */
 /******/ 	!function() {
 /******/ 		// define getter functions for harmony exports
 /******/ 		__webpack_require__.d = function(exports, definition) {
 /******/ 			for(var key in definition) {
 /******/ 				if(__webpack_require__.o(definition, key) && !__webpack_require__.o(exports, key)) {
 /******/ 					Object.defineProperty(exports, key, { enumerable: true, get: definition[key] });
 /******/ 				}
 /******/ 			}
 /******/ 		};
 /******/ 	}();
 /******/ 	
 /******/ 	/* webpack/runtime/hasOwnProperty shorthand */
 /******/ 	!function() {
 /******/ 		__webpack_require__.o = function(obj, prop) { return Object.prototype.hasOwnProperty.call(obj, prop); }
 /******/ 	}();
 /******/ 	
 /************************************************************************/
 var __webpack_exports__ = {};
 // This entry need to be wrapped in an IIFE because it need to be isolated against other modules in the chunk.
 !function() {
 /* harmony import */ var katex__WEBPACK_IMPORTED_MODULE_0__ = __webpack_require__(771);
 /* harmony import */ var katex__WEBPACK_IMPORTED_MODULE_0___default = /*#__PURE__*/__webpack_require__.n(katex__WEBPACK_IMPORTED_MODULE_0__);
 /**
 * renderA11yString returns a readable string.
 *
 * In some cases the string will have the proper semantic math
 * meaning,:
 *   renderA11yString("\\frac{1}{2}"")
 *   -> "start fraction, 1, divided by, 2, end fraction"
 *
 * However, other cases do not:
 *   renderA11yString("f(x) = x^2")
 *   -> "f, left parenthesis, x, right parenthesis, equals, x, squared"
 *
 * The commas in the string aim to increase ease of understanding
 * when read by a screenreader.
 */
 // NOTE: since we're importing types here these files won't actually be
 // included in the build.
 // $FlowIgnore: we import the types directly anyways

 var stringMap = {
  "(": "left parenthesis",
  ")": "right parenthesis",
  "[": "open bracket",
  "]": "close bracket",
  "\\{": "left brace",
  "\\}": "right brace",
  "\\lvert": "open vertical bar",
  "\\rvert": "close vertical bar",
  "|": "vertical bar",
  "\\uparrow": "up arrow",
  "\\Uparrow": "up arrow",
  "\\downarrow": "down arrow",
  "\\Downarrow": "down arrow",
  "\\updownarrow": "up down arrow",
  "\\leftarrow": "left arrow",
  "\\Leftarrow": "left arrow",
  "\\rightarrow": "right arrow",
  "\\Rightarrow": "right arrow",
  "\\langle": "open angle",
  "\\rangle": "close angle",
  "\\lfloor": "open floor",
  "\\rfloor": "close floor",
  "\\int": "integral",
  "\\intop": "integral",
  "\\lim": "limit",
  "\\ln": "natural log",
  "\\log": "log",
  "\\sin": "sine",
  "\\cos": "cosine",
  "\\tan": "tangent",
  "\\cot": "cotangent",
  "\\sum": "sum",
  "/": "slash",
  ",": "comma",
  ".": "point",
  "-": "negative",
  "+": "plus",
  "~": "tilde",
  ":": "colon",
  "?": "question mark",
  "'": "apostrophe",
  "\\%": "percent",
  " ": "space",
  "\\ ": "space",
  "\\$": "dollar sign",
  "\\angle": "angle",
  "\\degree": "degree",
  "\\circ": "circle",
  "\\vec": "vector",
  "\\triangle": "triangle",
  "\\pi": "pi",
  "\\prime": "prime",
  "\\infty": "infinity",
  "\\alpha": "alpha",
  "\\beta": "beta",
  "\\gamma": "gamma",
  "\\omega": "omega",
  "\\theta": "theta",
  "\\sigma": "sigma",
  "\\lambda": "lambda",
  "\\tau": "tau",
  "\\Delta": "delta",
  "\\delta": "delta",
  "\\mu": "mu",
  "\\rho": "rho",
  "\\nabla": "del",
  "\\ell": "ell",
  "\\ldots": "dots",
  // TODO: add entries for all accents
  "\\hat": "hat",
  "\\acute": "acute"
 };
 var powerMap = {
  "prime": "prime",
  "degree": "degrees",
  "circle": "degrees",
  "2": "squared",
  "3": "cubed"
 };
 var openMap = {
  "|": "open vertical bar",
  ".": ""
 };
 var closeMap = {
  "|": "close vertical bar",
  ".": ""
 };
 var binMap = {
  "+": "plus",
  "-": "minus",
  "\\pm": "plus minus",
  "\\cdot": "dot",
  "*": "times",
  "/": "divided by",
  "\\times": "times",
  "\\div": "divided by",
  "\\circ": "circle",
  "\\bullet": "bullet"
 };
 var relMap = {
  "=": "equals",
  "\\approx": "approximately equals",
  "≠": "does not equal",
  "\\geq": "is greater than or equal to",
  "\\ge": "is greater than or equal to",
  "\\leq": "is less than or equal to",
  "\\le": "is less than or equal to",
  ">": "is greater than",
  "<": "is less than",
  "\\leftarrow": "left arrow",
  "\\Leftarrow": "left arrow",
  "\\rightarrow": "right arrow",
  "\\Rightarrow": "right arrow",
  ":": "colon"
 };
 var accentUnderMap = {
  "\\underleftarrow": "left arrow",
  "\\underrightarrow": "right arrow",
  "\\underleftrightarrow": "left-right arrow",
  "\\undergroup": "group",
  "\\underlinesegment": "line segment",
  "\\utilde": "tilde"
 };

 var buildString = function buildString(str, type, a11yStrings) {
  if (!str) {
    return;
  }

  var ret;

  if (type === "open") {
    ret = str in openMap ? openMap[str] : stringMap[str] || str;
  } else if (type === "close") {
    ret = str in closeMap ? closeMap[str] : stringMap[str] || str;
  } else if (type === "bin") {
    ret = binMap[str] || str;
  } else if (type === "rel") {
    ret = relMap[str] || str;
  } else {
    ret = stringMap[str] || str;
  } // If the text to add is a number and there is already a string
  // in the list and the last string is a number then we should
  // combine them into a single number


  if (/^\d+$/.test(ret) && a11yStrings.length > 0 && // TODO(kevinb): check that the last item in a11yStrings is a string
  // I think we might be able to drop the nested arrays, which would make
  // this easier to type
  // $FlowFixMe
  /^\d+$/.test(a11yStrings[a11yStrings.length - 1])) {
    a11yStrings[a11yStrings.length - 1] += ret;
  } else if (ret) {
    a11yStrings.push(ret);
  }
 };

 var buildRegion = function buildRegion(a11yStrings, callback) {
  var regionStrings = [];
  a11yStrings.push(regionStrings);
  callback(regionStrings);
 };

 var handleObject = function handleObject(tree, a11yStrings, atomType) {
  // Everything else is assumed to be an object...
  switch (tree.type) {
    case "accent":
      {
        buildRegion(a11yStrings, function (a11yStrings) {
          buildA11yStrings(tree.base, a11yStrings, atomType);
          a11yStrings.push("with");
          buildString(tree.label, "normal", a11yStrings);
          a11yStrings.push("on top");
        });
        break;
      }

    case "accentUnder":
      {
        buildRegion(a11yStrings, function (a11yStrings) {
          buildA11yStrings(tree.base, a11yStrings, atomType);
          a11yStrings.push("with");
          buildString(accentUnderMap[tree.label], "normal", a11yStrings);
          a11yStrings.push("underneath");
        });
        break;
      }

    case "accent-token":
      {
        // Used internally by accent symbols.
        break;
      }

    case "atom":
      {
        var text = tree.text;

        switch (tree.family) {
          case "bin":
            {
              buildString(text, "bin", a11yStrings);
              break;
            }

          case "close":
            {
              buildString(text, "close", a11yStrings);
              break;
            }
          // TODO(kevinb): figure out what should be done for inner

          case "inner":
            {
              buildString(tree.text, "inner", a11yStrings);
              break;
            }

          case "open":
            {
              buildString(text, "open", a11yStrings);
              break;
            }

          case "punct":
            {
              buildString(text, "punct", a11yStrings);
              break;
            }

          case "rel":
            {
              buildString(text, "rel", a11yStrings);
              break;
            }

          default:
            {
              tree.family;
              throw new Error("\"" + tree.family + "\" is not a valid atom type");
            }
        }

        break;
      }

    case "color":
      {
        var color = tree.color.replace(/katex-/, "");
        buildRegion(a11yStrings, function (regionStrings) {
          regionStrings.push("start color " + color);
          buildA11yStrings(tree.body, regionStrings, atomType);
          regionStrings.push("end color " + color);
        });
        break;
      }

    case "color-token":
      {
        // Used by \color, \colorbox, and \fcolorbox but not directly rendered.
        // It's a leaf node and has no children so just break.
        break;
      }

    case "delimsizing":
      {
        if (tree.delim && tree.delim !== ".") {
          buildString(tree.delim, "normal", a11yStrings);
        }

