Order of subgroup on elliptic curve over $Z_p$

Question

I should determine the order of subgroup on elliptic curve over $\mathbb{Z}_p$ where $p$ is prime, and point $X$ is generator of some subgroup. While generating the subgroup by points addition I found out that $kX$ and $kX+X$ have the same value of $x$ coordinate (thus $kX=(x,y)$ and $kX+X = (x,y')$) and this happened for the first time while computing. Helps this observation to find out the order of subgroup?

Answer 1

If the infinity point is the additive identity of the elliptic curve, then it is known that the inverse of $(x,y)$ is $(x,-y)$.

If $(k+1)X$ has the same $x$-coordinate as $kX$, and that $X$ is not the infinity point, then you can rule out that $(k+1)X = kX$. This means that $(k+1)X = -kX$, or $(2k+1)X = 0$. If this is the first time you see $x$-coordinate repeats itself in multiples of $X$, then $2k+1$ would be the order of $X$.