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Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board Member: Blackburn, Simon R. / Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Schäge, Sven / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura

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Generic Subgroups of Group Amalgams

Benjamin Fine1 / Alexei Myasnikov2 / Gerhard Rosenberger3

1Department of Mathematics, Fairfield University, Fairfield, Connecticut 06430, USA

2Department of Mathematics, McGill University, Montreal, Canada

3Department of Mathematics, TU Dortmund, 44227 Dortmund, Germany

Citation Information: Groups – Complexity – Cryptology. Volume 1, Issue 1, Pages 51–61, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: 10.1515/GCC.2009.51, February 2010

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For many groups the structure of finitely generated subgroups is generically simple. That is with asymptotic density equal to one a randomly chosen finitely generated subgroup has a particular well-known and easily analyzed structure. For example a result of D. B. A. Epstein says that a finitely generated subgroup of GL(n, ℝ) is generically a free group. We say that a group G has the generic free group property if any finitely generated subgroup is generically a free group. Further G has the strong generic free group property if given randomly chosen elements g 1, . . . , gn in G then generically they are a free basis for the free subgroup they generate. In this paper we show that for any arbitrary free product of finitely generated infinite groups satisfies the strong generic free group property. There are also extensions to more general amalgams - free products with amalgamation and HNN groups. These results have implications in cryptography. In particular several cryptosystems use random choices of subgroups as hard cryptographic problems. In groups with the generic free group property any such cryptosystem may be attackable by a length based attack.

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