Elliptic Curve Cryptography Referencia
Referencia para elliptic curve groups, point operations, y estándar curves usado en ECC.
Group Operations
An elliptic curve group consists of points satisfying y^2 = x^3 + ax + b plus a point at infinity. Point addition combines two points geometrically via the chord-and-tangent method. Scalar multiplication (k*P) is the core operation, analogous to exponentiation in multiplicative groups. The discrete logarithm problem (given P and Q=kP, find k) is believed hard for well-chosen curves.
- point addition
- Geometric chord rule
- scalar mult
- Repeated addition
- ECDLP
- Discrete log is hard
P + Q = Rk*P core opno polynomial algoStandard Curves
NIST curves (P-256, P-384, P-521) are widely deployed in TLS and X.509 certificates. Curve25519 offers a secure, efficient Montgomery curve used in Ed25519 signatures and X25519 key exchange. Brainpool curves provide an alternative with verifiable random generation. The choice of curve affects security, performance, and implementation difficulty. RFC 7748 and 8032 standardize Curve25519 operations.
- P-256
- NIST prime curve
- Curve25519
- Montgomery, complete formulas
- Ed25519
- Twisted Edwards signatures
TLS defaultRFC 7748RFC 8032