Group-based cryptography is a relatively unexplored family in post-quantum cryptography, and the so-called Semidirect Discrete Logarithm Problem (SDLP) is one of its most central problems. However, the complexity of SDLP and its relationship to more well-known hardness problems, particularly with respect to its security against quantum adversaries, has not been well understood and was a significant open problem for researchers in this area. In this paper we give the first dedicated security analysis of SDLP. In particular, we provide a connection between SDLP and group actions, a context in which quantum subexponential algorithms are known to apply. We are therefore able to construct a subexponential quantum algorithm for solving SDLP, thereby classifying the complexity of SDLP and its relation to known computational problems.

翻译：基于群的后量子密码学是一个相对未充分探索的领域，而所谓的半群离散对数问题（SDLP）是其核心问题之一。然而，SDLP的复杂度及其与更广为人知的困难问题之间的关系，特别是针对量子对手的安全性，尚不明确，这已成为该领域研究人员面临的一个重大开放性问题。本文首次对SDLP进行了专门的安全性分析。具体而言，我们建立了SDLP与群作用之间的联系，而后者已知可通过量子子指数算法求解。因此，我们能够构造一种求解SDLP的子指数量子算法，从而明确了SDLP的复杂度及其与已知计算问题的关系。