Is the elliptic curve scalar multiplication $[n]G$ defined if $n=0$?
I saw multiple software implementations with multiple results such that, $[0]G=0$ or $[0]G=G$.
This made me wonder, how can i compute $[0]G$ on 25519 curve, where G is the base point. $G=(9,14781619447589544791020593568409986887264606134616475288964881837755586237401).$
Also, is [n]$\mathcal{O}$ defined?
I saw multiple software implementations with multiple results such that, $[0]G=0$ or $[0]G=G$.
As stated in the comments, we define $[0]G = 0$, anything else is incorrect
I thought I'd outline why we define things we did.
What we want is to have $[a+b]G = [a]G + [b]G$ be true for all integers $a, b$, and all points $G$.
If it is true for all integers $b$, it must be true for $b=0$, hence we must have:
$$[a+0]G = [a]G + [0]G$$
This obviously simplifies to $[a]G = [a]G + [0]G$, and this is true only if $[0]G$ is the group identity, that is, only if $[0]G = 0$
