The theory of finite simple groups is a (rather unexplored) area likely to provide interesting computational problems and modelling tools useful in a cryptographic context. In this note, we review some applications of finite non-abelian simple groups to cryptography and discuss different scenarios in which this theory is clearly central, providing the relevant definitions to make the material accessible to both cryptographers and group theorists, in the hope of stimulating further interaction between these two (non-disjoint) communities. In particular, we look at constructions based on various group-theoretic factorization problems, review group theoretical hash functions, and discuss fully homomorphic encryption using simple groups. The Hidden Subgroup Problem is also briefly discussed in this context.

翻译：有限单群理论是一个（尚未充分探索的）领域，可能为密码学提供有趣的计算问题与建模工具。本文综述了有限非阿贝尔单群在密码学中的若干应用，并探讨了该理论发挥核心作用的不同场景。为使密码学家与群论研究者均能理解相关内容，我们提供了必要的定义，旨在促进这两个（非互斥）学术群体间的进一步互动。具体而言，我们考察了基于群论因子分解问题的若干构造，综述了群论哈希函数，并讨论了利用单群实现的全同态加密方案。此外，文中还简要探讨了该语境下的隐子群问题。