New Residue Arithmetic Based Barrett Algorithms, Part I: Modular Integer Computations

TL;DR

This paper introduces new residue arithmetic-based Barrett algorithms that perform modular computations entirely within residue number systems, eliminating the need for large integer conversions and scaling operations, with applications in cryptography.

Contribution

The paper develops a novel RNS-based Barrett algorithm using two constants other than powers of two, including a complete mathematical framework and algorithms for cryptographic applications.

Findings

New RNS-based Barrett algorithm without large integer conversions

Mathematical proofs validating the algorithm's correctness

Algorithms for modular multiplication and exponentiation in RNS

Abstract

In this paper, we derive new computational techniques for residue number systems (RNS) based Barrett algorithm (BA). The focus of the work is an algorithm that carries out the entire computation using only modular arithmetic without conversion to large integers via the Chinese Remainder Theorem (CRT). It also avoids the computationally expensive scaling-rounding operation required in the earlier work. There are two parts to this work. First, we set up a new BA using two constants other than powers of two. Second, a RNS based BA is described. A complete mathematical framework is described including proofs of the various steps in the computations and the validity of results. Third, we present a computational algorithm for RNS based BA. Fourth, the RNS based BA is used as a basis for new RNS based algorithms for MoM and MoE. The applications we are dealing with are in the area of cryptography.