若干基于非交换群的密钥交换协议的密码分析
Cryptanalysis of some Nonabelian Group-Based Key Exchange Protocols
AI总结：
该研究针对基于非交换群的密码学中的共轭搜索问题，分别在双生成元有限多循环群和有限域矩阵群中给出多项式时间求解或规约方法，并对三类密钥交换协议实现了具体密码分析。
摘要：
在新兴的基于非交换群的密码学领域中，共轭搜索问题（Conjugacy Search Problem，CSP）是一种被广泛使用的单向函数，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]中的三个协议为切入点，探讨了这两类平台群中共轭搜索问题（CSP）的复杂度。我们给出了双生成元有限多循环群中CSP的多项式时间求解方法，并证明受限CSP可规约为离散对数问题（DLP）。在有限域上的矩阵群中，我们利用矩阵的若尔当分解，将A-受限CSP（其中A是一般线性群的循环子群）多项式时间规约为有限域Fq扩域上的一组DLP。我们利用这些通用方法与结果，给出了针对这三个系统的具体密码分析算法。特别地，我们证明在有限域上的可逆矩阵群以及双生成元多循环群中，共轭元限定为循环子群元素的CSP可规约为O(n²)个离散对数问题。基于我们的通用结果，我们为这三种方案分别设计了具体的密码分析算法。我们认为，本文的方法与结论有望为一般情形下的更多启发式攻击提供支撑。
英文摘要：
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.
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