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Journal of Mathematical Cryptology

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

Alexei G. Myasnikov1 / Alexander Ushakov2

1McGill University, 805 Sherbrooke West, Montreal, H3A 2K6, Canada. Email: alexeim@math.mcgill.ca

2Stevens Institute of Technology, Castle Point on Hudson, Hoboken NJ 07030-5991, USA. Email: aushakov@stevens.edu

Citation Information: Journal of Mathematical Cryptology. Volume 2, Issue 1, Pages 29–61, ISSN (Online) 1862-2984, ISSN (Print) 1862-2976, DOI: 10.1515/JMC.2008.003, May 2008

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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.

Keywords.: Braid group cryptography; random subgroup of a braid group; length-based attack; quotient attack; commutator key-exchange

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