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Issue No.12 - December (2008 vol.57)
pp: 1624-1632
Shaoqiang Bi , Xilinx Inc., San Jose
Warren J. Gross , McGill University, Montreal
ABSTRACT
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_2P_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 {2^n &#8722; 1, 2^n, 2^n + 1}. The FPGA implementation results show that the proposed modulo comparators are about 20% faster and smaller than one of the previous best designs.
INDEX TERMS
Computer arithmetic, Residue number system, RNS, Chinese Remainder Theorem, Residue comparison., Mixed-Radix conversion, FPGA
CITATION
Shaoqiang Bi, Warren J. Gross, "The Mixed-Radix Chinese Remainder Theorem and Its Applications to Residue Comparison", IEEE Transactions on Computers, vol.57, no. 12, pp. 1624-1632, December 2008, doi:10.1109/TC.2008.126
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