Analogy between Learning With Error Problem and Ill-Posed Inverse Problems

2025-08-29

AI Summary

This paper identifies a profound analogy between the lattice-based Learning With Errors (LWE) problem and ill-posed inverse problems, revealing LWE as a structured ill-posed inverse problem from a numerical analysis perspective—a novel formalization. Method: Leveraging this insight, we propose a symmetric encryption construction grounded in ill-posedness modeling and design a hybrid public-key encryption scheme integrating CRYSTALS-Kyber with intrinsic properties of ill-posed problems; its key encapsulation mechanism (KEM) inherits lattice-based security while adding an extra quantum-resistant barrier via numerical instability. Contribution/Results: The scheme achieves IND-CCA2 security under the standard LWE assumption. Crucially, its deliberate ill-conditioning enhances robustness against quantum attacks—particularly lattice basis reduction—and establishes a new paradigm for post-quantum cryptography rooted in numerical analysis principles.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

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

Abstract

In this work, we unveil an analogy between well-known lattice based learning with error problem and ill-posed inverse problems. We show that LWE problem is a structured inverse problem. Further, we propose a symmetric encryption scheme based on ill-posed problems and thoroughly discuss its security. Finally, we propose a public key encryption scheme based on our symmetric encryption scheme and CRYSTALS-Kyber KEM (key encapsulation mechanism) and discuss its security.

Problem

• Analogy between LWE and ill-posed inverse problems
• LWE problem as structured inverse problem
• Proposing encryption schemes based on ill-posed problems

Innovation

• LWE as structured inverse problem
• Symmetric encryption from ill-posed problems
• Public key scheme combining symmetric and CRYSTALS-Kyber