Chinese Remainder Theorem Approach to Montgomery-Type Algorithms

2024-02-01

AI Summary

This work addresses the long-standing challenge of correctness verification in Montgomery-type modular reduction algorithms. We propose a unified algebraic modeling framework grounded in the Chinese Remainder Theorem (CRT) and Qin Jiushao’s polynomial evaluation identity. First, we formally derive the standard Montgomery reduction algorithm directly via CRT, exposing its intrinsic algebraic structure. Second, we establish the first extensible verification paradigm covering diverse state-of-the-art variants—including CIOS and FIPS 186-4—enabling automated counterexample generation and defect localization. Rigorously verifying over ten mainstream variants, we identify and correct multiple algorithmic errors previously reported in authoritative cryptographic literature. Our framework not only provides a rigorous theoretical foundation for Montgomery-type reductions but also introduces a novel algebraic methodology for the formal verification of cryptographic algorithms, advancing both foundational understanding and practical assurance in cryptographic implementation.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Calibration & Uncertainty Quantification
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Applications of cryptography
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Model Montgomery algorithms using Chinese Remainder Theorem
• Unify framework for modular reduction algorithms
• Detect errors in Montgomery-type algorithm designs

Innovation

• Chinese Remainder Theorem
• Montgomery reduction algorithm
• unified framework