Bound of point's order on elliptic curve

Question

For a given elliptic curve over a finite field and a point $P$ on that curve, how can we bound its order (integer $k$, such that $k*P=O$).

Answer 1

The number of points in an elliptic curve $E$ over $\mathbb F_q$ is $\le q + 1 + 2\sqrt q$. So the order of every point is $\le q + 1 + 2\sqrt q$. That's about all you can say in the general case. If you can find the precise number of points $n$ in $E$, and if you can factorise $n$, then you can find the order of any point $P$ by checking, for every factor $k$ of $n$, whether $[k]P$ is the Point at Infinity.