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+# Elliptic curves and JavaScript
+
+At my company we regularly have talks and other knowledge events.
+After attending many of these events, I decided to give a talk
+about [elliptic curves](https://en.wikipedia.org/wiki/Elliptic_curve),
+which I worked with during my PhD.
+
+The intended audience consists of software developers, but their
+background is mixed: some have studied Maths in university, but many did
+not. Most are not familiar with abstract algebra other than polynomials
+and matrices, let alone algebraic geometry. Considering all of this,
+I decided to give a presentation about
+[Lenstra elliptic curve factorization](https://en.wikipedia.org/wiki/Lenstra_elliptic-curve_factorization),
+so I could also show some code and do a practical demo.
+
+Since I gave a presentation on this topic to a different audience a
+couple of years ago, I could have simply adapted the slides and used the
+same code.  Instead, I took this as an opportunity to experiment with
+new tools and languages. This post is a summary of what I have learned
+while preparing this talk.
+
+All the code I talk about in this post, including the source for the
+slides, can be found in [this git repository](https://git.tronto.net/ecm).
+
+## The code
+
+For the practical part of the talk I had to write some code to demonstrate
+the factorization algorithm. It's not much, maybe 200 lines or so, and
+I already had a working Python version. It was a rather straightforward
+implementation, and it used exceptions to handle the "a factor was found"
+part of the algorithm; although this is not amazing coding style, it
+was faithful to how the algorithm was originally explained - or to how
+I wanted to explain it, anyway.
+
+It was not bad code, but I wanted to experiment with something new.
+
+### Adventures in C++
+
+Recently I have been learning C++, and I wanted to experiment with some
+of its more advanced features. So I decided to rewrite the whole thing.
+
+I had a clear idea of where I wanted to start: a small
+[library for modular arithmetic](https://git.tronto.net/zmodn/file/README.md.html),
+with compile-time fixed
+[modulus](https://en.wikipedia.org/wiki/Modular_arithmetic) via templates
+and heavy use of
+[type inference](https://en.wikipedia.org/wiki/Type_inference)
+for seamless operations between regular integers and integers modulo N.
+This endeavor was quite successful, and it taught me how to use
+templates and concepts, which I have talked about in
+[my previous blog post](../2025-01-21-taming-cpp-templates).
+
+When working on the previous part, I made sure that any kind of integer
+could be used as a base type for the modular integers, so I could use
+some custom big integer types to show off the power of the factorization
+algorithm with very large numbers. Unfortunately, I did not take into
+account that with my setup I needed a big integer class that supported
+*compile-time constants* - for example in the form of `constexpr`
+constructors. I could not find any, so I decided to write
+[my own big integer class](https://git.tronto.net/zmodn/file/bigint.h.html).
+This was less successful: implementing an *efficient* big integer library
+was not as straightforward as the modular integers library. I decided
+not to care about efficiency for the time being, but that would come
+back to bite me very soon.
+
+Finally, I put these two libraries together and implemented the elliptic
+curve factorization algorithm. This was not hard.  The only problem was
+that it was excruciatingly slow when I used large numbers, undoubtedly
+due to my half-assed big integer implementation.  I could restrict
+myself to using regular 64-bit integers, but then I would have to use
+to relatively small numbers, making the demo less interesting.  I looked
+online for other big integer libraries that I could use and I found
+[ctbignum](https://github.com/niekbouman/ctbignum), but I could not make
+it work together with my modular arithmetic class. The day of the
+presentation was approaching quickly, so I decided to go back to
+my original Python implementation instead.
+
+### Back to Python
+
+When I looked back at my old Python implementation, I found it nicer
+and cleaner than how I remembered it. I did not have to do much
+cleanup, it was pretty much ready to go, and much more readable than
+the C++ verion for anyone who is not a C++ expert - and probably for C++
+experts too. Moreover, Python's seamless use of large integers was
+exactly what I was missing from the C++ version.
+
+One of the few changes I made to this code was reworking a little bit
+the "elliptic curve point" class I used. If anything, this was a good
+excuse to learn about
+[dataclasses](https://docs.python.org/3/library/dataclasses.html). I also
+decided to add
+[type hints](https://docs.python.org/3/library/typing.html), which I
+have recently found out about.
+
+And with little work, the old code was ready to go!
+
+## The slides
+
+Compared to the code, the slides needed a few more adjustments.
+When I gave this talk the first time, it was for an audience of
+Math students at the end of their Bachelor program. I could freely
+use all that Math jargon that we Mathematicians like, such as
+"let K be a
+[field](https://en.wikipedia.org/wiki/Field_(mathematics))"
+and "E is a
+[projective](https://en.wikipedia.org/wiki/Projective_space) curve".
+But this time I had to phrase things differently. The content itself
+didn't need much change, but I had to use a more approachable language,
+at the cost of being a little less rigorous. For example, I could
+get rid of all the projective plane business by just saying "let's
+pretend that there is a point *at infinity*; trust me, the Math
+works out".
+
+The problem with changing the old slides is that I did not want to touch
+[LaTeX](https://nl.wikipedia.org/wiki/LaTeX)
+anymore. As a Mathematician I like it because it can do pretty much everything
+you need (did you know you can
+[draw diagrams programmatically](https://www.youtube.com/watch?v=mWqhB6qOIk0)?),
+but as a computer scientist I'd rather not deal with the mess that is a
+LaTeX installation.
+So what did I decide to use instead? HTML, CSS and a bit of JavaScript!
+
+### Math formulas with MathJax
+
+Even without LaTeX, I still needed a way to write Math formulas in my
+slides. One way to achieve this is using [MathJax](https://www.mathjax.org),
+which allows you to write LaTeX or
+[MathML](https://en.wikipedia.org/wiki/MathML) formulas direclty in
+your HTML, and have them rendered dynamically. A minimal example
+looks like this:
+
+```
+<!doctype html>
+<head>
+        <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js">
+        </script>
+</head>
+
+<body>
+        <p>Hello! \[ e^{\pi i} + 1 = 0 \]</p>
+</body>
+```
+
+In the example above I am including the MathJax library directly from a
+URL. This means every time you load that page, a request is sent from
+your browser to the MathJax server to get the code for rendering the
+formulas. Among other things, this implies that my slides are going
+to be dependent on this external website, and that they won't work
+offline. The horror!
+
+In theory I could install MathJax locally (or on my server) and get rid
+of this dependency, and maybe at some point I'll do it. But for now
+it is just easier and faster to include the script like this. And while
+I was at it, I doubled down on the remote library thing and included 
+also [highlight.js](https://highlightjs.org),
+a package for rendering code blocks with syntax highlighting.
+
+### Scrolling with JavaScript
+
+The other big feature I wanted for the slides was for them to look and
+behave like actual slides. So the whole HTML page should be divided into
+single frames, and I wanted to be able to move back and forth between
+frames by clicking or pressing a key.
+
+In order to do this, I had to write my first piece of JavaScript.
+The main part looks more or less like this:
+
+```
+const slides = document.querySelectorAll(".slide");
+
+const keysNext = ["ArrowRight", "ArrowDown", " "];
+const keysPrev = ["ArrowLeft", "ArrowUp"];
+
+// Disable default action of the navigation keys (e.g. scrolling).
+document.addEventListener("keydown", function(e) {
+        if (keysNext.includes(e.key) || keysPrev.includes(e.key)) {
+                e.preventDefault();
+        }
+});
+
+function goto(slide) {
+        slide.focus();
+        slide.scrollIntoView({
+                behavior: "instant",
+                block: "start"
+        });
+}
+
+function onkeydown(i, e) {
+        if (keysNext.includes(e.key) && i+1 < slides.length) {
+                goto(slides[i+1]);
+        }
+        if (keysPrev.includes(e.key) && i > 0) {
+                goto(slides[i-1]);
+        }
+}
+
+function onclick(i, e) {
+        const w = slides[i].offsetWidth;
+        const x = e.clientX;
+
+        if (x > w/2 && i+1 < slides.length) {
+                goto(slides[i+1]);
+        }
+        if (x < w/2 && i > 0) {
+                goto(slides[i-1]);
+        }
+}
+
+for (let i = 0; i < slides.length; i++) {
+        slides[i].addEventListener("keydown", e => onkeydown(i, e));
+        slides[i].addEventListener("click", e => onclick(i, e));
+}
+
+goto(slides[0]); // Go to the first slide when the presentation starts
+```
+
+First of all, every slide is a `div` with `class="slide"`. This allows
+me to select all the slides with `document.querySelectorAll(".slide")`.
