Why equation Y^2=X^3 +AX+B can't work with finte field of charateristic 2? I know that we can't define $dx/dy$ with this equation because $2y = 0$ with finite field of charateristic $2$. But with $GF(2^n)$ (has characteristic by $2$) $2=x$ not $0$. Do I misunderstand here?
In $GF(2^n)$, $y+y = 0$ for all $y$. These elements are not integers; using something like “2” to denote a field element is merely a convention, and, as you demonstrate, a confusing one. In this case, $2y$ means $y+y$, not the product in the field of $y$ with the element conventionally represented by “2”.