New Residue Arithmetic Based Barrett Algorithms, Part II: Modular Polynomial Computations

TL;DR

This paper introduces a novel residue arithmetic-based Barrett algorithm for efficient modular polynomial multiplication and exponentiation, avoiding large polynomial conversions, with applications in cryptography and signal security.

Contribution

It presents a new BA-P algorithm and a residue arithmetic-based BA-MPM method, along with a complete mathematical framework and computational procedures.

Findings

Validated the correctness of the algorithms through proofs.

Developed a complete computational procedure for BA-MPM.

Applied the algorithms to modular polynomial exponentiation in cryptography.

Abstract

In this paper, we derive a new computational algorithm for Barrett technique for modular polynomial multiplication, termed BA-P. BA-P is then applied to a new residue arithmetic based Barrett algorithm for modular polynomial multiplication (BA-MPM). The focus of the work is an algorithm that carries out the entire computation using only modular arithmetic without conversion to large degree polynomials. There are several parts to this work. First, we set up a new BA-P using polynomials other than u^alfa. Second, residue arithmetic based BA-MPM is described. A complete mathematical framework is described including proofs of the steps in the computations and the validity of results. Third, we present a computational procedure for BA-MPM. Fourth, the BA-MPM is used as a basis for algorithms for modular polynomial exponentiation (MPE). Applications are in areas of signal security and cryptography.