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

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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.

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Abbreviations

- :

m 1,m 2,…,m n : set of pairwise relatively prime moduli (n integer, n ≥ 2)

- :

\( M={\displaystyle \prod_{i=1}^n{m}_i} \)

- :

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

- :

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

- :

\( {M}_i=\frac{M}{m_i} \), i = 1,2,…,n

- :

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

- :

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

- :

⌊a⌋: the largest integer not exceeding a

- :

⌈b⌉: rounding to a higher integer

- :

≡: modular congruence

- :

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

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Authors and Affiliations

  1. Dipartimento di Informatica, Università degli Studi di Bari Aldo Moro, Via Orabona, Bari, 70125, Italy

    Giuseppe Pirlo

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  1. Giuseppe Pirlo
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Correspondence to Giuseppe Pirlo .

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Editors and Affiliations

  1. Department of Computer Engineering, Islamic Azad University, Kerman Branch, Kerman, Iran

    Amir Sabbagh Molahosseini

  2. INESC-ID, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    Leonel Seabra de Sousa

  3. School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, Singapore

    Chip-Hong Chang

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Pirlo, G. (2017). Non-Modular Operations of the Residue Number System: Functions for Computing. In: Molahosseini, A., de Sousa, L., Chang, CH. (eds) Embedded Systems Design with Special Arithmetic and Number Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-49742-6_3

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  • DOI: https://doi.org/10.1007/978-3-319-49742-6_3

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