Elliptic Curves
Explore the geometry of elliptic curves over the reals. Adjust the curve parameters, click two points on the curve to see point addition — the geometric operation that underpins modern cryptography.
How it works
An elliptic curve over the reals is the set of points (x, y) satisfying y² = x³ + ax + b, together with a special “point at infinity” that acts as the identity element. For the curve to be non-singular (smooth, with no cusps or self-intersections), its discriminant Δ = −16(4a³ + 27b²) must be non-zero.
The remarkable property of elliptic curves is that their points form an abelian group under a geometric operation called point addition. To add two points P and Q: draw the line through them, find the third intersection point R with the curve, then reflect R across the x-axis to get P + Q. When P = Q, we use the tangent line at P instead. This simple geometric construction gives rise to a rich algebraic structure.
Elliptic curves are foundational to modern cryptography. The Elliptic Curve Discrete Logarithm Problem (ECDLP) — given points P and Q = nP, find n — is believed to be computationally intractable. This hardness assumption powers Elliptic Curve Diffie-Hellman key exchange, ECDSA signatures (used in Bitcoin and TLS), and many other protocols. Compared to RSA, elliptic curve cryptography achieves equivalent security with dramatically smaller key sizes: a 256-bit ECC key provides security comparable to a 3072-bit RSA key.