Coordinate geometry, point arithmetic equations, and curve standards (secp256k1 & Curve25519).
1. Elliptic Curve Coordinates & Point Arithmetic
2. The Discrete Logarithm Problem on Curves (ECDLP)
Given a starting point P (generator) and a scalar multiplier k (private key), we compute the public key point Q = k * P by repeatedly doubling and adding.
The security of ECC relies on the difficulty of finding the scalar k given only points P and Q. This is the Elliptic Curve Discrete Logarithm Problem (ECDLP), for which there are no known sub-exponential time solving algorithms.