In the recently emerging field of nonabelian group-based cryptography, a prominently used one-way function is the Conjugacy Search Problem (CSP), and two important classes of platform groups are polycyclic and matrix groups. In this paper, we discuss the complexity of the conjugacy search problem (CSP) in these two classes of platform groups using the three protocols in [10], [26], and [29] as our starting point. We produce a polynomial time solution for the CSP in a finite polycyclic group with two generators, and show that a restricted CSP is reducible to a DLP. In matrix groups over finite fields, we usedthe Jordan decomposition of a matrix to produce a polynomial time reduction of an A-restricted CSP, where A is a cyclic subgroup of the general linear group, to a set of DLPs over an extension of Fq. We use these general methods and results to describe concrete cryptanalysis algorithms for these three systems. In particular, we show that in the group of invertible matrices over finite fields and in polycyclic groups with two generators, a CSP where conjugators are restricted to a cyclic subgroup is reducible to a set of O(n2) discrete logarithm problems. Using our general results, we demonstrate concrete cryptanalysis algorithms for each of these three schemes. We believe that our methods and findings are likely to allow for several other heuristic attacks in the general case.

翻译：在近年来新兴的非阿贝尔群密码学领域中，共轭搜索问题（CSP）是一种被广泛使用的单向函数，而多循环群和矩阵群是两类重要的平台群。本文以文献[10]、[26]和[29]中的三种协议为出发点，讨论了这两类平台群中共轭搜索问题（CSP）的复杂性。我们提出了有限双生成元多循环群中CSP的多项式时间求解方法，并证明受限CSP可归约为离散对数问题（DLP）。在有限域上的矩阵群中，我们利用矩阵的Jordan分解，将A-受限CSP（其中A为一般线性群中的循环子群）多项式时间归约为有限域Fq扩域上的一组DLP。我们运用这些通用方法和结果，为这三个系统描述了具体的密码分析算法。特别地，我们证明在有限域上的可逆矩阵群和双生成元多循环群中，当共轭元被限制于循环子群时，CSP可归约为O(n²)个离散对数问题。基于这些通用结果，我们为每个方案展示了具体的密码分析算法。我们相信，在一般情况下，我们的方法和发现可能支持多种其他启发式攻击。