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There are about 2^256 points, then 2^255 different X-coordinates (k*G and -k*G have the same x-coordinate).
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Is this proved fact?
I mean why every private key leads to the unique X-coordinate? (except for symmetry point).
In other words, if
k and
(order-k) leads to x-coordinate Xk, how could we be sure that there are no other key
m leads to Xk as well? (m differs from k and (order-k) )
Because it is proved that:
1) for each private key there is only 1 point (n private keys, n points, n is the order of the curve)
2) for each x-coordinate, the y-coordinate must fulfil this relation: y^2 = x^3 + 7 mod n;
each equation of the form y^2 = a has or 2 opposite solutions (+y and -y -> k*G / -k*G), or has no solution.
Therefor there can't be 3 different points with the same x-coordinate and with 3 different y-coordinate, because there are no 3 different y solutions for the equation y^2 = a mod n.
We know that there are exactly (n-1)/2 different x-coordinates (and (n-1)/3 different y-coordinates)
That's how the math works. The points on the curve form a group (
) and the point G is a generator of that group.