Solving a group equation of length nine

Levin's conjecture for an equation of length nine

Levin's conjecture has been established to hold true for group equations of length up to seven. Recently, it is shown that Levin's conjecture is also true (modulo exceptional cases) for some group equations of length eight and nine. In this paper we consider a group equation of length nine and show that the Levin's conjecture is true for this equation modulo some exceptional cases.

Solving random equations in Garside groups using length functions

We give a systematic exposition of memory-length algorithms for solving equations in noncommutative groups. This exposition clarifies some points untouched in earlier expositions. We then focus on the main ingredient in these attacks: Length functions. After a self-contained introduction to Garside groups, we describe length functions induced by the greedy normal form and by the rational normal form in these groups, and compare their worst-case performances. Our main concern is Artin's braid groups, with their two known Garside presentations, due to Artin and due to Birman-Ko-Lee (BKL). We sho…

Permute and conjugate: the conjugacy problem in relatively hyperbolic groups

Modelled on efficient algorithms for solving the conjugacy problem in hyperbolic groups, we define and study the permutation conjugacy length function. This function estimates the length of a short conjugator between words $u$ and $v$, up to taking cyclic permutations. This function might be bounded by a constant, even in the case when the standard conjugacy length function is unbounded. We give applications to the complexity of the conjugacy problem, estimating conjugacy growth rates, and languages. Our main result states that for a relatively hyperbolic group, the permutation conjugacy lengt…

A fast solution to the conjugacy problem in the 4-strand braid group

We present an algorithm for solving the conjugacy search problem in the four strand braid group. The computational complexity is cubic with respect to the braid length.

Length-based cryptanalysis: The case of Thompson's Group

The length-based approach is a heuristic for solving randomly generated equations in groups which possess a reasonably behaved length function. We describe several improvements of the previously suggested length-based algorithms, that make them applicable to Thompson's group with significant success rates. In particular, this shows that the Shpilrain-Ushakov public key cryptosystem based on Thompson's group is insecure, and suggests that no practical public key cryptosystem based on this group can be secure.

虚拟阿贝尔群中的有效方程求解、约束与增长

Effective equation solving, constraints and growth in virtually abelian groups

本文研究虚拟阿贝尔群中带组合、代数和语言约束的群方程求解，将约束转化为有理集以归约为可判定问题，并证明增长序列可有效计算。