Aspects of Infinite Groups: A Festschrift in Honor of - download pdf or read online

By Benjamin Fine, Gerhard Rosenberger, Dennis Spellman

ISBN-10: 9812793402

ISBN-13: 9789812793409

This booklet is a festschrift in honor of Professor Anthony Gaglione's 60th birthday. This quantity offers an outstanding mixture of examine and expository articles on a number of elements of endless team thought. The papers provide a extensive assessment of current learn in endless workforce idea mostly, and combinatorial workforce concept and non-Abelian group-based cryptography particularly. in addition they pinpoint the interactions among combinatorial team idea and mathematical good judgment, specifically version conception.

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M. Gaglione on the occasion of his 60th birthday. Abstract: Recently there has been an active line of research on noncommutative algebraic cryptography. This involves the use of noncommutative algebraic objects as the platforms for encryption systems. Most of this work, such as the Anshel-Anshel-Goldfeld scheme, the Ko-Lee scheme and the Baumslag-FineXu Modular group scheme use non abelian groups as the basic algebraic object. Some of these encryption methods have been successful and some have been broken.

All of the proposed schemes in this paper are based in a general sense on free group cryptography. Consider a free group F on free generators Xl, ""xr' Then each element gin F has a unique expression as a word W(XI, ... , x r ). Let WI, ... , W k with Wi = Wi (Xl, ... , x r ) be a set of words in the generators Xl, ... , Xr of the free group F. At the most basic level, to construct a cryptosystem, suppose that we have a plaintext alphabet A. For example suppose A = {a, b, ... } are the symbols needed to construct meaningful messages in English.

5, Ref. 1). Every abelian subgroup of a free nilpotent group is free abelian. 1, Ref. 1). Let B be a subgroup of a free nilpotent group Nn,c of class c ~ 1 and rank n ~ 1. Then B is generated by a set of c subgroups where 48 (i) for k = 1,3, ... , c, the subgroup Bk is a free nilpotent group of class LfJ; (ii) for n = 2, that is when Nn,e is 2-generated, the subgroup B2 is infinite cyclic; otherwise the subgroup B2 is free nilpotent of class L~ J; (iii) for i + j ~ c, the subgroup [Bi' B j ] is contained in the subgroup (Bi+j,'" ,Be); (iv) for i + j > c, the subgroup [Bi' B j ] is trivial; and (v) for k = 1,2, ...

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Aspects of Infinite Groups: A Festschrift in Honor of Anthony Gaglione (Algebra and Discrete Mathematics) by Benjamin Fine, Gerhard Rosenberger, Dennis Spellman


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