Montgomery Reduction - what should the choice of R be?

Question

When we need to compute $z = xy \text{ mod } N$ and the Montgomery Reduction of $x$ is $xR^{-1}$ why should the choice of R be $2^l$ where $l$ is the length of $N$ to the base $2$? Why cannot we have a larger $R$?

Answer 1

Theoretically $R$ needs to be such that $2^{l-1} < N < R = 2^l$. But practically $l$ is chosen to be multiples of CPU word so if 1 word is 32 bits and $l$ comes out to be 30 as per above equation, l will instead be chosen as 32 and then $R = 2^{32}$.
$2^l$ is chosen because in computers, it is efficient to do operations like multiplication, division, modulus with powers of 2.
Multiplication by $2^l$ is equivalent to shifting first $l$ bits to the left (and put zero at the place of shifted bits).
Division by $2^l$ is equivalent to shifting first $l$ bits to the right.
Taking modulo $2^l$ is equivalent to choosing the first $l$ bits only.