What is the order of the Identity point on prime order elliptic curve groups? I'm trying to understand how the identity point is represented in a group of prime order.
What I think is correct:
If the group has even order, then the identity point is in the group, because the identity point has order 2.
If the group has prime order, like elliptic curves such as secp256 then the identity point can still be represented as a group, but is the point at infinity. If the point could not be represented in the group, then it would not be a group.
I do not believe I completely understand the second point. What order is the identity point if it is in the group?
For something to be a group, it must have an identity. This is a definition.
The order of an element, $g$, in a group is defined to be the smallest positive number $n$ such that $g^n = 1$, where $1$ is the identity of the group.
By this definition, the order of the identity in any group is $1$.