American Institute of Mathematical Sciences

Advanced
August  2010, 4(3): 307-321. doi: 10.3934/amc.2010.4.307

Invalid-curve attacks on (hyper)elliptic curve cryptosystems

Koray Karabina 1, and  Berkant Ustaoglu 2,

1. 

Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ON, Canada, N2L 3G1, Canada
2. 

NTT Information Sharing Platform Laboratories, 3-9-11, Midori-cho Musashino-shi, Tokyo 180-8585, Japan

Received  May 2009 Revised  January 2010 Published  August 2010

We extend the notion of an invalid-curve attack from elliptic curves to genus 2 hyperelliptic curves. We also show that invalid singular (hyper)elliptic curves can be used in mounting invalid-curve attacks on (hyper)elliptic curve cryptosystems, and make quantitative estimates of the practicality of these attacks. We thereby show that proper key validation is necessary even in cryptosystems based on hyperelliptic curves. As a byproduct, we enumerate the isomorphism classes of genus g hyperelliptic curves over a finite field by a new counting argument that is simpler than the previous methods.
Keywords: hyperelliptic curves., Invalid-curve attacks.
Mathematics Subject Classification: 94A6.
Citation: 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
[1]

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
[2]

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
[3]

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
[4]

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
[5]

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
[6]

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
[7]

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
[8]

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

[9]

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
[10]

Qinglei Zhang, Wenying Feng. Detecting coalition attacks in online advertising: A hybrid data mining approach. Big Data & Information Analytics, 2016, 1 (2&3) : 227-245. doi: 10.3934/bdia.2016006
[11]

Claude Carlet, Sylvain Guilley. Complementary dual codes for counter-measures to side-channel attacks. Advances in Mathematics of Communications, 2016, 10 (1) : 131-150. doi: 10.3934/amc.2016.10.131
[12]

Konstantinos A. Draziotis, Anastasia Papadopoulou. Improved attacks on knapsack problem with their variants and a knapsack type ID-scheme. Advances in Mathematics of Communications, 2018, 12 (3) : 429-449. doi: 10.3934/amc.2018026
[13]

Saide Zhu, Wei Li, Hong Li, Chunqiang Hu, Zhipeng Cai. A survey: Reward distribution mechanisms and withholding attacks in Bitcoin pool mining. Mathematical Foundations of Computing, 2018, 1 (4) : 393-414. doi: 10.3934/mfc.2018020
[14]

Wenjing Chen, Louis Dupaigne, Marius Ghergu. A new critical curve for the Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2469-2479. doi: 10.3934/dcds.2014.34.2469
[15]

Diego F. Aranha, Ricardo Dahab, Julio López, Leonardo B. Oliveira. Efficient implementation of elliptic curve cryptography in wireless sensors. Advances in Mathematics of Communications, 2010, 4 (2) : 169-187. doi: 10.3934/amc.2010.4.169
[16]

Huaiyu Jian, Hongjie Ju, Wei Sun. Traveling fronts of curve flow with external force field. Communications on Pure & Applied Analysis, 2010, 9 (4) : 975-986. doi: 10.3934/cpaa.2010.9.975
[17]

Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu. A curve of positive solutions for an indefinite sublinear Dirichlet problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 817-845. doi: 10.3934/dcds.2020063
[18]

Gian-Italo Bischi, Laura Gardini, Fabio Tramontana. Bifurcation curves in discontinuous maps. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 249-267. doi: 10.3934/dcdsb.2010.13.249
[19]

Stefano Marò. Relativistic pendulum and invariant curves. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1139-1162. doi: 10.3934/dcds.2015.35.1139
[20]

Carlos Munuera, Alonso Sepúlveda, Fernando Torres. Castle curves and codes. Advances in Mathematics of Communications, 2009, 3 (4) : 399-408. doi: 10.3934/amc.2009.3.399

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences