The Learning With Errors (LWE) problem forms the foundation of many post-quantum cryptographic systems. These systems depend on a critical assumption: that no one can distinguish between LWE-generated samples and truly random data. I wanted to test this assumption by building sophisticated machine learning models to see if they could break this fundamental security property.
My goal wasn’t to completely “break” LWE—that would be a monumental achievement. Instead, I aimed to map out where LWE’s security boundaries actually lie in practice. Could a neural network, with its pattern-recognition capabilities, successfully identify LWE samples and violate the core security assumption? Through multiple iterations of model improvements and data refinement, I discovered just how resilient LWE really is. The results highlight why cryptographers rely on standardized libraries with carefully chosen parameters backed by years of analysis.
Table of Contents
- The Learning With Errors (LWE) Problem
- Methodology: A Machine Learning-Based Auditor
- A Multi-Stage Analytical Process
- Results and Discussion: The Intractability of the LWE Signal
- Conclusion
The Learning With Errors (LWE) Problem
The LWE problem rests on the difficulty of recovering a secret vector $\mathbf{s} \in \mathbb{Z}_q^n$ from noisy linear equations. Each LWE sample consists of a pair $(\mathbf{a}, b) \in \mathbb{Z}_q^n \times \mathbb{Z}_q$, where $b$ gets calculated as:
$b = \langle \mathbf{a}, \mathbf{s} \rangle + e \mod q$
Here, $\mathbf{a}$ represents a publicly known random vector, while $e$ is a small error term drawn from a discrete Gaussian distribution. LWE-based systems rely on two key assumptions:
- Search-LWE: Finding $\mathbf{s}$ is computationally infeasible
- Decision-LWE: Distinguishing LWE samples from uniformly random pairs is impossible
My analysis focuses on this second assumption.
Methodology: A Machine Learning-Based Auditor
I built a machine learning distinguisher as my primary analytical tool—essentially a neural network trained as a binary classifier to tackle the Decision-LWE problem. The model receives two types of data:
- Positive Class (Label 1): Authentic LWE samples $(\mathbf{a}, b)$
- Negative Class (Label 0): Uniformly random pairs $(\mathbf{a}, u)$
The model’s accuracy on unseen test data directly measures how distinguishable LWE samples are for specific parameter sets. Any accuracy substantially above 50% would signal a practical weakness, suggesting the model found a generalizable statistical pattern. I designed the experiment to map this accuracy across different LWE parameters, varying both the dimension (n) and noise magnitude (sigma).
A Multi-Stage Analytical Process
This investigation didn’t follow a straight path—early failures actually proved crucial in refining my approach and revealing deeper insights about the problem’s complexity.
Stage I: The Baseline Model and Overfitting
My first attempt used a standard Multi-Layer Perceptron (MLP) trained on complete $(\mathbf{a}, b)$ pairs. The model consistently failed to generalize, with test accuracy stuck at 50%. The training logs showed classic severe overfitting: while training accuracy climbed, validation performance stayed at random chance levels. This meant the model was just memorizing training artifacts rather than learning the actual mathematical structure.
Stage II: Feature Engineering for Modular Arithmetic
I suspected the model’s failure stemmed from a fundamental data representation mismatch. Neural networks treat numbers linearly, but LWE operates in a modular ring where $q-1$ sits right next to $0$. To fix this, I implemented circular embeddings, transforming each integer x into a 2D vector (cos(2πx/q), sin(2πx/q)). This feature engineering explicitly gave the model an understanding of modular proximity.
Despite this major improvement in data representation, the model still couldn’t generalize—severe overfitting persisted. This suggested the problem wasn’t just about data format but something more fundamental.
Stage III: Architectural Refinement with a 1D CNN
The LWE problem has an inherently sequential structure—$b$ equals the sum of component-wise products. Standard MLPs aren’t architecturally suited for capturing these relationships. I switched to a 1D Convolutional Neural Network (CNN), specifically designed to identify local patterns in sequential data. I also added L2 Regularization to penalize model complexity and reduce overfitting.
The results were striking: training accuracy shot up dramatically, proving the CNN was far more capable. However, validation accuracy stayed flat at 50%. This was a critical discovery: even with proper data representation and a powerful, regularized architecture, the LWE secret’s signal was too diluted across the high-dimensional input space for the model to learn any generalizable pattern. This provided experimental evidence of the “curse of dimensionality” serving as LWE’s core security feature.
Stage IV: A Focused Analysis of the Output Distribution
After consistently failing to distinguish complete $(\mathbf{a}, b)$ pairs, I formed a final, more focused hypothesis. If the full problem proves intractable, maybe the LWE process leaves a detectable statistical bias in the distribution of b values alone.
I redesigned the experiment to train my most sophisticated model—the regularized 1D CNN with circular embeddings—on a simpler task: distinguishing collections of LWE-generated b values from collections of uniformly random integers.
Results and Discussion: The Intractability of the LWE Signal
The final experiment confirmed what cryptographers have long theorized. Across all tested parameters, including deliberately weakened ones, the model failed to distinguish the distribution of LWE b values from the uniform distribution. Test accuracy remained locked at 50%.
LWE’s resistance to machine learning attacks is well-established in theory, and this empirical evidence reinforces that understanding. Even with sophisticated neural architectures and optimized data representations, the statistical signature of the secret remains undetectable. The high-dimensional dot product combined with additive noise creates such effective diffusion that no learnable patterns emerge, even under conditions favorable to the attacker.
Conclusion
This investigation into LWE’s boundaries provides concrete evidence supporting the theoretical foundations of post-quantum cryptography. Even deliberately weakened LWE instances proved robust against sophisticated statistical analysis using modern machine learning techniques, including deep convolutional neural networks. The consistent inability of these models to generalize reinforces the mathematical principles underlying LWE’s security.
The results underscore a fundamental principle in applied cryptography: parameter selection matters enormously. The resilience observed even with suboptimal parameters demonstrates why cryptographers rely on standardized, peer-reviewed libraries where parameters undergo extensive analysis to ensure substantial security margins against known attack vectors.
Note: The full Python script for this analysis is available at this GitHub repo for review and further experimentation.