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Scalar Multiplication on Koblitz Curves Using Double Bases

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Progress in Cryptology - VIETCRYPT 2006 (VIETCRYPT 2006)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 4341))

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Abstract

The paper is an examination of double-base decompositions of integers n, namely expansions loosely of the form \(n = \sum_{i,j} \pm A^iB^j \) for some base {A,B}. This was examined in previous works [5,6], in the case when A,B lie in ℕ.

We show here how to extend the results of [5] to Koblitz curves over binary fields. Namely, we obtain a sublinear scalar algorithm to compute, given a generic positive integer n and an elliptic curve point P, the point nP in time \(O\left(\frac{\log n}{\log\log n}\right)\) elliptic curve operations with essentially no storage, thus making the method asymptotically faster than any know scalar multiplication algorithm on Koblitz curves. In view of combinatorial results, this is the best type of estimate with two bases, apart from the value of the constant in the O notation.

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Authors and Affiliations

  1. Institute for Cryptology and IT-Security, The Horst Görtz institute (HGI) of IT-Security, Ruhr-Universität Bochum, Universitätsstraße 150, D-44780, Bochum, Germany

    Roberto Avanzi

  2. Department of Mathematics and Computer Science, Mount Allison University – AceCrypt, 67 York Street, Sackville, NB, E4L 1E6, Canada

    Francesco Sica

Authors
  1. Roberto Avanzi
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  2. Francesco Sica
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Editors and Affiliations

  1. Département d’Informatique, École Normale Supérieure, 45 rue d’Ulm, 75230, Paris Cedex 05, France

    Phong Q. Nguyen

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Avanzi, R., Sica, F. (2006). Scalar Multiplication on Koblitz Curves Using Double Bases. In: Nguyen, P.Q. (eds) Progress in Cryptology - VIETCRYPT 2006. VIETCRYPT 2006. Lecture Notes in Computer Science, vol 4341. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11958239_9

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  • DOI: https://doi.org/10.1007/11958239_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-68799-3

  • Online ISBN: 978-3-540-68800-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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