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Let's take into account the Bitcoin curve. My questions are:

  1. What exactly is the "point at infinity"?
  2. Is there more than one "point at infinity"
  3. How can I identify if my EC generated x and y are the "point of infinity"?
  4. Is there a way to calculate the "point of infinity"?
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What exactly is the "point at infinity"?

It's a point that is added to the points on the curve. Together they form a group. It has the following properties:

  • (x,y) + (x,-y) = infinity
  • (x,y) + infinity = (x,y)
  • infinity + (x,y) = (x y)

In other words, the point at infinity is the identity element of the addition in the group. Therefore some people write the point at infinity as "0".

Is there more than one "point at infinity"

Just one.

How can I identify if my EC generated x and y are the "point of infinity"?

The point at infinity is not on the curve, so it does not have x or y coordinates. It appears whenever a point is added to its own negation (for which the normal addition rule has no answer).

Is there a way to calculate the "point of infinity"?

It simply is "the point at infinity", there is nothing to be calculated about it.

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  • But when I am deriving child public keys from a parent public key there is a step which is if Ki is the point at infinity the resulting key is invalid, I still miss how to figure it out. – Allan Romanato Apr 9 at 20:35
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    Ki is the sum of point(...) and Kpar. I've given the rules above: if those two have identical coordinates, except the y coordinate is negated, the result is the point at infinity. – Pieter Wuille Apr 9 at 20:47
  • I got it, so if x of Ki = x of point(...)+Kpar and y of Ki = -y of point(...)+Kpar so it is the point at infinity... Greattt – Allan Romanato Apr 9 at 20:56
  • The point of infinity DOES have coordinates because IT IS on the curve. It is when the point on the curve is on y=0. To know x just do the math. It is called the point of infinity because at this point, the tangent is a totally vertical line which means it has an "infinite" slop. – Oscar Serna Apr 9 at 20:56
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    @OscarSerna That's nonsense, sorry. The point is on the curve if you're using projective or Jacobian coordinates, but in affine coordinates there is no (x,y) that corresponds to it. – Pieter Wuille Apr 9 at 20:59

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