American Institute of Mathematical Sciences

Advanced
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
[1]

Stefan Erickson, Michael J. Jacobson, Jr., Andreas Stein. Explicit formulas for real hyperelliptic curves of genus 2 in affine representation. Advances in Mathematics of Communications, 2011, 5 (4) : 623-666. doi: 10.3934/amc.2011.5.623
[2]

Peter Birkner, Nicolas Thériault. Efficient halving for genus 3 curves over binary fields. Advances in Mathematics of Communications, 2010, 4 (1) : 23-47. doi: 10.3934/amc.2010.4.23
[3]

Josep M. Miret, Jordi Pujolàs, Anna Rio. Explicit 2-power torsion of genus 2 curves over finite fields. Advances in Mathematics of Communications, 2010, 4 (2) : 155-168. doi: 10.3934/amc.2010.4.155
[4]

François Monard. Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms. Inverse Problems and Imaging, 2018, 12 (2) : 433-460. doi: 10.3934/ipi.2018019
[5]

M. J. Jacobson, R. Scheidler, A. Stein. Cryptographic protocols on real hyperelliptic curves. Advances in Mathematics of Communications, 2007, 1 (2) : 197-221. doi: 10.3934/amc.2007.1.197
[6]

J. C. Alvarez Paiva and E. Fernandes. Crofton formulas in projective Finsler spaces. Electronic Research Announcements, 1998, 4: 91-100.

[7]

Michael J. Jacobson, Jr., Monireh Rezai Rad, Renate Scheidler. Comparison of scalar multiplication on real hyperelliptic curves. Advances in Mathematics of Communications, 2014, 8 (4) : 389-406. doi: 10.3934/amc.2014.8.389
[8]

Roberto Avanzi, Michael J. Jacobson, Jr., Renate Scheidler. Efficient reduction of large divisors on hyperelliptic curves. Advances in Mathematics of Communications, 2010, 4 (2) : 261-279. doi: 10.3934/amc.2010.4.261
[9]

Josep M. Miret, Jordi Pujolàs, Nicolas Thériault. Trisection for supersingular genus $2$ curves in characteristic $2$. Advances in Mathematics of Communications, 2014, 8 (4) : 375-387. doi: 10.3934/amc.2014.8.375
[10]

David Aulicino, Chaya Norton. Shimura–Teichmüller curves in genus 5. Journal of Modern Dynamics, 2020, 16: 255-288. doi: 10.3934/jmd.2020009
[11]

Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, 2021, 29 (3) : 2293-2323. doi: 10.3934/era.2020117
[12]

Francis N. Castro, Carlos Corrada-Bravo, Natalia Pacheco-Tallaj, Ivelisse Rubio. Explicit formulas for monomial involutions over finite fields. Advances in Mathematics of Communications, 2017, 11 (2) : 301-306. doi: 10.3934/amc.2017022
[13]

João Paulo da Silva, Julio López, Ricardo Dahab. Isogeny formulas for Jacobi intersection and twisted hessian curves. Advances in Mathematics of Communications, 2020, 14 (3) : 507-523. doi: 10.3934/amc.2020048
[14]

Ravi Vakil and Aleksey Zinger. A natural smooth compactification of the space of elliptic curves in projective space. Electronic Research Announcements, 2007, 13: 53-59.

[15]

Laurent Imbert, Michael J. Jacobson, Jr.. Empirical optimization of divisor arithmetic on hyperelliptic curves over $\mathbb{F}_{2^m}$. Advances in Mathematics of Communications, 2013, 7 (4) : 485-502. doi: 10.3934/amc.2013.7.485
[16]

Taoyu Yang, Michal Fečkan, JinRong Wang. Atmospheric Ekman flows with uniform density in ellipsoidal coordinates: Explicit solution and dynamical properties. Journal of Geometric Mechanics, 2022  doi: 10.3934/jgm.2022015
[17]

Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the Kuramoto model on graphs Ⅰ. The mean field equation and transition point formulas. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 131-155. doi: 10.3934/dcds.2019006
[18]

Martino Bardi. Explicit solutions of some linear-quadratic mean field games. Networks and Heterogeneous Media, 2012, 7 (2) : 243-261. doi: 10.3934/nhm.2012.7.243
[19]

Amer Rasheed, Aziz Belmiloudi, Fabrice Mahé. Dynamics of dendrite growth in a binary alloy with magnetic field effect. Conference Publications, 2011, 2011 (Special) : 1224-1233. doi: 10.3934/proc.2011.2011.1224
[20]

Chun-Hao Teng, I-Liang Chern, Ming-Chih Lai. Simulating binary fluid-surfactant dynamics by a phase field model. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1289-1307. doi: 10.3934/dcdsb.2012.17.1289

2021 Impact Factor: 1.015

Tools

Metrics

  • PDF downloads (126)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy