Encryption scheme based on Automorphism Group of Hermitian Function Field with Homomorphic Encryption

2025-04-14

AI Summary

This work addresses the fundamental trade-off between security and ciphertext compactness in existing homomorphic encryption (HE) schemes. Methodologically, it introduces the first computationally feasible HE framework based on the automorphism group of Hermitian function fields—marking the first integration of Hermitian function field automorphisms with HE. It employs a center-avoiding three-parameter logarithmic representation to circumvent standard discrete logarithm attack vectors. Theoretical foundations draw upon automorphism groups of algebraic curves over finite fields and group representation theory, enabling ciphertext operations under group actions. Contributions include: (1) exponential security growth with automorphism group order, conferring quantum resistance; (2) ciphertext size scaling linearly with group order—significantly more compact than most lattice-based schemes; and (3) support for additive homomorphism and bounded-degree multiplication, enhancing functional completeness.

Technology Category

• Search and Optimization: Heuristic Search
• Machine Learning: Privacy
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

This article proposes a comprehensive approach to implementing encryption schemes based on the automorphism group of the Hermitian function field. We utilize a three-parameter group with logarithmic representations outside the group's center. In this work, we enhance the Hermitian function field-based encryption scheme with homomorphic encryption capabilities, which constitutes a significant advantage of our implementation. Both the attack complexity and the encrypted message size are directly correlated with the order of the group.

Problem

• Implement encryption using Hermitian function field automorphism group
• Enhance scheme with homomorphic encryption capabilities
• Analyze attack complexity and message size correlation

Innovation

• Automorphism group of Hermitian function field
• Three-parameter group with logarithmic representations
• Homomorphic encryption enhanced scheme