Revisiting sum of residues modular multiplication

Abstract

the 1980s,when the introduction of public key cryptography spurred interest in modularmultiplication, many implementations performed modularmultiplication using a sumof residues. As the fieldmatured, sum of residues modularmultiplication lost favour to the extent that all recent surveys have either overlooked it or incorporated it within a larger class of reduction algorithms. In this paper, we present a new taxonomy of modular multiplication algorithms. We include sum of residues as one of four classes and argue why it should be considered different to the other, now more common, algorithms.We then apply techniques developed for other algorithms to reinvigorate sum of residues modular multiplication. We compare FPGA implementations of modular multiplication up to 24 bits wide. The Sum of Residues multipliers demonstrate reduced latency at nearly 50% compared to Montgomery architectures at the cost of nearly doubled circuit area. The new multipliers are useful for systems based on the Residue Number System (RNS).Yinan Kong and Braden Phillip