Skip to main content
SpringerLink
Log in

Fast Scalar Multiplication for ECC over GF(p) Using Division Chains

  • Conference paper
Information Security Applications (WISA 2010)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 6513))

Included in the following conference series:

  • 1063 Accesses

Abstract

There have been many recent developments in formulae for efficient composite elliptic curve operations of the form dP + Q for a small integer d and points P and Q where the underlying field is a prime field. To make best use of these in a scalar multiplication kP, it is necessary to generate an efficient “division chain” for the scalar where divisions of k are by the values of d available through composite operations.

An algorithm-generating algorithm for this is presented that takes into account the different costs of using various representations for curve points. This extends the applicability of methods presented by Longa & Gebotys at PKC 2009 to using specific characteristics of the target device. It also enables the transfer of some scalar recoding computation details to design time. An improved cost function also provides better evaluation of alternatives in the relevant addition chain.

One result of these more general and improved methods includes a slight increase over the scalar multiplication speeds reported at PKC. Furthermore, by the straightforward removal of rules for unusual cases, some particularly concise yet efficient presentations can be given for algorithms in the target device.

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. Bernstein, D., Lange, T.: Analysis and Optimization of Elliptic-Curve Single-Scalar Multiplication. Cryptology ePrint Archive, Report 2007/455, IACR 2007 (2007)

    Google Scholar 

  2. Billet, O., Joye, M.: The Jacobi Model of an Elliptic Curve and Side-Channel Analysis. In: Fossorier, M.P.C., Høholdt, T., Poli, A. (eds.) AAECC 2003. LNCS, vol. 2643, pp. 34–42. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  3. Ciet, M., Joye, M., Lauter, K., Montgomery, P.: Trading Inversions for Multiplications in Elliptic Curve Cryptography. Designs, Codes and Cryptography 39(2), 189–206 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dimitrov, V., Cooklev, T.: Two Algorithms for Modular Exponentiation using Non-Standard Arithmetics. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E78-A(1), 82–87 (1995)

    Google Scholar 

  5. Dimitrov, V.S., Jullien, G.A., Miller, W.C.: Theory and Applications for a Double-Base Number System. In: Proc. 13th IEEE Symposium on Computer Arithmetic, Monterey, July 6-9, pp. 44–51. IEEE, Los Alamitos (1997)

    Chapter  Google Scholar 

  6. Dimitrov, V.S., Imbert, L., Mishra, P.K.: Efficient and Secure Elliptic Curve Point Multiplication using Double-Base Chains. In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 59–78. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  7. Edwards, H.: A Normal Form for Elliptic Curves. Bull. Amer. Math. Soc. 44, 393–422 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Elmegaard-Fessel, L.: Efficient Scalar Multiplication and Security against Power Analysis in Cryptosystems based on the NIST Elliptic Curves over Prime Fields, Masters Thesis, University of Copenhagen (2006)

    Google Scholar 

  9. Fouque, P.-A., Valette, F.: The Doubling Attack – Why upwards is better than downwards. In: Walter, C.D., Koç, Ç.K., Paar, C. (eds.) CHES 2003. LNCS, vol. 2779, pp. 269–280. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  10. Doche, C., Icart, T., Kohel, D.R.: Efficient Scalar Multiplication by Isogeny Decompositions. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T.G. (eds.) PKC 2006. LNCS, vol. 3958, pp. 191–206. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  11. Giessmann, E.-G.: Ein schneller Algorithmus zur Punktevervielfachung, der gegen Seitenanalattacken resistent ist. In: Workshop über Theoretische und praktische Aspekte von Kryptographie mit Elliptischen Kurven, Berlin (2001)

    Google Scholar 

  12. Gordon, D.M.: A Survey of Fast Exponentiation Algorithms. Journal of Algorithms 27, 129–146 (1998)

    Article  MATH  Google Scholar 

  13. Hisil, H., Wong, K., Carter, G., Dawson, E.: Faster Group Operations on Elliptic Curves. Cryptology ePrint Archive, Report 2007/441, IACR (2007)

    Google Scholar 

  14. Knuth, D.E.: The Art of Computer Programming, 2nd edn. Seminumerical Algorithms, vol. 2, §4.6.3, pp. 441–466. Addison-Wesley, Reading (1981)

    Google Scholar 

  15. Longa, P.: Accelerating the Scalar Multiplication on Elliptic Curve Cryptosystems over Prime Fields, Masters Thesis, University of Ottawa (2007)

    Google Scholar 

  16. Longa, P., Miri, A.: New Composite Operations and Precomputation Scheme for Elliptic Curve Cryptosystems over Prime Fields. In: Cramer, R. (ed.) PKC 2008. LNCS, vol. 4939, pp. 229–247. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  17. Longa, P., Gebotys, C.: Fast Multibase Methods and Other Several Optimizations for Elliptic Curve Scalar Multiplication. In: Jarecki, S., Tsudik, G. (eds.) PKC 2009. LNCS, vol. 5443, pp. 443–462. Springer, Heidelberg (2009)

    Google Scholar 

  18. Mishra, P.K., Dimitrov, V.: Efficient Quintuple Formulas for Elliptic Curves and Efficient Scalar Multiplication using Multibase Number Representation. In: Garay, J.A., Lenstra, A.K., Mambo, M., Peralta, R. (eds.) ISC 2007. LNCS, vol. 4779, pp. 390–406. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  19. Walter, C.D.: Exponentiation using Division Chains. In: Proc. 13th IEEE Symposium on Computer Arithmetic, Monterey, CA, July 6-9, pp. 92–98. IEEE, Los Alamitos (1997)

    Chapter  Google Scholar 

  20. Walter, C.D.: Exponentiation using Division Chains. IEEE Transactions on Computers 47(7), 757–765 (1998)

    Article  MathSciNet  Google Scholar 

  21. Walter, C.D.: MIST: An Efficient, Randomized Exponentiation Algorithm for Resisting Power Analysis. In: Preneel, B. (ed.) CT-RSA 2002. LNCS, vol. 2271, pp. 53–66. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  22. Walter, C.D.: Some Security Aspects of the MIST Randomized Exponentiation Algorithm. In: Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 276–290. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  23. Yao, A.C.-C.: On the Evaluation of Powers. SIAM J. Comput. 5(1), 100–103 (1976)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Information Security Group, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, United Kingdom

    Colin D. Walter

Authors
  1. Colin D. Walter
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer and Information Science, Korea University, Chungnam-do, Korea

    Yongwha Chung

  2. Computer Science Department, Google Inc. and Columbia University, NewYork, USA

    Moti Yung

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Walter, C.D. (2011). Fast Scalar Multiplication for ECC over GF(p) Using Division Chains. In: Chung, Y., Yung, M. (eds) Information Security Applications. WISA 2010. Lecture Notes in Computer Science, vol 6513. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17955-6_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-17955-6_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-17954-9

  • Online ISBN: 978-3-642-17955-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation