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International Conference on Information Security and Cryptology

ICISC 2004: Information Security and Cryptology – ICISC 2004 pp 137–153Cite as

Fractional Windows Revisited:Improved Signed-Digit Representations for Efficient Exponentiation

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 3506))

Abstract

This paper extends results concerning efficient exponentiation in groups where inversion is easy (e.g. in elliptic curve cryptography). It examines the right-to-left and left-to-right signed fractional window (RL-SFW and LR-SFW) techniques and shows that both RL-SFW and LR-SFW representations have minimal weight among all signed-digit representations with digit set {±1, ±3, ..., ±m, 0}.(Fractional windows generalize earlier sliding-window techniques, providing more flexibility for exponentiation algorithms in order to make best use of the memory that is available for storing intermediate results.) Then it considers the length of representations: LR-SFW representations are an improvement over RL-SFW representations in that they tend to be shorter; further length improvements are possible by post-processing the representations.

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References

  1. Avanzi, R.M.: A note on the sliding window integer recoding and its left-to-right analogue. In: Handschuh, H., Hasan, M.A. (eds.) SAC 2004. LNCS, vol. 3357, pp. 130–143. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  2. Blake, I.F., Seroussi, G., Smart, N.P.: Elliptic Curves in Cryptography. London Mathematical Society Lecture Note Series, vol. 265. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  3. Cohen, H., Ono, T., Miyaji, A.: Efficient elliptic curve exponentiation using mixed coordinates. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 51–65. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  4. Gordon, D.M.: A survey of fast exponentiation methods. Journal of Algorithms 27, 129–146 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Goriac, I., Iftene, S.: Personal communication (2003)

    Google Scholar 

  6. Grabner, P.J., Heuberger, C., Prodinger, H., Thuswaldner, J.M.: Analysis of linear combination algorithms in cryptography. Preprint (2003), Available from http://www.opt.math.tu-graz.ac.at/~cheub/publications/

  7. Heuberger, C., Katti, R., Prodinger, H., Ruan, X.: The alternating greedy expansion and applications to left-to-right algorithms in cryptography. Preprint (2004), Available from http://www.opt.math.tu-graz.ac.at/~cheub/publications/

  8. Joye, M., Yen, S.-M.: Optimal left-to-right binary signed-digit recoding. IEEE Transactions on Computers 49, 740–748 (2000)

    Article  Google Scholar 

  9. Khabbazian, M., Gulliver, T.A.: A new minimal average weight representation for left-to-right point multiplication methods. Cryptology ePrint Archive Report 2004/266 (2004), Available from http://eprint.iacr.org/

  10. Knuth, D.E.: The Art of Computer Programming. Seminumerical Algorithms, vol. 2. Addison-Wesley, Reading (1969)

    MATH  Google Scholar 

  11. Knuth, D.E.: The Art of Computer Programming, 3rd edn. Seminumerical Algorithms, vol. 2. Addison-Wesley, Reading (1998)

    Google Scholar 

  12. Miyaji, A., Ono, T., Cohen, H.: Efficient elliptic curve exponentiation. In: Han, Y., Quing, S. (eds.) ICICS 1997. LNCS, vol. 1334, pp. 282–290. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  13. Möller, B.: Algorithms for multi-exponentiation. In: Vaudenay, S., Youssef, A.M. (eds.) SAC 2001. LNCS, vol. 2259, pp. 165–180. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  14. Möller, B.: Improved techniques for fast exponentiation. In: Lee, P.J., Lim, C.H. (eds.) ICISC 2002. LNCS, vol. 2587, pp. 298–312. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  15. Muir, J.A., Stinson, D.R.: Minimality and other properties of the width-w nonadjacent form. Mathematics of Computation. preprint (to appear), available from http://www.cacr.math.uwaterloo.ca/tech_reports.html

  16. Muir, J.A., Stinson, D.R.: New minimal weight representations for left-to-right window methods. In: Menezes, A. (ed.) CT-RSA 2005. LNCS, vol. 3376, pp. 366–383. Springer, Heidelberg (2005), preprint available from http://www.cacr.math.uwaterloo.ca/tech_reports.html

    Chapter  Google Scholar 

  17. Okeya, K., Schmidt-Samoa, K., Spahn, C., Takagi, T.: Signed binary representations revisited. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 123–139. Springer, Heidelberg (2004)

    Google Scholar 

  18. Reitwiesner, G.W.: Binary arithmetic. Advances in Computers 1, 231–308 (1960)

    Article  MathSciNet  Google Scholar 

  19. Schmidt-Samoa, K., Semay, O., Takagi, T.: Analysis of some efficient window methods and their application to elliptic curve cryptosystems. Technical Report TI-3/04 (2004), Available from http://www.informatik.tu-darmstadt.de/ftp/pub/TI/TR/

  20. Schroeppel, R., Orman, H., O’Malley, S., Spatscheck, O.: Fast key exchange with elliptic curve systems. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 43–56. Springer, Heidelberg (1995)

    Google Scholar 

  21. Solinas, J.A.: An improved algorithm for arithmetic on a family of elliptic curves. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 357–371. Springer, Heidelberg (1997)

    Google Scholar 

  22. Solinas, J.A.: Efficient arithmetic on Koblitz curves. Designs, Codes and Cryptography 19, 195–249 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Yao, A.C.-C.: On the evaluation of powers. SIAM Journal on Computing 5, 100–103 (1976)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. University of California, Berkeley

    Bodo Möller

Authors
  1. Bodo Möller
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Editor information

Editors and Affiliations

  1. National Security Research Institute, 305-348, Daejeon, Korea

    Choon-sik Park

  2. National Security Research Institute (NSRI), 161 Gajeong-dong, Yuseong-gu, 305-350, Daejeon, Korea

    Seongtaek Chee

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Möller, B. (2005). Fractional Windows Revisited:Improved Signed-Digit Representations for Efficient Exponentiation. In: Park, Cs., Chee, S. (eds) Information Security and Cryptology – ICISC 2004. ICISC 2004. Lecture Notes in Computer Science, vol 3506. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11496618_11

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  • DOI: https://doi.org/10.1007/11496618_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-26226-8

  • Online ISBN: 978-3-540-32083-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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