+Then, after disabling any default handling of the arrows and space keys,
+I add a new event listeners to every slide. These listeners uses the
+`scrollIntoView()` function to scroll to the next or previous slide.
+The `onclick()` function similarly handles clicks, where a click on the
+left half of the screen goes to the previous slide and a click on the
+right half goes to the next one.
+
+And by the way, highjacking the default scrolling behavior is another
+trend of modern web development that I hate. By I am fine with using
+it here, because these slides are not meant to be a regular web page.
+
+I also a function to add a footer to every slide:
+
+```
+function slideFooter() {
+        const start = "<div class=\"footer\"><table class=\"footer-table\"><tr>";
+        const title = "Elliptic Curves and the ECM algorithm"
+        const link = "<a href=https://tronto.net/talks/ecm>tronto.net/talks/ecm</a>";
+        const end = "</tr></table></div>";
+        const content =
+                "<td class=\"footer-title\">" + title + "</td>" +
+                "<td class=\"footer-link\">" + link + "</td>";
+
+        return start + content + end;
+}
+
+// In the main loop:
+// slides[i].innerHTML += slideFooter()
+```
+
+In an older version I also had a small slide counter in the footer,
+but I decided not to use it in the end.
+
+Of course there was also some CSS work to
+do. A new thing I learned in this regard is the
+[flex](https://developer.mozilla.org/en-US/docs/Web/CSS/flex)
+layout property, with which I was able to easily arrange
+pieces of text and pictures in the slides - shoutout to
+my friend [Jared](https://guissmo.com) for telling me about
+it. Apart from this I don't have anything interesting to comment
+about the CSS part of the slides. You can check the full code
+[here](https://git.tronto.net/ecm/file/index.html.html) if you want;
+everything is in a single HTML file.
+
+The result looks fine, but it does not work perfectly will every screen
+resolution. It's fine in 4:3 or 16:9, but with wider screens your
+mileage may vary. I could improve it and always use the device height
+(or width, but not both) as a reference. But this is quite some work
+for very little gain, and in the end all I care about is that I can
+show these slides from my laptop. I am sorry if you are viewing this
+presentation from a smartphone.
+
+You can view the slides
+[on this page](https://sebastiano.tronto.net/talks/ecm).
+
+## Conclusion
+
+I didn't need to do all this work for this presentation, but it was fun to
+learn new stuff - not only for the C++ part, that I did not end up using
+anyway, but also for the slides. It's good to know a bit of JavaScript,
+even if I don't plan to use it much in the future.
+
+I scheduled this post to go online on the same day as my presentation -
+wish me good luck :)
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+<!doctype html>
+<html lang="en">
+<head>
+        <title>Elliptic Curves and the ECM algorithm</title>
+        <meta name="viewport" content="width=device-width" />
+
+        <!-- Import MathJax script -->
+        <script id="MathJax-script" async src=
+                "https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"
+        ></script>
+
+        <!-- Import highlight.js script and style sheet -->
+        <script id="highlight.js" src="
+                https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.9.0/highlight.min.js"
+        ></script>
+        <link rel="stylesheet" href="
+                https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.9.0/styles/vs.css"
+        >
+
+        <!-- Custom style -->
+        <style>
+                html { height: 100vh; width: 98vw; margin: auto; font-family: sans-serif; }
+                h1 { font-size: 3.5vw; background-color: #eeeeee; margin: 0; }
+                body { font-size: 2.2vw; height: 100%; width: 100%; }
+                a, a:visited { color: #0f2899; text-decoration: none; }
+                a:hover { text-decoration: underline; }
