# A history of elliptic curves in tweets

**Kepler**: so, you see, the orbit of a planet is elliptical. To find where the Earth is, we need a method to calculate the arc length of an ellipse.

**Newton**: This is a fluent problem, and, as usual, it can be solved with an infinite series expansion.

**Leibniz**: Hold my beer, what's a fluent? The arc length of an ellipse is an integral problem. You want to compute $\int f(x)dx$ where $f(x) = \sqrt{\frac{1 - k^2x^2}{1 - x^2}}$ for a certain constant $k$. For that, you have to find a closed form function such that $F'(x) = f(x)$.

*100 years had passed. The search for Leibniz's closed-form solution for the elliptic integral, that is $f(x) = \int_{c}^{x} R \left(t, \sqrt{P(t)} \right) dt$ where $R$ is a rational function and $P$ is a polynomial of degree 3 or 4, had been fruitless.*

*By the end of the 17th century, Bernoulli -- which one is left as an exercise -- conjectured that the task is impossible. It was finally confirmed by Liouville in the 19th century who proved that elliptic integrals, as many others, are non-elementary.*

*In the mean time, Fagnano found a double law and later Euler discovered general addition laws for the elliptic integral. Elliptic, double, addition! You can see where history is heading to ;-)*

**Legendre** systematically reduced elliptic integrals to just three kinds, with their various addition and transformation laws. Poor Legendre, what happened next made much of his life's work obsolete as soon as it was published.

**Abel**: $\int \frac{1}{\sqrt(1 - t^4)}dt$ looks suspiciously similar to $\arcsin(x) = \int \frac{1}{\sqrt(1 - t^2)}dt$. But $sin$ is much more interesting than $arcsin$, isn't it? Well, then, why don't we invert the elliptic integrals?

**Jacobi**: Woohoo, elliptic functions ftw!

**Gauss**: The fact is I discovered elliptic functions before Herr Abel.

**Eisenstein**: All elliptic functions of a special form must satisfy $y^2 = p(x)$, where p is a cubic polynomial with no repeated roots.

**Weierstrass**: All your elliptic function are belong to mine, and mine satisfies $y^2 = 4 * x^3 − a * x^2 − b$.

**Poincaré**: The points on this curve form a group structure. Is it finitely generated?

**Mordell**: Yes, very well monsieur.

**Taniyama-Shimura**: All rational elliptic curves are modular. Well, maybe, maybe not.

**Frey**: If $a, b, c$ are solutions to $x^n + y^n = z^n$, then $y^2 = x(x - a^n)(x - b^n)$ looks very weird.

**Serr - Ribet**: Weird or not, we don't know yet, but it wouldn't be modular.

**Wiles - Taylor**: It is indeed so weird that it does not exist! Fermat's last theorem QED.

*Meanwhile in cryptography:*

**Diffie-Hellman**: Give us a group we shall encrypt the world.

**Lenstra**: Just don't encrypt with RSA. I can factor the modulus, with elliptic curves!

**Montgomery**: Mr.Lenstra, can I have my ladder back? Need to factor some big numbers this afternoon.

**Miller**: Isn't there a group in elliptic curves? Let's encrypt!

**Koblitz**: Of course. Here's another look at elliptic curves.

**NSA**: Use our curves. They were selected *randomly*. Promise, wink wink.

**Bernstein**: (conspiratorial music in the background) NSA curves were rigged, use mine!

**Nakamoto**: My funny money is on Koblitz's.

**Edwards**: The history so far is, unfortunately, not complete...

**De Feo - Jao - Plut**: ...until you take Shor for a random walk around the isogeny volcanoes.