LINE: Public-key encryption

2025-07-06

AI Summary

This paper addresses the challenge of achieving polynomial-time attack resistance in public-key encryption. It proposes a novel construction leveraging the inherent multiplicity of solutions in underdetermined linear equation systems. Methodologically, it employs homogeneous matrix transformations over the vector space $mathbb{F}_2^m$ to populate system parameters; integrates basis factorization substitution and homomorphic matrix transformations; and realizes predefined shared secrets via a one-way function over a univariate Abelian 2-group. A decomposable substitution mechanism ensures ciphertext generation depends on the solution-space structure—not on any unique solution. The scheme maintains low computational overhead while reducing key recovery to finding a specific sparse solution within a high-dimensional solution space—a problem proven to be intractable in polynomial time. Experimental evaluation confirms resilience against known cryptanalytic attacks.

Technology Category

• Machine Learning: Privacy
• 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
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

We propose a public key encryption cryptosystem based on solutions of linear equation systems with predefinition of input parameters through shared secret computation for factorizable substitutions. The existence of multiple equivalent solutions for an underdetermined system of linear equations determines the impossibility of its resolution by a cryptanalyst in polynomial time. The completion of input parameters of the equation system is implemented through secret homomorphic matrix transformation for substitutions factorized over the basis of a vector space of dimension m over the field F2. Encryption is implemented through computation of substitutions that are one-way functions on an elementary abelian 2-group of order 2"m. Decryption is implemented through completion of input parameters of the equation system. Homomorphic transformations are constructed based on matrix computations. Matrix computations enable the implementation of high security and low computational overhead for homomorphic transformations.

Problem

• Proposes public-key encryption using linear equation systems
• Ensures security via underdetermined equations and homomorphic transformations
• Implements efficient encryption/decryption with matrix computations

Innovation

• Public key encryption using linear equation systems
• Secret homomorphic matrix transformations for security
• Matrix computations ensure low overhead encryption