Elliptic Curve - X Coordinate

Question

I am currently working on a Koblitz curve. I have found the curve has two matching groups based on the base curve point and N-1 point. My question is as follows: Is there an algorithm to determine how many X coordinates are the same in an elliptic curve? This will tell me how many groups are inside the entire curve based on its parameters given.

Answer 1

As per the Hasse's theorem the total number of points can be computed for both the Finite field curve and binary field curve

For the Finite Field , the number of points on E(F p) is denoted by #E(F p). The Hasse Theorem states that: p+1-2√p≤ E(F p) ≤p+1+2√p.

For the Binary field, The number of points on E(F2^m)() is denoted by #E(F2^m). The Hasse Theorem states that: 2^m+1-2√2^m≤#E(F2^m) ≤2^m+1+2√2^m.

For Every x co-ordinate there are 2 Y co-ordinates. Hence the number of X co-ordinates can be deduced as per the calculation of Hasse's theorem points/2.