Elliptic Curve Cryptography: Point Multiplication by 3 on secp256k1 Curve

Question

Is there a direct non-iterative formula for point multiplication by 3 in the secp256k1 elliptic curve just like point multiplication by 2 (point doubling)? If such a formula exists, could you explain how to achieve this? If not, could you clarify why it's not possible?

Answer 1

A simple way to derive a point tripling method in Cartesian coordinates for secp256k1 is per $3P=(2P)+P$, and towards this

• computing $2P$ by point doubling in projective coordinates with Z=1 on input, and a=0
• computing $(2P)+P$ by point addition in projective coordinates with Z=1 on one input
• finally return to Cartesian coordinates with a single modular inversion.

p = 2**256-2**32-977
def triple(x:int, y:int):
    if x==0 or y==0:
        return 0,0 # point at infinity
    X = x2
    w = 3X
    R = 2y2 % p
    Z = 4yR % p
    u = RR
    B = ((x+R)2-X-u) % p
    h = (w2-2B) % p
    X = 2hy % p
    Y = (w(B-h)-2u) % p
    u = yZ-Y % p
    v = xZ-X % p
    w = v2
    R = wX % p
    w = vw % p
    A = (u2Z-w-2R) % p
    h = pow(wZ,-1,p)
    return vAh % p, (u(R-A)-wY)h % p

I don't claim that's optimal.