There are finitely many points on a curve over F2m .
Elements of the field F2m are m-bit strings. The rules for arithmetic in F2m can be defined by either polynomial representation or by optimal normal basis representation. Since F2m operates on bit strings, computers can perform arithmetic in this field very efficiently.
An elliptic curve with the underlying field F2m is formed by choosing the elements a and b within F2m (the only condition is that b is not 0). As a result of the field F2m having a characteristic 2, the elliptic curve equation is slightly adjusted for binary representation:
y2 + xy = x3 + ax2 + b
The elliptic curve includes all points (x,y) which satisfy the elliptic curve equation over F2m (where x and y are elements of F2m ). An elliptic curve group over F2m consists of the points on the corresponding elliptic curve, together with a point at infinity, O. There are finitely many points on such an elliptic curve.
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