ECC scalar multiplication

Question

I'm now trying to use scalar multiplication to recover v, but I have questions with ECC scalar multiplication.

I'm wondering if the following equations are correct.

If scalar * P = Q, then scalar * (-P) = (-Q) ?

For example, assume that 5 * P = 5 * (Px,Py) = (Qx, Qy)

Is 5 * (-P) = 5 * (x,-y) = (Qx, -Qy) correct?

If there is anyone familiar with elliptic curve cryptography, please help me. Thanks!

Answer 1

I believe this question it's more about cryptography than about Ethereum.

The scalar multiplication operation for negative factors works like that:

scalar = k = 5

(−k)*P = (−P) + (−P) + ⋯ + (−P)
-5*P = (−P) + (−P) + (−P) + (−P) + (−P)

Using the Ethereum curve secp256k1: y^2 = x^3 + 7, that uses the Weierstrass form (Koblitz curve).

Modulo p prime:

p = 115792089237316195423570985008687907853269984665640564039457584007908834671663

Using one point of the curve, you can verify the above:

Point P = (Px, Py):
Px: 88557899412428680052041839616894562461261334161833606533830754247259944108251 
Py: 34702325852084136078619568153466699496255363507872892365540796564166359798327

Point -P = (Px, -Py):
Px: 88557899412428680052041839616894562461261334161833606533830754247259944108251
-Py: 81089763385232059344951416855221208357014621157767671673916787443742474873336

Point Q = (Qx, Qy) = 5 * (Px, Py):
Qx: 52731728111805074880654456353252121798595467574662768547858124403109672375563
Qy: 39386080389503298834537707240645959527120091168125799865817705270770834912647

Point -Q = (Qx, -Qy):
Qx: 52731728111805074880654456353252121798595467574662768547858124403109672375563
-Qy: 76406008847812896589033277768041948326149893497514764173639878737137999759016

Point 5*(-P) = (Qx, -Qy):
Qx: 52731728111805074880654456353252121798595467574662768547858124403109672375563
-Qy: 76406008847812896589033277768041948326149893497514764173639878737137999759016