Mathematics
The Arithmetic of Elliptic Curves and the Group Law
Quick fact
On an elliptic curve, the operation of point addition is commutative and associative, forming an abelian group—a fact that was only rigorously proved in the 20th century. The group law's surprising structure turns a geometric curve into an algebraic object with deep number-theoretic and cryptographic significance.
Why this is interesting
What if you could add two points on a curve and get another point on the same curve? This simple idea, discovered in the 19th century, now secures your online banking and underpins a million-dollar conjecture.