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.
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.