Elliptic Curve Point Addition Calculator

About this calculator

An elliptic curve defined by y² = x³ + ax + b, together with a geometric rule for "adding" points on it, forms an algebraic group — the mathematical structure underneath modern public-key cryptography. To add two distinct points P and Q, you draw the line through them, find where it intersects the curve a third time, and reflect that point across the x-axis to get R = P + Q; this calculator carries out that construction and reports the resulting point along with the chord's slope.

This point-addition rule (and the related point-doubling rule for adding a point to itself) is the foundation of elliptic curve cryptography (ECC), used in ECDSA digital signatures, TLS/HTTPS key exchange, and Bitcoin and other cryptocurrency wallets — where the same group law is applied over a finite field rather than the real numbers, and the difficulty of reversing repeated point addition (the elliptic curve discrete logarithm problem) is what makes the cryptography secure. Working through it over the reals, as this calculator does, is exactly how cryptography and coding-theory courses introduce the geometry before moving to finite fields.