Perform raw elliptic curve coordinate math over the finite field Fp
secp256k1 Point Addition & Doubling Calculator
Compute secp256k1 Point Arithmetic
Resulting Coordinates
None
The secp256k1 Point Addition & Doubling Calculator executes raw elliptic curve arithmetic over the finite field Fp. By processing high-precision 256-bit coordinate values, this tool computes point doubling (2 · P1) and distinct point addition (P1 + P2), essential operations for blockchain key generation, ECDSA validation, and cryptographic research.
secp256k1 Curve Constants
p: FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
a: 0 | b: 7
Eq: y2 ≡ x3 + 7 (mod p)
a: 0 | b: 7
Eq: y2 ≡ x3 + 7 (mod p)
Formula:
Mathematical Formulas for secp256k1 Point Arithmetic
Elliptic Curve Cryptography (ECC) operations over the finite field Fp rely on finding modular slopes (λ) rather than standard coordinate addition. For the secp256k1 curve, the foundational parameter is a = 0, which simplifies the geometric operations.
1. Point Doubling (When P1 = P2)
When computing 2 · P1, the slope (λ) represents the tangent line passing through the coordinate:
λ = (3 · x12 + a) · (2 · y1)-1 (mod p)
Because a = 0 on secp256k1, this simplifies directly inside the code execution to: λ = (3 · x12) · (2 · y1)-1 (mod p).
2. Point Addition (When P1 ≠ P2)
When summing two distinct points to find P1 + P2, the slope (λ) represents the line intersecting both targets:
λ = (y2 - y1) · (x2 - x1)-1 (mod p)
Note: The exponent -1 denotes the modular multiplicative inverse calculated over field prime p using the Extended Euclidean Algorithm. If x1 = x2 but y1 ≠ y2, the intersection hits vertical infinity (POINT_AT_INFINITY).
3. Deriving Result Coordinates (x3, y3)
Once the correct slope (λ) is resolved from the routines above, the resulting intersection coordinates are computed identically:
x3 = λ2 - x1 - x2 (mod p)
y3 = λ · (x1 - x3) - y1 (mod p)
For point doubling calculations, substitute the variable value of x2 with x1 within the x3 equation structure.