Curves and Points

Cryptography · Axiom Academy

Discover the beautiful geometry that powers modern cryptography. Let's explore the equation y² = x³ + ax + b . Adjust the parameters a and b to see how the curve shape changes! Click anywhere on the curve to place a point. Notice something special about the geometry! Click two points on the curve to see how point addition works geometrically! Every group needs an identity element - a point that doesn't change anything when you add it. Imagine a point infinitely far away in the vertical direction - we call it O (the point at infinity). P + (-P) = O (every point has an inverse) Vertical lines intersect the curve at O Think of O as "zero" for point addition. It's what allows elliptic curve points to form a complete group structure! Try different curve parameters and see valid vs. invalid curves for cryptography! Elliptic curves provide exponential security with smaller key sizes. A 256-bit elliptic curve key offers similar security to a 3072-bit RSA key! Points on the curve form a group with point addition. This structure enables: Key generation (pick a random point, multiply it) Security relies on the Elliptic Curve Discrete Logarithm Problem : Given points P and Q = kP, finding k is computationally infeasible. It's easy to multiply k times, but very hard to reverse!

This is the written version of the interactive lesson above. See the full Cryptography course.