+                figcaption { text-align: center; font-size: 1.5vw; }
+                em { font-style: normal; color: blue; }
+
+                .slide { outline: none; height: 100vh; width: 100%; }
+                .slide {
+                        display: flex;
+                        flex-direction: column;
+                        justify-content: space-between;
+                }
+                .slide ul { margin-left: 1.5vw; }
+                .slide ol { margin-left: 1.5vw; }
+                .slide p { margin-left: 1.5vw; }
+
+                .slide.titlepage p, a { text-align: center; }
+                .slide.titlepage span.title { font-size: 3.6vw; font-weight: bold; }
+                .slide.titlepage span.author { }
+
+                .slide.ecsum img { width: 50%; display: block; margin: auto; }
+
+                .slide div.centertext { text-align: center; margin: auto; width: 60%; }
+
+                .columns {
+                        display: flex;
+                        flex-direction: row;
+                        justify-content: space-between;
+                        align-items: center;
+                }
+
+                .footer { font-size: 1.8vw; background-color: #eeeeee; }
+                .footer table { width: 100%; }
+                .footer-title { font-weight: bold; }
+                .footer-link { text-align: right; }
+        </style>
+        <meta charset="utf-8">
+</head>
+
+<body>
+
+<div class="slide titlepage" tabindex="-1"
+style="background-image: url('images/sum-2c.png');
+background-position: center;
+background-repeat: no-repeat;
+background-size: 70%;
+background-color: rgba(255, 255, 255, 0.85);
+background-blend-mode: overlay;">
+
+<p></p>
+
+<p><span class="title">Elliptic curves and the ECM algorithm<span></p>
+<p><span class="author">Sebastiano Tronto<span></p>
+<p><span class="event">ALTEN Scientific Software Evening<span></p>
+
+</div>
+
+<div class="slide titlepage" tabindex="-1"
+style="background-image: url('images/numbers.jpg');
+background-position: center;
+background-repeat: no-repeat;
+background-size: 100%;
+background-color: rgba(255, 255, 255, 0.75);
+background-blend-mode: overlay;">
+<p></p>
+<p><span class="title">Part I: Numbers<span></p>
+<p></p>
+</div>
+
+<div class="slide" tabindex="-1">
+<h1>The integers</h1>
+
+<img alt="The number line" src="images/number-line.png"
+        style="width: 70%; margin-left: 15%; margin-right: 15%;"/>
+
+<ul>
+<li>Operations: sum \(+\), difference \(-\) and multiplication \(\times\)</li>
+<li>Various properties: associativity, commutativity, etc...</li>
+<li>What about division (without remainder)?<br>
+If <tt>\(\frac ab\)</tt> is an integer we say that
+<tt>\(a\)</tt><em> divides </em><tt>\(b\)</tt> (in code:
+<tt>a % b == 0</tt>)</li>
+</ul>
+</div>
+
+<div class="slide" tabindex="-1">
+<h1>The integers modulo N</h1>
+
+<div class="columns">
+
+<ul>
+<!-- li>Integers modulo <tt>N</tt>: the possible remainders of
+division by <tt>N</tt><br-->
+<li>Two numbers are the same if they give the <em>same remainder</em>
+when divided by <tt>N</tt></li>
+<li>Think of <tt>int</tt>, but with <tt>% N</tt>
+after every operation</li>
+<li>Examples with \(N=12\):
+\[9+5\equiv 14\equiv 2\pmod{12}\]
+\[7-11\equiv-4\equiv 8\pmod{12}\]
+\[3\times 4\equiv 12\equiv 0\pmod{12}\]
+</li>
+</ul>
+
+<img alt="The number clock" src="images/clock.png"
+        style="width: 35%;"/>
+
+</div>
+
+</div>
+
+<div class="slide" tabindex="-1">
+<h1>The integers modulo N - Division</h1>
+
+<p style="text-align: center;"><strong>What about division?</strong></p>
+
+<ul>
+<li>
+Sometimes it works
+\[
+\frac 37\equiv 9 \pmod{12} \qquad
+\text{because} \qquad 9\times 7 \equiv 63 \equiv 3 \pmod{12}
+\]
+</li>
+
+<li>
+Sometimes it does not
+\[
+\frac 32\equiv \; ? \pmod{4} \qquad {\color{red}\text{Impossible!}}
+\]
+</li>
+</ul>
+</div>
+
+<div class="slide" tabindex="-1">
+<h1>Integers modulo N - Division</h1>
+
+<ul>
+<li>
+Sometimes it's... weird?
