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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

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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