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

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May 2010, 4(2): 115-139. doi: 10.3934/amc.2010.4.115

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

Rodrigo Abarzúa ^1, , Nicolas Thériault ^2, , Roberto Avanzi ^3, , Ismael Soto ^1, and Miguel Alfaro ^1,

1.	Depto de Ingeniería Industrial, Universidad de Santiago de Chile, Av. Ecuador 3769, Santiago, Chile, Chile, Chile
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, Germany

Received May 2009 Revised April 2010 Published May 2010

Abstract
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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.

Keywords: projective coordinates., explicit formulas, genus 4, Hyperelliptic curves, binary field.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Rodrigo Abarzúa, Nicolas Thériault, Roberto Avanzi, Ismael Soto, Miguel Alfaro. Optimization of the arithmetic of the ideal class group for genus 4 hyperelliptic curves over projective coordinates. Advances in Mathematics of Communications, 2010, 4 (2) : 115-139. doi: 10.3934/amc.2010.4.115

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