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Full-Text Articles in Physical Sciences and Mathematics
Some Model Theory Of Free Groups, Christopher James Natoli
Some Model Theory Of Free Groups, Christopher James Natoli
Dissertations, Theses, and Capstone Projects
There are two main sets of results, both pertaining to the model theory of free groups. In the first set of results, we prove that non-abelian free groups of finite rank at least 3 or of countable rank are not A-homogeneous. We then build on the proof of this result to show that two classes of groups, namely finitely generated free groups and finitely generated elementary free groups, fail to form A-Fraisse classes and that the class of non-abelian limit groups fails to form a strong A-Fraisse class.
The second main result is that if a countable group is elementarily …
Solving Algorithmic Problems In Finitely Presented Groups Via Machine Learning, Jonathan Gryak
Solving Algorithmic Problems In Finitely Presented Groups Via Machine Learning, Jonathan Gryak
Dissertations, Theses, and Capstone Projects
Machine learning and pattern recognition techniques have been successfully applied to algorithmic problems in free groups. In this dissertation, we seek to extend these techniques to finitely presented non-free groups, in particular to polycyclic and metabelian groups that are of interest to non-commutative cryptography.
As a prototypical example, we utilize supervised learning methods to construct classifiers that can solve the conjugacy decision problem, i.e., determine whether or not a pair of elements from a specified group are conjugate. The accuracies of classifiers created using decision trees, random forests, and N-tuple neural network models are evaluated for several non-free groups. …
Exploring Platform (Semi)Groups For Non-Commutative Key-Exchange Protocols, Ha Lam
Exploring Platform (Semi)Groups For Non-Commutative Key-Exchange Protocols, Ha Lam
Dissertations, Theses, and Capstone Projects
In this work, my advisor Delaram Kahrobaei, our collaborator David Garber, and I explore polycyclic groups generated from number fields as platform for the AAG key-exchange protocol. This is done by implementing four different variations of the length-based attack, one of the major attacks for AAG, and submitting polycyclic groups to all four variations with a variety of tests. We note that this is the first time all four variations of the length-based attack are compared side by side. We conclude that high Hirsch length polycyclic groups generated from number fields are suitable for the AAG key-exchange protocol.
Delaram Kahrobaei …