        break;
      }

    case "genfrac":
      {
        buildRegion(a11yStrings, function (regionStrings) {
          // genfrac can have unbalanced delimiters
          var leftDelim = tree.leftDelim,
              rightDelim = tree.rightDelim; // NOTE: Not sure if this is a safe assumption
          // hasBarLine true -> fraction, false -> binomial

          if (tree.hasBarLine) {
            regionStrings.push("start fraction");
            leftDelim && buildString(leftDelim, "open", regionStrings);
            buildA11yStrings(tree.numer, regionStrings, atomType);
            regionStrings.push("divided by");
            buildA11yStrings(tree.denom, regionStrings, atomType);
            rightDelim && buildString(rightDelim, "close", regionStrings);
            regionStrings.push("end fraction");
          } else {
            regionStrings.push("start binomial");
            leftDelim && buildString(leftDelim, "open", regionStrings);
            buildA11yStrings(tree.numer, regionStrings, atomType);
            regionStrings.push("over");
            buildA11yStrings(tree.denom, regionStrings, atomType);
            rightDelim && buildString(rightDelim, "close", regionStrings);
            regionStrings.push("end binomial");
          }
        });
        break;
      }

    case "hbox":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "kern":
      {
        // No op: we don't attempt to present kerning information
        // to the screen reader.
        break;
      }

    case "leftright":
      {
        buildRegion(a11yStrings, function (regionStrings) {
          buildString(tree.left, "open", regionStrings);
          buildA11yStrings(tree.body, regionStrings, atomType);
          buildString(tree.right, "close", regionStrings);
        });
        break;
      }

    case "leftright-right":
      {
        // TODO: double check that this is a no-op
        break;
      }

    case "lap":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "mathord":
      {
        buildString(tree.text, "normal", a11yStrings);
        break;
      }

    case "op":
      {
        var body = tree.body,
            name = tree.name;

        if (body) {
          buildA11yStrings(body, a11yStrings, atomType);
        } else if (name) {
          buildString(name, "normal", a11yStrings);
        }

        break;
      }

    case "op-token":
      {
        // Used internally by operator symbols.
        buildString(tree.text, atomType, a11yStrings);
        break;
      }

    case "ordgroup":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "overline":
      {
        buildRegion(a11yStrings, function (a11yStrings) {
          a11yStrings.push("start overline");
          buildA11yStrings(tree.body, a11yStrings, atomType);
          a11yStrings.push("end overline");
        });
        break;
      }

    case "phantom":
      {
        a11yStrings.push("empty space");
        break;
      }

    case "raisebox":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "rule":
      {
        a11yStrings.push("rectangle");
        break;
      }

    case "sizing":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "spacing":
      {
        a11yStrings.push("space");
        break;
      }

    case "styling":
      {
        // We ignore the styling and just pass through the contents
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "sqrt":
      {
        buildRegion(a11yStrings, function (regionStrings) {
          var body = tree.body,
              index = tree.index;

          if (index) {
            var indexString = flatten(buildA11yStrings(index, [], atomType)).join(",");

            if (indexString === "3") {
              regionStrings.push("cube root of");
              buildA11yStrings(body, regionStrings, atomType);
              regionStrings.push("end cube root");
              return;
            }

            regionStrings.push("root");
            regionStrings.push("start index");
            buildA11yStrings(index, regionStrings, atomType);
            regionStrings.push("end index");
            return;
          }

          regionStrings.push("square root of");
          buildA11yStrings(body, regionStrings, atomType);
          regionStrings.push("end square root");
        });
        break;
      }

    case "supsub":
      {
        var base = tree.base,
            sub = tree.sub,
            sup = tree.sup;
        var isLog = false;

        if (base) {
          buildA11yStrings(base, a11yStrings, atomType);
          isLog = base.type === "op" && base.name === "\\log";
        }

        if (sub) {
          var regionName = isLog ? "base" : "subscript";
          buildRegion(a11yStrings, function (regionStrings) {
            regionStrings.push("start " + regionName);
            buildA11yStrings(sub, regionStrings, atomType);
            regionStrings.push("end " + regionName);
          });
        }

        if (sup) {
          buildRegion(a11yStrings, function (regionStrings) {
            var supString = flatten(buildA11yStrings(sup, [], atomType)).join(",");

            if (supString in powerMap) {
              regionStrings.push(powerMap[supString]);
              return;
            }

            regionStrings.push("start superscript");
            buildA11yStrings(sup, regionStrings, atomType);
            regionStrings.push("end superscript");
          });
        }

        break;
      }

    case "text":
      {
        // TODO: handle other fonts
        if (tree.font === "\\textbf") {
          buildRegion(a11yStrings, function (regionStrings) {
            regionStrings.push("start bold text");
            buildA11yStrings(tree.body, regionStrings, atomType);
            regionStrings.push("end bold text");
          });
          break;
        }

        buildRegion(a11yStrings, function (regionStrings) {
          regionStrings.push("start text");
          buildA11yStrings(tree.body, regionStrings, atomType);
          regionStrings.push("end text");
        });
        break;
      }

    case "textord":
      {
        buildString(tree.text, atomType, a11yStrings);
        break;
      }

    case "smash":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "enclose":
      {
        // TODO: create a map for these.
        // TODO: differentiate between a body with a single atom, e.g.
        // "cancel a" instead of "start cancel, a, end cancel"
        if (/cancel/.test(tree.label)) {
          buildRegion(a11yStrings, function (regionStrings) {
            regionStrings.push("start cancel");
            buildA11yStrings(tree.body, regionStrings, atomType);
            regionStrings.push("end cancel");
          });
          break;
        } else if (/box/.test(tree.label)) {
          buildRegion(a11yStrings, function (regionStrings) {
            regionStrings.push("start box");
            buildA11yStrings(tree.body, regionStrings, atomType);
            regionStrings.push("end box");
          });
          break;
        } else if (/sout/.test(tree.label)) {
          buildRegion(a11yStrings, function (regionStrings) {
            regionStrings.push("start strikeout");
            buildA11yStrings(tree.body, regionStrings, atomType);
            regionStrings.push("end strikeout");
          });
          break;
        } else if (/phase/.test(tree.label)) {
          buildRegion(a11yStrings, function (regionStrings) {
            regionStrings.push("start phase angle");
            buildA11yStrings(tree.body, regionStrings, atomType);
            regionStrings.push("end phase angle");
          });
          break;
        }

        throw new Error("KaTeX-a11y: enclose node with " + tree.label + " not supported yet");
      }

    case "vcenter":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "vphantom":
      {
        throw new Error("KaTeX-a11y: vphantom not implemented yet");
      }

    case "hphantom":
      {
        throw new Error("KaTeX-a11y: hphantom not implemented yet");
      }

    case "operatorname":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "array":
      {
        throw new Error("KaTeX-a11y: array not implemented yet");
      }

    case "raw":
      {
        throw new Error("KaTeX-a11y: raw not implemented yet");
      }

    case "size":
      {
        // Although there are nodes of type "size" in the parse tree, they have
        // no semantic meaning and should be ignored.
        break;
      }

    case "url":
      {
        throw new Error("KaTeX-a11y: url not implemented yet");
      }

    case "tag":
      {
        throw new Error("KaTeX-a11y: tag not implemented yet");
      }

    case "verb":
      {
        buildString("start verbatim", "normal", a11yStrings);
        buildString(tree.body, "normal", a11yStrings);
        buildString("end verbatim", "normal", a11yStrings);
        break;
      }

    case "environment":
      {
        throw new Error("KaTeX-a11y: environment not implemented yet");
      }

    case "horizBrace":
      {
        buildString("start " + tree.label.slice(1), "normal", a11yStrings);
        buildA11yStrings(tree.base, a11yStrings, atomType);
        buildString("end " + tree.label.slice(1), "normal", a11yStrings);
        break;
      }

    case "infix":
      {
        // All infix nodes are replace with other nodes.
        break;
      }

    case "includegraphics":
      {
        throw new Error("KaTeX-a11y: includegraphics not implemented yet");
      }

    case "font":
      {
        // TODO: callout the start/end of specific fonts
        // TODO: map \BBb{N} to "the naturals" or something like that
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "href":
      {
        throw new Error("KaTeX-a11y: href not implemented yet");
      }

    case "cr":
      {
        // This is used by environments.
        throw new Error("KaTeX-a11y: cr not implemented yet");
      }

    case "underline":
      {
        buildRegion(a11yStrings, function (a11yStrings) {
          a11yStrings.push("start underline");
          buildA11yStrings(tree.body, a11yStrings, atomType);
          a11yStrings.push("end underline");
        });
        break;
      }

    case "xArrow":
      {
        throw new Error("KaTeX-a11y: xArrow not implemented yet");
      }

    case "cdlabel":
      {
        throw new Error("KaTeX-a11y: cdlabel not implemented yet");
      }

    case "cdlabelparent":
      {
        throw new Error("KaTeX-a11y: cdlabelparent not implemented yet");
      }

    case "mclass":
      {
        // \neq and \ne are macros so we let "htmlmathml" render the mathmal
        // side of things and extract the text from that.
        var _atomType = tree.mclass.slice(1); // $FlowFixMe: drop the leading "m" from the values in mclass


        buildA11yStrings(tree.body, a11yStrings, _atomType);
        break;
      }

    case "mathchoice":
      {
        // TODO: track which which style we're using, e.g. dispaly, text, etc.
        // default to text style if even that may not be the correct style
        buildA11yStrings(tree.text, a11yStrings, atomType);
        break;
      }

