Algorithm 2: Montgomery Multiplication

Input: $\overline{P}=P\times rmodN$ and $\overline{Q}=Q\times rmodN$ where $r=2^n$ and $n=\lceil {log}_{2}N\rceil$ Output: $\overline{R}=P\times Q\times rmodN$ $S\leftarrow \overline{P}\times \overline{Q}$ $W\leftarrow S\times N^{\prime }modr$ where $N^{\prime }=N^{-1}modr$ $T\leftarrow (S+W\times N)/r$ if $T\ge N$ then$\phantom{(}$$\phantom{(}$ $| T=T-N$ end $\phantom{(}$ $R\leftarrow T$ Return $\left(\overline{R}\right)$