Fast Exponentiation with Precomputation

Extended Abstract
  • Ernest F. Brickell
  • Daniel M. Gordon
  • Kevin S. McCurley
  • David B. Wilson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 658)

Abstract

In several cryptographic systems, a fixed element g of a group (generally \( \mathbb{Z}/q\mathbb{Z} \) ) is repeatedly raised to many different powers. In this paper we present a practical method of speeding up such systems, using precomputed values to reduce the number of multiplications needed. In practice this provides a substantial improvement over the level of performance that can be obtained using addition chains, and allows the computation of gn for n < N in O(log N/log log N) group multiplications. We also show how these methods can be parallelized, to compute powers in O(log log N) group multiplications with O(log N/log log N) processors.

References

  1. 1.
    A Proposed Federal Information Processing Standard for Digital Signature Standard, Federal Register, Volume 56, No. 169, August 31, 1991, pp. 42980–42982.Google Scholar
  2. 2.
    J. Bos and M. Coster, Addition Chain Heuristics, in Advances in Cryptology-Proceedings of Crypto’ 89, Lecture Notes in Computer Science, Volume 435, Springer-Verlag, New York, 1990, pp. 400–407.CrossRefGoogle Scholar
  3. 3.
    W. Diffie and M. Hellman, New Directions in Cryptography, IEEE Transactions on Information Theory 22 (1976), 472–492.CrossRefMathSciNetGoogle Scholar
  4. 4.
    E.F. Brickell and K.S. McCurley, An Interactive Identification Scheme Based on Discrete Logarithms and Factoring, to appear in Journal of Cryptology.Google Scholar
  5. 5.
    Ryo Fuji-Hara, Cipher Algorithms and Computational Complexity, Bit 17 (1985), 954–959 (in Japanese).Google Scholar
  6. 6.
    D.E. Knuth,The Art of Computer Programming, Vol. 2, Seminumerical Algorithms, Second Edition, Addison-Wesley, Massachusetts, 1981.Google Scholar
  7. 7.
    D.W. Matula, Basic digit sets for radix representation, Journal of the ACM, 29 (1982), pp. 1131–1143.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    C.P. Schnorr, Efficient signature generation by smart cards, to appear in Journal of Cryptology.Google Scholar
  9. 9.
    D.R. Stinson, Some observations on parallel algorithms for fast exponentiation in GF(2n), Siam. J. Comput., 19, (1990), pp. 711–717.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    J.S. Vitter and P. Flajolet, Average-case analysis of algorithms and data structures, in Handbook of Theoretical Computer Science, ed. J. van Leeuwen, Elsevier, Amsterdam, 1990, pp. 431–524.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Ernest F. Brickell
    • 1
  • Daniel M. Gordon
    • 2
  • Kevin S. McCurley
    • 1
  • David B. Wilson
    • 3
  1. 1.Division 1423Sandia National LaboratoriesAlbuquerque
  2. 2.Department of Computer ScienceUniversity of GeorgiaAthens
  3. 3.Department of MathematicsM.I.T.Cambridge

Personalised recommendations