Algorithmic problems in Engel groups and cryptographic applications

Algorithmic decidability of Engel's property for automaton groups

We consider decidability problems associated with Engel's identity ($[\cdots[[x,y],y],\dots,y]=1$ for a long enough commutator sequence) in groups generated by an automaton. We give a partial algorithm that decides, given $x,y$, whether an Engel identity is satisfied. It succeeds, importantly, in proving that Grigorchuk's $2$-group is not Engel. We consider next the problem of recognizing Engel elements, namely elements $y$ such that the map $x\mapsto[x,y]$ attracts to $\{1\}$. Although this problem seems intractable in general, we prove that it is decidable for Grigorchuk's group: Engel eleme…

学习字问题：测地线长度及其密码学应用

Learning the Word Problem: Geodesic Lengths and Cryptographic Applications

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。

使用辫群中的米哈伊洛娃子群进行密钥交换构造

A Key Exchange Construction using Mihailova Subgroups in Braid groups

研究利用辫群中米哈伊洛娃子群进行密钥交换构造，源于其成员问题不可解，引出商群公式，但代数事实不能提供完整安全证明，此构造是有代数动机的候选方案，完整密码安全性待进一步研究。

Group-based Cryptography in the Quantum Era

In this expository article we present an overview of the current state-of-the-art in post-quantum group-based cryptography. We describe several families of groups that have been proposed as platforms, with special emphasis in polycyclic groups and graph groups, dealing in particular with their algorithmic properties and cryptographic applications. We then, describe some applications of combinatorial algebra in fully homomorphic encryption. In the end we discussing several open problems in this direction.