Skip to main content
SpringerLink
Log in

Montgomery Ladder for All Genus 2 Curves in Characteristic 2

  • Conference paper
Arithmetic of Finite Fields (WAIFI 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5130))

Included in the following conference series:

Abstract

Using the Kummer surface, we generalize Montgomery ladder for scalar multiplication to the Jacobian of genus 2 curves in characteristic 2. Previously this method was known for elliptic curves and for genus 2 curves in odd characteristic. We obtain an algorithm that is competitive compared to usual methods of scalar multiplication and that has additional properties such as resistance to simple side-channel attacks. Moreover it provides a significant speed-up of scalar multiplication in many cases. This new algorithm has very important applications in cryptography using hyperelliptic curves and more particularly for people interested in cryptography on embedded systems (such as smart cards).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Brier, E., Joye, M.: Weierstrass Elliptic Curves and Side-Channel Attacks. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  2. Byramjee, B., Duquesne, S.: Classification of genus 2 curves over \(\mathbb{F}_{2^n}\) and optimization of their arithmetic. Cryptology ePrint Archive 107 (2004)

    Google Scholar 

  3. Cantor, D.G.: Computing on the Jacobian of a hyperelliptic curve. Math. Comp. 48, 95–101 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  4. Choie, Y., Yun, D.: Isomorphism classes of hyperelliptic curves of genus 2 over . In: Batten, L.M., Seberry, J. (eds.) ACISP 2002. LNCS, vol. 2384, pp. 190–202. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  5. Cohen, H., Frey, G.: Handbook of elliptic and hyperelliptic curve cryptography, Discrete Math. Appl. Chapman & Hall/CRC, Boca Raton (2006)

    Google Scholar 

  6. Duquesne, S.: Montgomery scalar multiplication for genus 2 curves. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 153–168. Springer, Heidelberg (2004)

    Google Scholar 

  7. Duquesne, S.: Traces of the group law on the Kummer surface of a curve of genus 2 in characteristic 2, preprint, available at [8]

    Google Scholar 

  8. Duquesne, S.: Formulas for traces of the group law on the Kummer surface of a curve of genus 2 in characteristic 2, http://www.math.univ-montp2.fr/~duquesne/articles/kummer2

  9. Flynn, E.V.: The group law on the Jacobian of a curve of genus 2. J. reine angew. Math. 439, 45–69 (1993)

    MATH  MathSciNet  Google Scholar 

  10. Galbraith, S.: Supersingular curves in cryptography. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 495–513. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  11. Gaudry, P., Hess, F., Smart, N.: Constructive and destructive facets of Weil descent on elliptic curves. J. Cryptology 15(1), 19–46 (2002)

    Article  MathSciNet  Google Scholar 

  12. Gaudry, P.: Fast genus 2 arithmetic based on Theta functions. Journal of Mathematical Cryptology 1, 243–265 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gaudry, P.: Variants of the Montgomery form based on Theta functions, Toronto (November 2006)

    Google Scholar 

  14. Gaudry, P., Lubicz, D.: The arithmetic of characteristic 2 Kummer surfaces. Cryptology ePrint Archive 133 (2008)

    Google Scholar 

  15. Harley, R.: Fast arithmetic on genus 2 curves (2000), http://cristal.inria.fr/~harley/hyper

  16. Imbert, L., Peirera, A., Tisserand, A.: A Library for Prototyping the Computer Arithmetic Level in Elliptic Curve Cryptography. In: Proc. SPIE, vol. 6697, 66970N (2007)

    Google Scholar 

  17. Koblitz, N.: Elliptic curve cryptosystems. Math. Comp. 48, 203–209 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kocher, P.C.: Timing attacks on implementations of DH, RSA, DSS and other systems. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 104–113. Springer, Heidelberg (1996)

    Google Scholar 

  19. Kocher, P.C., Jaffe, J., Jun, B.: Differential power analysis. In: Wiener, M.J. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 388–397. Springer, Heidelberg (1999)

    Google Scholar 

  20. Lange, T.: Arithmetic on binary genus 2 curves suitable for small devices. In: Proceedings ECRYPT Workshop on RFID and Lightweight Crypto., Graz, Austria, July 14-15 (2005)

    Google Scholar 

  21. Lange, T.: Formulae for arithmetic on genus 2 hyperelliptic curves. Appl. Algebra Engrg. Comm. Comput. 15(5), 295–328 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lange, T., Mishra, P.K.: SCA resistant parallel explicit formula for addition and doubling of divisors in the Jacobian of hyperelliptic curves of genus 2. In: Maitra, S., Veni Madhavan, C.E., Venkatesan, R. (eds.) INDOCRYPT 2005. LNCS, vol. 3797, pp. 403–416. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  23. Lopez, J., Dahab, R.: Improved algorithms for elliptic curve arithmetic in GF(2n). In: Tavares, S., Meijer, H. (eds.) SAC 1998. LNCS, vol. 1556, pp. 201–212. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  24. Lopez, J., Dahab, R.: Fast multiplication on elliptic curves over GF(2m) without precomputation. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 316–327. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  25. Menezes, A., Wu, Y.H., Zuccherato, R.: An elementary introduction to hyperelliptic curves. In: Koblitz, N. (ed.) Algebraic aspects of cryptography. Algorithms and Computation in Mathematics, vol. 3, pp. 155–178 (1998)

    Google Scholar 

  26. Miller, V.S.: Use of elliptic curves in cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986)

    Google Scholar 

  27. Montgomery, P.L.: Speeding the Pollard and elliptic curve methods of factorization. Math. Comp. 48, 164–243 (1987)

    Article  Google Scholar 

  28. Mumford, D.: Tata lectures on Theta II. Birkhäuser, Basel (1984)

    MATH  Google Scholar 

  29. Okeya, O., Sakurai, K.: Efficient Elliptic Curve Cryptosystems from a Scalar Multiplication Algorithm with Recovery of the y-Coordinate on a Montgomery-Form Elliptic Curve. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 126–141. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  30. Quisquater, J.J., Samyde, D.: ElectroMagnetic Analysis (EMA): Measures and Countermeasures for Smart Cards. In: Attali, S., Jensen, T. (eds.) E-smart 2001. LNCS, vol. 2140, pp. 200–210. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  31. Smart, N., Siksek, S.: A fast Diffe-Hellman protocol in genus 2. Journal of Cryptology 12, 67–73 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  32. Stam, M.: On Montgomery-Like Representations for Elliptic Curves over GF(2k). In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 240–253. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoires I3M, UMR CNRS 5149 and LIRMM, UMR CNRS 5506, Université Montpellier II, Place Eugène Bataillon CC 051, 34005, Montpellier Cedex, France

    Sylvain Duquesne

Authors
  1. Sylvain Duquesne
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Joachim von zur Gathen José Luis Imaña Çetin Kaya Koç

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Duquesne, S. (2008). Montgomery Ladder for All Genus 2 Curves in Characteristic 2. In: von zur Gathen, J., Imaña, J.L., Koç, Ç.K. (eds) Arithmetic of Finite Fields. WAIFI 2008. Lecture Notes in Computer Science, vol 5130. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69499-1_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-69499-1_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-69498-4

  • Online ISBN: 978-3-540-69499-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation