Index-Calculus Algorithms for Computing Class Groups of Quartic Number Fields

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dc.contributor.advisorJacobson Jr, Michael John
dc.contributor.authorMarquis, David
dc.contributor.committeememberScheidler, Renate
dc.contributor.committeememberBauer, Mark L
dc.contributor.committeememberNguyen, Dang Khoa
dc.contributor.committeememberFiori, Andrew
dc.date.accessioned2024-09-19T14:57:58Z
dc.date.available2024-09-19T14:57:58Z
dc.date.issued2024-09-17
dc.description.abstractWe address the problem of quickly computing the class group and unit group for quartic number fields of large discriminant. Index-calculus algorithms are the fastest way to solve this problem for arbitrary number fields. Our focus is primarily on improvements to relation generation, one of the main stages in an index-calculus algorithm. In the quadratic case, the self-initialization approach to relation generation developed by Jacobson has been very successful, but applying this idea to quartic fields has not been attempted. We present a novel generalization of this approach that is applicable to quartic number fields. Additionally, we characterize the efficiency of our method in terms of the size of the roots of the field’s defining polynomial. We discuss our implementation of a complete index-calculus algorithm using this approach. Our implementation's relation generation produces relations significantly faster than the current state-of-the-art, Magma. Our implementation of the complete algorithm, including the improved relation generation, is faster than Magma for number fields whose defining polynomial has small roots, and is comparable for typical number fields.
dc.identifier.citationMarquis, D. (2024). Index-calculus algorithms for computing class groups of quartic number fields (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.
dc.identifier.urihttps://hdl.handle.net/1880/119768
dc.language.isoen
dc.publisher.facultyGraduate Studies
dc.publisher.institutionUniversity of Calgary
dc.rightsUniversity of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.
dc.subjectcomputation
dc.subjectalgorithm
dc.subjectnumber theory
dc.subjectalgebraic number theory
dc.subjectclass group
dc.subjectnumber field
dc.subjectindex-calculus
dc.subjectperformance
dc.subject.classificationInformation Science
dc.subject.classificationEducation--Mathematics
dc.subject.classificationComputer Science
dc.titleIndex-Calculus Algorithms for Computing Class Groups of Quartic Number Fields
dc.typedoctoral thesis
thesis.degree.disciplineComputer Science
thesis.degree.grantorUniversity of Calgary
thesis.degree.nameDoctor of Philosophy (PhD)
ucalgary.thesis.accesssetbystudentI do not require a thesis withhold – my thesis will have open access and can be viewed and downloaded publicly as soon as possible.
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