Quote from: kTimesG on Today at 10:13:39 AM

Quote from: JonJonBon on Today at 06:06:10 AM

Hello everyone, can you tell me if there has been a message here for a long time? where it says "if it would be possible to divide a point on the curve by 2, then find the private key, etc.." I've seen this message here before and it was answered by someone. I can't seem to find this page.

Division by 2 just means to multiply by 1/2 mod N.
1/2 just means 2's inverse so that 2*x = 1 mod N.
Inverse of x mod N just means x^N-2 mod N (Fermat Little Theorem).

Let's take F = { y(x) = 2^x mod 11, 0 < x < 11 }

and some known y = 5: 2^x = 5 mod 11

"Divide" public key by 2: 2^x/2 = 5/2 mod 11

2^x-1 = 2^-1 * 5 mod 11
2^-1 = 2^11-2 mod 11 = 6

So: 2^x-1 = 5*6 mod 11 = 8

Great, now you have to find x - 1. Repeat? Sounds like nothing really changed.

Let's take F = { (x, y) = [k]G, 0 < k < N }

and some known (x, y) = Q: [k]G = Q

"Divide" public key by 2: [k/2]G = [1/2]Q

[k/2]G = [2^-1]Q
2^-1 = 2^N-2 mod N

So: [k/2 mod N]G = [2^N-2 mod N]Q

Great, now you have to find k / 2. Repeat? Sounds like nothing really changed.

If you fall into the trap that k is somehow half the size now, remember this:
1. Division of a field element means multiplying the element with the divisor's field inverse.
k is not an integer in an infinite field, but a finite field. You can't just half it's value, that only makes sense in an
infinite field, and only if such an inverse really exists.
2. Groups do not have multiplication operation, only addition. There's no such thing as multiplying or division of elliptic points, they form a group, not a field. "Point multiplication by k" just means adding the point to itself k times. "Division by k" means adding the same point to itself k^N-2 mod N times. You need to respect the definitions of what something can be called a "group" or "field", "finite" vs "infinite", not invent or borrow properties from different structures. It can't work.

Yes, I read about it here on the forum! thanks for the answer! But there was a message on this branch with a division by 2. and there was an answer to it, but I don't remember which one and on which page these messages are also I don't remember. But the question is, if it were possible to divide public keys by 2? would it make it easier to find the private key? what I remember seems to be the answer to that message was "if we could divide the public key by 2, then finding the private key would be easy, and it seems to have given the formula", if I was wrong, please correct me!