Skip to main content
Springer Nature Link
Log in

On Modular Decomposition of Integers

  • Conference paper
Progress in Cryptology – AFRICACRYPT 2009 (AFRICACRYPT 2009)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 5580))

Included in the following conference series:

Abstract

At Crypto 2001, Gallant et al. showed how to exploit fast endomorphisms on some specific classes of elliptic curves to obtain fast scalar multiplication. The GLV method works by decomposing scalars into two small portions using multiplications, divisions, and rounding operations in the rationals. We present a new simple method based on the extended Euclidean algorithm that uses notably different operations than that of traditional decomposition. We obtain strict bounds on each component. Additionally, we examine the use of random decompositions, useful for key generation or cryptosystems requiring ephemeral keys. Specifically, we provide a complete description of the probability distribution of random decompositions and give bounds for each component in such a way that ensures a concrete level of entropy. This is the first analysis on distribution of random decompositions in GLV allowing the derivation of the entropy and thus an answer to the question first posed by Gallant in 1999.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Gallant, R.P., Lambert, R.J., Vanstone, S.A.: Faster point multiplication on elliptic curves with efficient endomorphisms. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 190–200. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  2. Bellman, R., Straus, E.G.: Problems and Solutions: Solutions of Advanced Problems: 5125. Amer. Math. Monthly 71(7), 806–808 (1964)

    Article  MathSciNet  Google Scholar 

  3. ElGamal, T.: A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. Inform. Theory 31(4), 469–472 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Galbraith, S.D., Lin, X., Scott, M.: Endomorphisms for faster elliptic curve cryptography on a large class of curves. In: Advances in cryptology—EUROCRYPT 2009. LNCS, Springer, Heidelberg (2009) (to appear)

    Google Scholar 

  5. Gallant, R.: Faster elliptic curve cryptography using efficient endomorphisms. In: 3rd workshop on Elliptic Curve Cryptography—ECC 1999 (1999) (presentation slides)

    Google Scholar 

  6. SECG: Recommended elliptic curve domain parameters. Standards for Efficient Cryptography SEC 2 (September 20, 2000)

    Google Scholar 

  7. ANSI: Public key cryptography for the financial services industry: Key agreement and key transport using elliptical curve cryptography (2001) ANSI X9.63

    Google Scholar 

  8. Hankerson, D., Menezes, A.J., Vanstone, S.A.: Guide to Elliptic Curve Cryptography. Springer, New York (2004)

    MATH  Google Scholar 

  9. Babai, L.: On Lovász’ lattice reduction and the nearest lattice point problem. Combinatorica 6(1), 1–13 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kim, D., Lim, S.: Integer decomposition for fast scalar multiplication on elliptic curves. In: Nyberg, K., Heys, H.M. (eds.) SAC 2002. LNCS, vol. 2595, pp. 13–20. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  11. Sica, F., Ciet, M., Quisquater, J.J.: Analysis of the Gallant-Lambert-Vanstone method based on efficient endomorphisms: elliptic and hyperelliptic curves. In: Nyberg, K., Heys, H.M. (eds.) SAC 2002. LNCS, vol. 2595, pp. 21–36. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  12. Park, Y.H., Jeong, S., Kim, C.H., Lim, J.: An alternate decomposition of an integer for faster point multiplication on certain elliptic curves. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274, pp. 323–334. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  13. Grabner, P.J., Heuberger, C., Prodinger, H.: Distribution results for low-weight binary representations for pairs of integers. Theoret. Comput. Sci. 319(1-3), 307–331 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Antipa, A., Brown, D., Gallant, R., Lambert, R., Struik, R., Vanstone, S.: Accelerated verification of ECDSA signatures. In: Preneel, B., Tavares, S. (eds.) SAC 2005. LNCS, vol. 3897, pp. 307–318. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  15. Knuth, D.E.: The Art of Computer Programming, Volume II: Seminumerical Algorithms, 2nd edn. Addison-Wesley, Reading (1981)

    MATH  Google Scholar 

  16. Järvinen, K., Forsten, J., Skyttä, J.: Efficient circuitry for computing τ-adic non-adjacent form. In: Proceedings of the 13th IEEE International Conference on Electronics, Circuits and Systems, ICECS 2006, pp. 232–235. IEEE, Los Alamitos (2006)

    Chapter  Google Scholar 

  17. Lange, T., Shparlinski, I.E.: Distribution of some sequences of points on elliptic curves. J. Math. Cryptol. 1(1), 1–11 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cohen, H., Frey, G. (eds.): Handbook of elliptic and hyperelliptic curve cryptography. CRC Press, Boca Raton (2005)

    Google Scholar 

  19. IEEE: IEEE P1363 working group for public-key cryptography standards. meeting minutes (November 15, 2000), http://grouper.ieee.org/groups/1363/WorkingGroup/minutes/Nov00.txt

Download references

Author information

Authors and Affiliations

  1. Department of Information and Computer Science, Helsinki University of Technology, P.O. Box 5400, FI-02015, TKK, Finland

    Billy Bob Brumley & Kaisa Nyberg

  2. Nokia Research Center, Finland

    Kaisa Nyberg

Authors
  1. Billy Bob Brumley
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kaisa Nyberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dept. Electrical Engineering-ESAT/COSIC, Katholieke Universiteit Leuven, Kasteelpark Arenberg 10, Bus 2446, 3001, Leuven, Belgium

    Bart Preneel

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Brumley, B.B., Nyberg, K. (2009). On Modular Decomposition of Integers. In: Preneel, B. (eds) Progress in Cryptology – AFRICACRYPT 2009. AFRICACRYPT 2009. Lecture Notes in Computer Science, vol 5580. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02384-2_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-02384-2_24

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02383-5

  • Online ISBN: 978-3-642-02384-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics