Let $E:y^2-xy+y-x^3=0$ over a field $K$, $P=(0,0)$. If $\text{char}(K)\neq2$, $E$ is an elliptic curve for which I can easily get $n*P$. Now, something goes wrong with $\text{char}(K)=2$: Basically, I draw the tangent line in $P$ (a horizontal line) which should then intersect $E$ in $P'$ and I would get $2*P=-P'$.
Unfortunately, this horizontal line never intersects $E$ except in $P$ itself. Who can help? Probably, there is something that $\text{char}(K)=2$ hints at?