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?

  • $\begingroup$ You can construct one, however, it will be a bit more complex. Why do you need this? You need complete and side-channel free addition Is there any "exception-free" coordinates system for Weierstrass curves? $\endgroup$
    – kelalaka
    Mar 19 at 14:15
  • $\begingroup$ @kelalaka I followed the link you provided but I wasn't able to come up with anything. I'll have only been able to construct a direct secp256k1 elliptic curve subtraction formula other than negating and adding , assuming you want to subtract $G^x$ from $Q^x$ you can use the below formula. s = (Qy + Gy) * pow(Gx - Qx, -1, p) % p Rx = (s**2 - Qx - Gx) % p Ry = (s * (Rx - Qx) - Qy) % p $\endgroup$
    – Favour
    Mar 19 at 14:59
  • $\begingroup$ My point is, we don't need this formula for general scalar multiplication. Why do you need? $\endgroup$
    – kelalaka
    Mar 19 at 15:04
  • $\begingroup$ It will help me understand better. If you don't mind can you provide the formula for point multiplication by 3 $\endgroup$
    – Favour
    Mar 19 at 15:18
  • 2
    $\begingroup$ @Favour What's wrong with combining a doubling with an addition ? Like computing (2*P)+P? $\endgroup$
    – Ruggero
    Mar 20 at 10:14

1 Answer 1


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

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 = x**2
    w = 3*X
    R = 2*y**2 % p
    Z = 4*y*R % p
    u = R*R
    B = ((x+R)**2-X-u) % p
    h = (w**2-2*B) % p
    X = 2*h*y % p
    Y = (w*(B-h)-2*u) % p
    u = y*Z-Y % p
    v = x*Z-X % p
    w = v**2
    R = w*X % p
    w = v*w % p
    A = (u**2*Z-w-2*R) % p
    h = pow(w*Z,-1,p)
    return v*A*h % p, (u*(R-A)-w*Y)*h % p

I don't claim that's optimal.


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