An elliptic curve is a set of coordinate pairs satisfying an equation such as y² = x³ + ax + b, with 4a³ + 27b² ≠ 0. A specified point at infinity completes the structure into a group, making elliptic curves useful for cryptographic key exchange. section 5.3.1
The geometric addition rule works like this:
- Draw a line through two curve points; its third intersection determines the sum after the prescribed vertical-line step. section 5.3.1
- Add a point to itself by drawing its tangent instead of a line through two distinct points. section 5.3.1
- Use the point at infinity when two points cancel; it acts as the identity for the additive group. section 5.3.1
From geometry to ECDH
Elliptic-curve groups use additive notation: repeatedly adding a base point produces another curve point, and Diffie–Hellman can be built on that group. Common standardized choices include P-256 and Curve25519. section 5.3.1 section 5.3.3
