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International Conference on Cryptology in Africa

AFRICACRYPT 2013: Progress in Cryptology – AFRICACRYPT 2013 pp 424–432Cite as

On the Expansion Length Of Triple-Base Number Systems

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 7918))

Abstract

Triple-base number systems are mainly used in elliptic curve cryptography to speed up scalar multiplication. We give an upper bound on the length of the canonical triple-base representation with base {2, 3, 5} of an integer x, which is \(\mathcal{O}(\frac{\log x}{\log\log x})\) by the greedy algorithm, and show that there are infinitely many integers x whose shortest triple-base representations with base {2, 3, 5} have length greater than \(\frac{c\log x}{\log\log x\log\log\log x},\) where c is a positive constant, using the universal exponent method. This analysis gives a limit how much scalar multiplication on elliptic curves may be made faster.

Supported in part by National Basic Research Program of China(973) under Grant No.2013CB338002, in part by National Research Foundation of China under Grant No. 61272040 and 61070171, and in part by the Strategic Priority Research Program of Chinese Academy of Sciences under Grant XDA06010702.

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Author information

Authors and Affiliations

  1. Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, 230027, China

    Wei Yu

  2. Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, China

    Wei Yu, Kunpeng Wang, Bao Li & Song Tian

Authors
  1. Wei Yu
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  2. Kunpeng Wang
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  3. Bao Li
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  4. Song Tian
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Editor information

Editors and Affiliations

  1. Concordia Institute for Information Systems Engineering, Concordia University, 1515 St. Catherine Street West, H3G 2W1, Montreal, QC, Canada

    Amr Youssef

  2. Laboratoire de Mathématiques Nicolas Oresme, Université de Caen Basse-Normandie, BP 5186, 14032, Caen, France

    Abderrahmane Nitaj

  3. Department of Information Technology, Cairo University, 5 Dr. Ahmed Zewil Street, 12613, Cairo, Giza, Egypt

    Aboul Ella Hassanien

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Yu, W., Wang, K., Li, B., Tian, S. (2013). On the Expansion Length Of Triple-Base Number Systems. In: Youssef, A., Nitaj, A., Hassanien, A.E. (eds) Progress in Cryptology – AFRICACRYPT 2013. AFRICACRYPT 2013. Lecture Notes in Computer Science, vol 7918. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38553-7_25

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  • DOI: https://doi.org/10.1007/978-3-642-38553-7_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-38552-0

  • Online ISBN: 978-3-642-38553-7

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