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

TL;DR

This paper introduces a universal architecture for residue generators mod 2n+1 that efficiently produce residues in diminished-1 form, optimizing hardware for residue number systems used in low-cost odd moduli computations.

Contribution

It proposes a flexible, hardware-efficient residue generator architecture for mod 2n+1 in D1 form, extendable to bi-residue generators for moduli 2n±1.

Findings

Universal architecture adaptable for any p ≥ 4n.

Efficient hardware sharing reduces resource usage.

Direct generation of residues in D1 form enhances computational speed.

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.