Montgomery Modular Multiplication via Single-Base Residue Number Systems

Montgomery modular multiplication (MMM) in residue number systems (RNS) uses a base extension (BE) technique. This is to avoid division, which is hard, slow and costly in RNS. It is somewhat less costly and faster than the reverse conversion, via Chinese remainder theorem (CRT) and reduction factor method. However, it is used one after the other, for each of the equally large bases. In this work, we modify the conventional RNS-MMM algorithm via replacing the two unparalleled BE undertakings with three parallel CRT-like operations with the same complexity, as BE. As for the reduction factors, we use a special case of the Kawamura’s algorithm that leads to definitive result. The proposed RNS-MMM method allows for squaring the working dynamic range, or halving the bit-width of the balanced residue channels. Moreover, the common practice of dynamically changing the working moduli set in security and crypto applications is less critical due to doubled size of the pool of available moduli. The proposed circuits are simulated, tested and synthesized via Synopsys Design Compiler on the TSMC 65-nm technology, to show 69% less delay and 28% less area-time-product at the cost of 14% more energy consumption, with respect to the most relevant reference work.