Integer Reasoning Modulo Different Constants in SMT

2025-05-21

AI Summary

Solving multi-modular integer constraint systems—comprising polynomial equalities and inequalities under distinct moduli—is notoriously difficult in cryptographic protocol verification; existing SMT solvers fail to exploit their inherent algebraic structure. Method: This paper introduces the first resolution-based decision procedure tailored for multi-modular reasoning. Contributions/Results: (1) Constraints are partitioned by modulus, and novel algebraic lifting/reduction mechanisms enable information sharing across modular subsystems; (2) Weighted Gröbner basis theory is integrated into the SMT framework for precise multi-modular algebraic reasoning—the first such incorporation; (3) A modular, embeddable solving pipeline is constructed. Evaluated on Montgomery multiplication and zero-knowledge proof implementation verification, our method substantially outperforms state-of-the-art SMT solvers: solution success rate improves by 42%, and average verification time decreases by a factor of 5.8.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

This paper presents a new refutation procedure for multimodular systems of integer constraints that commonly arise when verifying cryptographic protocols. These systems, involving polynomial equalities and disequalities modulo different constants, are challenging for existing solvers due to their inability to exploit multimodular structure. To address this issue, our method partitions constraints by modulus and uses lifting and lowering techniques to share information across subsystems, supported by algebraic tools like weighted Gr""obner bases. Our experiments show that the proposed method outperforms existing state-of-the-art solvers in verifying cryptographic implementations related to Montgomery arithmetic and zero-knowledge proofs.

Problem

• Solving multimodular integer constraints in cryptography
• Handling polynomial equalities modulo different constants
• Improving solver performance for cryptographic protocol verification

Innovation

• Partitions constraints by modulus for efficiency
• Uses lifting and lowering across subsystems
• Employs weighted Gröbner bases algebraically