Random subgroups and analysis of the length-based and quotient attacks

• Main Authors: Myasnikov Alexei G., Ushakov Alexander
• Format: Article
• Language: English
• Published: De Gruyter 2008-04-01
• Series: Journal of Mathematical Cryptology
• Subjects: braid group cryptography; random subgroup of a braid group; length-based attack; quotient attack; commutator key-exchange
• Online Access: https://doi.org/10.1515/JMC.2008.003

In this paper we discuss generic properties of “random subgroups” of a given group G. It turns out that in many groups G (even in most exotic of them) the random subgroups have a simple algebraic structure and they “sit” inside G in a very particular way. This gives a strong mathematical foundation for cryptanalysis of several group-based cryptosystems and indicates on how to chose “strong keys”. To illustrate our technique we analyze the Anshel-Anshel-Goldfeld (AAG) cryptosystem and give a mathematical explanation of recent success of some heuristic length-based attacks on it. Furthermore, we design and analyze a new type of attack, which we term the quotient attacks. Mathematical methods we develop here also indicate how one can try to choose “parameters” in AAG to foil the attacks.