    case "htmlmathml":
      {
        buildA11yStrings(tree.mathml, a11yStrings, atomType);
        break;
      }

    case "middle":
      {
        buildString(tree.delim, atomType, a11yStrings);
        break;
      }

    case "internal":
      {
        // internal nodes are never included in the parse tree
        break;
      }

    case "html":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    default:
      tree.type;
      throw new Error("KaTeX a11y un-recognized type: " + tree.type);
  }
 };

 var buildA11yStrings = function buildA11yStrings(tree, a11yStrings, atomType) {
  if (a11yStrings === void 0) {
    a11yStrings = [];
  }

  if (tree instanceof Array) {
    for (var i = 0; i < tree.length; i++) {
      buildA11yStrings(tree[i], a11yStrings, atomType);
    }
  } else {
    handleObject(tree, a11yStrings, atomType);
  }

  return a11yStrings;
 };

 var flatten = function flatten(array) {
  var result = [];
  array.forEach(function (item) {
    if (item instanceof Array) {
      result = result.concat(flatten(item));
    } else {
      result.push(item);
    }
  });
  return result;
 };

 var renderA11yString = function renderA11yString(text, settings) {
  var tree = katex__WEBPACK_IMPORTED_MODULE_0___default().__parse(text, settings);

  var a11yStrings = buildA11yStrings(tree, [], "normal");
  return flatten(a11yStrings).join(", ");
 };

 /* harmony default export */ __webpack_exports__["default"] = (renderA11yString);
 }();
 __webpack_exports__ = __webpack_exports__["default"];
 /******/ 	return __webpack_exports__;
 /******/ })()
 ;
 });
--- a/public/js/katex/contrib/render-a11y-string.min.js
+++ b/public/js/katex/contrib/render-a11y-string.min.js
--- a/public/js/katex/contrib/render-a11y-string.mjs
+++ b/public/js/katex/contrib/render-a11y-string.mjs
@ -0,0 +1,794 @@
 import katex from '../katex.mjs';

 /**
 * renderA11yString returns a readable string.
 *
 * In some cases the string will have the proper semantic math
 * meaning,:
 *   renderA11yString("\\frac{1}{2}"")
 *   -> "start fraction, 1, divided by, 2, end fraction"
 *
 * However, other cases do not:
 *   renderA11yString("f(x) = x^2")
 *   -> "f, left parenthesis, x, right parenthesis, equals, x, squared"
 *
 * The commas in the string aim to increase ease of understanding
 * when read by a screenreader.
 */
 var stringMap = {
  "(": "left parenthesis",
  ")": "right parenthesis",
  "[": "open bracket",
  "]": "close bracket",
  "\\{": "left brace",
  "\\}": "right brace",
  "\\lvert": "open vertical bar",
  "\\rvert": "close vertical bar",
  "|": "vertical bar",
  "\\uparrow": "up arrow",
  "\\Uparrow": "up arrow",
  "\\downarrow": "down arrow",
  "\\Downarrow": "down arrow",
  "\\updownarrow": "up down arrow",
  "\\leftarrow": "left arrow",
  "\\Leftarrow": "left arrow",
  "\\rightarrow": "right arrow",
  "\\Rightarrow": "right arrow",
  "\\langle": "open angle",
  "\\rangle": "close angle",
  "\\lfloor": "open floor",
  "\\rfloor": "close floor",
  "\\int": "integral",
  "\\intop": "integral",
  "\\lim": "limit",
  "\\ln": "natural log",
  "\\log": "log",
  "\\sin": "sine",
  "\\cos": "cosine",
  "\\tan": "tangent",
  "\\cot": "cotangent",
  "\\sum": "sum",
  "/": "slash",
  ",": "comma",
  ".": "point",
  "-": "negative",
  "+": "plus",
  "~": "tilde",
  ":": "colon",
  "?": "question mark",
  "'": "apostrophe",
  "\\%": "percent",
  " ": "space",
  "\\ ": "space",
  "\\$": "dollar sign",
  "\\angle": "angle",
  "\\degree": "degree",
  "\\circ": "circle",
  "\\vec": "vector",
  "\\triangle": "triangle",
  "\\pi": "pi",
  "\\prime": "prime",
  "\\infty": "infinity",
  "\\alpha": "alpha",
  "\\beta": "beta",
  "\\gamma": "gamma",
  "\\omega": "omega",
  "\\theta": "theta",
  "\\sigma": "sigma",
  "\\lambda": "lambda",
  "\\tau": "tau",
  "\\Delta": "delta",
  "\\delta": "delta",
  "\\mu": "mu",
  "\\rho": "rho",
  "\\nabla": "del",
  "\\ell": "ell",
  "\\ldots": "dots",
  // TODO: add entries for all accents
  "\\hat": "hat",
  "\\acute": "acute"
 };
 var powerMap = {
  "prime": "prime",
  "degree": "degrees",
  "circle": "degrees",
  "2": "squared",
  "3": "cubed"
 };
 var openMap = {
  "|": "open vertical bar",
  ".": ""
 };
 var closeMap = {
  "|": "close vertical bar",
  ".": ""
 };
 var binMap = {
  "+": "plus",
  "-": "minus",
  "\\pm": "plus minus",
  "\\cdot": "dot",
  "*": "times",
  "/": "divided by",
  "\\times": "times",
  "\\div": "divided by",
  "\\circ": "circle",
  "\\bullet": "bullet"
 };
 var relMap = {
  "=": "equals",
  "\\approx": "approximately equals",
  "≠": "does not equal",
  "\\geq": "is greater than or equal to",
  "\\ge": "is greater than or equal to",
  "\\leq": "is less than or equal to",
  "\\le": "is less than or equal to",
  ">": "is greater than",
  "<": "is less than",
  "\\leftarrow": "left arrow",
  "\\Leftarrow": "left arrow",
  "\\rightarrow": "right arrow",
  "\\Rightarrow": "right arrow",
  ":": "colon"
 };
 var accentUnderMap = {
  "\\underleftarrow": "left arrow",
  "\\underrightarrow": "right arrow",
  "\\underleftrightarrow": "left-right arrow",
  "\\undergroup": "group",
  "\\underlinesegment": "line segment",
  "\\utilde": "tilde"
 };

 var buildString = (str, type, a11yStrings) => {
  if (!str) {
    return;
  }

  var ret;

  if (type === "open") {
    ret = str in openMap ? openMap[str] : stringMap[str] || str;
  } else if (type === "close") {
    ret = str in closeMap ? closeMap[str] : stringMap[str] || str;
  } else if (type === "bin") {
    ret = binMap[str] || str;
  } else if (type === "rel") {
    ret = relMap[str] || str;
  } else {
    ret = stringMap[str] || str;
  } // If the text to add is a number and there is already a string
  // in the list and the last string is a number then we should
  // combine them into a single number


  if (/^\d+$/.test(ret) && a11yStrings.length > 0 && // TODO(kevinb): check that the last item in a11yStrings is a string
  // I think we might be able to drop the nested arrays, which would make
  // this easier to type
  // $FlowFixMe
  /^\d+$/.test(a11yStrings[a11yStrings.length - 1])) {
    a11yStrings[a11yStrings.length - 1] += ret;
  } else if (ret) {
    a11yStrings.push(ret);
  }
 };

 var buildRegion = (a11yStrings, callback) => {
  var regionStrings = [];
  a11yStrings.push(regionStrings);
  callback(regionStrings);
 };

 var handleObject = (tree, a11yStrings, atomType) => {
  // Everything else is assumed to be an object...
  switch (tree.type) {
    case "accent":
      {
        buildRegion(a11yStrings, a11yStrings => {
          buildA11yStrings(tree.base, a11yStrings, atomType);
          a11yStrings.push("with");
          buildString(tree.label, "normal", a11yStrings);
          a11yStrings.push("on top");
        });
        break;
      }

    case "accentUnder":
      {
        buildRegion(a11yStrings, a11yStrings => {
          buildA11yStrings(tree.base, a11yStrings, atomType);
          a11yStrings.push("with");
          buildString(accentUnderMap[tree.label], "normal", a11yStrings);
          a11yStrings.push("underneath");
        });
        break;
      }

    case "accent-token":
      {
        // Used internally by accent symbols.
        break;
      }

    case "atom":
      {
        var {
          text
        } = tree;

        switch (tree.family) {
          case "bin":
            {
              buildString(text, "bin", a11yStrings);
              break;
            }

          case "close":
            {
              buildString(text, "close", a11yStrings);
              break;
            }
          // TODO(kevinb): figure out what should be done for inner

          case "inner":
            {
              buildString(tree.text, "inner", a11yStrings);
              break;
            }

          case "open":
            {
              buildString(text, "open", a11yStrings);
              break;
            }

          case "punct":
            {
              buildString(text, "punct", a11yStrings);
              break;
            }

          case "rel":
            {
              buildString(text, "rel", a11yStrings);
              break;
            }

          default:
            {
              tree.family;
              throw new Error("\"" + tree.family + "\" is not a valid atom type");
            }
        }

