Is scalar division in finite field elliptic curve not multiply by inverse of 2?

Question

Using my understanding of finite field elliptic curve arithmetics, I've come to the conclusion that it is possible to divide a point by a scalar the same was we can multiply a point by a scalar. My method for division of 2 is simply multiply by the inverse of 2, however, I then tried to multiply the resulting point by 2 and I did not get the original point back. What am I missing here?

Answer 1

You can do this, as long as the elliptic curve group has odd order.

In general $E(\Bbb F_q)$ has order $m$ with $(\sqrt q-1)^2\le m\le(\sqrt q+1)^2$ (Hasse's theorem). This order can be calculated by Schoof's algorithm. If $m$ is odd then $Q=[\frac{m+1}2]P$ is a solution of $[2]Q=P$. If $m$ is even then the equation $[2]Q=P$ may or may not be soluble. But, when $m$ is even, this trick will not find a solution.