Beyond the Finite Variant Property: Extending Symbolic Diffie-Hellman Group Models (Extended Version)

2026-01-29

AI Summary

Existing symbolic protocol verification tools struggle to support full operations in Diffie-Hellman groups—particularly exponent addition—limiting precise modeling and verification of related protocols. This work proposes a computationally sound approximation of the complete Diffie-Hellman theory and designs a corresponding semi-decision procedure, enabling, for the first time, mainstream symbolic verification tool Tamarin to handle full group operations including multiplication (i.e., exponent addition), thereby overcoming limitations imposed by the finite variant property. The approach successfully verifies the security of the ElGamal encryption scheme and reproduces a known attack on the MQV protocol, demonstrating its effectiveness and practical utility in real-world protocol analysis.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Representing, Reasoning, and Using Provenance, Trust
• 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
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data

Abstract

Diffie-Hellman groups are commonly used in cryptographic protocols. While most state-of-the-art, symbolic protocol verifiers support them to some degree, they do not support all mathematical operations possible in these groups. In particular, they lack support for exponent addition, as these tools reason about terms using unification, which is undecidable in the theory describing all Diffie-Hellman operators. In this paper we approximate such a theory and propose a semi-decision procedure to determine whether a protocol, which may use all operations in such groups, satisfies user-defined properties. We implement this approach by extending the Tamarin prover to support the full Diffie-Hellman theory, including group element multiplication and hence addition of exponents. This is the first time a state-of-the-art tool can model and reason about such protocols. We illustrate our approach's effectiveness with different case studies: ElGamal encryption and MQV. Using Tamarin, we prove security properties of ElGamal, and we rediscover known attacks on MQV.

Problem

• Diffie-Hellman groups
• symbolic protocol verification
• exponent addition
• unification
• cryptographic protocols

Innovation

• Diffie-Hellman groups
• symbolic protocol verification
• exponent addition
• Tamarin prover
• semi-decision procedure