Graduation Year
2024
Document Type
Dissertation
Degree
Ph.D.
Degree Name
Doctor of Philosophy (Ph.D.)
Degree Granting Department
Mathematics and Statistics
Major Professor
Jean-Francois Biasse, Ph.D.
Committee Member
Dmytro Savchuk, Ph.D.
Committee Member
Natasha Jonoska, Ph.D.
Committee Member
Giacomo Micheli, Ph.D.
Committee Member
Ruthmae Sears, Ph.D.
Keywords
Imaginary quadratic field, quadratic forms, cyclotomic field, norm relations, principal ideal problem
Abstract
The ideal class group is a fundamental concept in algebraic number theory, providing insights into the structure and factorization properties of the ring of integers of a number field. It measures the extent to which unique factorization fails in the ring of integers. Efficiently computing the ideal class group is crucial for exploring unproved heuristics in number theory and solving Diophantine equations. Additionally, the computation of the ideal class group has several cryptographic applications, such as schemes based on the Discrete Logarithm Problem (DLP), the computation of isogenies, and groups of unknown order. Despite its importance, much about the ideal class group and its order, known as the class number, remains enigmatic due to the computational challenges involved.
In the first part of this thesis, we present a modified version of the Hafner and McCurley class group algorithm~\cite{Hafner1989ARS} for the Ideal Class Group computation of imaginary quadratic fields. Our modified version improves the asymptotic run time and achieves the conjectured run time of the Hafner and McCurley class group algorithm~\cite[Sec. 5]{Hafner1989ARS}. This improvement relies on recent results regarding the properties of the Cayley graph of the ideal class group~\cite{jmv}.
In the second part of this thesis, we introduce a new approach for unconditional class number computation of the maximal real subfield of cyclotomic fields. Our method for computing class numbers in the maximal real subfield of cyclotomic fields closely adheres to the methodology outlined by John C. Miller~\cite{MillerThesis,MillerComposite,MillerPrime}. The cornerstone of our novel approach for unconditional class number computation of the maximal real subfield of cyclotomic fields lies in the \textit{norm relations} techniques introduced by Biasse, Fieker, Hofmann, and Page~\cite{Biasse2020NormRA}. In particular, we employ norm relation-based Principal Ideal Problem (PIP) technique as outlined in~\cite{BiasseWilliam}.
Scholar Commons Citation
Erukulangara, Muhammed Rashad, "Improvements in Computational Techniques For Determining Ideal Class Groups and Class Numbers" (2024). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/10505
