The Mixed-Radix Chinese Remainder Theorem and Its Applications to Residue Comparison

The Chinese remainder theorem (CRT) and mixed-radix conversion (MRC) are two classic theorems used to convert a residue number to its binary correspondence for a given moduli set {P _n,...,P ₂, P ₁}. The MRC is a weighted number system and it requires operations modulo P _i only and hence magnitude comparison is easily performed. However, the calculation of the mixed-radix coefficients in the MRC is a strictly sequential process and involves complex divisions. Thus the residue-to-binary (R/B) conversions and residue comparisons based on the MRC require large delay. In contrast, the R/B conversion and residue comparison based on the CRT are fully parallel processes. However, the CRT requires large operations modulo M = P _n,...,P ₂ P ₁. In this paper, a new mixed-radix CRT is proposed which possesses both the advantages of the CRT and the MRC, which are parallel processing, small operations modulo P _i only, and the efficiency of making modulo comparison. Based on the proposed CRT, new residue comparators are developed for the three-moduli set {2ⁿ - 1, 2ⁿ, 2ⁿ + 1}. The FPGA implementation results show that the proposed modulo comparators are about 20% faster and smaller than one of the previous best designs.