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
kN-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.
