# montgomery reduction multiplicative identity

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'$$`$$ ?

• Did you try to transfer 1 into Montgomery Domain by $1 R \mod n$ – kelalaka Apr 23 '19 at 23:17

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 $$x\cdot R \bmod N$$.