        break;
      }

    case "color":
      {
        var color = tree.color.replace(/katex-/, "");
        buildRegion(a11yStrings, regionStrings => {
          regionStrings.push("start color " + color);
          buildA11yStrings(tree.body, regionStrings, atomType);
          regionStrings.push("end color " + color);
        });
        break;
      }

    case "color-token":
      {
        // Used by \color, \colorbox, and \fcolorbox but not directly rendered.
        // It's a leaf node and has no children so just break.
        break;
      }

    case "delimsizing":
      {
        if (tree.delim && tree.delim !== ".") {
          buildString(tree.delim, "normal", a11yStrings);
        }

        break;
      }

    case "genfrac":
      {
        buildRegion(a11yStrings, regionStrings => {
          // genfrac can have unbalanced delimiters
          var {
            leftDelim,
            rightDelim
          } = tree; // NOTE: Not sure if this is a safe assumption
          // hasBarLine true -> fraction, false -> binomial

          if (tree.hasBarLine) {
            regionStrings.push("start fraction");
            leftDelim && buildString(leftDelim, "open", regionStrings);
            buildA11yStrings(tree.numer, regionStrings, atomType);
            regionStrings.push("divided by");
            buildA11yStrings(tree.denom, regionStrings, atomType);
            rightDelim && buildString(rightDelim, "close", regionStrings);
            regionStrings.push("end fraction");
          } else {
            regionStrings.push("start binomial");
            leftDelim && buildString(leftDelim, "open", regionStrings);
            buildA11yStrings(tree.numer, regionStrings, atomType);
            regionStrings.push("over");
            buildA11yStrings(tree.denom, regionStrings, atomType);
            rightDelim && buildString(rightDelim, "close", regionStrings);
            regionStrings.push("end binomial");
          }
        });
        break;
      }

    case "hbox":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "kern":
      {
        // No op: we don't attempt to present kerning information
        // to the screen reader.
        break;
      }

    case "leftright":
      {
        buildRegion(a11yStrings, regionStrings => {
          buildString(tree.left, "open", regionStrings);
          buildA11yStrings(tree.body, regionStrings, atomType);
          buildString(tree.right, "close", regionStrings);
        });
        break;
      }

    case "leftright-right":
      {
        // TODO: double check that this is a no-op
        break;
      }

    case "lap":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "mathord":
      {
        buildString(tree.text, "normal", a11yStrings);
        break;
      }

    case "op":
      {
        var {
          body,
          name
        } = tree;

        if (body) {
          buildA11yStrings(body, a11yStrings, atomType);
        } else if (name) {
          buildString(name, "normal", a11yStrings);
        }

        break;
      }

    case "op-token":
      {
        // Used internally by operator symbols.
        buildString(tree.text, atomType, a11yStrings);
        break;
      }

    case "ordgroup":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "overline":
      {
        buildRegion(a11yStrings, function (a11yStrings) {
          a11yStrings.push("start overline");
          buildA11yStrings(tree.body, a11yStrings, atomType);
          a11yStrings.push("end overline");
        });
        break;
      }

    case "phantom":
      {
        a11yStrings.push("empty space");
        break;
      }

    case "raisebox":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "rule":
      {
        a11yStrings.push("rectangle");
        break;
      }

    case "sizing":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "spacing":
      {
        a11yStrings.push("space");
        break;
      }

    case "styling":
      {
        // We ignore the styling and just pass through the contents
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "sqrt":
      {
        buildRegion(a11yStrings, regionStrings => {
          var {
            body,
            index
          } = tree;

          if (index) {
            var indexString = flatten(buildA11yStrings(index, [], atomType)).join(",");

            if (indexString === "3") {
              regionStrings.push("cube root of");
              buildA11yStrings(body, regionStrings, atomType);
              regionStrings.push("end cube root");
              return;
            }

            regionStrings.push("root");
            regionStrings.push("start index");
            buildA11yStrings(index, regionStrings, atomType);
            regionStrings.push("end index");
            return;
          }

          regionStrings.push("square root of");
          buildA11yStrings(body, regionStrings, atomType);
          regionStrings.push("end square root");
        });
        break;
      }

    case "supsub":
      {
        var {
          base,
          sub,
          sup
        } = tree;
        var isLog = false;

        if (base) {
          buildA11yStrings(base, a11yStrings, atomType);
          isLog = base.type === "op" && base.name === "\\log";
        }

        if (sub) {
          var regionName = isLog ? "base" : "subscript";
          buildRegion(a11yStrings, function (regionStrings) {
            regionStrings.push("start " + regionName);
            buildA11yStrings(sub, regionStrings, atomType);
            regionStrings.push("end " + regionName);
          });
        }

        if (sup) {
          buildRegion(a11yStrings, function (regionStrings) {
            var supString = flatten(buildA11yStrings(sup, [], atomType)).join(",");

            if (supString in powerMap) {
              regionStrings.push(powerMap[supString]);
              return;
            }

            regionStrings.push("start superscript");
            buildA11yStrings(sup, regionStrings, atomType);
            regionStrings.push("end superscript");
          });
        }

        break;
      }

    case "text":
      {
        // TODO: handle other fonts
        if (tree.font === "\\textbf") {
          buildRegion(a11yStrings, function (regionStrings) {
            regionStrings.push("start bold text");
            buildA11yStrings(tree.body, regionStrings, atomType);
            regionStrings.push("end bold text");
          });
          break;
        }

        buildRegion(a11yStrings, function (regionStrings) {
          regionStrings.push("start text");
          buildA11yStrings(tree.body, regionStrings, atomType);
          regionStrings.push("end text");
        });
        break;
      }

    case "textord":
      {
        buildString(tree.text, atomType, a11yStrings);
        break;
      }

    case "smash":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "enclose":
      {
        // TODO: create a map for these.
        // TODO: differentiate between a body with a single atom, e.g.
        // "cancel a" instead of "start cancel, a, end cancel"
        if (/cancel/.test(tree.label)) {
          buildRegion(a11yStrings, function (regionStrings) {
            regionStrings.push("start cancel");
            buildA11yStrings(tree.body, regionStrings, atomType);
            regionStrings.push("end cancel");
          });
          break;
        } else if (/box/.test(tree.label)) {
          buildRegion(a11yStrings, function (regionStrings) {
            regionStrings.push("start box");
            buildA11yStrings(tree.body, regionStrings, atomType);
            regionStrings.push("end box");
          });
          break;
        } else if (/sout/.test(tree.label)) {
          buildRegion(a11yStrings, function (regionStrings) {
            regionStrings.push("start strikeout");
            buildA11yStrings(tree.body, regionStrings, atomType);
            regionStrings.push("end strikeout");
          });
          break;
        } else if (/phase/.test(tree.label)) {
          buildRegion(a11yStrings, function (regionStrings) {
            regionStrings.push("start phase angle");
            buildA11yStrings(tree.body, regionStrings, atomType);
            regionStrings.push("end phase angle");
          });
          break;
        }

        throw new Error("KaTeX-a11y: enclose node with " + tree.label + " not supported yet");
      }

    case "vcenter":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "vphantom":
      {
        throw new Error("KaTeX-a11y: vphantom not implemented yet");
      }

    case "hphantom":
      {
        throw new Error("KaTeX-a11y: hphantom not implemented yet");
      }

    case "operatorname":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "array":
      {
        throw new Error("KaTeX-a11y: array not implemented yet");
      }

    case "raw":
      {
        throw new Error("KaTeX-a11y: raw not implemented yet");
      }

    case "size":
      {
        // Although there are nodes of type "size" in the parse tree, they have
        // no semantic meaning and should be ignored.
        break;
      }

    case "url":
      {
        throw new Error("KaTeX-a11y: url not implemented yet");
      }

    case "tag":
      {
        throw new Error("KaTeX-a11y: tag not implemented yet");
      }

    case "verb":
      {
        buildString("start verbatim", "normal", a11yStrings);
        buildString(tree.body, "normal", a11yStrings);
        buildString("end verbatim", "normal", a11yStrings);
        break;
      }

    case "environment":
      {
        throw new Error("KaTeX-a11y: environment not implemented yet");
      }

    case "horizBrace":
      {
        buildString("start " + tree.label.slice(1), "normal", a11yStrings);
        buildA11yStrings(tree.base, a11yStrings, atomType);
        buildString("end " + tree.label.slice(1), "normal", a11yStrings);
        break;
      }

    case "infix":
      {
        // All infix nodes are replace with other nodes.
        break;
      }

    case "includegraphics":
      {
        throw new Error("KaTeX-a11y: includegraphics not implemented yet");
      }

    case "font":
      {
        // TODO: callout the start/end of specific fonts
        // TODO: map \BBb{N} to "the naturals" or something like that
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    case "href":
      {
        throw new Error("KaTeX-a11y: href not implemented yet");
      }

    case "cr":
      {
        // This is used by environments.
        throw new Error("KaTeX-a11y: cr not implemented yet");
      }

    case "underline":
      {
        buildRegion(a11yStrings, function (a11yStrings) {
          a11yStrings.push("start underline");
          buildA11yStrings(tree.body, a11yStrings, atomType);
          a11yStrings.push("end underline");
        });
        break;
      }

    case "xArrow":
      {
        throw new Error("KaTeX-a11y: xArrow not implemented yet");
      }

    case "cdlabel":
      {
        throw new Error("KaTeX-a11y: cdlabel not implemented yet");
      }