+\[
+\frac 62 \equiv \; ? \pmod{8} \quad
+\begin{array}{l}
+\rightarrow {\color{red}3} \times 2 \equiv 6\pmod{8}\\
+\rightarrow {\color{red}7} \times 2 \equiv 14 \equiv 6 \pmod{8}
+\end{array}
+\]
+</li>
+</div>
+
+<div class="slide" tabindex="-1">
+<h1>Integers modulo N - Division</h1>
+<div class="columns">
+<ul>
+<li>Can divide by \(a\) when \[\operatorname{GCD}(a, N)=1\]</li>
+<li>With the
+<a href="https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm">
+extended GCD algorithm</a> find \(x\) and \(y\) such that
+\[
+ax+Ny=1
+\]
+<li>
+This means \(\frac{1}{a}\equiv x\pmod{N}\)
+</li>
+</ul>
+<pre><code class="language-python"
+style="border: 0.2vw solid; font-size: 2.2vw;">
+def extended_gcd(a, b):
+    if b == 0:
+        return a, 1, 0
+    g, x, y = extended_gcd(b, a % b)
+    return g, y, x - y*(a // b)
+</code></pre>
+</div>
+<p style="text-align: center;">
+Division always works if \(N\) is a <em>prime</em> number!</p>
+</div>
+
+<div class="slide" tabindex="-1">
+<h1>Modular arithmetic - recap</h1>
+<div class="columns">
+<ul>
+<li>Integers modulo \(N\) are (almost) like numbers</li>
+<li>Normal operations like \(+\), \(-\) and \(\times\) work</li>
+<li>Division sometimes works, sometimes not</li>
+<li>Division always works if \(N\) is <em>prime</em></li>
+</ul>
+<img alt="The number line" src="images/clock2.png" style="width: 40%;"/>
+</div>
+</div>
+
+<div class="slide titlepage" tabindex="-1"
+style="background-image: url('images/sum-2c.png');
+background-position: center;
+background-repeat: no-repeat;
+background-size: 70%;
+background-color: rgba(255, 255, 255, 0.85);
+background-blend-mode: overlay;">
+<p></p>
+<p><span class="title">Part II: Elliptic Curves<span></p>
+<p></p>
+</div>
+
+<div class="slide" tabindex="-1">
+<h1>Elliptic curves</h1>
+
+<div class="columns">
+
+<p>
+An <em>elliptic curve</em> is a curve with equation
+\[ y^2 = x^3+Ax+B \]
+Where \(A\) and \(B\) are numbers with \[4A^3+27B^2\neq 0\]
+</p>
+
+<figure style="width: 45%;">
+<figcaption>\(y^2=x^3-x+1\) <br> \((A=-1, B=1\))</figcaption>
+<img alt="An elliptic curve" src="images/ec1.png" style="width: 100%;"/>
+</figure>
+
+</div>
+</div>
+
+<div class="slide" tabindex="-1">
+<h1>Elliptic curves</h1>
+
+<div class="columns">
+
+<figure style="width: 45%;">
+<figcaption>\(y^2=x^3+13x-34\) <br> \((A=13, B=-34\))</figcaption>
+<img alt="An elliptic curve" src="images/ec3.png" style="width: 100%;"/>
+</figure>
+
+<figure style="width: 45%;">
+<figcaption>\(y^2=x^3-x\) <br> \((A=-1, B=0\))</figcaption>
+<img alt="An elliptic curve" src="images/ec2.png" style="width: 100%;"/>
+</figure>
+
+</div>
+</div>
+
+<div class="slide" tabindex="-1">
+<h1>Elliptic curves</h1>
+
+<div class="columns">
+<ul>
+<li>There is a "sum" operation for points of a curve
+(NOT the sum of coordinates)</li>
+<li>To make things work out nicely, we pretend the curve has
+a <em>point at infinity</em> that acts as \(0\):
+\[P+0=0\qquad 0+P=0\qquad P-P=0\]</li>
+</ul>
+
+<img alt="Elliptic curve sum" src="images/sum-2c.png" style="width: 80%;"/>
+
+</div>
+</div>
+
+<div class="slide ecsum" tabindex="-1">
+<h1>Elliptic curves - sum operation - example 1</h2>
+<img alt="Elliptic curve sum" src="images/sum-2a.png"/>
+</div>
+<div class="slide ecsum" tabindex="-1">
+<h1>Elliptic curves - sum operation - example 1</h2>
