Elliptic curves are pure and applied, concrete and abstract, simple and complex.
Elliptic curves have been studied for many years by pure mathematicians with no intention to apply the results to anything outside math itself. And yet elliptic curves have become a critical part of applied cryptography.
Elliptic curves are very concrete. There are some subtleties in the definition—more on that in a moment—but they’re essentially the set of points satisfying a simple equation. And yet a lot of extremely abstract mathematics has been developed out of necessity to study these simple objects. And while the objects are in some sense simple, the questions that people naturally ask about them are far from simple.
Preliminary definition
A preliminary definition of an elliptic curve is the set of points satisfying
y² = x³ + ax + b.
This is a theorem, not a definition, and it requires some qualifications. The values x, y, a, and b come from some field, and that field is an important part of the definition of an elliptic curve. If that field is the real numbers, then all elliptic curves do have the form above, known as the Weierstrass form. For fields of characteristic 2 or 3, the Weierstrass form isn’t general enough. Also, we require that
4a³ + 27b² ≠ 0.
The other day I wrote about Curve1174, a particular elliptic curve used in cryptography. The points on this curve satisfy
x² + y² = 1 – 1174 x² y²
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