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<!doctype html>
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<title>Elliptic Curves and the ECM Algorithm</title>
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<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(f) + 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>
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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>
|
