Quote from: mamuu on January 27, 2024, 10:07:45 PM

Hello

Can a public key be generated without knowing the private key? yes

https://doc.sagemath.org/html/en/reference/arithmetic_curves/sage/schemes/elliptic_curves/ell_generic.html#

Can the coordinate of a desired number in the curve be generated pubkey at the same time? yes.

After creating the curve, it is possible to create it with the "lift_x" function.
If the source you mentioned has private keys
and if the diagram you show is produced, we can talk about a security vulnerability.
However, it is possible to calculate the points that will form this diagram without private keys.

Thank you.

Thanks mamuu, this was my first thought that Satoshi generated those public keys and put $72 Billion in them without knowing private keys.
However, would anyone do that? I would not. I would prefer to design a game on math, rather than design a game on randomness.
But, of course, this can be an option too.

Btw, for an update, this formula "(P * (P^m-2 % m)) % m" lost its validity after several tests. So, my new approach is to identify if point is on the left or right side of the curve, which means P > Half Curve Point (57896044618658097711785492504343953926418782139537452191302581570759080747169) or P < Half Curve Point. Because if the point is Odd, when you apply halve on the point, halved P will be greater than curve half point ALWAYS. However if it is Even and when you apply halve, the halved P will be lower than curve half point ALWAYS. I am now training an AI to identify this, do not have success yet, and searching the ways to identify if point P is greater or lower than the curve half.