With the development of Lie theory, Lie groups have profound significance in many branches of mathematics and physics. In Lie theory, matrix exponential plays a crucial role between Lie groups and Lie algebras. Meanwhile, as finite analogues of Lie groups, finite groups of Lie type also have wide application scenarios in mathematics and physics owning to their unique mathematical structures. In this context, it is meaningful to explore the potential applications of finite groups of Lie type in cryptography. In this paper, we firstly built the relationship between matrix exponential and discrete logarithmic problem (DLP) in finite groups of Lie type. Afterwards, we proved that the complexity of solving non-abelian factorization (NAF) problem is polynomial with the rank of the finite group of Lie type. Furthermore, combining with the Algebraic Span, we proposed an efficient algorithm for solving group factorization problem (GFP) in finite groups of Lie type. Therefore, it's still an open problem to devise secure cryptosystems based on Lie theory.

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有限李型群中的群分解问题

随着李理论的发展，李群在数学和物理学的许多分支中具有深远的意义。在李理论中，矩阵指数在李群和李代数之间起着至关重要的作用。同时，作为李群的有限类比，李型有限群由于其独特的数学结构，在数学和物理中也有着广泛的应用场景。在此背景下，探索Lie型有限群在密码学中的潜在应用是有意义的。在本文中，我们首先建立了有限李型群中矩阵指数与离散对数问题（DLP）之间的关系。随后，我们证明了求解非阿贝尔因式分解（NAF）问题的复杂度与Lie型有限群的秩成多项式。此外，结合代数跨度，我们提出了一种求解有限李型群中群因式分解问题（GFP）的有效算法。因此，设计基于李理论的安全密码系统仍然是一个悬而未决的问题。