+<img alt="Elliptic curve sum" src="images/sum-2b.png"/>
+</div>
+<div class="slide ecsum" tabindex="-1">
+<h1>Elliptic curves - sum operation - example 1</h2>
+<img alt="Elliptic curve sum" src="images/sum-2c.png"/>
+</div>
+
+<div class="slide ecsum" tabindex="-1">
+<h1>Elliptic curves - sum operation - example 2</h2>
+<img alt="Elliptic curve sum" src="images/sum-3a.png"/>
+</div>
+<div class="slide ecsum" tabindex="-1">
+<h1>Elliptic curves - sum operation - example 2</h2>
+<img alt="Elliptic curve sum" src="images/sum-3b.png"/>
+</div>
+<div class="slide ecsum" tabindex="-1">
+<h1>Elliptic curves - sum operation - example 2</h2>
+<img alt="Elliptic curve sum" src="images/sum-3c.png"/>
+</div>
+
+<div class="slide ecsum" tabindex="-1">
+<h1>Elliptic curves - sum operation - example 3</h2>
+<img alt="Elliptic curve sum" src="images/sum-4a.png"/>
+</div>
+<div class="slide ecsum" tabindex="-1">
+<h1>Elliptic curves - sum operation - example 3</h2>
+<img alt="Elliptic curve sum" src="images/sum-4b.png"/>
+</div>
+<div class="slide ecsum" tabindex="-1">
+<h1>Elliptic curves - sum operation - example 3</h2>
+<img alt="Elliptic curve sum" src="images/sum-4c.png"/>
+</div>
+
+<div class="slide" tabindex="-1">
+<h1>Elliptic curves - sum operation - code</h1>
+
+<div class="columns">
+<pre><code class="language-python"
+style="font-size: 2.8vh;">
+# Computes p+q on the elliptic curve y^2 = x^3 + Ax + B
+def ec_sum(p: Point, q: Point, A: double) -> Point:
+    if p.is_zero:
+        return q
+    if q.is_zero:
+        return p
+    if p.x == q.x and p.y == -q.y:
+        return Point(is_zero = True)
+
+    if p.x != q.x:
+        k = (p.y - q.y) / (p.x - q.x)
+    else:
+        k = (3 * p.x**2 + A) / (p.y + q.y)
+
+    new_x = k**2 - p.x - q.x
+    new_y = k * (p.x - new_x) - p.y
+    return Point(x = new_x, y = new_y)
+</code></pre>
+
+<pre><code class="language-python"
+style="border: 0.2vw solid; font-size: 2.8vh;">
+@dataclass
+class Point:
+    x: int = 0
+    y: int = 0
+    is_zero: bool = False
+</code></pre>
+
+</div>
+</div>
+
+<div class="slide" tabindex="-1">
+<h1>Elliptic curves - recap</h1>
+<div class="columns">
+<ul>
+<li>Curves of equation \(y^2=x^3+Ax+B\)</li>
+<li>Can "sum" points of the same curve</li>
+<li>Nice properties: associativity, commutativity...</li>
+<li>The sum operation can be easily implemented</li>
+</ul>
+<img alt="The number line" src="images/sum-2c.png" style="width: 40%;"/>
+</div>
+</div>
+
+<div class="slide titlepage" tabindex="-1"
+style="background-image: url('images/factorization.webp');
+background-position: center;
+background-repeat: no-repeat;
+background-size: 50%;
+background-color: rgba(255, 255, 255, 0.90);
+background-blend-mode: overlay;">
+<p></p>
+<p><span class="title">Part III: The Elliptic Curve Factorization Method<span></p>
+<p></p>
+</div>
+
+<div class="slide" tabindex="-1">
+<h1>Integer factorization</h1>
+<p style="text-align: center;"><strong>
+Every positive integer can be written as the product of prime numbers
+</strong></p>
+<div class="columns">
+<ul>
+<li>Example: \(69420 = 2\times 2\times 3\times 5\times 13\times 89\)</li>
+<li>Computationally hard</li>
+<li>Important for cryptography!</li>
+</ul>
+<img src="images/euclid.png" style="height: 60vh;"/>
+</div>
+</div>
+
+<div class="slide" tabindex="-1">
