An online cryptographic tool to compute the doubling of a public key point 2P on an algebraic elliptic curve over a finite field mod p.
Formula:
The secp256k1 Point Doubling Formula
Because the secp256k1 parameter choice sets a = 0, the equation simplifies slightly. First, the gradient slope (lambda) of the tangent is determined:
The resulting 64-character hex key outputs are extracted following finite-field arithmetic computations:
Ry = (lambda * (Px - Rx) - Py) mod p
Understanding ECDSA Public Key Point Doubling
In Elliptic Curve Cryptography (ECC) and the Elliptic Curve Digital Signature Algorithm (ECDSA), Point Doubling is the operational foundation for performing scalar point multiplication. Given a Public Key point defined by coordinates (Px, Py) on an elliptic curve equation y^2 = x^3 + ax + b (mod p), doubling the point generates a completely new public point infrastructure R, where R = 2P = (Rx, Ry).
The Mathematical Equations
To double a point, you first find the slope (lambda) of the tangent line intersect crossing through the public key coordinate (Px, Py):
Using the calculated slope lambda value, the new coordinate targets (Rx, Ry) are resolved through:
Ry = (lambda * (Px - Rx) - Py) mod p
Edge Cases & System Constraints
- Point at Infinity: If Py mod p equals 0, the computed tangent line runs completely vertical. The doubling operation yields the computational Point at Infinity (O).
- Modular Inverse Division: Standard division cannot be executed directly in finite fields. The application computes the modular multiplicative inverse of (2 * Py) mod p via the Extended Euclidean Algorithm. This inverse only exists if the greatest common divisor gcd(2*Py, p) equals 1.