Contracting Self-similar Groups in Group-Based Cryptography
This work addresses the security of group-based cryptography for cryptographers by exploring a new class of groups that may resist existing attacks due to their non-linear properties.
This paper proposes self-similar contracting groups, including the Grigorchuk group, as a platform for cryptographic schemes based on the simultaneous conjugacy search problem (SCSP). The authors discuss the benefits and drawbacks of using these groups and provide a computational analysis of length-based attacks on SCSP for specific groups like the Grigorchuk and Basilica groups.
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.