Vol. 4, No. 1, 2020

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Divisor class group arithmetic on $C_{3,4}$ curves

Evan MacNeil, Michael J. Jacobson Jr. and Renate Scheidler

Vol. 4 (2020), No. 1, 317–334
Abstract

We present novel explicit formulas for arithmetic in the divisor class group of a C3,4 curve. Our formulas handle all cases of inputs and outputs without having to fall back on a generic method. We also improve on the most commonly occurring case by reducing the number of required field inversions to one at the cost of a small number of additional field operations, resulting in running times that are between 11 and 21% faster than the prior state of the art depending on the field size, and even more for small field sizes when nontypical cases frequently arise.

Keywords
computational number theory, computational algebraic geometry, divisor arithmetic, C34 curves, genus 3 nonhyperelliptic curves
Mathematical Subject Classification 2010
Primary: 11R65, 14H45, 14Q05
Milestones
Received: 28 February 2020
Accepted: 29 April 2020
Published: 29 December 2020
Authors
Evan MacNeil
Department of Mathematics and Statistics
University of Calgary
Calgary AB
Canada
Michael J. Jacobson Jr.
Department of Computer Science
University of Calgary
Calgary AB
Canada
Renate Scheidler
Department of Mathematics and Statistics
University of Calgary
Calgary AB
Canada