Point Operation Examples

Cryptography · Axiom Academy

EXAMPLE Point Operation Examples Worked examples of elliptic curve point operations on a small finite field Excellent work! You've completed three fundamental elliptic curve operations. Here's what we learned: Point Addition: When adding two distinct points P and Q, we calculate the slope m = (y₂ - y₁) / (x₂ - x₁), then use it to find the coordinates of P + Q. All arithmetic is performed modulo p. Point Doubling: When doubling a point P, we use the tangent line slope m = (3x₁² + a) / (2y₁), where a is the curve parameter. This is a special case of point addition. Scalar Multiplication: To compute kP efficiently, we use the double-and-add algorithm based on the binary representation of k. This reduces the number of operations from O(k) to O(log k). Modular Arithmetic: All operations are performed in a finite field F_p, meaning we take results modulo p. Division is performed using the modular multiplicative inverse. These operations form the foundation of elliptic curve cryptography (ECC). The difficulty of reversing scalar multiplication (the discrete logarithm problem) is what makes ECC secure!

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