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Efficient Implementations of Residue Generators Mod 2n + 1 Providing Diminished-1 Representation

Stanisław J. Piestrak, Piotr Patronik

TL;DR

This paper addresses efficient input conversion for residue number systems whose moduli include the odd form $2^n+1$, focusing on delivering operands in the diminished-1 (D1) representation. The authors propose a universal residue generator architecture for mod $2^n+1$ that accepts a $p$-bit input and, for $p \ge 4n$, partitions it into $q=\lceil p/(2n) \rceil$ blocks, processes them with a $q$-operand carry-save adder (CSA) with end-around carry, and yields a residue in D1 form via a final 4-operand adder modulo $2^n+1$ (where COR$_{2^n+1}=0$). They also extend the design to bi-residue generators for conjugate moduli $2^n \pm 1$ by sharing the same hardware to produce residues for both moduli directly in D1 form. The approach provides scalable, low-cost input converters suitable for RNS implementations in DSP, cryptography, and communications, with hardware sharing across conjugate moduli and preserved D1 efficiency.

Abstract

The moduli of the form 2n + 1 belong to a class of low-cost odd moduli, which have been frequently selected to form the basis of various residue number systems (RNS). The most efficient computations modulo (mod) 2n + 1 are performed using the so-called diminished-1 (D1) representation. Therefore, it is desirable that the input converter from the positional number system to RNS (composed of a set of residue generators) could generate the residues mod 2n + 1 in D1 form. In this paper, we propose the basic architecture of the residue generator mod 2n + 1 with D1 output. It is universal, because its initial part can be easily designed for an arbitrary p >= 4n, whereas its final block-the 4-operand adder mod 2n + 1-preserves the same structure for any p. If a pair of conjugate moduli 2n +/- 1 belongs to the RNS moduli set, the latter architecture can be easily extended to build p-input bi-residue generators mod 2n+/-1, which not only save hardware by sharing p - 4n full-adders, but also generate the residue mod 2n + 1 directly in D1 form.

Efficient Implementations of Residue Generators Mod 2n + 1 Providing Diminished-1 Representation

TL;DR

This paper addresses efficient input conversion for residue number systems whose moduli include the odd form , focusing on delivering operands in the diminished-1 (D1) representation. The authors propose a universal residue generator architecture for mod that accepts a -bit input and, for , partitions it into blocks, processes them with a -operand carry-save adder (CSA) with end-around carry, and yields a residue in D1 form via a final 4-operand adder modulo (where COR). They also extend the design to bi-residue generators for conjugate moduli by sharing the same hardware to produce residues for both moduli directly in D1 form. The approach provides scalable, low-cost input converters suitable for RNS implementations in DSP, cryptography, and communications, with hardware sharing across conjugate moduli and preserved D1 efficiency.

Abstract

The moduli of the form 2n + 1 belong to a class of low-cost odd moduli, which have been frequently selected to form the basis of various residue number systems (RNS). The most efficient computations modulo (mod) 2n + 1 are performed using the so-called diminished-1 (D1) representation. Therefore, it is desirable that the input converter from the positional number system to RNS (composed of a set of residue generators) could generate the residues mod 2n + 1 in D1 form. In this paper, we propose the basic architecture of the residue generator mod 2n + 1 with D1 output. It is universal, because its initial part can be easily designed for an arbitrary p >= 4n, whereas its final block-the 4-operand adder mod 2n + 1-preserves the same structure for any p. If a pair of conjugate moduli 2n +/- 1 belongs to the RNS moduli set, the latter architecture can be easily extended to build p-input bi-residue generators mod 2n+/-1, which not only save hardware by sharing p - 4n full-adders, but also generate the residue mod 2n + 1 directly in D1 form.
Paper Structure (9 sections, 1 theorem, 19 equations, 3 figures, 1 table)

This paper contains 9 sections, 1 theorem, 19 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Modulo $2^n+1$ Diminished-1 (D1) Representation
  4. Periodicity Properties of the Series of $|2^k|_{\hbox{$2^n \pm 1$}}$
  5. Design of Residue Generators mod $2^n+1$ with D1 Output
  6. Architecture Designed According to Piestrak199468
  7. New Universal Architecture
  8. Design of Bi-residue Generators mod $2^n \pm 1$
  9. Conclusions

Key Result

Theorem 1

For the circuit of Fig. new-genmod-dim1 the following two equations hold:

Figures (3)

  • Figure 1: Shorthand notation of the CSA tree for the residue generator mod 9 with: (a) $p=16$; (b) $p=17$; and (c) $p=18$ inputs.
  • Figure 2: New residue generator mod $2^n+1$ with D1 output.
  • Figure 3: New bi-residue generator mod $2^n \pm 1$ with D1 output.

Theorems & Definitions (2)

  • Example 1
  • Theorem 1