Efficient setup for a Montgomery multiplication Montgomery described an efficient method to compute a modular multiplication. This works by using a special constant $R$ and assumes the inputs $a$ and $b$ have been made into a special representation (residues $aR\mod N$ and $bR\mod N$) and produces the value $abR^{-1}\mod N$. Thus to pursue the computations, one needs the value $ab$ to be also in the special representation, which requires an additional (Montgomery) multiplication with the constant $R^2\mod N$. 

This is especially useful to compute modular exponentiations with a large modulus $N$ and a big exponent such as in RSA. 

Every step such as computing the special representation of $a$ and $b$ are costly, and so is the computation of $R^2\mod N$ (note that it has to be computed once, but still is costly). 

What are the different ways to do the computation of $R^2 \mod N$ efficienty?
The usual way is to calculate $2^k R \bmod N$ for a small divisor $k$ of $l$ where $R = 2^l$ and use Montgomery multiplication in a Square-and-Multiply algorithm. This does require a division, but as $R$ is usually chosen to be just a little bit longer than $N$, the division doesn't have to be optimized much.

But you should also consider the fact that in some occasions (like RSA key generation) one can do also without knowing $R^2 \bmod N$. You neither have to Montgomery transform the random bases for a Fermat or Miller-Rabin test nor a number you want to invert using Fermat's little theorem (in the second case just Montgomery multiply with $1$ afterwards).