Benjamin Fine

Abstract algebra : applications to Galois theory, algebraic geometry and cryptography

1 Groups, Rings and Fields 2 Maximal and Prime Ideals 3 Prime Elements and Unique Factorization Domains 4 Polynomials and Polynomial Rings 5 Field Extensions 6 Field Extensions and Compass and Straightedge Constructions 7 Kroneckers Theorem and Algebraic Closures 8 Splitting Fields and Normal Extensions 9 Groups, Subgroups and Examples 10 Normal Subgroups, Factor Groups and Direct Products 11 Symmetric and Alternating Groups 12 Solvable Groups 13 Groups Actions and the Sylow Theorems 14 Free Groups and Group Presentations 15 Finite Galois Extensions 16 Separable Field Extensions 17 Applications of Galois Theory 18 The Theory of Modules 19 Finitely Generated Abelian Groups 20 Integral and Transcendental Extensions 21 The Hilbert Basis Theorem and the Nullstellensatz 22 Algebraic Cryptography.

Combinatorial Group Theory, Discrete Groups, and Number Theory

A description of the arithmetic Fuchsian groups with signature

Generic Subgroups of Group Amalgams

(2 --)

A secret sharing scheme based on the Closest Vector Theorem and a modification to a private key cryptosystem

by P. Ackermann Outer automorphism groups of certain orientable Seifert 3-manifold groups by R. B. J. T. Allenby, G. Kim, and C. Y. Tang The search for origins of the commutator calculus by M. Anshel and A. M. Gaglione A note on nondiscrimination of nilpotent groups and Malcev completions by G. Baumslag, B. Fine, A. M. Gaglione, and D. Spellman A proposed public key cryptosystem using the modular group by G. Baumslag, B. Fine, and X. Xu Grobner basis techniques in the computation of two-sided syzygies by H. Bluhm and M. Kreuzer Normal subgroups of the modular group and other Hecke groups by M. Conder and P. Dobcsanyi Commutativity of units in group rings by O. B. Cristo and C. P. Milies Presentations of groups involving more generators than are necessary. II. by M. J. Evans Unions of varieties and quasivarieties by B. Fine, A. M. Gaglione, and D. Spellman Finitely presented infinite torsion groups and a question of V. H. Dyson by B. Fine, A. M. Gaglione, and D. Spellman Context-free irreducible word problems in groups by A. Fonseca and R. M. Thomas Autocommutators and the autocommutator subgroup by D. Garrison, L.-C. Kappe, and D. Yull Informative words and discreteness by J. Gilman An algorithm for potentially positive words in

Orderable groups, elementary theory, and the Kaplansky conjecture

F_2

Discrimination in a General Algebraic Setting

by R. Goldstein Using group theory for knowledge representation and discovery by G. Kern-Isberner Quotient tests and Grobner bases by M. Kreuzer, A. Myasnikov, G. Rosenberger, and A. Ushakov Transitivity of normality and pronormal subgroups by L. A. Kurdachenko and I. Y. Subbotin Torsion in maximal arithmetic Fuchsian groups by C. Maclachlan Nilpotent

Reflections on some aspects of infinite groups

mathbb{Q}[x]

11 Symmetric and Alternating Groups

-powered groups by S. Majewicz On the Rosenberger monster by R. F. Morse Density of test elements in finite abelian groups by C. F. Rocca, Jr. Adjoining a root does not decrease the rank by R. Weidmann.

On asymptotic densities and generic properties in finitely generated groups

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 g1,...,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.

Discriminating groups: a comprehensive overview

Abstract. We explain and perform the steps for an (n,t) secret sharing scheme based on the closest vector theorem. We then compare this scheme and its complexity to the secret sharing schemes of both Shamir and Panagopoulos. Finally we modify the (n,t) secret sharing scheme to a private key cryptosystem.