发表机构
Middlesex University（密德萨斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出新型图神经网络架构WPNet，可启发式求解字问题并预测测地线长度，成功攻击Wagner-Magyarik公钥密码系统，为后量子密码学提供了新的安全分析思路。
AI 中文摘要
字问题作为研究对象已有一个多世纪的数学研究历史，最初推动了组合群论的发展，近年成为后量子密码学（PQC）的基础困难假设。尽管字问题普遍不可判定，但多类无限非阿贝尔群的字问题可解或算法运行快速，使其成为密码设计的理想平台。本文提出WPNet，一种新型图神经网络架构，可启发式求解字问题，在Baumslag-Solitar群$BS(1,2)$和Artin群上得到验证。模型通过将未归约字映射为动态图结构，在连续嵌入空间中学习聚类代数等价元素，无需执行离散归约步骤即可有效识别字的测地线代表。作为应用，开发的模型变体可预测两类群中未归约字的测地线长度。为验证该结构泄露的密码学严重性，WPNet被成功用于攻击Wagner-Magyarik公钥密码系统。
英文摘要
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.
Comments21 pages, 3 figures, 4 tables
