Abstract
In this paper, we deal with the critical problems in residue arithmetic. The reverse conversion from a Residue Number System (RNS) to positional notation is a main non-modular operation, and it constitutes a basis of other non-modular procedures used to implement various computational algorithms. We present a novel approach to the parallel reverse conversion from the residue code into a weighted number representation in the Mixed-Radix System (MRS). In our proposed method, the calculation of mixed-radix digits reduces to a parallel summation of the small word-length residues in the independent modular channels corresponding to the primary RNS moduli. The computational complexity of the developed method concerning both required modular addition operations and one-input lookup tables is estimated as , where k equals the number of used moduli. The time complexity is modular clock cycles. In pipeline mode, the throughput rate of the proposed algorithm is one reverse conversion in one modular clock cycle.
Highlights
In this paper, we deal with the critical problems in residue arithmetic
We present a novel approach to the parallel reverse conversion from the residue code into the mixed-radix representation
The calculation of mixed-radix digits reduces to a parallel summation of the small word-length residues in the independent modular channels corresponding to the primary Residue Number System (RNS) moduli
Summary
The abstract algebra and number theory create the theoretical basis of the residue arithmetic [12,13]. Compared with the conventional WNS, the RNS simplifies and speeds up the addition and multiplication operations This fundamental advantage of the residue arithmetic strongly appears in the case of implementing computational procedures, which mainly contain long segments consisting of only sequences of modular arithmetic operations. In this case, the primary moduli set is chosen so that the final results of the computational procedure always belong to the used dynamic range for any allowed values of input operands. Due to a lack of efficient methods and algorithms for non-modular operations implementation, the residue arithmetic is mainly suitable when the modular additions and multiplications make up the bulk of required computations In this case, the number of used non-modular operations is relatively small. This circumstance bounds the widespread use of the RNS to a narrow class of specific tasks
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