# Non-Modular Operations of the Residue Number System: Functions for Computing

Chapter

## Abstract

This chapter introduces the problem of non-modular operations in the Residue Number System (RNS) and presents some recent approaches for their effective implementation. The approaches are based on specific functions defined from the RNS to the Integers that show mathematical properties useful to support the implementation of non-modular operations, like magnitude comparison and residue-to-binary conversion. In particular, two different functions defined from the RNS to the Integers are discussed: the ‘diagonal functions’ and the ‘quotient functions’. Through the paper the new implementations of non-modular operations in the RNS are described and their effectiveness is analysed with respect to traditional techniques in the literature.

## Keywords

Chinese remainder theorem Core function Diagonal function Magnitude comparison Mixed-radix conversion Multi-operand modular adder Quotient function Residue number system Residue-to-binary conversion

## Nomenclature

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m1,m2,…,m n : set of pairwise relatively prime moduli (n integer, n ≥ 2)

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$$M=\prod_{i=1}^n{m}_i}$$

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I = [0,M − 1]: dynamic range of the RNS (set of integers)

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$${X}_i={\left|X\right|}_{m_i}$$: the least positive residue of X modulo m i , i = 1,2,..,n

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$${M}_i=\frac{M}{m_i}$$, i = 1,2,…,n

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$$\mathit{SQ}=\sum_{i=1}^n{M}_i}$$: the ‘diagonal modulus’ of the RNS

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$${J}_{a,b}={\left|\frac{1}{a}\right|}_b$$: the multiplicative inverse of a modulo b (i.e. $$\vert a\cdot {\left|\frac{1}{a}\right|}_b{\vert}_b=1$$)

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a⌋: the largest integer not exceeding a

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b⌉: rounding to a higher integer

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≡: modular congruence

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MOMA (n, W): multi-operand modulo W adder for n operands.

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