RNS Reverse Conversion Algorithm and Parity Detection for the Wide Arbitrary Moduli Set

Abstract

Due to absence of long carry propagation logic in its modular adders and multipliers Residue Number System (RNS) provides advantages for performing arithmetic computations in the hardware accelerators targeting multiplications and additions only. However, a conversion from RNS to Binary Number System (BNS) is a well-known complex operation requiring larger hardware area and consuming more power than modular multipliers which significantly limits applicability of RNS. Many non-modular operation (e.g. comparison, division) implementations are also based on reverse conversion. In this article we demonstrate a variation of Chinese Reminder Theorem (CRT) based algorithm for RNS to BNS reverse conversion for an arbitrary moduli set. We propose an approximate method to overcome the heaviest $$mod\;M$$ operation via the rank of number calculation and demonstrate its correctness on the covered number range. We show that algorithm parameters selection has no restrictions on moduli set and needed parameters can always be found assuming that covered maximum value is less than $$M - 1$$ . Hardware implementation of the new RNS reverse conversion algorithm based on CRT is more than 30% faster and consumes up to 80% less power than implementations of CRT based reverse conversion algorithms using other known ways to compute the final reduction operation.