+<h1>Integer factorization - high-level procedure</h1>
+
+<div class="columns">
+<pre><code class="language-python"
+style="border: 0.2vw solid; font-size: 1.4vw;">
+# Returns the list of prime factors of n
+def factorize(n: int) -> list:
+    if n == 1:
+        return []
+
+    if is_prime(n):
+        return [n]
+
+    f = find_factor(n)
+
+    return factorize(n) + factorize(n//f)
+</code></pre>
+
+<ul>
+<li><tt>is_prime(n)</tt> can be implemented
+efficiently
+(<a href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test">Miller-Rabin</a>,
+<a href="https://en.wikipedia.org/wiki/AKS_primality_test">AKS</a>
+or <a href="https://en.wikipedia.org/wiki/Elliptic_curve_primality">ECPP</a>)
+</li>
+<li>Naive implementation of <tt>find_factor(n):</tt>
+<pre><code class="language-python" style="font-size: 2vw;">
+def find_factor(n: int) -> int:
+    for i in range(2,floor(sqrt(n))+1):
+        if n % i == 0:
+            return i
+</code></pre>
+</li>
+</ul>
+
+</div>
+</div>
+
+<div class="slide" tabindex="-1">
+<h1>Find Factor - Elliptic Curve Method</h1>
+<div>
+<p><strong>To find a factor of \(n\):</strong><p>
+<ol>
+<li>Take a random Elliptic Curve \(E\)
+and a random point \(P\) of \(E\)</li>
+<li>Take a <em>suitable number \(m\)</em></li>
+<li>Try to compute \(m\cdot P = P+P+\cdots+P\quad\) (\(m\) times)
+with <em>coordinates modulo \(n\)</em></li>
+<li>If you attempt an <em>impossible division</em> by some number \(d\),
+<em>return \(\operatorname{GCD}(n,d)\)</em></li>
+<li>Go back to 1.</li>
+</ol>
+</div>
+</div>
+
+<div class="slide" tabindex="-1" style="font-size: 2vw;">
+<h1>Find Factor - Elliptic Curve Method - Example</h1>
+<ul>
+<li>Take \(n = 91\)</li>
+<li>Take \(E: y^2 = x^3 + 51x -371\) and \(P = (11, 39)\) and \(M = 2\)</li>
+<li>
+Try to compute \(M\cdot P=P+P\pmod n\):
+\[ k = \frac{3x_p^2+51}{2y_p} \pmod n\]
+Is \(2y_p=78\) invertible modulo \(91\)?
+\[ \operatorname{GCD}(78, 91) = 13 \neq 1 \quad \implies \quad \text{NO!} \]
+</li>
+<li>Found factor: \(13\)</li>
+</div>
+
+<div class="slide titlepage" tabindex="-1"
+style="background-image: url('images/demo.jpg');
+background-position: center;
+background-repeat: no-repeat;
+background-size: 100%;
+background-color: rgba(255, 255, 255, 0.60);
+background-blend-mode: overlay;">
+<p></p>
+<p><span class="title">Demo time!<span></p>
+<a href="https://git.tronto.net/ecm">git.tronto.net/ecm</a>
+<p></p>
+</div>
+
+<div class="slide" tabindex="-1">
+<h1>Elliptic Curve Method - Questions</h1>
+<div class="centertext">
+<p><strong>
+Q: Aren't we just computing the \(\operatorname{GCD}\) with random numbers?
+</strong></p>
+<p>
+A: Yes, but Elliptic Curve operations produce "good candidates"
+for these random numbers.
+</p>
+</div>
+</div>
+
+<div class="slide" tabindex="-1">
+<h1>Elliptic Curve Method - Questions</h1>
+<div class="centertext">
+<p><strong>Q: Can we do the same without elliptic curves?</strong></p>
+<p>A: Yes, with
+<a href="https://en.wikipedia.org/wiki/Pollard%27s_p_%E2%88%92_1_algorithm">
+Pollard's \(p-1\) Algorithm</a>, but ECM is faster.</p>
+</div>
+</div>
+
+<div class="slide" tabindex="-1">
+<h1>Elliptic Curve Method - Questions</h1>
+<div class="centertext">
+<p><strong>
+Q: Are there objects that are more complicated than Elliptic Curves
+and can make the method even faster?
