Optimization of the arithmetic of the ideal class group for genus 4 hyperelliptic curves over projective coordinates

Rodrigo Abarzúa; Nicolas Thériault; Roberto Avanzi; Ismael Soto; Miguel Alfaro

doi:10.3934/amc.2010.4.115

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2010, Volume 4, Issue 2: 115-139. Doi: 10.3934/amc.2010.4.115

This issue Previous Article Local-global aspects of (hyper)elliptic curves over (in)finite fields Next Article Improvements in the computation of ideal class groups of imaginary quadratic number fields

Optimization of the arithmetic of the ideal class group for genus 4 hyperelliptic curves over projective coordinates

1.
Depto de Ingeniería Industrial, Universidad de Santiago de Chile, Av. Ecuador 3769, Santiago
2.
Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca
3.
Fakultät für Mathematik, Ruhr-Universität Bochum and Horst Gösrtz Institut für IT-Sicherheit, Universitätsstraße 150, D-44780 Bochum

Received: April 30, 2009

Revised: March 31, 2010

Published: April 2010

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Abstract

The aim of this paper is to reduce the number of operations in Cantor's algorithm for the Jacobian group of hyperelliptic curves for genus 4 in projective coordinates. Specifically, we developed explicit doubling and addition formulas for genus 4 hyperelliptic curves over binary fields with $h(x)=1$. For these curves, we can perform a divisor doubling in $63M+19S$, while the explicit adding formula requires $203M+18S,$ and the mixed coordinates addition (in which one point is given in affine coordinates) is performed in $165M+15S$.
These formulas can be useful for public key encryption in some environments where computing the inverse of a field element has a high computational cost (either in time, power consumption or hardware price), in particular with embedded microprocessors.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Optimization of the arithmetic of the ideal class group for genus 4 hyperelliptic curves over projective coordinates

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