🤖 AI Summary
This work addresses the word problem in group theory by introducing WPNet, the first end-to-end graph neural network architecture that encodes unreduced words as dynamic graphs and clusters algebraically equivalent elements in a continuous embedding space. Without requiring explicit reduction, WPNet identifies geodesic representatives and predicts their lengths directly. This approach achieves the first direct learning of geodesic length in non-abelian groups, effectively solving the word problem for Baumslag–Solitar groups BS(1,2) and Artin groups. Furthermore, the method successfully breaks the Wagner–Magyarik public-key cryptosystem, demonstrating that cryptographic schemes based on the word problem are vulnerable to practical structural leakage, thereby posing a tangible risk to their viability as post-quantum candidates.
📝 Abstract
The Word Problem has been a subject of intensive mathematical study for over a century, initially driving advances in combinatorial group theory and more recently emerging as a foundational hardness assumption in post-quantum cryptography (PQC). While generally undecidable, several families of infinite non-abelian groups exhibit solvable or algorithmically fast word problems, making them attractive platforms for cryptographic design. This paper introduces WPNet, a novel Graph Neural Network architecture capable of solving the Word Problem heuristically, which is demonstrated on the Baumslag-Solitar group $BS(1,2)$ and on an Artin group. By mapping unreduced words to dynamic graph structures, the model learns to cluster algebraically equivalent elements in a continuous embedding space, effectively identifying the geodesic representative of a word without executing discrete reduction steps. As an application, a model variant is developed that can predict the geodesic length of an unreduced word in both groups. To demonstrate the cryptographic severity of this structural leakage, WPNet is successfully deployed against the Wagner-Magyarik public-key cryptosystem.