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Scalar Multiplication on Koblitz Curves Using the Frobenius Endomorphism and Its Combination with Point Halving: Extensions and Mathematical Analysis

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Abstract

In this paper we prove the optimality and other properties of the τ-adic nonadjacent form: this expansion has been introduced in order to compute scalar multiplications on Koblitz curves efficiently. We also refine and extend results about double expansions of scalars introduced by Avanzi, Ciet and Sica in order to improve scalar multiplications further. Our double expansions are optimal and their properties are carefully analysed. In particular, we provide first- and second-order terms for the expected weight, determine the variance and prove a central limit theorem. Transducers for all the involved expansions are provided, as well as automata accepting all expansions of minimal weight.

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Authors and Affiliations

  1. Fakultat fur Mathematik, Ruhr-Universitat Bochum, Universitatsstrasse 150, D-44780, Bochum, Germany

    Roberto M. Avanzi

  2. Institut fur Mathematik B, Technische Universitat Graz, Steyrergasse 30, 8010, Graz, Austria

    Clemens Heuberger

  3. Mathematics Department, Stellenbosch University, 7602, Stellenbosch, South Africa

    Helmut Prodinger

Authors
  1. Roberto M. Avanzi
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  2. Clemens Heuberger
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  3. Helmut Prodinger
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Corresponding authors

Correspondence to Roberto M. Avanzi, Clemens Heuberger or Helmut Prodinger.

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Avanzi, R., Heuberger, C. & Prodinger, H. Scalar Multiplication on Koblitz Curves Using the Frobenius Endomorphism and Its Combination with Point Halving: Extensions and Mathematical Analysis. Algorithmica 46, 249–270 (2006). https://doi.org/10.1007/s00453-006-0105-9

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  • DOI: https://doi.org/10.1007/s00453-006-0105-9

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