    case "cdlabelparent":
      {
        throw new Error("KaTeX-a11y: cdlabelparent not implemented yet");
      }

    case "mclass":
      {
        // \neq and \ne are macros so we let "htmlmathml" render the mathmal
        // side of things and extract the text from that.
        var _atomType = tree.mclass.slice(1); // $FlowFixMe: drop the leading "m" from the values in mclass


        buildA11yStrings(tree.body, a11yStrings, _atomType);
        break;
      }

    case "mathchoice":
      {
        // TODO: track which which style we're using, e.g. dispaly, text, etc.
        // default to text style if even that may not be the correct style
        buildA11yStrings(tree.text, a11yStrings, atomType);
        break;
      }

    case "htmlmathml":
      {
        buildA11yStrings(tree.mathml, a11yStrings, atomType);
        break;
      }

    case "middle":
      {
        buildString(tree.delim, atomType, a11yStrings);
        break;
      }

    case "internal":
      {
        // internal nodes are never included in the parse tree
        break;
      }

    case "html":
      {
        buildA11yStrings(tree.body, a11yStrings, atomType);
        break;
      }

    default:
      tree.type;
      throw new Error("KaTeX a11y un-recognized type: " + tree.type);
  }
 };

 var buildA11yStrings = function buildA11yStrings(tree, a11yStrings, atomType) {
  if (a11yStrings === void 0) {
    a11yStrings = [];
  }

  if (tree instanceof Array) {
    for (var i = 0; i < tree.length; i++) {
      buildA11yStrings(tree[i], a11yStrings, atomType);
    }
  } else {
    handleObject(tree, a11yStrings, atomType);
  }

  return a11yStrings;
 };

 var flatten = function flatten(array) {
  var result = [];
  array.forEach(function (item) {
    if (item instanceof Array) {
      result = result.concat(flatten(item));
    } else {
      result.push(item);
    }
  });
  return result;
 };

 var renderA11yString = function renderA11yString(text, settings) {
  var tree = katex.__parse(text, settings);

  var a11yStrings = buildA11yStrings(tree, [], "normal");
  return flatten(a11yStrings).join(", ");
 };

 export { renderA11yString as default };
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--- a/public/kzg-batch-proof.html
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@ -21,7 +21,7 @@

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@ -67,47 +67,39 @@

 <h4>Batch proof</h4>

 <p>The benefit of <em>batch proof</em> is that allows us to proof multiple points while the proof size remains constant to one $\mathbb{G}_1$ point.
 Let $(z_0, y_0), (z_1, y_1), &hellip;, (z_k, y_k)$ be the points that we want to proof, and that we have a polynomial $p(x)$ that goes through the points.
 The <em>commitment</em> to the polynomial stands the same than for single proofs: $c=[p(\tau)]_1$.</p>
 <p>The benefit of <em>batch proof</em> is that allows us to proof multiple points while the proof size remains constant to one <span class="math inline">\(\mathbb{G}_1\)</span> point.
 Let <span class="math inline">\((z_0, y_0), (z_1, y_1), ..., (z_k, y_k)\)</span> be the points that we want to proof, and that we have a polynomial <span class="math inline">\(p(x)\)</span> that goes through the points.
 The <em>commitment</em> to the polynomial stands the same than for single proofs: <span class="math inline">\(c=[p(\tau)]_1\)</span>.</p>

 <p>For the evaluation proof, while in the single proofs we compute $q(x) = \frac{p(x)-y}{x-z}$, we will replace $y$ and $x-z$ by the following two polynomials.
 The constant $y$ is replaced by a polynomial that has roots at all the points that we want to prove. This is achieved by computing the <a href="/blog/shamir-secret-sharing.html#lagrange-polynomial%20interpolation">Lagrange interpolation</a> for the given set of points:</p>

 <p>$$
 <p>For the evaluation proof, while in the single proofs we compute <span class="math inline">\(q(x) = \frac{p(x)-y}{x-z}\)</span>, we will replace <span class="math inline">\(y\)</span> and <span class="math inline">\(x-z\)</span> by the following two polynomials.
 The constant <span class="math inline">\(y\)</span> is replaced by a polynomial that has roots at all the points that we want to prove. This is achieved by computing the <a href="/blog/shamir-secret-sharing.html#lagrange-polynomial%20interpolation">Lagrange interpolation</a> for the given set of points:</p>
 <p><span class="math display">\[
 I(x) = \sum_{j=0}^k y_j l_j(x)\newline
 where \space\space\space l_j(x) = \prod_{0\leq m \leq k} \frac{x-x_m}{x_j - x_m}
 $$</p>

 <p>And the $x-z$, which was to ensure that $q(x)$ had a root at $z$, now, as we want to ensure that $q(x)$ has roots at all the points of the commitment, we will use the <em>zero polynomial</em>:
 $$
 where \space\space\space l_j(x) = \prod\_{0\leq m \leq k} \frac{x-x_m}{x_j - x_m}
 \]</span></p><p>And the <span class="math inline">\(x-z\)</span>, which was to ensure that <span class="math inline">\(q(x)\)</span> had a root at <span class="math inline">\(z\)</span>, now, as we want to ensure that <span class="math inline">\(q(x)\)</span> has roots at all the points of the commitment, we will use the <em>zero polynomial</em>:</p>
 <p><span class="math display">\[
 Z(x) = \prod_{i=0}^{k} x-z_i =\newline
 =(x-z_0)(x-z_1)&hellip;(x-z_k)
 $$</p>
 =(x-z_0)(x-z_1)...(x-z_k)
 \]</span></p><p>This polynomial ensures that when <span class="math inline">\(x=z_i\)</span> (<span class="math inline">\(z_i\)</span> being one of our points), the polynomial evaluation will be zero.</p>

 <p>This polynomial ensures that when $x=z_i$ ($z_i$ being one of our points), the polynomial evaluation will be zero.</p>
 <p>Now we can put <span class="math inline">\(I(x)\)</span> and <span class="math inline">\(Z(x)\)</span> in place, obtaining <span class="math inline">\(q(x)=\frac{p(x)-I(x)}{Z(x)}\)</span>. And the batch proof evaluation is obtained by <span class="math inline">\(\pi=[q(\tau)]_1\)</span>.</p>

 <p>Now we can put $I(x)$ and $Z(x)$ in place, obtaining $q(x)=\frac{p(x)-I(x)}{Z(x)}$. And the batch proof evaluation is obtained by $\pi=[q(\tau)]_1$.</p>

 <p>The verification is quite similar than what we did for single proofs, but using the mentioned $z(x)$ and $I(x)$:
 $$
 <p>The verification is quite similar than what we did for single proofs, but using the mentioned <span class="math inline">\(z(x)\)</span> and <span class="math inline">\(I(x)\)</span>:</p>
 <p><span class="math display">\[
 \hat{e}(\pi, [Z(\tau)]_2) == \hat{e}(c - [I(\tau)]_1, H)
 $$</p>

 <p>Which, as we did with the single proofs in the previous post, we can unroll it and see that:
 $$
 \]</span></p><p>Which, as we did with the single proofs in the previous post, we can unroll it and see that:</p>
 <p><span class="math display">\[
 \hat{e}(\pi, [Z(\tau)]_2) == \hat{e}(c - [I(\tau)]_1, H)\newline
 \Rightarrow \hat{e}([q(\tau)]_1, [Z(\tau)]_2) == \hat{e}([p(\tau)]_1 - [I(\tau)]_1, H)\newline
 \Rightarrow [q(\tau) \cdot Z(\tau)]_T == [p(\tau) - I(\tau)]_T
 $$
 From where we see that is the equation $q(x)\cdot Z(x)=p(x)-I(x)$, which can be expressed as $q(x) = \frac{p(x) - I(x)}{Z(x)}$, evaluated at $\tau$ from the trusted setup, which is not known: $q(\tau) = \frac{p(\tau) - I(\tau)}{Z(\tau)}$.</p>
 \]</span></p><p>From where we see that is the equation <span class="math inline">\(q(x)\cdot Z(x)=p(x)-I(x)\)</span>, which can be expressed as <span class="math inline">\(q(x) = \frac{p(x) - I(x)}{Z(x)}\)</span>, evaluated at <span class="math inline">\(\tau\)</span> from the trusted setup, which is not known: <span class="math inline">\(q(\tau) = \frac{p(\tau) - I(\tau)}{Z(\tau)}\)</span>.</p>

 <h4>Vector commitments</h4>

 <p>As mentioned earlier, this scheme can be used as a <em>vector commitment</em> scheme.</p>

 <p>A vector commitment allows a prover to commit to a vector and later proof that a certain value belongs to that vector. As a traditional example, we can think of <em>Merkle Trees</em>, where we can commit to a vector of values, which are placed in the tree leafs. Then we compute the <em>root</em> which acts as the commitment. Then we can provide a proof that a certain leaf belongs to the vector for the commitment (<em>root</em>) that we showed earlier.
 The problem with Merkle Trees is that the proof size grows linearly with the size of the tree, as it contains the siblings from the leaf to the root. Here is where <em>KZG Commitments</em> can be benefitial, as the proof always stands with the same size, one $\mathbb{G}_1$ point, no matter of how many points we are batching.</p>
 The problem with Merkle Trees is that the proof size grows linearly with the size of the tree, as it contains the siblings from the leaf to the root. Here is where <em>KZG Commitments</em> can be benefitial, as the proof always stands with the same size, one <span class="math inline">\(\mathbb{G}_1\)</span> point, no matter of how many points we are batching.</p>

