What is the difference between order of base point and curve order in EC? When I was read about the elliptic curve cryptography I found some definition about domain parameter of  elliptic curve like the follow. But I did not understand something
$p$: prime number.
$a, b$: field elements, they specify the equation of the elliptic curve $ E$ over $F_P$,
$y^2  ≡ x^3+a • x+b  $
$G$: A base point represented by $G= (xg, yg)$ on $E (F_P)$
$n$: Order of point $G$ , that is $n$ is the smallest positive integer such that $nG = O$.
$h$: cofactor, and is equal to the ratio #E($F_P$)/$n$, where #E($F_P$) is the curve order.
My question
What's diffrence between  $n$ &amp; #E($F_P$)? also
I think two are same value.
because #E($F_P$) is Curve Order where The number of points on the elliptic curve is called its curve order. and when we do #E * $G$ = $O$.
Is this right or not right?
What's difference between n &amp; #E(FP)?


The difference is that $n$ is the smallest positive integer where $nG = O$; while you correctly state that $\#E \cdot G = O$, that doesn't mean that $\#E$ is the smallest integer that makes this happen.  There may be a smaller integer $n$; $n$ will always be a factor of $\#E$, however it can be smaller.

As for why we would want to make it smaller, that is, why would we want to have an $h = \#E/n &gt; 1$, well, that's exactly what this question addresses.