Elliptic Curve Mathematics Reference
Reference for elliptic curve group law, point operations, and discrete logarithm problem.
Group Law
An elliptic curve y^2 = x^3 + ax + b (with discriminant non-zero) forms an abelian group with the point at infinity as identity. Point addition: draw a line through P and Q; the third intersection reflected across the x-axis is P+Q. Point doubling uses the tangent line. Scalar multiplication k*P is repeated addition, computable in O(log k) via double-and-add. The group order is given by Hasse's theorem: |#E - (p+1)| <= 2*sqrt(p). The discrete logarithm problem (given P, Q=kP, find k) is believed hard with no subexponential classical algorithm.
- point add
- Geometric chord rule
- scalar mult
- Double-and-add
- ECDLP
- Discrete log hard
third intersectionO(log k)no subexp algoCurve Selection
Prime field curves (y^2 = x^3 + ax + b over Fp) are most common. Binary field curves (over F2^m) exist but are less used now. Montgomery curves (By^2 = x^3 + Ax^2 + x) enable efficient x-only scalar multiplication (Montgomery ladder). Twisted Edwards curves (ax^2 + y^2 = 1 + dx^2y^2) have complete addition formulas (no edge cases). Curve parameters must be verified: non-singular, large prime order subgroup, embedding degree large (MOV attack), twist secure. NIST, Brainpool, and Curve25519 are standard curves with different trust models.
- prime field
- Fp curves
- Montgomery
- X-only mult
- Edwards
- Complete formulas
most commonCurve25519no edge cases