American Institute of Mathematical Sciences

Advanced
November  2014, 8(4): 389-406. doi: 10.3934/amc.2014.8.389

Comparison of scalar multiplication on real hyperelliptic curves

Michael J. Jacobson, Jr. 1, , Monireh Rezai Rad 2, and  Renate Scheidler 2,

1. 

Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4, Canada
2. 

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

Received  March 2014 Revised  September 2014 Published  November 2014

Real hyperelliptic curves admit two structures suitable for cryptography --- the Jacobian (a finite abelian group) and the infrastructure. Mireles Morales described precisely the relationship between these two structures, and made the assertion that when implemented with balanced divisor arithmetic, the Jacobian generically yields more efficient arithmetic than the infrastructure for cryptographic applications. We confirm that this assertion holds for genus two curves, through rigorous analysis and the first detailed numerical performance comparisons, showing that cryptographic key agreement can be performed in the Jacobian without any extra operations beyond those required for basic scalar multiplication. We also present a modified version of Mireles Morales' map that more clearly reveals the algorithmic relationship between the two structures.
Keywords: scalar multiplication, Jacobian, infrastructure, Real hyperelliptic curve, cryptographic key agreement., balanced divisor.
Mathematics Subject Classification: Primary: 11G20, 11R11, 11R29, 11Y40; Secondary: 11Y16, 14G5.
Citation: 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
References:
[1]

NIST Special Publication 800-57, 2007. Google Scholar

[2]

Chapman & Hall/CRC, Boca Raton, 2006. doi: 10.1201/9781420034981.  Google Scholar

[3]

IEEE Trans. Inf. Theory, 22 (1976), 472-492. doi: 10.1109/TIT.1976.1055638.  Google Scholar

[4]

Adv. Math. Commun., 5 (2011), 623-666. doi: 10.3934/amc.2011.5.623.  Google Scholar

[5]

Adv. Math. Commun., 2 (2008), 293-307. doi: 10.3934/amc.2008.2.293.  Google Scholar

[6]

F. Fontein, Holes in the infrastructure of global hyperelliptic function fields,, preprint, ().   Google Scholar

[7]

Théorie des Nombres (Québec, PQ), de Gruyter, Berlin, 1989, 227-239.  Google Scholar

[8]

in Algorithmic Number Theory - ANTS 2008 (Berlin), Springer, 2008, 342-356. doi: 10.1007/978-3-540-79456-1_23.  Google Scholar

[9]

Adv. Math. Commun., 1 (2007), 197-221. doi: 10.3934/amc.2007.1.197.  Google Scholar

[10]

in Advances in Coding Theory and Cryptology (eds. T. Shaska, W.C. Huffman, D. Joyner and V. Ustimenko), World Scientific Publishing, 2007, 201-244. doi: 10.1142/9789812772022_0013.  Google Scholar

[11]

Tatra Mountains Math. Publ., 40 (2010), 1-35. doi: 10.2478/v10127-010-0030-9.  Google Scholar

[12]

J. Cryptology, 1 (1989), 139-150. doi: 10.1007/BF02252872.  Google Scholar

[13]

Appl. Algebra Eng. Commun. Comput., 15 (2005), 295-328. doi: 10.1007/s00200-004-0154-8.  Google Scholar

[14]

Cryptology eprint archive no. 2008/299, 2008. Google Scholar

[15]

J. Cryptology, 7 (1994), 171-199. doi: 10.1007/BF02318548.  Google Scholar

[16]

Des. Codes Crypt., 7 (1996), 153-174. doi: 10.1007/BF00125081.  Google Scholar

[17]

http://www.shoup.net, 2008. Google Scholar

[18]

Glas. Mat. Ser. III, 44(64) (2009), 89-126. doi: 10.3336/gm.44.1.05.  Google Scholar

show all references

References:
[1]

NIST Special Publication 800-57, 2007. Google Scholar

[2]

Chapman & Hall/CRC, Boca Raton, 2006. doi: 10.1201/9781420034981.  Google Scholar

[3]

IEEE Trans. Inf. Theory, 22 (1976), 472-492. doi: 10.1109/TIT.1976.1055638.  Google Scholar

[4]

Adv. Math. Commun., 5 (2011), 623-666. doi: 10.3934/amc.2011.5.623.  Google Scholar

[5]

Adv. Math. Commun., 2 (2008), 293-307. doi: 10.3934/amc.2008.2.293.  Google Scholar

[6]

F. Fontein, Holes in the infrastructure of global hyperelliptic function fields,, preprint, ().   Google Scholar

[7]

Théorie des Nombres (Québec, PQ), de Gruyter, Berlin, 1989, 227-239.  Google Scholar

[8]

in Algorithmic Number Theory - ANTS 2008 (Berlin), Springer, 2008, 342-356. doi: 10.1007/978-3-540-79456-1_23.  Google Scholar

[9]

Adv. Math. Commun., 1 (2007), 197-221. doi: 10.3934/amc.2007.1.197.  Google Scholar

[10]

in Advances in Coding Theory and Cryptology (eds. T. Shaska, W.C. Huffman, D. Joyner and V. Ustimenko), World Scientific Publishing, 2007, 201-244. doi: 10.1142/9789812772022_0013.  Google Scholar

[11]

Tatra Mountains Math. Publ., 40 (2010), 1-35. doi: 10.2478/v10127-010-0030-9.  Google Scholar

[12]

J. Cryptology, 1 (1989), 139-150. doi: 10.1007/BF02252872.  Google Scholar

[13]

Appl. Algebra Eng. Commun. Comput., 15 (2005), 295-328. doi: 10.1007/s00200-004-0154-8.  Google Scholar

[14]

Cryptology eprint archive no. 2008/299, 2008. Google Scholar

[15]

J. Cryptology, 7 (1994), 171-199. doi: 10.1007/BF02318548.  Google Scholar

[16]

Des. Codes Crypt., 7 (1996), 153-174. doi: 10.1007/BF00125081.  Google Scholar

[17]

http://www.shoup.net, 2008. Google Scholar

[18]

Glas. Mat. Ser. III, 44(64) (2009), 89-126. doi: 10.3336/gm.44.1.05.  Google Scholar

[1]

Iris Anshel, Derek Atkins, Dorian Goldfeld, Paul E. Gunnells. Ironwood meta key agreement and authentication protocol. Advances in Mathematics of Communications, 2021, 15 (3) : 397-413. doi: 10.3934/amc.2020073
[2]

Frank Sottile. The special Schubert calculus is real. Electronic Research Announcements, 1999, 5: 35-39.

[3]

Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145.

[4]

Yves Capdeboscq, Shaun Chen Yang Ong. Quantitative jacobian determinant bounds for the conductivity equation in high contrast composite media. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3857-3887. doi: 10.3934/dcdsb.2020228
[5]

Peng Zhang, Yongquan Zeng, Guotai Chi. Time-consistent multiperiod mean semivariance portfolio selection with the real constraints. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1663-1680. doi: 10.3934/jimo.2020039
[6]

Xianjun Wang, Huaguang Gu, Bo Lu. Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron. Electronic Research Archive, , () : -. doi: 10.3934/era.2021023
[7]

Christophe Zhang. Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021006
[8]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

2019 Impact Factor: 0.734

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences