montgomery reduction multiplicative identity

Question

How do you figure out the multiplicative and additive identity with respects to R?

I pick some R such that gcd(R, N) = 1 where N is the size of the group.

Given some field element x in the group, I do x' = xR mod N

How would I figure out the multiplicative identity m such that x' * m = x'

and the additive identity a such that x' + a = x'` ?

Answer 1

The additive identity is $0$, as usual.

The multiplicative identity is the Montgomery representative for $1$, namely $1\cdot R \bmod N = R \bmod N$, just like the Montgomery representative for any other element $x$ is $x\cdot R \bmod N$.