 <p>We can use KZG Commitments as a vector commitment scheme by mapping the <em>vector</em> as a batch of points that build the polynomial, so when commiting to the polynomial we are commiting to the vector. Then, it&rsquo;s a matter of using the <em>batch proof</em> approach explained above in order to proof multiple elements of the vector in a single proof that can be verified later.</p>

@ -144,8 +136,8 @@ The problem with Merkle Trees is that the proof size grows linearly with the siz
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@ -155,6 +147,8 @@ The problem with Merkle Trees is that the proof size grows linearly with the siz
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--- a/public/kzg-commitments.html
+++ b/public/kzg-commitments.html
@ -21,7 +21,7 @@

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@ -83,67 +83,53 @@

 <p>First of all, we need to generate a <em>Trusted Setup</em> that will be used later in the rest of steps. Here, the concept of <em>Trusted Setup</em> is quite similar to what we are familiar when dealing with other zk protocols such zkSNARKs, but with the advantage that for the <em>KZG Commitments</em> the nature of its <em>Trusted Setup</em> allows to have some kind of &lsquo;global&rsquo; <em>Trusted Setup</em> that can be used for different polynomials.</p>

 <p>It should be computed in a <em>Multi-Party Computation</em> (<em>MPC</em>) fashion, and ensuring that at least one of the participants is honest, in order to ensure that the original parameter $\tau$ can not be restored.</p>
 <p>It should be computed in a <em>Multi-Party Computation</em> (<em>MPC</em>) fashion, and ensuring that at least one of the participants is honest, in order to ensure that the original parameter <span class="math inline">\(\tau\)</span> can not be restored.</p>

 <p>The parameters of the <em>Trusted Setup</em> are generated by generating a random $\tau \in \mathbb{F}_p$, and from this parameter we can compute $[\tau^i]_1$ and $[\tau^i]_2$ for $i=0,&hellip;,n-1$:</p>

 <p>$$
 [\tau^i]_1 = ([\tau^0]_1, [\tau^1]_1, [\tau^2]_1, &hellip;, [\tau^{n-1}]_1)\newline
 [\tau^i]_2 = ([\tau^0]_2, [\tau^1]_2, [\tau^2]_2, &hellip;, [\tau^{n-1}]_2)
 $$</p>

 <p>Which in additive representation is:
 $$
 (G, \tau G, \tau^2 G, &hellip;, \tau^{n-1} G) \in \mathbb{G}_1\newline
 (H, \tau H, \tau^2 H, &hellip;, \tau^{n-1} H) \in \mathbb{G}_2
 $$</p>

 <p>The &lsquo;intuition&rsquo; about the <em>Trusted Setup</em> is that is like encrypting a secret value ($\tau$) that later will be used in the &lsquo;encrypted&rsquo; form to evaluate the polynomials.</p>
 <p>The parameters of the <em>Trusted Setup</em> are generated by generating a random <span class="math inline">\(\tau \in \mathbb{F}_p\)</span>, and from this parameter we can compute <span class="math inline">\([\tau^i]_1\)</span> and <span class="math inline">\([\tau^i]_2\)</span> for <span class="math inline">\(i=0,...,n-1\)</span>:</p>
 <p><span class="math display">\[
 [\tau^i]_1 = ([\tau^0]_1, [\tau^1]_1, [\tau^2]_1, ..., [\tau^{n-1}]_1)\newline
 [\tau^i]_2 = ([\tau^0]_2, [\tau^1]_2, [\tau^2]_2, ..., [\tau^{n-1}]_2)
 \]</span></p><p>Which in additive representation is:</p>
 <p><span class="math display">\[
 (G, \tau G, \tau^2 G, ..., \tau^{n-1} G) \in \mathbb{G}_1\newline
 (H, \tau H, \tau^2 H, ..., \tau^{n-1} H) \in \mathbb{G}_2
 \]</span></p><p>The &lsquo;intuition&rsquo; about the <em>Trusted Setup</em> is that is like encrypting a secret value (<span class="math inline">\(\tau\)</span>) that later will be used in the &lsquo;encrypted&rsquo; form to evaluate the polynomials.</p>

 <h4>Commitments</h4>

 <p>A commitment to a polynomial $p(x) = \sum^n_{i=0} p_i x^i$ is done by computing</p>

 <p>$$c=[p(\tau)]_1$$</p>

 <p>which is computed by $c = \sum^{deg(p(x))}_{i=0} [\tau^i] \cdot p_i$.</p>
 <p>A commitment to a polynomial <span class="math inline">\(p(x) = \sum^n_{i=0} p_i x^i\)</span> is done by computing</p>
 <p><span class="math display">\[c=[p(\tau)]_1\]</span></p><p>which is computed by <span class="math inline">\(c = \sum^{deg(p(x))}_{i=0} [\tau^i] \cdot p_i\)</span>.</p>

 <p>The prover would send the commitment to the polynomial $c$, and then the verifier would choose a value $z \in \mathbb{F}_p$, where $\mathbb{F}_p$ is the finite field of the polynomial.</p>
 <p>The prover would send the commitment to the polynomial <span class="math inline">\(c\)</span>, and then the verifier would choose a value <span class="math inline">\(z \in \mathbb{F}_p\)</span>, where <span class="math inline">\(\mathbb{F}_p\)</span> is the finite field of the polynomial.</p>

 <h4>Evalutaion proofs</h4>

 <p>To prove an evaluation of the polynomial at the choosen value $z$ such that $p(z)=y$, a quotient polynomial is computed: $q(x) = \frac{p(x)-y}{x-z}$. This polynomial is the proof that $p(z)=y$, as if $q$ exists it means that $p(x)-y$ is divisible by $x-z$, which means that it has a root at $z$, being $p(z)-y=0$.</p>
 <p>To prove an evaluation of the polynomial at the choosen value <span class="math inline">\(z\)</span> such that <span class="math inline">\(p(z)=y\)</span>, a quotient polynomial is computed: <span class="math inline">\(q(x) = \frac{p(x)-y}{x-z}\)</span>. This polynomial is the proof that <span class="math inline">\(p(z)=y\)</span>, as if <span class="math inline">\(q\)</span> exists it means that <span class="math inline">\(p(x)-y\)</span> is divisible by <span class="math inline">\(x-z\)</span>, which means that it has a root at <span class="math inline">\(z\)</span>, being <span class="math inline">\(p(z)-y=0\)</span>.</p>

 <p>Then, the evaluation proof is</p>
 <p><span class="math display">\[\pi = [q(\tau)]_1\]</span></p><p>which, as when computing <span class="math inline">\(c\)</span>, is computed by <span class="math inline">\(\pi=\sum^{deg(q(x))}_{i=0} [\tau^i] \cdot q_i\)</span>.</p>

 <p>$$\pi = [q(\tau)]_1$$</p>

 <p>which, as when computing $c$, is computed by $\pi=\sum^{deg(q(x))}_{i=0} [\tau^i] \cdot q_i$.</p>

 <p>Once computed, the prover would send this evaluation proof $\pi$ to the verifier.</p>
 <p>Once computed, the prover would send this evaluation proof <span class="math inline">\(\pi\)</span> to the verifier.</p>

 <h4>Verifying an evaluation proof</h4>

 <p>In order to verify an evaluation proof, the verifier has the commitment $c=[p(\tau)]_1$, the evaluation $y=p(z)$, and the proof $\pi=[q(\tau)]_1$.</p>
 <p>In order to verify an evaluation proof, the verifier has the commitment <span class="math inline">\(c=[p(\tau)]_1\)</span>, the evaluation <span class="math inline">\(y=p(z)\)</span>, and the proof <span class="math inline">\(\pi=[q(\tau)]_1\)</span>.</p>

 <p>So, the verifier can check the <a href="https://en.wikipedia.org/wiki/Pairing-based_cryptography">pairing</a> evaluation:
 $$\hat{e}(\pi, [\tau]_2 - [z]_2) == \hat{e}(c - [y]_1, H)$$</p>
 $<span class="math inline">\(\hat{e}(\pi, [\tau]_2 - [z]_2) == \hat{e}(c - [y]_1, H)\)</span>$</p>

 <p>Where $[\tau]_2$ comes from the Trusted Setup, $[z]_2$ is point at which the polynomial is evaluated, and $[y]_1$ is the claimed value p(z). And $\pi$ and $c$ are given by the prover.</p>
 <p>Where <span class="math inline">\([\tau]_2\)</span> comes from the Trusted Setup, <span class="math inline">\([z]_2\)</span> is point at which the polynomial is evaluated, and <span class="math inline">\([y]_1\)</span> is the claimed value p(z). And <span class="math inline">\(\pi\)</span> and <span class="math inline">\(c\)</span> are given by the prover.</p>

 <p>We can unroll that last equivalence, and see that:</p>

 <p>$$
 <p><span class="math display">\[
 \hat{e}(\pi, [\tau]_2 - [z]_2) == \hat{e}(c - [y]_1, H)\newline
 \Rightarrow \hat{e}([q(\tau)]_1, [\tau-z]_2) == \hat{e}([p(\tau)]_1 - [y]_1, H)\newline
 \Rightarrow [q(\tau) \cdot (\tau-z)]_T == [p(\tau) - y]_T
 $$</p>