+</strong></p>
+<p>A: Yes, there are higher-dimensional
+<a href="https://en.wikipedia.org/wiki/Abelian_variety">
+Abelian Varieties</a> and other
+<a href="https://en.wikipedia.org/wiki/Algebraic_group">
+Algebraic Groups</a>, but they are much harder (if not impossible)
+to implement efficiently.</p>
+</div>
+</div>
+
+<div class="slide titlepage" tabindex="-1"
+style="background-image: url('images/questions.png');
+background-position: center;
+background-repeat: no-repeat;
+background-size: 60%;
+background-color: rgba(255, 255, 255, 0.85);
+background-blend-mode: overlay;">
+<p></p>
+<p><span class="title">More questions?<span></p>
+<p></p>
+</div>
+
+<div class="slide titlepage" tabindex="-1"
+style="background-image: url('images/beer.jpg');
+background-position: center;
+background-repeat: no-repeat;
+background-size: 80%;
+background-color: rgba(255, 255, 255, 0.65);
+background-blend-mode: overlay;">
+<p></p>
+<p><span class="title">Drinks!<span></p>
+<p></p>
+</div>
+
+<script>
+        // The list of all slides of the presentation.
+        const slides = document.querySelectorAll(".slide");
+
+        // Navigation keys.
+        const keysNext = ["ArrowRight", "ArrowDown", " "];
+        const keysPrev = ["ArrowLeft", "ArrowUp"];
+
+        // Function to move to a given slide.
+        // This also focuses the slide, so any key press will be
+        // handled by the correct slide's event handler.
+        function goto(slide) {
+                slide.focus();
+                slide.scrollIntoView({
+                        behavior: "instant",
+                        block: "start"
+                });
+        }
+
+        // Handle key press events.
+        function onkeydown(i, e) {
+                if (keysNext.includes(e.key) && i+1 < slides.length) {
+                        goto(slides[i+1]);
+                }
+                if (keysPrev.includes(e.key) && i > 0) {
+                        goto(slides[i-1]);
+                }
+        }
+
+        // Handle click or tap events.
+        // Tapping on the right half of the screen scrolls forwards,
+        // tapping on the left half scrolls backwards.
+        function onclick(i, e) {
+                const w = slides[i].offsetWidth;
+                const x = e.clientX;
+
+                if (x > w/2 && i+1 < slides.length) {
+                        goto(slides[i+1]);
+                }
+                if (x < w/2 && i > 0) {
+                        goto(slides[i-1]);
+                }
+        }
+
+        // Disable default action of the navigation keys (e.g. scrolling).
+        document.addEventListener("keydown", function(e) {
+                if (keysNext.includes(e.key) || keysPrev.includes(e.key)) {
+                        e.preventDefault();
+                }
+        });
+
+        // Function to add a footer to every slide.
+        function slideFooter() {
+                const start = "<div class=\"footer\"><table class=\"footer-table\"><tr>";
+                const title = "Elliptic Curves and the ECM algorithm"
+                const link = "<a href=https://tronto.net/talks/ecm>tronto.net/talks/ecm</a>";
+                const end = "</tr></table></div>";
+                const content =
+                        "<td class=\"footer-title\">" + title + "</td>" +
+                        "<td class=\"footer-link\">" + link + "</td>";
+
+                return start + content + end;
+        }
+
+        // Add slide footers and event handlers.
+        for (let i = 0; i < slides.length; i++) {
+                slides[i].innerHTML += slideFooter();
+                slides[i].addEventListener("keydown", e => onkeydown(i, e));
+                slides[i].addEventListener("click", e => onclick(i, e));
+        }
+
+        // Focus and scroll into view the first slide.
+        goto(slides[0]);
+
+        // Call highlight.js
+        hljs.highlightAll();
+</script>
+
+</body>
+</html>
diff --git a/src/talks/talks.md b/src/talks/talks.md
index b7cea5b..595f49a 100644
--- a/src/talks/talks.md
+++ b/src/talks/talks.md
@@ -12,6 +12,10 @@
 
 ## Math
 
+* *Elliptic Curves and the ECM algorithm*.
+  ALTEN Scientific Software Evening, February 2025.
+  [Slides: [html, 2.4Mb](./ecm)]
+
 * *Kummer theory for elliptic curves*.
   Short talk for Tarrach Prize 2023, March 2023.
   [Slides: [pdf, 288Kb](kummer-tarrach.pdf)]