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Myasnikov and V.Shpilrain, Measuring Sets in Infinite Groups, Cont
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The Conjugacy Search Problem in Public Key Cryptography; Unnnecessary and Insufficient, Applicable Algebra in Engineering,Communication and computing
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Malnormal subgroups of free groups, in: Computational and Statistical Group Theory, R
A. Myasnikov, V. Shpilrain, A. Ushakov (2005)
A Practical Attack on a Braid Group Based Cryptographic Protocol
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Genericity of the Class of Groups in Which Subgroups with a Lesser Number of Generators are Free
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A Practical Attack on Some Braid Group Based Cryptographic Protocols
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A. Borovik, A. Myasnikov, V. Shpilrain (2002)
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The class of groups all of whose subgroups with lesser number of generators are free is genericMathematical Notes, 59
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Malnormal subgroups of free groups, in: Computational and Statistical Group Theory
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Ushakov: Length based attack and braid groups: cryptanalysis of Anshel-Anshel-Goldfeld key exchange protocol
A. Myasnikov, A. Ushakov (2007)
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B. Fine, A. Myasnikov, G. Rosenberger (2002)
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Iris Anshel, M. Anshel, D. Goldfeld (1999)
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Г Аржанцева, G. Arzhantseva, Александр Ольшанский, Aleksandr Ol'shanskii (1996)
Общность класса групп, в которых подгруппы с меньшим числом порождающих свободны@@@The class of groups all of whose subgroups with lesser number of generators are free is generic, 59
(2007)
Ushakov : A practical attack on some braid group based cryptographic protocols
(1996)
Olshanskii: Genericity of the class of groups in which subgroups with a lesser number of generators are free
Osin: Bounded Nielsen property in hyperbolic groups
D. Epstein (1971)
Almost all subgroups of a Lie group are freeJournal of Algebra, 19
V. Shpilrain, A. Ushakov (2004)
The Conjugacy Search Problem in Public Key Cryptography: Unnecessary and InsufficientApplicable Algebra in Engineering, Communication and Computing, 17
(1996)
Genericity of the class of groups in which subgroups with a lesser number of generators are free, Mat
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 , . . . , g n 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.
Groups - Complexity - Cryptology – de Gruyter
Published: Apr 1, 2009
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