Chinese Remainder Theorem Approach to Montgomery-Type Algorithms

TL;DR

This paper presents a unified CRT-based framework for Montgomery-type algorithms, enabling analysis, validation, and detection of errors in existing modular reduction methods, thereby advancing the understanding of these algorithms.

Contribution

It introduces a CRT formalism derived from Qin's Identity that models Montgomery reduction algorithms, unifying various variants and identifying errors in prior designs.

Findings

CRT framework models Montgomery algorithms effectively

Recent variants are validated within the CRT framework

Errors in some existing algorithms are detected and corrected

Abstract

This paper explores the ability of the Chinese Remainder Theorem formalism to model Montgomery-type algorithms. A derivation of CRT based on Qin's Identity gives Montgomery reduction algorithm immediately. This establishes a unified framework to treat modular reduction algorithms of Montgomery-type. Several recent notable variants of Montgomery algorithm are analyzed, validation of these methods are performed within the framework. Problems in some erroneous design of reduction algorithms of Montgomery-type in the literature are detected and counter examples are easily generated by using the CRT formulation.