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What's difference between n & #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 ...


In a group of size $n$ (e.g. an elliptic curve), the order of a subgroup generated by a group element necessarily divides $n$. We usually choose curves so that their order $n$ is prime; in that case, the order of a point must be either $1$ (the point is the "point at infinity") or $n$ (all other points). Thus, if $n$ is prime, then every non-zero point is ...

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