 <p>We can see that is the equation $q(x)(x-z)=p(x)-y$, which can be expressed as $q(x) = \frac{p(x) - y}{x-z}$, evaluated at $\tau$ from the <em>trusted setup</em>, which is not known: $q(\tau) = \frac{p(\tau) - y}{\tau-z}$.</p>
 \]</span></p><p>We can see that is the equation <span class="math inline">\(q(x)(x-z)=p(x)-y\)</span>, which can be expressed as <span class="math inline">\(q(x) = \frac{p(x) - y}{x-z}\)</span>, evaluated at <span class="math inline">\(\tau\)</span> from the <em>trusted setup</em>, which is not known: <span class="math inline">\(q(\tau) = \frac{p(\tau) - y}{\tau-z}\)</span>.</p>

 <h3>Conclusions</h3>

 <p>The content covered in this notes is just a quick overview, but allows us to see the potential of the scheme. One next iteration from what we&rsquo;ve seen is the approach to do batch proofs, which allows us to evaluate at multiple points with a single evaluation proof. This scheme can be used as a <em>vector commitment</em>, using a polynomial where the $p(i) = x_i$ for all values of $x_i$ of the vector, which can be obtained from the $x_i$ values and computing the <a href="https://en.wikipedia.org/wiki/Lagrange_polynomial">Lagrange interpolation</a>. This is quite useful combined with the mentioned batch proofs. The <em>batch proofs</em> logic can be found at the <a href="https://arnaucube.com/blog/kzg-batch-proof.html">blog/kzg-batch-proof</a> notes (kind of the continuation of the current notes).</p>
 <p>The content covered in this notes is just a quick overview, but allows us to see the potential of the scheme. One next iteration from what we&rsquo;ve seen is the approach to do batch proofs, which allows us to evaluate at multiple points with a single evaluation proof. This scheme can be used as a <em>vector commitment</em>, using a polynomial where the <span class="math inline">\(p(i) = x_i\)</span> for all values of <span class="math inline">\(x_i\)</span> of the vector, which can be obtained from the <span class="math inline">\(x_i\)</span> values and computing the <a href="https://en.wikipedia.org/wiki/Lagrange_polynomial">Lagrange interpolation</a>. This is quite useful combined with the mentioned batch proofs. The <em>batch proofs</em> logic can be found at the <a href="https://arnaucube.com/blog/kzg-batch-proof.html">blog/kzg-batch-proof</a> notes (kind of the continuation of the current notes).</p>

 <p>As a final note, in order to try to digest the notes, I&rsquo;ve did a <em>toy implementation</em> of this scheme at <a href="https://github.com/arnaucube/kzg-commitments-study">https://github.com/arnaucube/kzg-commitments-study</a>. It&rsquo;s quite simple, but contains the logic overviewed in this notes.</p>

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 </blockquote>

 <p>Imagine that you have a <em>secret</em> (for example a <em>private key</em> that can decrypt a file), and you want to backup that <em>secret</em>. You can split the <em>secret</em> and give each slice to a different person, so when you need to reconstruct the <em>secret</em> you just need to put together all the parts. But, what happens if one of the parts gets corrupted, or is lost? The secret would not be recoverable.
 A better solution can be done if we use <em>Shamir Secret Sharing</em>, which allows us to split the <em>secret</em> in $k$ different parts, and set a minimum threshold $n$, which defines the number of required parts to recover the <em>secret</em>, so just by putting together any $n$ parts we will recover the original secret.</p>
 A better solution can be done if we use <em>Shamir Secret Sharing</em>, which allows us to split the <em>secret</em> in <span class="math inline">\(k\)</span> different parts, and set a minimum threshold <span class="math inline">\(n\)</span>, which defines the number of required parts to recover the <em>secret</em>, so just by putting together any <span class="math inline">\(n\)</span> parts we will recover the original secret.</p>

 <p>This has interesting applications, such as social recovery of keys or distributing a secret and ensuring that cooperation is needed in order to recover it. In the following lines we will overview the concepts behind this scheme.</p>

@ -72,76 +72,68 @@ A better solution can be done if we use Shamir Secret Sharing, which al

 <p>Lagrange interpolation is also used in many schemes that work with polynomials, for example in <a href="https://arnaucube.com/blog/kzg-batch-proof.html">KZG Commitments</a> (an actual implementation <a href="https://github.com/arnaucube/kzg-commitments-study/blob/master/arithmetic.go#L272">can be found here</a>).</p>

 <p>The main idea behind is the following: for any $n$ distinct points over $\mathbb{R}^2$, there is a unique polynomial $p(x) \in \mathbb{R[x]}$ of degree $n-1$ which goes through all of them.
 From the &lsquo;other side&rsquo; point of view, this means that if we have a polynomial of degree $n-1$, we can take $n$ points (or more) from it, and we will be able to recover the original polynomial from those $n$ points.</p>
 <p>The main idea behind is the following: for any <span class="math inline">\(n\)</span> distinct points over <span class="math inline">\(\mathbb{R}^2\)</span>, there is a unique polynomial <span class="math inline">\(p(x) \in \mathbb{R[x]}\)</span> of degree <span class="math inline">\(n-1\)</span> which goes through all of them.
 From the &lsquo;other side&rsquo; point of view, this means that if we have a polynomial of degree <span class="math inline">\(n-1\)</span>, we can take <span class="math inline">\(n\)</span> points (or more) from it, and we will be able to recover the original polynomial from those <span class="math inline">\(n\)</span> points.</p>

 <p>We can see this starting with a line. If we are given any two points $P_0=(x_0, y_0)$ and $P_1=(x_1, y_1)$ from that line, we are able to recover the original line.</p>
 <p>We can see this starting with a line. If we are given any two points <span class="math inline">\(P_0=(x_0, y_0)\)</span> and <span class="math inline">\(P_1=(x_1, y_1)\)</span> from that line, we are able to recover the original line.</p>

 <div style="text-align:center;">
    <img style="width:300px;margin-bottom:20px;" src="img/posts/shamir-secret-sharing/line.png" />
 </div>

 <p>We can map this into the previous idea, seeing that our line is a degree $1$ polynomial, so, if we pick $2$ points from it, we later can recover the original line.</p>
 <p>We can map this into the previous idea, seeing that our line is a degree <span class="math inline">\(1\)</span> polynomial, so, if we pick <span class="math inline">\(2\)</span> points from it, we later can recover the original line.</p>

 <p>Same happens with polynomials of degree $2$. Let $p(x)$ be a polynomial of degree $2$ defined by $p(x)= x^2 - 5x - 6$. We can create infinity of polynomials of degree $2$ that go through $2$ points, but with 3 points there is a unique polynomial degree $2$</p>
 <p>Same happens with polynomials of degree <span class="math inline">\(2\)</span>. Let <span class="math inline">\(p(x)\)</span> be a polynomial of degree <span class="math inline">\(2\)</span> defined by <span class="math inline">\(p(x)= x^2 - 5x - 6\)</span>. We can create infinity of polynomials of degree <span class="math inline">\(2\)</span> that go through <span class="math inline">\(2\)</span> points, but with 3 points there is a unique polynomial degree <span class="math inline">\(2\)</span></p>

 <p>As the degree is $2$, if we pick $3$ points from the polynomial, we will be able to reconstruct it.
 <p>As the degree is <span class="math inline">\(2\)</span>, if we pick <span class="math inline">\(3\)</span> points from the polynomial, we will be able to reconstruct it.
 <div style="text-align:center;">
    <img style="width:300px;margin-bottom:20px;" src="img/posts/shamir-secret-sharing/degree2.png" />
 </div></p>

 <p>This is generalized by using <em>Lagrange polynomial interpolation</em>, which defines:</p>

 <p>For a set of points $(x_0, y_0), (x_1, y_1), &hellip;, (x_n, x_n)$,</p>

 <p>$$
 <p>For a set of points <span class="math inline">\((x_0, y_0), (x_1, y_1), ..., (x_n, x_n)\)</span>,</p>
 <p><span class="math display">\[
 I(x) = \sum_{i=0}^n y_i l_i(x)\newline
 where \space\space\space l_i(x) = \prod_{0\leq j \leq n, j\neq i} \frac{x-x_j}{x_i - x_j}
 $$</p>

 where \space\space\space l_i(x) = \prod\_{0\leq j \leq n, j\neq i} \frac{x-x_j}{x_i - x_j}
 \]</span></p>
 <h3>Shamir&rsquo;s secret sharing</h3>

 <p>As we&rsquo;ve seen, for a degree $n-1$ polynomial we can pick $n$ or more points and we will be able to reconstruct the original polynomial from it. This is the main idea used in <em>Shamir&rsquo;s secret sharing</em>.</p>
 <p>As we&rsquo;ve seen, for a degree <span class="math inline">\(n-1\)</span> polynomial we can pick <span class="math inline">\(n\)</span> or more points and we will be able to reconstruct the original polynomial from it. This is the main idea used in <em>Shamir&rsquo;s secret sharing</em>.</p>

