Contracting Self-similar Groups in Group-Based Cryptography

Delaram Kahrobaei, Arsalan Akram Malik, Dmytro Savchuk
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Abstract

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.
基于组的密码学中的自相似组契约
我们提出将自相似契约群作为基于同步共轭搜索问题(SCSP)的加密算法平台。这类群包含一些特殊的例子,如 Grigorchuk 群,众所周知,该群是非线性的,因此现有的一些针对 SCSP 的攻击都不适用。该类群有一个基于核肖像运动的自然正则表达式,这在我们的方法中起着关键作用。虽然对该类中的一些群的共轭搜索问题已有研究,但仍有许多群没有已知的求解算法。此外,还有一些自相似群的共轭问题无法解决。我们讨论了在基于群的密码学中使用这些群的优点和缺点,并针对该类中的一些群,包括格里高丘克群、巴西利卡群和其他群,提供了基于长度的 SCSP 攻击变体的计算分析。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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