Contracting Self-similar Groups in Group-Based Cryptography

TL;DR

This paper introduces self-similar contracting groups as a new cryptographic platform, leveraging their unique properties like non-linearity and normal forms to enhance security against certain attacks.

Contribution

It presents a novel cryptographic framework based on self-similar contracting groups, analyzing their security features and computational aspects, including resistance to length-based attacks.

Findings

Grigorchuk group's non-linearity makes some attacks inapplicable.

Normal form based on nucleus portrait aids cryptographic operations.

Length-based attacks show varying effectiveness on different groups.

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.