American Institute of Mathematical Sciences

Advanced
May  2010, 4(2): 261-279. doi: 10.3934/amc.2010.4.261

Efficient reduction of large divisors on hyperelliptic curves

Roberto Avanzi 1, , Michael J. Jacobson, Jr. 2, and  Renate Scheidler 3,

1. 

Fakultät für Mathematik, Ruhr-Universität Bochum and Horst Gösrtz Institut für IT-Sicherheit, Universitätsstraße 150, D-44780 Bochum
2. 

Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4, Canada
3. 

Department of Mathematics & Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4, Canada

Received  June 2009 Revised  March 2010 Published  May 2010

We present an algorithm for reducing a divisor on a hyperelliptic curve of arbitrary genus over any finite field. Our method is an adaptation of a procedure for reducing ideals in quadratic number fields due to Jacobson, Sawilla and Williams, and shares common elements with both the Cantor and the NUCOMP algorithms for divisor arithmetic. Our technique is especially suitable for the rapid reduction of a divisor with very large Mumford coefficients, obtained for example through an efficient tupling technique. Results of numerical experiments are presented, showing that our algorithm is superior to the standard reduction algorithm in many cases.
Keywords: Hyperelliptic curve, reduction, scalar multiplication., continued fraction expansion, divisor.
Mathematics Subject Classification: Primary: 11R58, 14H45; Secondary: 14G5.
Citation: 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
[1]

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
[2]

Kwang Ho Kim, Junyop Choe, Song Yun Kim, Namsu Kim, Sekung Hong. Speeding up regular elliptic curve scalar multiplication without precomputation. Advances in Mathematics of Communications, 2020, 14 (4) : 703-726. doi: 10.3934/amc.2020090
[3]

Deepak Kumar, Ahmad Jazlan, Victor Sreeram, Roberto Togneri. Partial fraction expansion based frequency weighted model reduction for discrete-time systems. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 329-337. doi: 10.3934/naco.2016015
[4]

Svetlana Katok, Ilie Ugarcovici. Theory of $(a,b)$-continued fraction transformations and applications. Electronic Research Announcements, 2010, 17: 20-33. doi: 10.3934/era.2010.17.20
[5]

Svetlana Katok, Ilie Ugarcovici. Structure of attractors for $(a,b)$-continued fraction transformations. Journal of Modern Dynamics, 2010, 4 (4) : 637-691. doi: 10.3934/jmd.2010.4.637
[6]

Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281
[7]

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
[8]

Roberto Avanzi, Nicolas Thériault. A filtering method for the hyperelliptic curve index calculus and its analysis. Advances in Mathematics of Communications, 2010, 4 (2) : 189-213. doi: 10.3934/amc.2010.4.189
[9]

Pierre Arnoux, Thomas A. Schmidt. Commensurable continued fractions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4389-4418. doi: 10.3934/dcds.2014.34.4389
[10]

Laura Luzzi, Stefano Marmi. On the entropy of Japanese continued fractions. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 673-711. doi: 10.3934/dcds.2008.20.673
[11]

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
[12]

D. Novikov and S. Yakovenko. Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian systems. Electronic Research Announcements, 1999, 5: 55-65.

[13]

Bertrand Lods. Variational characterizations of the effective multiplication factor of a nuclear reactor core. Kinetic & Related Models, 2009, 2 (2) : 307-331. doi: 10.3934/krm.2009.2.307
[14]

Claudio Bonanno, Carlo Carminati, Stefano Isola, Giulio Tiozzo. Dynamics of continued fractions and kneading sequences of unimodal maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1313-1332. doi: 10.3934/dcds.2013.33.1313
[15]

Élise Janvresse, Benoît Rittaud, Thierry de la Rue. Dynamics of $\lambda$-continued fractions and $\beta$-shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1477-1498. doi: 10.3934/dcds.2013.33.1477
[16]

Koray Karabina, Berkant Ustaoglu. Invalid-curve attacks on (hyper)elliptic curve cryptosystems. Advances in Mathematics of Communications, 2010, 4 (3) : 307-321. doi: 10.3934/amc.2010.4.307
[17]

Robert L. Devaney, Daniel M. Look. Buried Sierpinski curve Julia sets. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1035-1046. doi: 10.3934/dcds.2005.13.1035
[18]

Sebastian J. Schreiber. Expansion rates and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 433-438. doi: 10.3934/dcds.1997.3.433
[19]

Meina Gao. Small-divisor equation with large-variable coefficient and periodic solutions of DNLS equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 173-204. doi: 10.3934/dcds.2015.35.173
[20]

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

2019 Impact Factor: 0.734

Tools

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences