Contracting Self-similar Groups in Group-Based Cryptography

TL;DR

This work investigates contracting self-similar automaton groups as platforms for cryptographic schemes based on the Simultaneous Conjugacy Search Problem (SCSP). It introduces nucleus portraits as a practical normal form and shows how they enable efficient word reconstruction in contracting regular branch groups, while potentially obscuring secret conjugators. A length-based attack (LBA) variant on SCSP is developed and extensively tested across a spectrum of groups, revealing a clear dependence on group structure: some virtually abelian cases (e.g., $\mathbb Z\wr C_2$) are vulnerable under modest radii, whereas many non-virtually-abelian examples (notably Grigorchuk-type and Basilica groups) exhibit strong resistance and long runtimes. The results highlight both cryptographic promise and risk: nucleus-portrait representations can improve obfuscation, but the LBA can still exploit certain algebraic weaknesses in specific contracting groups, guiding secure platform selection for SCSP-based schemes.

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.

Figures (13)

• Figure 1: Binary tree
• Figure 2: Five-state Automaton $\mathcal{A}_\mathcal{G}$ generating the Grigorchuk group $\mathcal{G}$
• Figure 3: Nucleus portrait for $acab \in \mathcal{G}$
• Figure 4: An example of nucleus portraits of inverse group elements
• Figure 5: Nucleus portrait of Alice's private key $A = a_1a_2a_3$.

Theorems & Definitions (27)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Definition 3.6
• Definition 3.7
• Definition 3.8
• Proposition 3.9
• Definition 4.1