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+# Elliptic curve factorization method
+
+Slides and code for the a presentation about
+[Lenstra's elliptic-curve factorization](https://en.wikipedia.org/wiki/Lenstra_elliptic-curve_factorization).
+
+See also [this blog post](https://sebastiano.tronto.net/blog/2025-02-27-ecm).
+
+## Abstract
+
+Elliptic curves are mathematical objects that have both a geometric and an
+arithmetic side. They turn out to be useful for real-world applications
+because they sit in a sweet spot: they are complicated enough to have
+interesting and useful arithmetic properties, but simple enough to be
+implemented in software in an efficient way. For example, they are used in
+cryptographic schemes, such as the Elliptic-curve Diffie-Hellman scheme,
+to obtain greater security with smaller keys.
+
+After introducing elliptic curves and modular arithmetic, we will take a
+look at the elliptic curve factorization method (ECM), one of the most
+efficient method to find the prime factors of an integer number. We
+will see in practice how much faster this method is compared to a naive
+algorithm, and we'll see that the implementation of this method is not
+that hard at all.
+
+## Slides
+
+The slides are a single html file, `index.html`. They rely on a couple
+of external JavaScript libraries. They are also hosted
+[here](https://sebastiano.tronto.net/talks/ecm).
+
+## Code
+
+The folder `code/python` contains two files:
+
+* `ecm.py`: an implementation of the ECM algorithm.
+* `naive.py`: an implementation of the simple O(√n) algorithm for finding
+  a factor of a number, for comparison.
+
+To use any of the two, pass the number to factor as a command-line argument,
+for example:
+
+```
+$ ./ecm.py 255000007030000033
+255000007030000033 = 510000011 * 500000003
+```
+
+Some benchmarks (note: the ECM is randomized, the time can vary a lot):
+
+```
+$ time ./ecm.py 255000007030000033
+255000007030000033 = 510000011 * 500000003
+    0m01.45s real     0m01.43s user     0m00.01s system
+$ time ./naive.py 255000007030000033
+255000007030000033 = 500000003 * 510000011
+    0m26.62s real     0m26.50s user     0m00.01s system
+```
+
+### C++ code (experimental)
+
+The folder `code/cpp` contains an experimental implementation of the ECM
+algorithm in C++. It works, but it is very slow: it requires suppport
+for compile-time big integers, which I implement in an inefficient way. I
+may optimize this code in the future.