How to add two points on an elliptic curve with a ruler
On an elliptic curve you can add points using nothing but straight lines. Draw a line through two points, find where it hits the curve a third time, and flip that point across the horizontal axis. The result is their sum. This odd arithmetic helped prove Fermat's Last Theorem and now secures a great deal of cryptography.
Despite the name, an elliptic curve is not an ellipse; the term comes from elliptic integrals. Over the real numbers it is usually written in Weierstrass form, where y squared equals x cubed plus a times x plus b. The curve must be smooth, with no sharp points, self-crossings or isolated dots, which a quantity called the discriminant can check. When the discriminant is positive, the real graph falls into two separate pieces; when negative, it is one piece.
The addition rule needs a few patches. Adding a point to itself uses the tangent line at that point instead of a line through two points. A point added to its mirror image gives a special point imagined at infinity, which acts like zero. With these rules, the points form a group, and the order of addition does not matter. If the curve's coefficients are rational numbers, the rational points form a smaller group of their own, because a line through two rational points meets the curve a third time at another rational point.
Viewed with complex numbers instead of real ones, the picture changes shape entirely. A complex elliptic curve is topologically a torus, the surface of a doughnut, while a complex ellipse is a sphere. The doughnut also carries a natural addition, and the two match perfectly.
These curves are a major area of current research in number theory. Andrew Wiles used them in his proof of Fermat's Last Theorem, and they underpin elliptic curve cryptography as well as techniques for factoring large whole numbers.
Source: Elliptic curve