 <p>Let $s$ be our secret. We want to generate $k$ pieces and set a threshold $n$ which is the minimum number of pieces that are needed to reconstruct the secret $s$. We can define a polynomial of degree $n-1$, and pick $k$ points from that polynomial, so in this way with just putting together $n$ points of $k$ we will be able to reconstruct the original polynomial. And, we can place our secret $s$ in the <em>constant term</em> of the polynomial (the one that has $x^0$), in this way, when we reconstruct the polynomial using $n$ out of $k$ points, we will be able to recover the secret $s$.</p>
 <p>Let <span class="math inline">\(s\)</span> be our secret. We want to generate <span class="math inline">\(k\)</span> pieces and set a threshold <span class="math inline">\(n\)</span> which is the minimum number of pieces that are needed to reconstruct the secret <span class="math inline">\(s\)</span>. We can define a polynomial of degree <span class="math inline">\(n-1\)</span>, and pick <span class="math inline">\(k\)</span> points from that polynomial, so in this way with just putting together <span class="math inline">\(n\)</span> points of <span class="math inline">\(k\)</span> we will be able to reconstruct the original polynomial. And, we can place our secret <span class="math inline">\(s\)</span> in the <em>constant term</em> of the polynomial (the one that has <span class="math inline">\(x^0\)</span>), in this way, when we reconstruct the polynomial using <span class="math inline">\(n\)</span> out of <span class="math inline">\(k\)</span> points, we will be able to recover the secret <span class="math inline">\(s\)</span>.</p>

 <p>We can see this with an example with actual numbers (we will use small numbers):
 Imagine that we want to generate $5$ pieces from our secret, and define that just by putting together $3$ of the pieces we can recover the secret, this means setting $n=3$ and $k=5$. Then we will generate a polynomial of degree $n-1=2$, by $p(x) = \alpha_0 + \alpha_1 x  + \alpha_2 x^2$, where $\alpha_0 = s$ (the secret).</p>
 Imagine that we want to generate <span class="math inline">\(5\)</span> pieces from our secret, and define that just by putting together <span class="math inline">\(3\)</span> of the pieces we can recover the secret, this means setting <span class="math inline">\(n=3\)</span> and <span class="math inline">\(k=5\)</span>. Then we will generate a polynomial of degree <span class="math inline">\(n-1=2\)</span>, by <span class="math inline">\(p(x) = \alpha_0 + \alpha_1 x  + \alpha_2 x^2\)</span>, where <span class="math inline">\(\alpha_0 = s\)</span> (the secret).</p>

 <p>We will work over a finite field of size $p$, where $p$ is a prime number. For our example we will work over $\mathbb{F}_{19}$, in real world we would work with much more bigger field. You can find an <a href="https://en.wikipedia.org/wiki/Shamir%27s_Secret_Sharing#Example">example without finite fields in Wikipedia</a>.</p>
 <p>We will work over a finite field of size <span class="math inline">\(p\)</span>, where <span class="math inline">\(p\)</span> is a prime number. For our example we will work over <span class="math inline">\(\mathbb{F}_{19}\)</span>, in real world we would work with much more bigger field. You can find an <a href="https://en.wikipedia.org/wiki/Shamir%27s_Secret_Sharing#Example">example without finite fields in Wikipedia</a>.</p>

 <p>Let our secret be $s=14$. We now generate our polynomial of degree $n-1=2$, where $s$ will be the constant coefficient: $p(x)= s + \alpha_1 x^1 + \alpha_2 x^2$. We can set $\alpha_1$ and $\alpha_2$ into any random value, as example $\alpha_1=4$ and $\alpha_2=6$. So we have our polynomial: $p(x) = 14 + 4 x + 6 x^2$.</p>
 <p>Let our secret be <span class="math inline">\(s=14\)</span>. We now generate our polynomial of degree <span class="math inline">\(n-1=2\)</span>, where <span class="math inline">\(s\)</span> will be the constant coefficient: <span class="math inline">\(p(x)= s + \alpha_1 x^1 + \alpha_2 x^2\)</span>. We can set <span class="math inline">\(\alpha_1\)</span> and <span class="math inline">\(\alpha_2\)</span> into any random value, as example <span class="math inline">\(\alpha_1=4\)</span> and <span class="math inline">\(\alpha_2=6\)</span>. So we have our polynomial: <span class="math inline">\(p(x) = 14 + 4 x + 6 x^2\)</span>.</p>

 <p>Now that we have the polynomial, we can pick $k$ points from it, using incremental indexes for the $x$ coordinate: $P_1=(1, p(1)), P_2=(2, p(2)), \space\ldots\space, P_k=(k, p(k))$. With the numbers of our example this is (remember, we work over $\mathbb{F}_{19}$):
 $$
 <p>Now that we have the polynomial, we can pick <span class="math inline">\(k\)</span> points from it, using incremental indexes for the <span class="math inline">\(x\)</span> coordinate: <span class="math inline">\(P_1=(1, p(1)), P_2=(2, p(2)), \space\ldots\space, P_k=(k, p(k))\)</span>. With the numbers of our example this is (remember, we work over <span class="math inline">\(\mathbb{F}\_{19}\)</span>):</p>
 <p><span class="math display">\[
 p(x) = 14 + 4 x + 6 x^2,\newline
 p(1)=14 + 4 \cdot 1 + 6 \cdot 1^2 = 24 \space (mod \space 19) = 5\newline
 p(2)=14 + 4 \cdot 2 + 6 \cdot 2^2 = 46 \space (mod \space 19) = 8\newline
 p(3)=14 + 4 \cdot 3 + 6 \cdot 3^2 = 80 \space (mod \space 19) = 4\newline
 p(4)=14 + 4 \cdot 4 + 6 \cdot 4^2 = 126 \space (mod \space 19) = 12\newline
 p(5)=14 + 4 \cdot 5 + 6 \cdot 5^2 = 184 \space (mod \space 19) = 13
 $$
 So our $k$ points are: $(1,5), (2,8), (3,4), (4,12), (5,13)$. We can distribute these points as our &lsquo;secret parts&rsquo;.
 In order to recover the secret, we need at least $n=3$ points, for example $P_1$, $P_3$, $P_5$, and we compute the <em>Lagrange polynomial interpolation</em> to recover the original polynomial (remember, we work over $\mathbb{F}_{19}$):</p>

 <p>$$
 \]</span></p><p>So our <span class="math inline">\(k\)</span> points are: <span class="math inline">\((1,5), (2,8), (3,4), (4,12), (5,13)\)</span>. We can distribute these points as our &lsquo;secret parts&rsquo;.
 In order to recover the secret, we need at least <span class="math inline">\(n=3\)</span> points, for example <span class="math inline">\(P_1\)</span>, <span class="math inline">\(P_3\)</span>, <span class="math inline">\(P_5\)</span>, and we compute the <em>Lagrange polynomial interpolation</em> to recover the original polynomial (remember, we work over <span class="math inline">\(\mathbb{F}\_{19}\)</span>):</p>
 <p><span class="math display">\[
 I(x) = \sum_{i=0}^n y_i l_i(x) \space\space
 where \space\space\space l_i(x) = \prod_{0 \leq j \leq n \ j\neq i} \frac{x-x_j}{x_i - x_j}
 $$
 $$
 where \space\space\space l_i(x) = \prod\_{0 \leq j \leq n \\ j\neq i} \frac{x-x_j}{x_i - x_j}
 \]</span></p><p><span class="math display">\[
 l_1(x) = \frac{x-3}{1-3} \cdot \frac{x-5}{1-5} = \frac{x-3}{17} \cdot \frac{x-5}{15}=\frac{x^2+11x+15}{8}\newline
 l_3(x) = \frac{x-1}{3-1} \cdot \frac{x-5}{3-5} = \frac{x-1}{2} \cdot \frac{x-5}{17} =\frac{x^2+13x+5}{15}\newline
 l_5(x) = \frac{x-1}{5-1} \cdot \frac{x-3}{5-3} = \frac{x-1}{4} \cdot \frac{x-3}{2} = \frac{x^2 + 15x + 3}{8}\newline
 $$
 $$
 \]</span></p><p><span class="math display">\[
 I(x) = y_2 \cdot l_2(x) + y_4 \cdot l_4(x) + y_5 \cdot l_5(x)\newline
 = 5 \cdot (\frac{x^2+11x+15}{8}) + 4 \cdot (\frac{x^2+13x+5}{15}) + 13 \cdot (\frac{x^2 +15x + 3}{8})\newline
 = \frac{5x^2+17x+18}{8} + \frac{4x^2+14x+1}{15} + \frac{13x^2+5x+1}{8}\newline
 = 3x^2+14x+7 + 18x^2+6x+14 + 4x^2+3x+12\newline
 = 6x^2 + 4x + 14
 $$</p>

 <p>We can now take the <em>constant coefficient</em>, or just evaluate the obtained polynomial at 0, $p(0) = 6 \cdot 0^2 + 4 \cdot 0 + 14 = 14$, and we obtain our original secret $s=14$.</p>
 \]</span></p><p>We can now take the <em>constant coefficient</em>, or just evaluate the obtained polynomial at 0, <span class="math inline">\(p(0) = 6 \cdot 0^2 + 4 \cdot 0 + 14 = 14\)</span>, and we obtain our original secret <span class="math inline">\(s=14\)</span>.</p>

 <h3>Conclusions</h3>

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