Divisor Class Group Arithmetic on C3,4 Curves

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dc.contributor.advisorScheidler, Renate
dc.contributor.advisorJacobson, Michael J.
dc.contributor.authorMacNeil, Evan
dc.contributor.committeememberBauer, Mark L.
dc.contributor.committeememberGreenberg, Matthew
dc.date2020-06
dc.date.accessioned2020-02-20T15:44:44Z
dc.date.available2020-02-20T15:44:44Z
dc.date.issued2020-01-31
dc.description.abstractComputing in the divisor class group of an algebraic curve is a non-trivial component in computing L-series. L-series in turn are at the heart of the Sato-Tate conjecture and related conjectures. The Sato-Tate conjecture has been proven for elliptic curves with complex multiplication, but remains open for other families of algebraic curves. In order to test these conjectures against other curve families, it is desirable to have efficient algorithms to perform divisor class group arithmetic. Fast explicit formulas exist to perform divisor class group arithmetic for genus 1 and genus 2 curves. However, the picture for genus 3 curves is incomplete. Existing explicit formulas for arithmetic on non-hyperelliptic genus 3 curves (C3,4 curves) have been developed with cryptographic applications in mind. They make certain genericity assumptions on their inputs that hold with high probability in cryptographic settings, but are unsuited for number theoretic use cases. More general algorithms exist that can perform divisor class arithmetic over any curve, but they are slow. In this thesis, that gap is bridged. Fast explicit formulae are developed that may be used to add any pair of reduced divisors on any C3,4 curve. Formulae optimized for the generic case considered by previous authors are produced, allowing one to add divisors in 1I+111M+3S+99A and double divisors in 1I+135M+3S+116A (inversions, multiplications, squarings, and additions in a field). The formulae are implemented in Sage. Benchmark tests find that these new formulae allow one to add and double 13.2% and 11.1% faster, respectively, that the previous state-of-the-art in C3,4 curve arithmetic.en_US
dc.identifier.citationMacNeil, E. (2020). Divisor Class Group Arithmetic on C3,4 Curves (Master's thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.en_US
dc.identifier.doihttp://dx.doi.org/10.11575/PRISM/37582
dc.identifier.urihttp://hdl.handle.net/1880/111659
dc.language.isoengen_US
dc.publisher.facultyScienceen_US
dc.publisher.institutionUniversity of Calgaryen
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.en_US
dc.subjectAlgebraic Geometryen_US
dc.subjectComputational Algebraic Geometryen_US
dc.subjectPlane Curvesen_US
dc.subjectDivisor Class Groupen_US
dc.subjectJacobianen_US
dc.subjectGlobal Fieldsen_US
dc.subject.classificationMathematicsen_US
dc.titleDivisor Class Group Arithmetic on C3,4 Curvesen_US
dc.typemaster thesisen_US
thesis.degree.disciplineMathematics & Statisticsen_US
thesis.degree.grantorUniversity of Calgaryen_US
thesis.degree.nameMaster of Science (MSc)en_US
ucalgary.item.requestcopytrueen_US
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