Are there references or prove to say that multiplying two points in the elliptic curve cryptography ECC is not allowed, as an example below? Multiply the public key PKA by a point (Z) on the ECC since these two parameters (the public key and the point) are both points on the ECC.

  • $C=𝐶′\oplus h(Z.PK_A\mathbin\|T_1)$
  • $Pk=[SK]P$
  • $Z=[a]P$

where $P$ is a base point on an EC and $a\in\mathbb Z_q^*$.

  • 1
    $\begingroup$ How do I multiply two points on an elliptic curve?. It is scalar multiplication and points of EC can only form a Z-Module! $\endgroup$
    – kelalaka
    Dec 12, 2021 at 21:17
  • $\begingroup$ The purpose of the elliptic curve is that it provides a trapdoor function. Why is that trapdoor function (scalar multiplication) not enough for your purposes? Anything that you devised with two points would be an entirely different kind of "multiplication". $\endgroup$
    – knaccc
    Dec 12, 2021 at 22:04
  • 2
    $\begingroup$ Does this answer your question? How do I multiply two points on an elliptic curve? $\endgroup$
    – kelalaka
    Dec 12, 2021 at 23:09

1 Answer 1


Adding points within an elliptic curve group will give another point to the curve and all multiples of the points within the group will also be contained in the elliptical curve. There are three rules for adding points within an elliptical curve group that are followed:

  1. ∞ + ∞ = ∞
  2. (χ, γ) + ∞ = (χ, γ)
  3. (χ, γ) + (χ, -γ) = ∞

scalar multiplication of points in elliptic curves above GF (p) are calculated by the following formulas A) Addition of points Let be two points on the curve P = (x1, y1) and Q = (x2, y2) and their sum is R = (x3, y3). P and Q are distinguished if P and -Q are not the same (x1 ≠ x2). Adding the points, P + Q = R is defined as: (x1, y1) + (x2, y2) = (x3, y3) λ = (y2 - y1) (x1 - x1) -1 x3 = λ2 –x1 –x2 y3 = λ (x1- x3) - y1

Β) Doubling of a point Let the point P = (x1, x2) exist in the curve where x1 ≠ 0. The doubling of the point, 2P = R is defined as: (x1, y1) + (x1, y1) = (x3, y3) λ = (3x12 + a) (2y1) -1 x3 = λ2 –2x1 y3 = λ (x1- x3) - y1

C) Scalar multiplication of points Let P be a point and d be a bit string of an integer. In order to calculate the point Q = dP, combined methods of adding and doubling the points are used. The multiplication of a point, dP = Q, follows the following algorithm: if dn-1 = 1, then Q: = P else Q: =  for i = n-2 to 0 Q: = Q + Q if di = 1 then Q: = Q + P return Q

  • $\begingroup$ This doesn't address the misinformation of bitcoin books etc. OP intension is to multiply points. $\endgroup$
    – kelalaka
    Dec 12, 2021 at 23:13
  • $\begingroup$ I’ve agree regarding the misinformation. I’ve was trying to cover numerical functions in elliptic curves. $\endgroup$
    – Pegasus
    Dec 13, 2021 at 0:22
  • $\begingroup$ We have many answers containing that... $\endgroup$
    – kelalaka
    Dec 13, 2021 at 0:23

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