We propose self-similar contracting groups as a platform for cryptographic schemes based on simultaneous conjugacy search problem (SCSP). The class of these groups contains extraordinary examples like Grigorchuk group, which is known to be non-linear, thus making some of existing attacks against SCSP inapplicable. The groups in this class admit a natural normal form based on the notion of a nucleus portrait, that plays a key role in our approach. While for some groups in the class the conjugacy search problem has been studied, there are many groups for which no algorithms solving it are known. Moreover, there are some self-similar groups with undecidable conjugacy problem. We discuss benefits and drawbacks of using these groups in group-based cryptography and provide computational analysis of variants of the length-based attack on SCSP for some groups in the class, including Grigorchuk group, Basilica group, and others.

翻译：我们提出将自相似收缩群作为基于同时共轭搜索问题（SCSP）的密码学方案平台。这类群包含如Grigorchuk群等非凡实例，该群已知具有非线性特征，从而使得现有针对SCSP的某些攻击方法失效。该类群基于核肖像概念具有自然的正则形式，该概念在我们的方法中起关键作用。尽管该类中部分群的共轭搜索问题已被研究，但许多群仍缺乏已知的求解算法。此外，某些自相似群存在不可判定的共轭问题。我们讨论了这些群在群论密码学中的应用优势与局限，并对该类群（包括Grigorchuk群、Basilica群等）中基于长度的SCSP攻击变体进行了计算分析。