Exponentiation in finite fields: Theory and practice

  • Joachim von zur Gathen
  • Michael Nöcker
Conference paper

DOI: 10.1007/3-540-63163-1_8

Part of the Lecture Notes in Computer Science book series (LNCS, volume 1255)
Cite this paper as:
von zur Gathen J., Nöcker M. (1997) Exponentiation in finite fields: Theory and practice. In: Mora T., Mattson H. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1997. Lecture Notes in Computer Science, vol 1255. Springer, Berlin, Heidelberg

Abstract

Finally we want to outline the main properties for a fast software exponentiation algorithm in \(\mathbb{F}_{2^n }\)for large n∈ℕ:

  1. 1.

    The algorithm should use fast polynomial multiplication. Neither multiplication by multiplication tensors nor classical polynomial arithmetic is fast enough.

     
  2. 2.

    The algorithm should be based upon an addition chain for the exponent e with a small number of non-doubling steps.

     
  3. 3.

    The algorithm should offer a cheap way to compute \(\alpha ^{2^m }\)for m∈ℕ and \(\alpha \in \mathbb{F}_{2^n }\). Both Shoup's and Gao et al.'s algorithm achieve this.

     

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Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Joachim von zur Gathen
    • 1
  • Michael Nöcker
    • 1
  1. 1.Fachbereich 17 Mathematik-InformatikUniversität-GH PaderbornPaderbornGermany

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